https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Hung&feedformat=atomUW-Math Wiki - User contributions [en]2024-03-29T14:51:17ZUser contributionsMediaWiki 1.39.5https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=26235PDE Geometric Analysis seminar2024-02-27T22:46:50Z<p>Htran24: </p>
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<div>All talks will be ''in person'' unless specified otherwise.<br />
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(Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.) <br />
===[[Previous PDE/GA seminars]]===<br />
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= Spring 2024 =<br />
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The first day of class is Tuesday, January 23, 2023.<br />
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<br />
'''January 23, 2024 (special date/time)'''<br />
<br />
Donghyun Lee (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: 901 Van Vleck Hall<br />
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Title: DYNAMICAL BILLIARD AND A LONG-TIME BEHAVIOR OF THE BOLTZMANN EQUATION IN GENERAL 3D TOROIDAL DOMAINS<br />
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Abstract: Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant progress in this question when domains are strictly convex, the same question without the strict convexity of domains is still totally open in 3D. The major difficulty arises when a billiard map has an infinite number of bounces in a finite time interval or when the map fails to be Lipschitz continuous, both of which happen generically when the domain is non-convex. In this paper, we develop a new method to control a billiard map on a surface of revolution generated by revolving any planar analytic convex closed curve (e.g., typical shape of tokamak reactors’ chamber). In particular, we classify and measure the size (to be small) of a pullback set (along the billiard trajectory) of the infinite-bouncing and singular- bouncing cases. As a consequence, we solve the open question affirmatively in such domains. To the best of our knowledge, this work is the first construction of global solutions to the hard-sphere Boltzmann equation in generic non-convex 3-dimensional domains. This is joint work with Gyounghun Ko and Chanwoo Kim.<br />
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'''January 29, 2024'''<br />
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Minh-Binh Tran (TAMU). Host: Hung Tran<br />
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Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
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Title: Some Results On the Kinetic Theory for Classical and Quantum Waves<br />
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Abstract: Kinetic equations can be used to describe the dynamics of nonlinear classical and quantum waves out of thermal equilibrium, as well as the propagation of waves in a random medium. In this talk, I will present some of our recent results on the kinetic theory of waves. I will discuss the analysis of those kinetic equations for waves. Next, I will focus on the numerical schemes we have been developing to resolve those equations. I will also address some control problems concerning kinetic equations for waves. The last part is devoted to some physical applications of wave kinetic theory for Bose-Einstein Condensates.<br />
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'''January 30, 2024 (special date/time)'''<br />
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Jin Woo Jang (POSTECH). Host: Chanwoo Kim<br />
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Format: in-person. Time: 3:00-4:00PM, Location: 901 Van Vleck Hall<br />
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Title: On the Relativistic Boltzmann Equation with Long Range Interactions<br />
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Abstract: In this talk, I will discuss three recent, interrelated results concerning the special relativistic Boltzmann equation without angular cutoff. In the non-relativistic situation without angular cutoff, the change of variables from $v \to v'$ is a crux of the widely used "cancellation lemma". Firstly, in collaboration with James Chapman and Robert M. Strain, we calculate this very complex ten variable Jacobian determinant in the special relativisticsituation and illustrate some numerical results which show that it has a large number of distinct points where it is machine zero. Secondly, with Strain, we prove the sharp pointwise asymptotics for the frequency multiplier of the linearized relativistic Boltzmann collision operator that has not been previously established. As a consequence of these calculations, we further explain why the well known change of variables p \to p' is not well defined in the special relativisticcontext. Finally, also with Strain, we will present our recent proof of global-in-time existence and uniqueness of the solutions nearby the relativistic Maxwellian to the special relativistic Boltzmann equation without any angular cutoff and its asymptotic stability. We work in the case of a spatially periodic box. We assume the generic hard-interaction and mildly-soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Jeżewska (in 1985). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. <br />
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'''February 5, 2024'''<br />
<br />
Thierry Laurens (UW-Madison) <br />
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Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
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Title: Continuum Calogero--Moser models<br />
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Abstract: The focusing CCM model is a dispersive equation that describes a continuum limit of a particle gas interacting pairwise through an inverse square potential. Recently, Gérard and Lenzmann discovered solutions to this equation that exhibit frequency cascades.<br />
<br />
In this talk, we will discuss a scaling-critical well-posedness result for the focusing and defocusing CCM models on the line. In the focusing case, this requires solutions to have mass less than that of the soliton. This is joint work with Rowan Killip and Monica Visan.<br />
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'''February 12, 2024'''<br />
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Laurel Ohm (UW-Madison)<br />
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Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
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Title: Free boundary dynamics of an elastic filament in 3D Stokes flow<br />
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Abstract: We consider a free boundary problem for a thin elastic filament immersed in 3D Stokes flow. The 3D fluid is coupled to the quasi-1D filament dynamics via a novel type of angle-averaged Neumann-to-Dirichlet operator. Much of the difficulty in the analysis lies in understanding this operator. We show that the principal part of this NtD map is the corresponding operator about a straight, periodic filament, for which we derive an explicit symbol. It is then possible to establish local well-posedness for an immersed filament evolving via a simple elasticity law. This establishes a mathematical foundation for the myriad computational results based on slender body approximations for thin immersed elastic structures.<br />
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'''February 19, 2024'''<br />
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No seminar<br />
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'''February 26, 2024'''<br />
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Jiwoong Jang (UW-Madison)<br />
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Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
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Title: Periodic homogenization of geometric equations without perturbed correctors.<br />
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Abstract: Proving homogenization has been a subtle issue for geometric equations due to the discontinuity when the gradient vanishes. To overcome the difficulty and conclude homogenization, the work of Caffarelli-Monneau suggests a sufficient condition using perturbed correctors. However, some noncoercive equations do not satisfy this condition. In this talk, we discuss homogenization of geometric equations without using perturbed correctors, and we conclude homogenization for the noncoercive equations. Also, we derive a rate of periodic homogenization of coercive geometric equations by utilizing the fact that they remain coercive under perturbation. We also present an example that homogenizes with a rate Ω(ε| log ε|).<br />
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'''March 4, 2024'''<br />
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Yuming Paul Zhang (Auburn). Host: Hung Tran.<br />
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Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
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Title: Convergence of Policy Iteration for Deterministic Control<br />
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Abstract: We study the convergence of policy iteration for (deterministic) optimal control problems. To overcome the problem of ill-posedness due to lack of regularity, we consider both discrete and semi-discrete schemes by adding a viscosity term via finite differences in space. We prove that PI for the schemes converges exponentially fast, and provide a bound on the error induced by the schemes. If time permits, I will also discuss the convergence of exploratory Hamilton--Jacobi--Bellman (HJB) equations arising from the entropy-regularized exploratory control problem. These are joint works with Wenpin Tang, Hung Tran and Xunyu Zhou.<br />
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'''March 5, 2024 (special date/time and room)'''<br />
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Wei Xiang (City U. of Hong Kong). Host: Mikhail Feldman<br />
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Format: in-person. Time: 3:00-4:00PM, Location: Birge B302<br />
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Title: Convexity, uniqueness, and stability of the regular shock reflection-diffraction problem. <br />
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Abstract: We will talk about our recent results on the convexity, uniqueness, and stability of regular reflection solutions for the potential flow equation in a natural class of self-similar solutions. The approach is based on a nonlinear version of the method of continuity. <br />
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'''March 11, 2024'''<br />
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Rajendra Beekie (Duke). Host: Dallas Albritton<br />
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Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
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Title:<br />
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'''March 18, 2024'''<br />
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'''March 25, 2024'''<br />
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Spring Break. No seminar.<br />
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'''April 1, 2024'''<br />
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Dominic Wynter (Cambridge). Host: Chanwoo Kim<br />
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Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
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Title:<br />
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'''April 8, 2024'''<br />
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Fizay-Noah Lee (Vanderbilt). Host: Dallas Albritton<br />
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Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
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Title:<br />
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'''April 15, 2024'''<br />
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Myoungjean Bae (KAIST). Host: Mikhail Feldman<br />
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Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
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Title:<br />
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'''April 22, 2024'''<br />
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Sarah Strikwerda (Penn). Host: Hung Tran.<br />
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Format: in-person. Time: 3:30-4:30PM, VV 901<br />
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'''April 29, 2024'''<br />
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Joonhyun La (Princeton). Host: Dallas Albritton<br />
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Format: in-person. Time: 3:30-4:30PM, VV 901<br />
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The last day of class is Friday, May 3, 2023.<br />
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== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
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=Spring 2023= <br />
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'''January 30, 2023 '''<br />
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[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
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Time: 3:30 PM -4:30 PM, in person in VV901 <br />
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Title: Mean curvature flows in the sphere via phase transitions.<br />
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Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
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This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
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'''February 6, 2023'''<br />
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[[No seminar]] <br />
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'''February 13, 2023'''<br />
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[[Trinh Tien Nguyen]] (UW Madison) <br />
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Format: In person, Time: 3:30-4:30PM. <br />
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'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
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'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
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'''February 20, 2023'''<br />
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[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
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Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
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'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
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'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
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'''February 27, 2023'''<br />
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[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
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'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
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'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
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'''March 6, 2023'''<br />
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Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
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Format: In-person, Time: 3:30-4:30PM. <br />
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'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
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'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
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'''March 13, 2023'''<br />
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[[Spring Recess: No Seminar]] <br />
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'''March 20, 2023''' <br />
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Format: , Time: <br />
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'''Title:''' <br />
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'''Abstract:'''<br />
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'''March 27, 2023'''<br />
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[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
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Format: In person, Time: 3:30-4:30PM. <br />
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'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
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'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
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An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
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In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
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'''April 3, 2023'''<br />
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Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
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Format: In person, Time: 3:30-4:30PM. <br />
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'''Title:''' ''Translating mean curvature flow with simple end.''<br />
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'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
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'''April 10, 2023'''<br />
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Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
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Format: In person, Time: 3:30-4:30PM. <br />
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'''Title:''' ''Large amplitude solution of BGK model''<br />
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'''Abstract:''' Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time.<br />
This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.<br />
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'''April 17, 2023'''<br />
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[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
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Format: In person, Time: 3:30-4:30PM. <br />
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'''Title:''' ''Interior W^{2,p} estimates for complex Monge-Ampere equations'' <br />
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'''Abstract:''' The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.<br />
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'''April 24, 2023'''<br />
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[[Ben Pineau| Ben Pineau]] (Berkeley)<br />
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Format: In person, Time: 3:30-4:30PM. <br />
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'''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion<br />
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'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.<br />
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'''May 1, 2023'''<br />
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[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
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Time: 3:30 PM -4:30 PM, in person in VV901 <br />
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'''Title:'''The Boltzmann equation with large data<br />
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'''Abstract:''' The Boltzmann equation is a nonlocal, nonlinear equation arising in gas dynamics for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk I will discuss a recent program to understand a more tractable, related question: what is the largest space in which local well-posedness holds and what quantities prevent blow-up at finite times? This program covers several papers and intertwines with a recent push to understand the regularity theory of kinetic Fokker-Planck-type equations. I will give a broad outline of the proof, and then, instead of slogging through the whole thing, I will focus on a simple proof of a technically important and physically interesting lower bound -- how ``vacuum regions'' fill in. This is a joint work with Snelson and Tarfulea.<br />
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'''May 8, 2023'''<br />
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[[TBA|Lei Wu (Lehigh)]]. Host: Chanwoo Kim<br />
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Time: 3:30 PM -4:30 PM, in person in '''VV B223 (special room)''' <br />
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'''Title:'''Ghost Effect from Boltzmann Theory<br />
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'''Abstract:'''It is a classical and fundamental problem to study the hydrodynamic limits of kinetic equations as the Knudsen number \varepsilon\rightarrow 0. In this talk, we focus on the stationary Boltzmann equation with diffuse-reflection boundary condition where the Mach number is O(\varepsilon) but the temperature variance is O(1). We rigorously derived the limiting fluid system, the so-called ghost-effect equations, in which an infinitesimal boundary variance will have macroscopic observable effects. This pure kinetic effect reveals the incompleteness of traditional fluid theory. Our proof relies on newly developed kernel estimates with conservation laws and novel BV estimates for the cutoff boundary layer.<br />
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=Fall 2022= <br />
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'''September 12, 2022'''<br />
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[[No Seminar]]<br />
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'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
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[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
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Format: in-person. Time: 4-5PM, VV B139. <br />
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Title: Homogenization in front propagation models<br />
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Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
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'''September 26, 2022 ''' <br />
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[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
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Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
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<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
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<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
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'''October 3, 2022'''<br />
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[[No Seminar]]<br />
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Title:<br />
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Abstract:<br />
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'''October 10, 2022'''<br />
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[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
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Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
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Speaker: Sasha Kiselev (Duke)<br />
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Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
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'''October 17, 2022'''<br />
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[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
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Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
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'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
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'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
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'''October 24, 2022'''<br />
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[[No seminar.]]<br />
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Format: , Time: <br />
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Title:<br />
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<br />
<br />
<br />
<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=26165PDE Geometric Analysis seminar2024-02-19T23:48:46Z<p>Htran24: /* Spring 2024 */</p>
<hr />
<div>All talks will be ''in person'' unless specified otherwise.<br />
<br />
(Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.) <br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Spring 2024 =<br />
<br />
<br />
<br />
The first day of class is Tuesday, January 23, 2023.<br />
<br />
<br />
<br />
'''January 23, 2024 (special date/time)'''<br />
<br />
Donghyun Lee (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: 901 Van Vleck Hall<br />
<br />
Title: DYNAMICAL BILLIARD AND A LONG-TIME BEHAVIOR OF THE BOLTZMANN EQUATION IN GENERAL 3D TOROIDAL DOMAINS<br />
<br />
Abstract: Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant progress in this question when domains are strictly convex, the same question without the strict convexity of domains is still totally open in 3D. The major difficulty arises when a billiard map has an infinite number of bounces in a finite time interval or when the map fails to be Lipschitz continuous, both of which happen generically when the domain is non-convex. In this paper, we develop a new method to control a billiard map on a surface of revolution generated by revolving any planar analytic convex closed curve (e.g., typical shape of tokamak reactors’ chamber). In particular, we classify and measure the size (to be small) of a pullback set (along the billiard trajectory) of the infinite-bouncing and singular- bouncing cases. As a consequence, we solve the open question affirmatively in such domains. To the best of our knowledge, this work is the first construction of global solutions to the hard-sphere Boltzmann equation in generic non-convex 3-dimensional domains. This is joint work with Gyounghun Ko and Chanwoo Kim.<br />
<br />
<br />
<br />
'''January 29, 2024'''<br />
<br />
Minh-Binh Tran (TAMU). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Some Results On the Kinetic Theory for Classical and Quantum Waves<br />
<br />
Abstract: Kinetic equations can be used to describe the dynamics of nonlinear classical and quantum waves out of thermal equilibrium, as well as the propagation of waves in a random medium. In this talk, I will present some of our recent results on the kinetic theory of waves. I will discuss the analysis of those kinetic equations for waves. Next, I will focus on the numerical schemes we have been developing to resolve those equations. I will also address some control problems concerning kinetic equations for waves. The last part is devoted to some physical applications of wave kinetic theory for Bose-Einstein Condensates.<br />
<br />
<br />
<br />
'''January 30, 2024 (special date/time)'''<br />
<br />
Jin Woo Jang (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: 901 Van Vleck Hall<br />
<br />
Title: On the Relativistic Boltzmann Equation with Long Range Interactions<br />
<br />
Abstract: In this talk, I will discuss three recent, interrelated results concerning the special relativistic Boltzmann equation without angular cutoff. In the non-relativistic situation without angular cutoff, the change of variables from $v \to v'$ is a crux of the widely used "cancellation lemma". Firstly, in collaboration with James Chapman and Robert M. Strain, we calculate this very complex ten variable Jacobian determinant in the special relativisticsituation and illustrate some numerical results which show that it has a large number of distinct points where it is machine zero. Secondly, with Strain, we prove the sharp pointwise asymptotics for the frequency multiplier of the linearized relativistic Boltzmann collision operator that has not been previously established. As a consequence of these calculations, we further explain why the well known change of variables p \to p' is not well defined in the special relativisticcontext. Finally, also with Strain, we will present our recent proof of global-in-time existence and uniqueness of the solutions nearby the relativistic Maxwellian to the special relativistic Boltzmann equation without any angular cutoff and its asymptotic stability. We work in the case of a spatially periodic box. We assume the generic hard-interaction and mildly-soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Jeżewska (in 1985). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. <br />
<br />
<br />
<br />
'''February 5, 2024'''<br />
<br />
Thierry Laurens (UW-Madison) <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Continuum Calogero--Moser models<br />
<br />
Abstract: The focusing CCM model is a dispersive equation that describes a continuum limit of a particle gas interacting pairwise through an inverse square potential. Recently, Gérard and Lenzmann discovered solutions to this equation that exhibit frequency cascades.<br />
<br />
In this talk, we will discuss a scaling-critical well-posedness result for the focusing and defocusing CCM models on the line. In the focusing case, this requires solutions to have mass less than that of the soliton. This is joint work with Rowan Killip and Monica Visan.<br />
<br />
<br />
<br />
'''February 12, 2024'''<br />
<br />
Laurel Ohm (UW-Madison)<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Free boundary dynamics of an elastic filament in 3D Stokes flow<br />
<br />
Abstract: We consider a free boundary problem for a thin elastic filament immersed in 3D Stokes flow. The 3D fluid is coupled to the quasi-1D filament dynamics via a novel type of angle-averaged Neumann-to-Dirichlet operator. Much of the difficulty in the analysis lies in understanding this operator. We show that the principal part of this NtD map is the corresponding operator about a straight, periodic filament, for which we derive an explicit symbol. It is then possible to establish local well-posedness for an immersed filament evolving via a simple elasticity law. This establishes a mathematical foundation for the myriad computational results based on slender body approximations for thin immersed elastic structures.<br />
<br />
'''February 19, 2024'''<br />
<br />
<br />
<br />
'''February 26, 2024'''<br />
<br />
Jiwoong Jang (UW-Madison)<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Periodic homogenization of geometric equations without perturbed correctors.<br />
<br />
Abstract: Proving homogenization has been a subtle issue for geometric equations due to the discontinuity when the gradient vanishes. To overcome the difficulty and conclude homogenization, the work of Caffarelli-Monneau suggests a sufficient condition using perturbed correctors. However, some noncoercive equations do not satisfy this condition. In this talk, we discuss homogenization of geometric equations without using perturbed correctors, and we conclude homogenization for the noncoercive equations. Also, we derive a rate of periodic homogenization of coercive geometric equations by utilizing the fact that they remain coercive under perturbation. We also present an example that homogenizes with a rate Ω(ε| log ε|).<br />
<br />
<br />
'''March 4, 2024'''<br />
<br />
Yuming Paul Zhang (Auburn). Host: Hung Tran.<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''March 5, 2024 (special date/time)'''<br />
<br />
Wei Xiang (City U. of Hong Kong). Host: Mikhail Feldman<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: Birge B302<br />
<br />
Title: <br />
<br />
'''March 11, 2024'''<br />
<br />
Rajendra Beekie (Duke). Host: Dallas Albritton<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''March 18, 2024'''<br />
<br />
'''March 25, 2024'''<br />
<br />
Spring Break. No seminar.<br />
<br />
'''April 1, 2024'''<br />
<br />
Dominic Wynter (Cambridge). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''April 8, 2024'''<br />
<br />
Fizay-Noah Lee (Vanderbilt). Host: Dallas Albritton<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
'''April 15, 2024'''<br />
<br />
Myoungjean Bae (KAIST). Host: Mikhail Feldman<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
'''April 22, 2024'''<br />
<br />
Sarah Strikwerda (Penn). Host: Hung Tran.<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901<br />
<br />
'''April 29, 2024'''<br />
<br />
Joonhyun La (Princeton). Host: Dallas Albritton<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901<br />
<br />
The last day of class is Friday, May 3, 2023.<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
<br />
'''March 20, 2023''' <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
<br />
'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
<br />
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
<br />
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Large amplitude solution of BGK model''<br />
<br />
'''Abstract:''' Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time.<br />
This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.<br />
<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Interior W^{2,p} estimates for complex Monge-Ampere equations'' <br />
<br />
'''Abstract:''' The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] (Berkeley)<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion<br />
<br />
'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''The Boltzmann equation with large data<br />
<br />
'''Abstract:''' The Boltzmann equation is a nonlocal, nonlinear equation arising in gas dynamics for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk I will discuss a recent program to understand a more tractable, related question: what is the largest space in which local well-posedness holds and what quantities prevent blow-up at finite times? This program covers several papers and intertwines with a recent push to understand the regularity theory of kinetic Fokker-Planck-type equations. I will give a broad outline of the proof, and then, instead of slogging through the whole thing, I will focus on a simple proof of a technically important and physically interesting lower bound -- how ``vacuum regions'' fill in. This is a joint work with Snelson and Tarfulea.<br />
<br />
<br />
<br />
<br />
'''May 8, 2023'''<br />
<br />
[[TBA|Lei Wu (Lehigh)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in '''VV B223 (special room)''' <br />
<br />
'''Title:'''Ghost Effect from Boltzmann Theory<br />
<br />
'''Abstract:'''It is a classical and fundamental problem to study the hydrodynamic limits of kinetic equations as the Knudsen number \varepsilon\rightarrow 0. In this talk, we focus on the stationary Boltzmann equation with diffuse-reflection boundary condition where the Mach number is O(\varepsilon) but the temperature variance is O(1). We rigorously derived the limiting fluid system, the so-called ghost-effect equations, in which an infinitesimal boundary variance will have macroscopic observable effects. This pure kinetic effect reveals the incompleteness of traditional fluid theory. Our proof relies on newly developed kernel estimates with conservation laws and novel BV estimates for the cutoff boundary layer.<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
<br />
<br />
<br />
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<br />
<br />
<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=26006PDE Geometric Analysis seminar2024-01-30T13:42:31Z<p>Htran24: /* Spring 2024 */</p>
<hr />
<div>All talks will be ''in person'' unless specified otherwise.<br />
<br />
(Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.) <br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Spring 2024 =<br />
<br />
<br />
<br />
The first day of class is Tuesday, January 23, 2023.<br />
<br />
<br />
<br />
'''January 23, 2024 (special date/time)'''<br />
<br />
Donghyun Lee (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: 901 Van Vleck Hall<br />
<br />
Title: DYNAMICAL BILLIARD AND A LONG-TIME BEHAVIOR OF THE BOLTZMANN EQUATION IN GENERAL 3D TOROIDAL DOMAINS<br />
<br />
Abstract: Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant progress in this question when domains are strictly convex, the same question without the strict convexity of domains is still totally open in 3D. The major difficulty arises when a billiard map has an infinite number of bounces in a finite time interval or when the map fails to be Lipschitz continuous, both of which happen generically when the domain is non-convex. In this paper, we develop a new method to control a billiard map on a surface of revolution generated by revolving any planar analytic convex closed curve (e.g., typical shape of tokamak reactors’ chamber). In particular, we classify and measure the size (to be small) of a pullback set (along the billiard trajectory) of the infinite-bouncing and singular- bouncing cases. As a consequence, we solve the open question affirmatively in such domains. To the best of our knowledge, this work is the first construction of global solutions to the hard-sphere Boltzmann equation in generic non-convex 3-dimensional domains. This is joint work with Gyounghun Ko and Chanwoo Kim.<br />
<br />
<br />
<br />
'''January 29, 2024'''<br />
<br />
Minh-Binh Tran (TAMU). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Some Results On the Kinetic Theory for Classical and Quantum Waves<br />
<br />
Abstract: Kinetic equations can be used to describe the dynamics of nonlinear classical and quantum waves out of thermal equilibrium, as well as the propagation of waves in a random medium. In this talk, I will present some of our recent results on the kinetic theory of waves. I will discuss the analysis of those kinetic equations for waves. Next, I will focus on the numerical schemes we have been developing to resolve those equations. I will also address some control problems concerning kinetic equations for waves. The last part is devoted to some physical applications of wave kinetic theory for Bose-Einstein Condensates.<br />
<br />
<br />
<br />
'''January 30, 2024 (special date/time)'''<br />
<br />
Jin Woo Jang (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: 901 Van Vleck Hall<br />
<br />
Title: On the Relativistic Boltzmann Equation with Long Range Interactions<br />
<br />
Abstract: In this talk, I will discuss three recent, interrelated results concerning the special relativistic Boltzmann equation without angular cutoff. In the non-relativistic situation without angular cutoff, the change of variables from $v \to v'$ is a crux of the widely used "cancellation lemma". Firstly, in collaboration with James Chapman and Robert M. Strain, we calculate this very complex ten variable Jacobian determinant in the special relativisticsituation and illustrate some numerical results which show that it has a large number of distinct points where it is machine zero. Secondly, with Strain, we prove the sharp pointwise asymptotics for the frequency multiplier of the linearized relativistic Boltzmann collision operator that has not been previously established. As a consequence of these calculations, we further explain why the well known change of variables p \to p' is not well defined in the special relativisticcontext. Finally, also with Strain, we will present our recent proof of global-in-time existence and uniqueness of the solutions nearby the relativistic Maxwellian to the special relativistic Boltzmann equation without any angular cutoff and its asymptotic stability. We work in the case of a spatially periodic box. We assume the generic hard-interaction and mildly-soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Jeżewska (in 1985). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. <br />
<br />
<br />
<br />
'''February 5, 2024'''<br />
<br />
Thierry Laurens (UW-Madison) <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: <br />
<br />
<br />
<br />
'''February 12, 2024'''<br />
<br />
Laurel Ohm (UW-Madison)<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''February 19, 2024'''<br />
<br />
<br />
<br />
'''February 26, 2024'''<br />
<br />
Jiwoong Jang (UW-Madison)<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
'''March 4, 2024'''<br />
<br />
Yuming Paul Zhang (Auburn). Host: Hung Tran.<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
'''March 11, 2024'''<br />
<br />
Rajendra Beekie (Duke). Host: Dallas Albritton<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''March 18, 2024'''<br />
<br />
'''March 25, 2024'''<br />
<br />
Spring Break. No seminar.<br />
<br />
'''April 1, 2024'''<br />
<br />
Dominic Wynter (Cambridge). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''April 8, 2024'''<br />
<br />
Fizay-Noah Lee (Vanderbilt). Host: Dallas Albritton<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
'''April 15, 2024'''<br />
<br />
Myoungjean Bae (KAIST). Host: Mikhail Feldman<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''April 22, 2024'''<br />
<br />
Sarah Strikwerda (Penn). Host: Hung Tran.<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
'''April 29, 2024'''<br />
<br />
The last day of class is Friday, May 3, 2023.<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
<br />
'''March 20, 2023''' <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
<br />
'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
<br />
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
<br />
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Large amplitude solution of BGK model''<br />
<br />
'''Abstract:''' Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time.<br />
This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.<br />
<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Interior W^{2,p} estimates for complex Monge-Ampere equations'' <br />
<br />
'''Abstract:''' The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] (Berkeley)<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion<br />
<br />
'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''The Boltzmann equation with large data<br />
<br />
'''Abstract:''' The Boltzmann equation is a nonlocal, nonlinear equation arising in gas dynamics for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk I will discuss a recent program to understand a more tractable, related question: what is the largest space in which local well-posedness holds and what quantities prevent blow-up at finite times? This program covers several papers and intertwines with a recent push to understand the regularity theory of kinetic Fokker-Planck-type equations. I will give a broad outline of the proof, and then, instead of slogging through the whole thing, I will focus on a simple proof of a technically important and physically interesting lower bound -- how ``vacuum regions'' fill in. This is a joint work with Snelson and Tarfulea.<br />
<br />
<br />
<br />
<br />
'''May 8, 2023'''<br />
<br />
[[TBA|Lei Wu (Lehigh)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in '''VV B223 (special room)''' <br />
<br />
'''Title:'''Ghost Effect from Boltzmann Theory<br />
<br />
'''Abstract:'''It is a classical and fundamental problem to study the hydrodynamic limits of kinetic equations as the Knudsen number \varepsilon\rightarrow 0. In this talk, we focus on the stationary Boltzmann equation with diffuse-reflection boundary condition where the Mach number is O(\varepsilon) but the temperature variance is O(1). We rigorously derived the limiting fluid system, the so-called ghost-effect equations, in which an infinitesimal boundary variance will have macroscopic observable effects. This pure kinetic effect reveals the incompleteness of traditional fluid theory. Our proof relies on newly developed kernel estimates with conservation laws and novel BV estimates for the cutoff boundary layer.<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
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<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
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<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
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<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
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<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=26004PDE Geometric Analysis seminar2024-01-30T01:39:13Z<p>Htran24: /* Spring 2024 */</p>
<hr />
<div>All talks will be ''in person'' unless specified otherwise.<br />
<br />
(Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.) <br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Spring 2024 =<br />
<br />
<br />
<br />
The first day of class is Tuesday, January 23, 2023.<br />
<br />
<br />
<br />
'''January 23, 2024 (special date/time)'''<br />
<br />
Donghyun Lee (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: 901 Van Vleck Hall<br />
<br />
Title: DYNAMICAL BILLIARD AND A LONG-TIME BEHAVIOR OF THE BOLTZMANN EQUATION IN GENERAL 3D TOROIDAL DOMAINS<br />
<br />
Abstract: Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant progress in this question when domains are strictly convex, the same question without the strict convexity of domains is still totally open in 3D. The major difficulty arises when a billiard map has an infinite number of bounces in a finite time interval or when the map fails to be Lipschitz continuous, both of which happen generically when the domain is non-convex. In this paper, we develop a new method to control a billiard map on a surface of revolution generated by revolving any planar analytic convex closed curve (e.g., typical shape of tokamak reactors’ chamber). In particular, we classify and measure the size (to be small) of a pullback set (along the billiard trajectory) of the infinite-bouncing and singular- bouncing cases. As a consequence, we solve the open question affirmatively in such domains. To the best of our knowledge, this work is the first construction of global solutions to the hard-sphere Boltzmann equation in generic non-convex 3-dimensional domains. This is joint work with Gyounghun Ko and Chanwoo Kim.<br />
<br />
<br />
<br />
'''January 29, 2024'''<br />
<br />
Minh-Binh Tran (TAMU). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Some Results On the Kinetic Theory for Classical and Quantum Waves<br />
<br />
Abstract: Kinetic equations can be used to describe the dynamics of nonlinear classical and quantum waves out of thermal equilibrium, as well as the propagation of waves in a random medium. In this talk, I will present some of our recent results on the kinetic theory of waves. I will discuss the analysis of those kinetic equations for waves. Next, I will focus on the numerical schemes we have been developing to resolve those equations. I will also address some control problems concerning kinetic equations for waves. The last part is devoted to some physical applications of wave kinetic theory for Bose-Einstein Condensates.<br />
<br />
<br />
<br />
'''January 30, 2024 (special date/time)'''<br />
<br />
Jin Woo Jang (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: 901 Van Vleck Hall<br />
<br />
Title: On the Relativistic Boltzmann Equation with Long Range Interactions<br />
<br />
Abstract: In this talk, I will discuss three recent, interrelated results concerning the special relativistic Boltzmann equation without angular cutoff. In the non-relativistic situation without angular cutoff, the change of variables from $v \to v'$ is a crux of the widely used "cancellation lemma". Firstly, in collaboration with James Chapman and Robert M. Strain, we calculate this very complex ten variable Jacobian determinant in the special relativisticsituation and illustrate some numerical results which show that it has a large number of distinct points where it is machine zero. Secondly, with Strain, we prove the sharp pointwise asymptotics for the frequency multiplier of the linearized relativistic Boltzmann collision operator that has not been previously established. As a consequence of these calculations, we further explain why the well known change of variables p \to p' is not well defined in the special relativisticcontext. Finally, also with Strain, we will present our recent proof of global-in-time existence and uniqueness of the solutions nearby the relativistic Maxwellian to the special relativistic Boltzmann equation without any angular cutoff and its asymptotic stability. We work in the case of a spatially periodic box. We assume the generic hard-interaction and mildly-soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Jeżewska (in 1985). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. <br />
<br />
<br />
<br />
'''February 5, 2024'''<br />
<br />
Thierry Laurens (UW-Madison) <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: <br />
<br />
<br />
<br />
'''February 12, 2024'''<br />
<br />
Laurel Ohm (UW-Madison)<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''February 19, 2024'''<br />
<br />
<br />
<br />
'''February 26, 2024'''<br />
<br />
(open; tentative talk cancelled)<br />
<br />
<br />
'''March 4, 2024'''<br />
<br />
Yuming Paul Zhang (Auburn). Host: Hung Tran.<br />
<br />
'''March 11, 2024'''<br />
<br />
Rajendra Beekie (Duke). Host: Dallas Albritton<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''March 18, 2024'''<br />
<br />
'''March 25, 2024'''<br />
<br />
Spring Break. No seminar.<br />
<br />
'''April 1, 2024'''<br />
<br />
Dominic Wynter (Cambridge). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''April 8, 2024'''<br />
<br />
Fizay-Noah Lee (Vanderbilt). Host: Dallas Albritton<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
'''April 15, 2024'''<br />
<br />
Myoungjean Bae (KAIST). Host: Mikhail Feldman<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''April 22, 2024'''<br />
<br />
Sarah Strikwerda (Penn). Host: Hung Tran.Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
'''April 29, 2024'''<br />
<br />
The last day of class is Friday, May 3, 2023.<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
<br />
'''March 20, 2023''' <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
<br />
'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
<br />
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
<br />
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Large amplitude solution of BGK model''<br />
<br />
'''Abstract:''' Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time.<br />
This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.<br />
<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Interior W^{2,p} estimates for complex Monge-Ampere equations'' <br />
<br />
'''Abstract:''' The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] (Berkeley)<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion<br />
<br />
'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''The Boltzmann equation with large data<br />
<br />
'''Abstract:''' The Boltzmann equation is a nonlocal, nonlinear equation arising in gas dynamics for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk I will discuss a recent program to understand a more tractable, related question: what is the largest space in which local well-posedness holds and what quantities prevent blow-up at finite times? This program covers several papers and intertwines with a recent push to understand the regularity theory of kinetic Fokker-Planck-type equations. I will give a broad outline of the proof, and then, instead of slogging through the whole thing, I will focus on a simple proof of a technically important and physically interesting lower bound -- how ``vacuum regions'' fill in. This is a joint work with Snelson and Tarfulea.<br />
<br />
<br />
<br />
<br />
'''May 8, 2023'''<br />
<br />
[[TBA|Lei Wu (Lehigh)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in '''VV B223 (special room)''' <br />
<br />
'''Title:'''Ghost Effect from Boltzmann Theory<br />
<br />
'''Abstract:'''It is a classical and fundamental problem to study the hydrodynamic limits of kinetic equations as the Knudsen number \varepsilon\rightarrow 0. In this talk, we focus on the stationary Boltzmann equation with diffuse-reflection boundary condition where the Mach number is O(\varepsilon) but the temperature variance is O(1). We rigorously derived the limiting fluid system, the so-called ghost-effect equations, in which an infinitesimal boundary variance will have macroscopic observable effects. This pure kinetic effect reveals the incompleteness of traditional fluid theory. Our proof relies on newly developed kernel estimates with conservation laws and novel BV estimates for the cutoff boundary layer.<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
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<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
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Format: , Time: <br />
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Title:<br />
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Abstract:<br />
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<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
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<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
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Format: , Time: <br />
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Title:<br />
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Abstract:<br />
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<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
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<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
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<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
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<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
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===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=Research_at_UW-Madison_in_DifferentialEquations&diff=25984Research at UW-Madison in DifferentialEquations2024-01-29T04:22:07Z<p>Htran24: /* Recent former Postdocs in PDE */</p>
<hr />
<div>==Seminars of interest==<br />
<br />
The weekly [http://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE & Geometric Analysis seminar] is held on Monday afternoons, 3:30-4:30pm. <br />
<br />
Other seminars that will feature PDE related material are the [http://www.math.wisc.edu/wiki/index.php/Geometry_and_Topology_Seminar Geometry and Topology seminar], the [https://www.math.wisc.edu/wiki/index.php/Analysis_Seminar Analysis seminar], and the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math seminar].<br />
<br />
[https://sites.google.com/view/vapseminar Virtual Analysis and PDE Seminar (VAPS)]: organized jointly by 9 US universities including UW.<br />
<br />
==Faculty==<br />
<br />
[https://sites.google.com/view/albri050/dallas-albritton Dallas Albritton] (University of Minnesota, 2020) Pure and applied aspects of PDE and fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~feldman Mikhail Feldman] (UC Berkeley, 1994) Nonlinear PDE, Calculus of Variations.<br />
<br />
[http://www.math.wisc.edu/~ifrim/Home.html Mihaela Ifrim] (UC Davis, 2012) Nonlinear Dispersive Equations (water-wave equations and related dispersive models), Fluid Mechanics, Elastodynamics, Harmonic Analysis, General Relativity.<br />
<br />
[https://sites.google.com/view/ckim Chanwoo Kim] (Brown, 2011) Applied PDE, Kinetic theory, Fluid dynamics.<br />
<br />
[https://sites.google.com/a/umn.edu/math-laurelohm/home Laurel Ohm] (University of Minnesota, 2020) Analysis of PDE arising in biofluid mechanics.<br />
<br />
[http://www.math.wisc.edu/~hung Hung Vinh Tran] (UC Berkeley, 2012) Nonlinear PDE.<br />
<br />
==Faculty in related areas==<br />
<br />
[http://www.math.wisc.edu/~denissov Sergey Denisov] (Moscow State University, 1999) Analysis, PDE.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li] (UW Madison, 2013) Numerical analysis and scientific computing.<br />
<br />
[http://www.math.wisc.edu/~spagnolie/ Saverio Spagnolie] (Courant Institute, 2008) Fluid dynamics, complex fluids, soft matter, computation.<br />
<br />
[http://www.math.wisc.edu/~stechmann/ Sam Stechmann] (Courant Institute, 2008) Applied math, computational math, fluid dynamics, atmospheric science, climate.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (UT Austin, 1998) Mixing in fluids, optimization of mixing.<br />
<br />
[https://www.nicolasgarciat.com/ Nicolas Garcia Trillos] (CMU, 2015) Optimal transport, calculus of variations, and data analysis.<br />
==Current Postdocs (Van Vleck Assistant Professor) in PDE==<br />
<br />
[https://www.math.wisc.edu/~aai/ Albert Ai]<br />
<br />
[https://sites.google.com/view/tnguyen65 Trinh Nguyen]<br />
<br />
[https://people.math.wisc.edu/~laurens/ Thierry Laurens]<br />
<br />
==Recent former Postdocs in PDE==<br />
<br />
[https://blog.nus.edu.sg/yyao/ Yao Yao] (VV assist prof 2012-2015). Current position: Dean’s Chair Associate Professor, National University of Singapore.<br />
<br />
[https://sites.google.com/view/KYUDONGCHOI Kyudong Choi] (VV assist prof 2012-2015). Current position: Associate Professor, UNIST (2015-).<br />
<br />
[https://sites.google.com/view/jessicalin-math/home Jessica Lin] (VV assist prof 2014-2017). Current position: Assistant Professor, McGill University (2017-).<br />
<br />
[https://sites.google.com/site/donghyunlee295/ Donghyun Lee] (VV assist prof 2015-2018). Current position: Associate Professor and Mueunjae Distinguished Professor, Postech (2018-). <br />
<br />
[https://minhbinhtran.org/ Minh-Binh Tran] (VV assist prof 2015-2018). Current position: Associate Professor at Texas A&M University (2022-). <br />
<br />
[https://cam.uchicago.edu/people/profile/eric-baer/ Eric Baer] (VV assist prof 2015-2018). Current position: Senior Lecturer, University of Chicago (2019-).<br />
<br />
[https://sites.google.com/site/guoxx097/welcome Xiaoqin Guo] (VV assist prof 2017-2020). Current position: Assistant Professor, University of Cincinnati (2020-).<br />
<br />
[http://www.homepages.ucl.ac.uk/~ucahms0/index.htm Matthew Schrecker] (VV assist prof 2018-2020). Current position: EPSRC Postdoctoral Research Fellow, University College London (2020-).<br />
<br />
[https://sites.google.com/site/dhkwonmath/home?authuser=0 Dohyun Kwon] (VV assist prof 9/2020-1/2023). Current position: Assistant Professor, University of Seoul.<br />
<br />
[https://schulzmath.wordpress.com/ Simon Schulz] (VV assist prof 9/2021-8/2023). Current position: Junior Visiting Position at the Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale Superiore.<br />
<br />
==Emeriti==<br />
<br />
[http://www.math.wisc.edu/~angenent Sigurd Angenent] (Leiden, 1986) Nonlinear PDE, differential geometry, medical imaging, math biology.<br />
<br />
[http://www.math.wisc.edu/~bolotin Sergey Bolotin] (Moscow State University, 1982) Dynamical Systems, Variational Methods, Celestial Mechanics.<br />
<br />
[http://www.math.wisc.edu/~rabinowi Paul Rabinowitz]<br />
PDE, Calculus of Variations, Dynamical Systems, Nonlinear Analysis.<br />
<br />
[http://www.math.wisc.edu/~robbin Joel Robbin]<br />
Global Analysis, Differential Equations.<br />
<br />
[http://www.math.wisc.edu/~turner Robert Turner]<br />
Partial Differential Equations, Fluid Mechanics, Mathematical Biology.<br />
<br />
==Previous events==<br />
<br />
The 81st Midwest PDE seminar '''[https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]''' was held in Madison on April 21/22 (2018).</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=25964PDE Geometric Analysis seminar2024-01-23T23:59:03Z<p>Htran24: /* Previous PDE/GA seminars */</p>
<hr />
<div>All talks will be ''in person'' unless specified otherwise.<br />
<br />
(Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.) <br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Spring 2024 =<br />
<br />
<br />
<br />
The first day of class is Tuesday, January 23, 2023.<br />
<br />
<br />
<br />
'''January 23, 2024 (special date/time)'''<br />
<br />
Donghyun Lee (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: 901 Van Vleck Hall<br />
<br />
Title: DYNAMICAL BILLIARD AND A LONG-TIME BEHAVIOR OF THE BOLTZMANN EQUATION IN GENERAL 3D TOROIDAL DOMAINS<br />
<br />
Abstract: Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant progress in this question when domains are strictly convex, the same question without the strict convexity of domains is still totally open in 3D. The major difficulty arises when a billiard map has an infinite number of bounces in a finite time interval or when the map fails to be Lipschitz continuous, both of which happen generically when the domain is non-convex. In this paper, we develop a new method to control a billiard map on a surface of revolution generated by revolving any planar analytic convex closed curve (e.g., typical shape of tokamak reactors’ chamber). In particular, we classify and measure the size (to be small) of a pullback set (along the billiard trajectory) of the infinite-bouncing and singular- bouncing cases. As a consequence, we solve the open question affirmatively in such domains. To the best of our knowledge, this work is the first construction of global solutions to the hard-sphere Boltzmann equation in generic non-convex 3-dimensional domains. This is joint work with Gyounghun Ko and Chanwoo Kim.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''January 29, 2024'''<br />
<br />
Minh-Binh Tran (TAMU). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Some Results On the Kinetic Theory for Classical and Quantum Waves<br />
<br />
Abstract: Kinetic equations can be used to describe the dynamics of nonlinear classical and quantum waves out of thermal equilibrium, as well as the propagation of waves in a random medium. In this talk, I will present some of our recent results on the kinetic theory of waves. I will discuss the analysis of those kinetic equations for waves. Next, I will focus on the numerical schemes we have been developing to resolve those equations. I will also address some control problems concerning kinetic equations for waves. The last part is devoted to some physical applications of wave kinetic theory for Bose-Einstein Condensates.<br />
<br />
<br />
<br />
<br />
'''January 30, 2024 (special date/time)'''<br />
<br />
Jin Woo Jang (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: 901 Van Vleck Hall<br />
<br />
Title:<br />
<br />
<br />
<br />
<br />
<br />
'''February 5, 2024''' <br />
<br />
Thierry Laurens (UW-Madison) <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: <br />
<br />
<br />
<br />
'''February 12, 2024'''<br />
<br />
Laurel Ohm (UW-Madison)<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''February 19, 2024'''<br />
<br />
<br />
<br />
'''February 26, 2024'''<br />
<br />
Katy Craig (UCSB). Host: Nicolas Garcia Trillos, Hung Tran.<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
'''March 4, 2024'''<br />
<br />
Yuming Paul Zhang (Auburn). Host: Hung Tran.<br />
<br />
<br />
<br />
'''March 11, 2024'''<br />
<br />
Rajendra Beekie (Duke). Host: Dallas Albritton<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''March 18, 2024'''<br />
<br />
'''March 25, 2024'''<br />
<br />
Spring Break. No seminar.<br />
<br />
'''April 1, 2024'''<br />
<br />
Dominic Wynter (Cambridge). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''April 8, 2024'''<br />
<br />
<br />
<br />
'''April 15, 2024'''<br />
<br />
Myoungjean Bae (KAIST). Host: Mikhail Feldman<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''April 22, 2024'''<br />
<br />
Sarah Strikwerda (Penn). Host: Hung Tran.Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
'''April 29, 2024'''<br />
<br />
The last day of class is Friday, May 3, 2023.<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
<br />
'''March 20, 2023''' <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
<br />
'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
<br />
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
<br />
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Large amplitude solution of BGK model''<br />
<br />
'''Abstract:''' Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time.<br />
This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.<br />
<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Interior W^{2,p} estimates for complex Monge-Ampere equations'' <br />
<br />
'''Abstract:''' The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] (Berkeley)<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion<br />
<br />
'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''The Boltzmann equation with large data<br />
<br />
'''Abstract:''' The Boltzmann equation is a nonlocal, nonlinear equation arising in gas dynamics for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk I will discuss a recent program to understand a more tractable, related question: what is the largest space in which local well-posedness holds and what quantities prevent blow-up at finite times? This program covers several papers and intertwines with a recent push to understand the regularity theory of kinetic Fokker-Planck-type equations. I will give a broad outline of the proof, and then, instead of slogging through the whole thing, I will focus on a simple proof of a technically important and physically interesting lower bound -- how ``vacuum regions'' fill in. This is a joint work with Snelson and Tarfulea.<br />
<br />
<br />
<br />
<br />
'''May 8, 2023'''<br />
<br />
[[TBA|Lei Wu (Lehigh)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in '''VV B223 (special room)''' <br />
<br />
'''Title:'''Ghost Effect from Boltzmann Theory<br />
<br />
'''Abstract:'''It is a classical and fundamental problem to study the hydrodynamic limits of kinetic equations as the Knudsen number \varepsilon\rightarrow 0. In this talk, we focus on the stationary Boltzmann equation with diffuse-reflection boundary condition where the Mach number is O(\varepsilon) but the temperature variance is O(1). We rigorously derived the limiting fluid system, the so-called ghost-effect equations, in which an infinitesimal boundary variance will have macroscopic observable effects. This pure kinetic effect reveals the incompleteness of traditional fluid theory. Our proof relies on newly developed kernel estimates with conservation laws and novel BV estimates for the cutoff boundary layer.<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia&diff=25957Colloquia2024-01-23T13:59:39Z<p>Htran24: /* Spring 2024 */</p>
<hr />
<div>__NOTOC__<br />
<br />
<br />
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm in Van Vleck B239 unless otherwise noted.</b><br />
<br />
Contacts for the colloquium are Simon Marshall and Dallas Albritton.<br />
<br />
<br />
<br />
==Spring 2024==<br />
{| cellpadding="8"<br />
! align="left" |date<br />
! align="left" |speaker<br />
! align="left" |title<br />
! align="left" | host(s)<br />
|-<br />
|<b>Monday Jan 22 at 4pm in B239</b><br />
|[https://www.mathematik.tu-darmstadt.de/fb/personal/details/yingkun_li.en.jsp Yingkun Li] (Darmstadt Tech U, Germany)<br />
|[[#Li|Arithmetic of real-analytic modular forms]]<br />
|Yang<br />
|-<br />
|'''Thursday Jan 25 at 4pm in VV911'''<br />
|[https://chimeraki.weebly.com/scientificresearch.html Sanjukta Krishnagopal] (UCLA/UC Berkeley)<br />
|Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes<br />
|Smith<br />
|-<br />
|Jan 26<br />
|[https://www.math.ucla.edu/~jacob/ Jacob Bedrossian] (UCLA)<br />
|Lyapunov exponents in stochastic systems<br />
|Tran<br />
|-<br />
|Feb 2<br />
|William Chen (to be confirmed)<br />
|<br />
|<br />
|-<br />
|Feb 9<br />
|(held for town hall)<br />
|<br />
|<br />
|-<br />
|Feb 16<br />
|[https://jacklutz.com/ Jack Lutz] (Iowa State)<br />
|<br />
|Guo<br />
|-<br />
|Feb 23<br />
|<br />
|<br />
|<br />
|-<br />
|Mar 1<br />
|[https://users.oden.utexas.edu/~pgm/ Per-Gunnar Martinsson] (UT-Austin)<br />
|TBA<br />
|Li<br />
|-<br />
|Mar 8<br />
|Anton Izosimov (U of Arizona)<br />
|<br />
|Gloria Mari-Beffa<br />
|-<br />
|Mar 15<br />
|[https://sites.google.com/view/peterhumphries/ Peter Humphries] (Virginia)<br />
|<br />
|Marshall<br />
|-<br />
|Mar 20<br />
|[https://www.math.wustl.edu/~wanlin/index.html Wanlin Li] (Washington U St Louis)<br />
|<br />
|Dymarz, GmMaW<br />
|-<br />
|Mar 29<br />
|Spring break<br />
|<br />
|<br />
|-<br />
|Apr 5<br />
|[https://www.math.columbia.edu/~savin/ Ovidiu Savin] (Columbia)<br />
|<br />
|Tran<br />
|-<br />
|Apr 12<br />
|[https://www.mikaylakelley.com/about Mikayla Kelley] (U Chicago Philosophy)<br />
|Math And... seminar, title TBA<br />
|Ellenberg, Marshall<br />
|-<br />
|Apr 19<br />
|[https://sites.math.rutgers.edu/~yyli/ Yanyan Li] (Rutgers)<br />
|<br />
|Tran<br />
|-<br />
|Apr 26<br />
|[https://sites.google.com/view/chris-leiningers-webpage/home Chris Leininger] (Rice)<br />
|TBA<br />
|Uyanik<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<div id="Li">'''Monday, January 22. Yingkun Li''' <br />
<br />
'''Arithmetic of real-analytic modular forms'''<br />
<br />
Modular form is a classical mathematical object dating back to the 19th century. Because of its connections to and appearances in many different areas of math and physics, it remains a popular subject today. Since the work of Hans Maass in 1949, real-analytic modular form has found important applications in arithmetic geometry and number theory. In this talk, I will discuss the amazing works in this area over the past 20 years, and give a glimpse of its fascinating future directions. <br />
<br />
'''Thursday, January 25. Sanjukta Krinshagopal''' <br />
<br />
'''Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes'''<br />
<br />
In this talk I will discuss some aspects at the intersection of mathematics, machine learning, and networks to introduce interdisciplinary methods with wide application. <br />
<br />
First, I will discuss some recent advances in mathematical machine learning for prediction on graphs. Machine learning is often a black box. Here I will present some exact theoretical results on the dynamics of weights while training graph neural networks using graphons - a graph limit or a graph with infinitely many nodes. I will use these ideas to present a new method for predictive and personalized medicine applications with remarkable success in prediction of Parkinson's subtype five years in advance.<br />
<br />
Then, I will discuss some work on higher-order models of graphs: simplicial complexes - that can capture simultaneous many-body interactions. I will present some recent results on spectral theory of simplicial complexes, as well as introduce a mathematical framework for studying the topology and dynamics of ''multilayer'' simplicial complexes using Hodge theory, and discuss applications of such interdisciplinary methods to studying bias in society, opinion dynamics, and hate speech in social media.<br />
<br />
<br />
<br />
'''Friday, January 26. Jacob Bedrossian'''<br />
<br />
'''Lyapunov exponents in stochastic systems'''<br />
<br />
In this overview talk we discuss several results regarding positive Lyapunov exponents in stochastic systems. First we discuss proving "Lagrangian chaos" in stochastic fluid mechanics, that is, demonstrating a positive Lyapunov exponent for the motion of a particle in the velocity field arising from the stochastic Navier-Stokes equations. We describe how this chaos can be used to deduce qualitatively optimal almost-sure exponential mixing of passive scalars. Next we describe more recently developed methods for obtaining strictly positive lower bounds and some quantitative estimates on the top Lyapunov exponent of weakly-damped stochastic differential equations, such as Lorenz-96 model or Galerkin truncations of the 2d Navier-Stokes equations (called "Eulerian chaos" in fluid mechanics). Further applications of the ideas to the chaotic motion of charged particles in fluctuating magnetic fields and the non-uniqueness of stationary measures for Lorenz 96 in degenerate forcing situations will be discussed if time permits. All of the work except for the charged particles (joint with Chi-Hao Wu) is joint with Alex Blumenthal and Sam Punshon-Smith.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Spring2024|Spring 2024]]<br />
<br />
[[Colloquia/Fall 2023|Fall 2023]]<br />
<br />
[[Colloquia/Spring2023|Spring 2023]]<br />
<br />
[[Colloquia/Fall2022|Fall 2022]]<br />
<br />
[[Spring 2022 Colloquiums|Spring 2022]]<br />
<br />
[[Colloquia/Fall2021|Fall 2021]]<br />
<br />
[[Colloquia/Spring2021|Spring 2021]]<br />
<br />
[[Colloquia/Fall2020|Fall 2020]]<br />
<br />
[[Colloquia/Spring2020|Spring 2020]]<br />
<br />
[[Colloquia/Fall2019|Fall 2019]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]<br />
<br />
[[WIMAW]]</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=25868PDE Geometric Analysis seminar2024-01-12T14:27:28Z<p>Htran24: /* Spring 2024 */</p>
<hr />
<div>All talks will be ''in person'' unless specified otherwise.<br />
<br />
(Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.) <br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Spring 2024 =<br />
<br />
<br />
<br />
The first day of class is Tuesday, January 23, 2023.<br />
<br />
<br />
<br />
'''January 23, 2024 (special date/location)'''<br />
<br />
Donghyun Lee (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: TBA<br />
<br />
Title: DYNAMICAL BILLIARD AND A LONG-TIME BEHAVIOR OF THE BOLTZMANN EQUATION IN GENERAL 3D TOROIDAL DOMAINS<br />
<br />
Abstract : Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant progress in this question when domains are strictly convex, the same question without the strict convexity of domains is still totally open in 3D. The major difficulty arises when a billiard map has an infinite number of bounces in a finite time interval or when the map fails to be Lipschitz continuous, both of which happen generically when the domain is non-convex. In this paper, we develop a new method to control a billiard map on a surface of revolution generated by revolving any planar analytic convex closed curve (e.g., typical shape of tokamak reactors’ chamber). In particular, we classify and measure the size (to be small) of a pullback set (along the billiard trajectory) of the infinite-bouncing and singular- bouncing cases. As a consequence, we solve the open question affirmatively in such domains. To the best of our knowledge, this work is the first construction of global solutions to the hard-sphere Boltzmann equation in generic non-convex 3-dimensional domains. This is joint work with Gyounghun Ko and Chanwoo Kim.<br />
<br />
<br />
<br />
<br />
<br />
'''January 29, 2024'''<br />
<br />
Minh-Binh Tran (TAMU). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''January 30, 2024 (special date/location)'''<br />
<br />
Jin Woo Jang (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: TBA<br />
<br />
Title:<br />
<br />
<br />
<br />
<br />
<br />
'''February 5, 2024''' <br />
<br />
Thierry Laurens (UW-Madison) <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: <br />
<br />
<br />
<br />
'''February 12, 2024'''<br />
<br />
Laurel Ohm (UW-Madison)<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''February 19, 2024'''<br />
<br />
<br />
<br />
'''February 26, 2024'''<br />
<br />
Katy Craig (UCSB). Host: Nicolas Garcia Trillos, Hung Tran.<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
'''March 4, 2024'''<br />
<br />
Yuming Paul Zhang (Auburn). Host: Hung Tran.<br />
<br />
<br />
<br />
'''March 11, 2024'''<br />
<br />
Rajendra Beekie (Duke). Host: Dallas Albritton<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''March 18, 2024'''<br />
<br />
'''March 25, 2024'''<br />
<br />
Spring Break. No seminar.<br />
<br />
'''April 1, 2024'''<br />
<br />
Dominic Wynter (Cambridge). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''April 8, 2024'''<br />
<br />
<br />
<br />
'''April 15, 2024'''<br />
<br />
Myoungjean Bae (KAIST). Host: Mikhail Feldman<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''April 22, 2024'''<br />
<br />
Sarah Strikwerda (Penn). Host: Hung Tran.Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
'''April 29, 2024'''<br />
<br />
The last day of class is Friday, May 3, 2023.<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
<br />
'''March 20, 2023''' <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
<br />
'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
<br />
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
<br />
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Large amplitude solution of BGK model''<br />
<br />
'''Abstract:''' Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time.<br />
This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.<br />
<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Interior W^{2,p} estimates for complex Monge-Ampere equations'' <br />
<br />
'''Abstract:''' The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] (Berkeley)<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion<br />
<br />
'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.<br />
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<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''The Boltzmann equation with large data<br />
<br />
'''Abstract:''' The Boltzmann equation is a nonlocal, nonlinear equation arising in gas dynamics for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk I will discuss a recent program to understand a more tractable, related question: what is the largest space in which local well-posedness holds and what quantities prevent blow-up at finite times? This program covers several papers and intertwines with a recent push to understand the regularity theory of kinetic Fokker-Planck-type equations. I will give a broad outline of the proof, and then, instead of slogging through the whole thing, I will focus on a simple proof of a technically important and physically interesting lower bound -- how ``vacuum regions'' fill in. This is a joint work with Snelson and Tarfulea.<br />
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'''May 8, 2023'''<br />
<br />
[[TBA|Lei Wu (Lehigh)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in '''VV B223 (special room)''' <br />
<br />
'''Title:'''Ghost Effect from Boltzmann Theory<br />
<br />
'''Abstract:'''It is a classical and fundamental problem to study the hydrodynamic limits of kinetic equations as the Knudsen number \varepsilon\rightarrow 0. In this talk, we focus on the stationary Boltzmann equation with diffuse-reflection boundary condition where the Mach number is O(\varepsilon) but the temperature variance is O(1). We rigorously derived the limiting fluid system, the so-called ghost-effect equations, in which an infinitesimal boundary variance will have macroscopic observable effects. This pure kinetic effect reveals the incompleteness of traditional fluid theory. Our proof relies on newly developed kernel estimates with conservation laws and novel BV estimates for the cutoff boundary layer.<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
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<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
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<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
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<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
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Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
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<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
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<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
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Format: , Time: <br />
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Title:<br />
<br />
Abstract:<br />
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<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
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<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
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===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=25744PDE Geometric Analysis seminar2023-12-26T15:16:19Z<p>Htran24: </p>
<hr />
<div>All talks will be ''in person'' unless specified otherwise.<br />
<br />
(Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.) <br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Spring 2024 =<br />
<br />
<br />
<br />
The first day of class is Tuesday, January 23, 2023.<br />
<br />
<br />
<br />
'''January 23, 2024 (special date/location)'''<br />
<br />
Donghyun Lee (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: TBA<br />
<br />
Title:<br />
<br />
<br />
<br />
<br />
<br />
'''January 29, 2024'''<br />
<br />
Minh-Binh Tran (TAMU). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''January 30, 2024 (special date/location)'''<br />
<br />
Jin Woo Jang (POSTECH). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:00-4:00PM, Location: TBA<br />
<br />
Title:<br />
<br />
<br />
<br />
<br />
<br />
'''February 5, 2024''' <br />
<br />
Thierry Laurens (UW-Madison) <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: <br />
<br />
<br />
<br />
'''February 12, 2024'''<br />
<br />
Laurel Ohm (UW-Madison)<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''February 19, 2024'''<br />
<br />
<br />
<br />
'''February 26, 2024'''<br />
<br />
Katy Craig (UCSB). Host: Nicolas Garcia Trillos, Hung Tran.<br />
<br />
<br />
<br />
'''March 4, 2024'''<br />
<br />
Yuming Paul Zhang (Auburn). Host: Hung Tran.<br />
<br />
<br />
'''March 11, 2024'''<br />
<br />
Rajendra Beekie (Duke). Host: Dallas Albritton<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''March 18, 2024'''<br />
<br />
'''March 25, 2024'''<br />
<br />
Spring Break. No seminar.<br />
<br />
'''April 1, 2024'''<br />
<br />
Dominic Wynter (Cambridge). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''April 8, 2024'''<br />
<br />
<br />
<br />
'''April 15, 2024'''<br />
<br />
Myoungjean Bae (KAIST). Host: Mikhail Feldman<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''April 22, 2024'''<br />
<br />
<br />
<br />
'''April 29, 2024'''<br />
<br />
The last day of class is Friday, May 3, 2023.<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
<br />
'''March 20, 2023''' <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
<br />
'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
<br />
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
<br />
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Large amplitude solution of BGK model''<br />
<br />
'''Abstract:''' Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time.<br />
This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.<br />
<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Interior W^{2,p} estimates for complex Monge-Ampere equations'' <br />
<br />
'''Abstract:''' The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] (Berkeley)<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion<br />
<br />
'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''The Boltzmann equation with large data<br />
<br />
'''Abstract:''' The Boltzmann equation is a nonlocal, nonlinear equation arising in gas dynamics for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk I will discuss a recent program to understand a more tractable, related question: what is the largest space in which local well-posedness holds and what quantities prevent blow-up at finite times? This program covers several papers and intertwines with a recent push to understand the regularity theory of kinetic Fokker-Planck-type equations. I will give a broad outline of the proof, and then, instead of slogging through the whole thing, I will focus on a simple proof of a technically important and physically interesting lower bound -- how ``vacuum regions'' fill in. This is a joint work with Snelson and Tarfulea.<br />
<br />
<br />
<br />
<br />
'''May 8, 2023'''<br />
<br />
[[TBA|Lei Wu (Lehigh)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in '''VV B223 (special room)''' <br />
<br />
'''Title:'''Ghost Effect from Boltzmann Theory<br />
<br />
'''Abstract:'''It is a classical and fundamental problem to study the hydrodynamic limits of kinetic equations as the Knudsen number \varepsilon\rightarrow 0. In this talk, we focus on the stationary Boltzmann equation with diffuse-reflection boundary condition where the Mach number is O(\varepsilon) but the temperature variance is O(1). We rigorously derived the limiting fluid system, the so-called ghost-effect equations, in which an infinitesimal boundary variance will have macroscopic observable effects. This pure kinetic effect reveals the incompleteness of traditional fluid theory. Our proof relies on newly developed kernel estimates with conservation laws and novel BV estimates for the cutoff boundary layer.<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
<br />
<br />
<br />
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<br />
<br />
<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
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{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=25686PDE Geometric Analysis seminar2023-12-05T16:40:01Z<p>Htran24: /* Spring 2024 */</p>
<hr />
<div>All talks will be ''in person'' unless specified otherwise.<br />
<br />
(Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.) <br />
===[[Previous PDE/GA seminars]]===<br />
<br />
=== [[Fall 2023-Spring 2024|Schedule for Fall 2023-Spring 2024]]===<br />
<br />
<br />
<br />
'''September 11, 2023 '''<br />
<br />
Dallas Albritton (UW-Madison)<br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
Title: Kinetic shock profiles for the Landau equation <br />
<br />
Abstract: Compressible Euler solutions develop jump discontinuities known as shocks. However, physical shocks are not, strictly speaking, discontinuous. Rather, they exhibit an internal structure which, in certain regimes, can be represented by a smooth function, the shock profile. We demonstrate the existence of weak shock profiles to the kinetic Landau equation. Joint work with Matthew Novack (Purdue University) and Jacob Bedrossian (UCLA). <br />
<br />
'''September 18, 2023'''<br />
<br />
Hongjie Dong (Brown). Host: Hung Tran<br />
<br />
Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Recent results about the insulated conductivity problem.<br />
<br />
Abstract: In the first part of the talk, I will present our work about the insulated conductivity problem with closely spaced inclusions in a bounded domain in $R^n$. A noteworthy phenomenon in this context is the potential for the gradient of solutions to blow up as the distance between inclusions tends to zero. We obtained an optimal gradient estimate of solutions in terms of the distance, which settled down a major open problem in this area. In the second part, I will discuss recent results about the insulated conductivity problem when the current-electric field relation is a power law. New results for the perfect conductivity problem will also be mentioned.<br />
<br />
Based on joint work with Yanyan Li (Rutgers University), Zhuolun Yang and Hanye Zhu (Brown University).<br />
<br />
'''September 25, 2023'''<br />
<br />
Olga Turanova (MSU). Host: Hung Tran<br />
<br />
Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Approximating degenerate diffusion via nonlocal equations<br />
<br />
Abstract: In this talk, I'll describe a deterministic particle method for the weighted porous medium equation. The key idea behind the method is to approximate the PDE via certain highly nonlocal continuity equations. The formulation of the method and the proof of its convergence rely on the Wasserstein gradient flow formulation of the aforementioned PDEs. This is based on joint work with Katy Craig, Karthik Elamvazhuthi, and Matt Haberland.<br />
<br />
'''October 2, 2023 '''<br />
<br />
Edriss S. Titi (University of Cambridge, Texas A&M), a Distinguished Lecture<br />
<br />
Time: 4PM - 5PM, VVB239<br />
<br />
Title: On the Solvability of the Navier-Stokes and Euler Equations, where do we stand?<br />
<br />
'''October 9, 2023 '''<br />
<br />
Montie Avery (BU). Host: Dallas Albritton<br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
Title: Universality in spreading into unstable states <br />
<br />
Abstract: Front propagation into unstable states plays an important role in organizing structure formation in many spatially extended systems. When a trivial background state is pointwise unstable, localized perturbations typically grow and spread with a selected speed, leaving behind a selected state in their wake. A fundamental question of interest is to predict the propagation speed and the state selected in the wake. The marginal stability conjecture postulates that speeds can be universally predicted via a marginal spectral stability criterion. In this talk, we will present background on the marginal stability conjecture and present some ideas of our recent conceptual proof of the conjecture in a model-independent framework focusing on systems of parabolic equations.<br />
<br />
'''October 16, 2023 '''<br />
<br />
[[Ian Tice]] (CMU). Host: Chanwoo Kim <br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
Title: Stationary and slowly traveling solutions to the free boundary Navier-Stokes equations <br />
<br />
Abstract: The stationary problem for the free boundary incompressible Navier-Stokes equations lies at the confluence of two distinct lines of inquiry in fluid mechanics. The first views the dynamic problem as an initial value problem. In this context, the stationary problem arises naturally as a special type of global-in-time solution with stationary sources of force and stress. One then expects solutions to the stationary problem to play an essential role in the study of long-time asymptotics or attractors for the dynamic problem. The second line of inquiry, which dates back essentially to the beginning of mathematical fluid mechanics, concerns the search for traveling wave solutions. In this context, a huge literature exists for the corresponding inviscid problem, but progress on the viscous problem was initiated much more recently in the work of the speaker and co-authors. For technical reasons, these results were only able to produce traveling solutions with nontrivial wave speed. In this talk we will discuss the well-posedness theory for the stationary problem and show that the solutions thus obtained lie along a one-parameter family of slowly traveling wave solutions. This is joint work with Noah Stevenson.<br />
<br />
'''October 23, 2023 '''<br />
<br />
Raghav Venkatraman (Courant). Hosts: Dallas Albritton and Laurel Ohm<br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
Title: Interaction energies in liquid crystal colloids. <br />
<br />
Abstract: In this talk we discuss some recent results on nematic liquid crystal colloids. The first half of the talk represents some recent progress on justification of the so-called "electrostatic analogy" proposed by Brochard and De Gennes as an approximate model for dilute suspensions of particles in a nematic background. This analogy is based on approximating the far-field behavior of the nematic (away from the colloids) by far-field expansions of the associated linearized problem. <br />
<br />
In the second part of the talk, I'll present a setting on interaction energies in ''para'' nematic colloids. In this setting, nematic ordering is only induced by boundary conditions on the colloids since the bulk potential prefers the isotropic phase. Thus, particles exhibit a very short-ranged interaction, whose character we clarify, since in this setting a far-field based treatment is inadequate. We derive expressions for the leading order interaction energies between particles. <br />
<br />
The first part represents joint work with Alama, Bronsard and Lamy, while the second is joint work with Golovaty, Taylor and Zarnescu.<br />
<br />
'''October 30, 2023 '''<br />
<br />
[[Sung-Jin Oh]] (UC Berkeley). Host: Chanwoo Kim <br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
Title: Late time tail of waves on dynamic asymptotically flat spacetimes of odd space dimensions <br />
<br />
Abstract: I will present an upcoming work with J. Luk (Stanford), where we develop a general method for understanding the late time tail for solutions to wave equations on asymptotically flat spacetimes with odd spatial dimensions, which is applicable to nonlinear problems on dynamical backgrounds. In addition to its inherent interest, such information is crucial for studying problems involving the interaction of waves with a spatially localized object; indeed, our motivation for developing this method comes from the Strong Cosmic Censorship Conjecture. I will explain how our method recovers and refines Price's law for linear problems on stationary backgrounds, and also how it shows that the late time tails are in general different(!) from the linear stationary case in the presence of nonlinearity and/or a dynamical background.<br />
<br />
'''November 6, 2023 '''<br />
<br />
Vera Hur (UIUC). Host: Dallas Albritton<br />
<br />
Time: 3:30 PM-4:30 PM, VV901<br />
<br />
Title: Stable undular bores: rigorous analysis and validated numerics <br />
<br />
Abstract: I will discuss the ‘global’ nonlinear asymptotic stability of the traveling front solutions to the Korteweg-de Vries–Burgers equation, and other dispersive-dissipative perturbations of the Burgers equation. Earlier works made strong use of the monotonicity of the profile, for relatively weak dispersion effects. We exploit the modulation of the translation parameter, establishing a new stability criterion that does not require monotonicity. Instead, a certain Schrodinger operator in one dimension must have exactly one negative eigenvalue, so that a rank-one perturbation of the operator can be made positive definite. Counting the number of bound states of the Schrodinger equation, we find a sufficient condition in terms of the ’width’ of a front. We analytically verify that our stability criterion is met for an open set in the parameter regime including all monotone fronts. Our numerical experiments, revealing more stable fronts, suggest a computer-assisted proof. Joint with Blake Barker, Jared Bronski, and Zhao Yang.<br />
<br />
'''November 13, 2023 '''<br />
<br />
Nicola De Nitti (EPFL). Host: Dallas Albritton<br />
<br />
Time: 3:30 PM-4:30 PM, VV901<br />
<br />
Title: Scalar conservation laws modeling supply-chains under constraints<br />
<br />
Abstract: We consider a conservation law with strictly positive wave velocity and study the well-posedness of the associated initial-value problem under a flux constraint active in the half-line $\mathbb{R}_+$. The strict positivity of the wave velocity allows for the dynamics in the unconstrained region $\mathbb{R}_-$ to be fully determined by the restriction of the initial data to $\mathbb{R}_-$. On the other hand, the solution in the constrained region is dictated by the assumption that the total mass of the initial datum is conserved along the evolution: the boundary datum for the initial-boundary value problem posed on $\mathbb{R}_+$ is given by the largest incoming flux that is admissible under the constraint, while the exceeding mass is accumulated (as an atomic measure) in a ``buffer<nowiki>''</nowiki> at the interface $\{x=0\}$. This talk is based on a joint work with D. Serre and E. Zuazua. <br />
<br />
'''November 20, 2023 '''<br />
<br />
Trinh Nguyen (UW-Madison). <br />
<br />
Time: 3:00 PM-4:00 PM, VV901 ''(Note the earlier time!)''<br />
<br />
Title: Boundary Layers in Fluid Dynamics: Prandtl Theory and Hydrodynamics Limits <br />
<br />
Abstract: This talk addresses the challenge of the inviscid limit in Navier-Stokes equations, focusing on domains with no-slip boundaries and for less regular initial data in R^2. I will discuss Prandtl boundary layer theory on the half-space and bounded domains. Additionally, the discussion extends to hydrodynamics limit problems, deriving singular layers like point vortices and Prandtl layers from the Boltzmann equations.<br />
<br />
'''November 27, 2023 '''([[ No seminar|First Monday after Thanksgiving]])<br />
<br />
'''December 4, 2023'''<br />
<br />
'''December 11, 2023'''<br />
<br />
Timur Yastrzhembskiy (Brown University). Host: Dallas Albritton<br />
<br />
Time: 3:30 PM-4:30 PM, VV901<br />
<br />
= Spring 2024 =<br />
<br />
<br />
<br />
The first day of class is Tuesday, January 23, 2023.<br />
<br />
'''January 29, 2024'''<br />
<br />
Minh-Binh Tran (TAMU). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''February 5, 2024''' <br />
<br />
Thierry Laurens (UW-Madison) <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: <br />
<br />
<br />
<br />
'''February 12, 2024'''<br />
<br />
Laurel Ohm (UW-Madison)<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
'''February 19, 2024'''<br />
<br />
<br />
<br />
'''February 26, 2024'''<br />
<br />
Katy Craig (UCSB). Host: Nicolas Garcia Trillos, Hung Tran.<br />
<br />
<br />
'''March 4, 2024'''<br />
<br />
<br />
'''March 11, 2024'''<br />
<br />
Rajendra Beekie (Duke). Host: Dallas Albritton<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
'''March 18, 2024'''<br />
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'''March 25, 2024'''<br />
<br />
Spring Break. No seminar.<br />
<br />
'''April 1, 2024'''<br />
<br />
Dominic Wynter (Cambridge). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
'''April 8, 2024'''<br />
<br />
<br />
'''April 15, 2024'''<br />
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<br />
'''April 22, 2024'''<br />
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<br />
'''April 29, 2024'''<br />
<br />
<br />
The last day of class is Friday, May 3, 2023.<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
<br />
'''March 20, 2023''' <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
<br />
'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
<br />
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
<br />
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Large amplitude solution of BGK model''<br />
<br />
'''Abstract:''' Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time.<br />
This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.<br />
<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Interior W^{2,p} estimates for complex Monge-Ampere equations'' <br />
<br />
'''Abstract:''' The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] (Berkeley)<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion<br />
<br />
'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''The Boltzmann equation with large data<br />
<br />
'''Abstract:''' The Boltzmann equation is a nonlocal, nonlinear equation arising in gas dynamics for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk I will discuss a recent program to understand a more tractable, related question: what is the largest space in which local well-posedness holds and what quantities prevent blow-up at finite times? This program covers several papers and intertwines with a recent push to understand the regularity theory of kinetic Fokker-Planck-type equations. I will give a broad outline of the proof, and then, instead of slogging through the whole thing, I will focus on a simple proof of a technically important and physically interesting lower bound -- how ``vacuum regions'' fill in. This is a joint work with Snelson and Tarfulea.<br />
<br />
<br />
<br />
<br />
'''May 8, 2023'''<br />
<br />
[[TBA|Lei Wu (Lehigh)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in '''VV B223 (special room)''' <br />
<br />
'''Title:'''Ghost Effect from Boltzmann Theory<br />
<br />
'''Abstract:'''It is a classical and fundamental problem to study the hydrodynamic limits of kinetic equations as the Knudsen number \varepsilon\rightarrow 0. In this talk, we focus on the stationary Boltzmann equation with diffuse-reflection boundary condition where the Mach number is O(\varepsilon) but the temperature variance is O(1). We rigorously derived the limiting fluid system, the so-called ghost-effect equations, in which an infinitesimal boundary variance will have macroscopic observable effects. This pure kinetic effect reveals the incompleteness of traditional fluid theory. Our proof relies on newly developed kernel estimates with conservation laws and novel BV estimates for the cutoff boundary layer.<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
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<br />
'''October 3, 2022'''<br />
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[[No Seminar]]<br />
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Format: , Time: <br />
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Title:<br />
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Abstract:<br />
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'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
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'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
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Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
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'''October 24, 2022'''<br />
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[[No seminar.]]<br />
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Format: , Time: <br />
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Title:<br />
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Abstract:<br />
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'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
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Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
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'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
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'''November 14, 2022 '''<br />
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Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
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Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
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<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
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'''November 28, 2022 '''<br />
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[[No Seminar]]- Thanksgiving <br />
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'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
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Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
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'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
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Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
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===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
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'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
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Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
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<br />
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<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
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<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
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<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
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<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia/Spring2024&diff=25678Colloquia/Spring20242023-12-01T19:40:10Z<p>Htran24: /* Spring 2024 */</p>
<hr />
<div>== Spring 2024 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
| Jan 19<br />
|<br />
|<br />
| <br />
|-<br />
| Jan 26<br />
|Jacob Bedrossian (UCLA)<br />
|<br />
|Tran<br />
|-<br />
| Feb 2<br />
|<br />
|<br />
|<br />
|-<br />
| Feb 9<br />
|<br />
|<br />
|<br />
|-<br />
| Feb 16<br />
|<br />
|<br />
|<br />
|-<br />
| Feb 23<br />
|<br />
|<br />
|<br />
|-<br />
| Mar 1<br />
|[https://users.oden.utexas.edu/~pgm/ Per-Gunnar Martinsson] (UT-Austin)<br />
|TBA<br />
|Li<br />
|-<br />
| Mar 8<br />
|Anton Izosimov (U of Arizona)<br />
|<br />
|Gloria Mari-Beffa<br />
|-<br />
| Mar 15<br />
|<br />
|<br />
|<br />
|-<br />
|Mar 20<br />
|[https://www.math.wustl.edu/~wanlin/index.html Wanlin Li] (Washington U St Louis)<br />
|<br />
|Dymarz, GmMaW<br />
|-<br />
| Mar 22<br />
|[https://jacklutz.com/ Jack Lutz] (Iowa State)<br />
|<br />
|Guo<br />
|-<br />
| Mar 29<br />
|Spring break<br />
|<br />
|<br />
|-<br />
| Apr 5<br />
|[https://www.math.columbia.edu/~savin/ Ovidiu Savin] (Columbia)<br />
|<br />
|Tran<br />
|-<br />
|Apr 12<br />
|[https://www.mikaylakelley.com/about Mikayla Kelley] (U Chicago Philosophy)<br />
|Math And... seminar, title TBA<br />
|Ellenberg, Marshall<br />
<br />
|-<br />
| Apr 19<br />
|[https://sites.math.rutgers.edu/~yyli/ Yanyan Li] (Rutgers)<br />
|<br />
|Tran<br />
|-<br />
| Apr 26<br />
|[https://sites.google.com/view/chris-leiningers-webpage/home Chris Leininger] (Rice)<br />
|TBA<br />
|Uyanik<br />
|}</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia/Spring2024&diff=25508Colloquia/Spring20242023-10-26T16:13:12Z<p>Htran24: /* Spring 2024 */</p>
<hr />
<div>== Spring 2024 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
| Jan 19<br />
|<br />
|<br />
| <br />
|-<br />
| Jan 26<br />
|<br />
|<br />
|<br />
|-<br />
| Feb 2<br />
|<br />
|<br />
|<br />
|-<br />
| Feb 9<br />
|<br />
|<br />
|<br />
|-<br />
| Feb 16<br />
|<br />
|<br />
|<br />
|-<br />
| Feb 23<br />
|<br />
|<br />
|<br />
|-<br />
| Mar 1<br />
|<br />
|<br />
|<br />
|-<br />
| Mar 8<br />
|Anton Izosimov (U of Arizona)<br />
|<br />
|Gloria Mari-Beffa<br />
|-<br />
| Mar 15<br />
|<br />
|<br />
|<br />
|-<br />
|Mar 20<br />
|[https://www.math.wustl.edu/~wanlin/index.html Wanlin Li] (Washington U St Louis)<br />
|<br />
|Dymarz, GmMaW<br />
|-<br />
| Mar 22<br />
|[https://jacklutz.com/ Jack Lutz] (Iowa State)<br />
|<br />
|Guo<br />
|-<br />
| Mar 29<br />
|Spring break<br />
|<br />
|<br />
|-<br />
| Apr 5<br />
|[https://www.math.columbia.edu/~savin/ Ovidiu Savin] (Columbia)<br />
|<br />
|Hung Tran<br />
|-<br />
| Apr 12<br />
|[https://users.oden.utexas.edu/~pgm/ Per-Gunnar Martinsson] (UT-Austin)<br />
|TBA<br />
|Qin Li<br />
|-<br />
| Apr 19<br />
|[https://sites.math.rutgers.edu/~yyli/ Yanyan Li] (Rutgers)<br />
|<br />
|Hung Tran<br />
|-<br />
| Apr 26<br />
|[https://sites.google.com/view/chris-leiningers-webpage/home Chris Leininger] (Rice)<br />
|TBA<br />
|Uyanik<br />
|}</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia/Spring2024&diff=25451Colloquia/Spring20242023-10-16T18:30:58Z<p>Htran24: /* Spring 2024 */</p>
<hr />
<div>== Spring 2024 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
| Jan 19<br />
|<br />
|<br />
| <br />
|-<br />
| Jan 26<br />
|<br />
|<br />
|<br />
|-<br />
| Feb 2<br />
|<br />
|<br />
|<br />
|-<br />
| Feb 9<br />
|<br />
|<br />
|<br />
|-<br />
| Feb 16<br />
|<br />
|<br />
|<br />
|-<br />
| Feb 23<br />
|<br />
|<br />
|<br />
|-<br />
| Mar 1<br />
|<br />
|<br />
|<br />
|-<br />
| Mar 8<br />
|Anton Izosimov (U of Arizona)<br />
|<br />
|Gloria Mari-Beffa<br />
|-<br />
| Mar 15<br />
|<br />
|<br />
|<br />
|-<br />
|Mar 20<br />
|[https://www.math.wustl.edu/~wanlin/index.html Wanlin Li] (Washington U St Louis)<br />
|<br />
|Dymarz, GmMaW<br />
|-<br />
| Mar 22<br />
|[https://jacklutz.com/ Jack Lutz] (Iowa State)<br />
|<br />
|Guo<br />
|-<br />
| Mar 29<br />
|Spring break<br />
|<br />
|<br />
|-<br />
| Apr 5<br />
|Reserved<br />
|<br />
|Hung Tran<br />
|-<br />
| Apr 12<br />
|[https://users.oden.utexas.edu/~pgm/ Per-Gunnar Martinsson] (UT-Austin)<br />
|TBA<br />
|Qin Li<br />
|-<br />
| Apr 19<br />
|[https://sites.math.rutgers.edu/~yyli/ Yanyan Li] (Rutgers)<br />
|<br />
|Hung Tran<br />
|-<br />
| Apr 26<br />
|[https://sites.google.com/view/chris-leiningers-webpage/home Chris Leininger] (Rice)<br />
|TBA<br />
|Uyanik<br />
|}</div>Htran24https://wiki.math.wisc.edu/index.php?title=Research_at_UW-Madison_in_DifferentialEquations&diff=25260Research at UW-Madison in DifferentialEquations2023-09-19T01:24:56Z<p>Htran24: /* Current Postdocs (Van Vleck Assistant Professor) in PDE */</p>
<hr />
<div>==Seminars of interest==<br />
<br />
The weekly [http://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE & Geometric Analysis seminar] is held on Monday afternoons, 3:30-4:30pm. <br />
<br />
Other seminars that will feature PDE related material are the [http://www.math.wisc.edu/wiki/index.php/Geometry_and_Topology_Seminar Geometry and Topology seminar], the [https://www.math.wisc.edu/wiki/index.php/Analysis_Seminar Analysis seminar], and the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math seminar].<br />
<br />
[https://sites.google.com/view/vapseminar Virtual Analysis and PDE Seminar (VAPS)]: organized jointly by 9 US universities including UW.<br />
<br />
==Faculty==<br />
<br />
[https://sites.google.com/view/albri050/dallas-albritton Dallas Albritton] (University of Minnesota, 2020) Pure and applied aspects of PDE and fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~feldman Mikhail Feldman] (UC Berkeley, 1994) Nonlinear PDE, Calculus of Variations.<br />
<br />
[http://www.math.wisc.edu/~ifrim/Home.html Mihaela Ifrim] (UC Davis, 2012) Nonlinear Dispersive Equations (water-wave equations and related dispersive models), Fluid Mechanics, Elastodynamics, Harmonic Analysis, General Relativity.<br />
<br />
[https://sites.google.com/view/ckim Chanwoo Kim] (Brown, 2011) Applied PDE, Kinetic theory, Fluid dynamics.<br />
<br />
[https://sites.google.com/a/umn.edu/math-laurelohm/home Laurel Ohm] (University of Minnesota, 2020) Analysis of PDE arising in biofluid mechanics.<br />
<br />
[http://www.math.wisc.edu/~hung Hung Vinh Tran] (UC Berkeley, 2012) Nonlinear PDE.<br />
<br />
==Faculty in related areas==<br />
<br />
[http://www.math.wisc.edu/~denissov Sergey Denisov] (Moscow State University, 1999) Analysis, PDE.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li] (UW Madison, 2013) Numerical analysis and scientific computing.<br />
<br />
[http://www.math.wisc.edu/~spagnolie/ Saverio Spagnolie] (Courant Institute, 2008) Fluid dynamics, complex fluids, soft matter, computation.<br />
<br />
[http://www.math.wisc.edu/~stechmann/ Sam Stechmann] (Courant Institute, 2008) Applied math, computational math, fluid dynamics, atmospheric science, climate.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (UT Austin, 1998) Mixing in fluids, optimization of mixing.<br />
<br />
[https://www.nicolasgarciat.com/ Nicolas Garcia Trillos] (CMU, 2015) Optimal transport, calculus of variations, and data analysis.<br />
==Current Postdocs (Van Vleck Assistant Professor) in PDE==<br />
<br />
[https://www.math.wisc.edu/~aai/ Albert Ai]<br />
<br />
[https://sites.google.com/view/tnguyen65 Trinh Nguyen]<br />
<br />
==Recent former Postdocs in PDE==<br />
<br />
[https://blog.nus.edu.sg/yyao/ Yao Yao] (VV assist prof 2012-2015). Current position: Dean’s Chair Associate Professor, National University of Singapore.<br />
<br />
[https://sites.google.com/view/KYUDONGCHOI Kyudong Choi] (VV assist prof 2012-2015). Current position: Associate Professor, UNIST (2015-)<br />
<br />
[https://sites.google.com/view/jessicalin-math/home Jessica Lin] (VV assist prof 2014-2017). Current position: Assistant Professor, McGill University (2017-)<br />
<br />
[https://sites.google.com/site/donghyunlee295/ Donghyun Lee] (VV assist prof 2015-2018). Current position: Associate Professor and Mueunjae Distinguished Professor, Postech (2018-) <br />
<br />
[https://cam.uchicago.edu/people/profile/eric-baer/ Eric Baer] (VV assist prof 2015-2018). Current position: Senior Lecturer, University of Chicago (2019-)<br />
<br />
[https://sites.google.com/site/guoxx097/welcome Xiaoqin Guo] (VV assist prof 2017-2020). Current position: Assistant Professor, University of Cincinnati (2020-).<br />
<br />
[http://www.homepages.ucl.ac.uk/~ucahms0/index.htm Matthew Schrecker] (VV assist prof 2018-2020). Current position: EPSRC Postdoctoral Research Fellow, University College London (2020-).<br />
<br />
[https://sites.google.com/site/dhkwonmath/home?authuser=0 Dohyun Kwon] (VV assist prof 9/2020-1/2023). Current position: Assistant Professor, University of Seoul.<br />
<br />
[https://schulzmath.wordpress.com/ Simon Schulz] (VV assist prof 9/2021-8/2023). Current position: Junior Visiting Position at the Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale Superiore.<br />
<br />
==Emeriti==<br />
<br />
[http://www.math.wisc.edu/~angenent Sigurd Angenent] (Leiden, 1986) Nonlinear PDE, differential geometry, medical imaging, math biology.<br />
<br />
[http://www.math.wisc.edu/~bolotin Sergey Bolotin] (Moscow State University, 1982) Dynamical Systems, Variational Methods, Celestial Mechanics.<br />
<br />
[http://www.math.wisc.edu/~rabinowi Paul Rabinowitz]<br />
PDE, Calculus of Variations, Dynamical Systems, Nonlinear Analysis.<br />
<br />
[http://www.math.wisc.edu/~robbin Joel Robbin]<br />
Global Analysis, Differential Equations.<br />
<br />
[http://www.math.wisc.edu/~turner Robert Turner]<br />
Partial Differential Equations, Fluid Mechanics, Mathematical Biology.<br />
<br />
==Previous events==<br />
<br />
The 81st Midwest PDE seminar '''[https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]''' was held in Madison on April 21/22 (2018).</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=25214PDE Geometric Analysis seminar2023-09-12T19:50:22Z<p>Htran24: </p>
<hr />
<div>All talks will be ''in person'' unless specified otherwise.<br />
<br />
(Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.) <br />
===[[Previous PDE/GA seminars]]===<br />
<br />
=== [[Fall 2023-Spring 2024|Schedule for Fall 2023-Spring 2024]]===<br />
<br />
<br />
<br />
'''September 11, 2023 '''<br />
<br />
Dallas Albritton (UW-Madison)<br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
Title: Kinetic shock profiles for the Landau equation <br />
<br />
Abstract: Compressible Euler solutions develop jump discontinuities known as shocks. However, physical shocks are not, strictly speaking, discontinuous. Rather, they exhibit an internal structure which, in certain regimes, can be represented by a smooth function, the shock profile. We demonstrate the existence of weak shock profiles to the kinetic Landau equation. Joint work with Matthew Novack (Purdue University) and Jacob Bedrossian (UCLA). <br />
<br />
'''September 18, 2023'''<br />
<br />
Hongjie Dong (Brown). Host: Hung Tran<br />
<br />
Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Recent results about the insulated conductivity problem.<br />
<br />
Abstract: In the first part of the talk, I will present our work about the insulated conductivity problem with closely spaced inclusions in a bounded domain in $R^n$. A noteworthy phenomenon in this context is the potential for the gradient of solutions to blow up as the distance between inclusions tends to zero. We obtained an optimal gradient estimate of solutions in terms of the distance, which settled down a major open problem in this area. In the second part, I will discuss recent results about the insulated conductivity problem when the current-electric field relation is a power law. New results for the perfect conductivity problem will also be mentioned.<br />
<br />
Based on joint work with Yanyan Li (Rutgers University), Zhuolun Yang and Hanye Zhu (Brown University).<br />
<br />
<br />
<br />
<br />
<br />
'''September 25, 2023'''<br />
<br />
Olga Turanova (MSU). Host: Hung Tran<br />
<br />
Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Approximating degenerate diffusion via nonlocal equations<br />
<br />
Abstract: In this talk, I'll describe a deterministic particle method for the weighted porous medium equation. The key idea behind the method is to approximate the PDE via certain highly nonlocal continuity equations. The formulation of the method and the proof of its convergence rely on the Wasserstein gradient flow formulation of the aforementioned PDEs. This is based on joint work with Katy Craig, Karthik Elamvazhuthi, and Matt Haberland.<br />
<br />
'''October 2, 2023 '''<br />
<br />
No seminar. (Edriss Titi will give a Distinguished Lecture today.)<br />
<br />
'''October 9, 2023 '''<br />
<br />
Montie Avery (BU). Host: Dallas Albritton<br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
'''October 16, 2023 '''<br />
<br />
[[Ian Tice]] (CMU). Host: Chanwoo Kim <br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
'''October 23, 2023 '''<br />
<br />
Raghav Venkatraman (Courant). Host: Dallas Albritton<br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
'''October 30, 2023 '''<br />
<br />
[[ Sung-Jin Oh| Sung-Jin Oh]] (UC Berkeley). Host: Chanwoo Kim <br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
'''November 6, 2023 '''<br />
<br />
Vera Hur (UIUC). Host: Dallas Albritton<br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
'''November 13, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, VV901 <br />
<br />
'''November 20, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, VV901 <br />
<br />
'''November 27, 2023 '''([[ No seminar|First Monday after Thanksgiving]])<br />
<br />
'''December 4, 2023'''<br />
<br />
'''December 11, 2023'''<br />
<br />
= Spring 2024 =<br />
<br />
<br />
'''January 29, 2024'''<br />
<br />
Minh-Binh Tran (TAMU). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
'''April 1, 2024'''<br />
<br />
Dominic Wynter (Cambridge). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
<br />
'''March 20, 2023''' <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
<br />
'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
<br />
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
<br />
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Large amplitude solution of BGK model''<br />
<br />
'''Abstract:''' Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time.<br />
This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.<br />
<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Interior W^{2,p} estimates for complex Monge-Ampere equations'' <br />
<br />
'''Abstract:''' The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] (Berkeley)<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion<br />
<br />
'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''The Boltzmann equation with large data<br />
<br />
'''Abstract:''' The Boltzmann equation is a nonlocal, nonlinear equation arising in gas dynamics for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk I will discuss a recent program to understand a more tractable, related question: what is the largest space in which local well-posedness holds and what quantities prevent blow-up at finite times? This program covers several papers and intertwines with a recent push to understand the regularity theory of kinetic Fokker-Planck-type equations. I will give a broad outline of the proof, and then, instead of slogging through the whole thing, I will focus on a simple proof of a technically important and physically interesting lower bound -- how ``vacuum regions'' fill in. This is a joint work with Snelson and Tarfulea.<br />
<br />
<br />
<br />
<br />
'''May 8, 2023'''<br />
<br />
[[TBA|Lei Wu (Lehigh)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in '''VV B223 (special room)''' <br />
<br />
'''Title:'''Ghost Effect from Boltzmann Theory<br />
<br />
'''Abstract:'''It is a classical and fundamental problem to study the hydrodynamic limits of kinetic equations as the Knudsen number \varepsilon\rightarrow 0. In this talk, we focus on the stationary Boltzmann equation with diffuse-reflection boundary condition where the Mach number is O(\varepsilon) but the temperature variance is O(1). We rigorously derived the limiting fluid system, the so-called ghost-effect equations, in which an infinitesimal boundary variance will have macroscopic observable effects. This pure kinetic effect reveals the incompleteness of traditional fluid theory. Our proof relies on newly developed kernel estimates with conservation laws and novel BV estimates for the cutoff boundary layer.<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=25139PDE Geometric Analysis seminar2023-09-05T15:13:25Z<p>Htran24: </p>
<hr />
<div>All talks will be ''in person'' unless specified otherwise.<br />
<br />
(Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.) <br />
===[[Previous PDE/GA seminars]]===<br />
<br />
=== [[Fall 2023-Spring 2024|Schedule for Fall 2023-Spring 2024]]===<br />
<br />
<br />
<br />
'''September 11, 2023 '''<br />
<br />
Dallas Albritton (UW-Madison)<br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
Title: <br />
<br />
'''September 18, 2023'''<br />
<br />
Hongjie Dong (Brown). Host: Hung Tran<br />
<br />
Time: 3:30-4:30PM, VV 901.<br />
<br />
Title: Recent results about the insulated conductivity problem.<br />
<br />
Abstract: In the first part of the talk, I will present our work about the insulated conductivity problem with closely spaced inclusions in a bounded domain in $R^n$. A noteworthy phenomenon in this context is the potential for the gradient of solutions to blow up as the distance between inclusions tends to zero. We obtained an optimal gradient estimate of solutions in terms of the distance, which settled down a major open problem in this area. In the second part, I will discuss recent results about the insulated conductivity problem when the current-electric field relation is a power law. New results for the perfect conductivity problem will also be mentioned.<br />
<br />
Based on joint work with Yanyan Li (Rutgers University), Zhuolun Yang and Hanye Zhu (Brown University).<br />
<br />
<br />
<br />
<br />
'''September 25, 2023'''<br />
<br />
Olga Turanova (MSU). Host: Hung Tran<br />
<br />
Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
'''October 2, 2023 '''<br />
<br />
No seminar. (Edriss Titi will give a Distinguished Lecture today.)<br />
<br />
'''October 9, 2023 '''<br />
<br />
Montie Avery (BU). Host: Dallas Albritton<br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
'''October 16, 2023 '''<br />
<br />
[[Ian Tice]] (CMU). Host: Chanwoo Kim <br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
'''October 23, 2023 '''<br />
<br />
Raghav Venkatraman (Courant). Host: Dallas Albritton<br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
'''October 30, 2023 '''<br />
<br />
[[ Sung-Jin Oh| Sung-Jin Oh]] (UC Berkeley). Host: Chanwoo Kim <br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
'''November 6, 2023 '''<br />
<br />
Vera Hur (UIUC). Host: Dallas Albritton<br />
<br />
Time: 3:30 PM-4:30 PM, VV901 <br />
<br />
'''November 13, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, VV901 <br />
<br />
'''November 20, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, VV901 <br />
<br />
'''November 27, 2023 '''([[ No seminar|First Monday after Thanksgiving]])<br />
<br />
'''December 4, 2023'''<br />
<br />
'''December 11, 2023'''<br />
<br />
= Spring 2024 =<br />
<br />
<br />
'''January 29, 2024'''<br />
<br />
Minh-Binh Tran (TAMU). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
'''April 1, 2024'''<br />
<br />
Dominic Wynter (Cambridge). Host: Chanwoo Kim<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
<br />
'''March 20, 2023''' <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
<br />
'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
<br />
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
<br />
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Large amplitude solution of BGK model''<br />
<br />
'''Abstract:''' Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time.<br />
This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.<br />
<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Interior W^{2,p} estimates for complex Monge-Ampere equations'' <br />
<br />
'''Abstract:''' The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] (Berkeley)<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion<br />
<br />
'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''The Boltzmann equation with large data<br />
<br />
'''Abstract:''' The Boltzmann equation is a nonlocal, nonlinear equation arising in gas dynamics for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk I will discuss a recent program to understand a more tractable, related question: what is the largest space in which local well-posedness holds and what quantities prevent blow-up at finite times? This program covers several papers and intertwines with a recent push to understand the regularity theory of kinetic Fokker-Planck-type equations. I will give a broad outline of the proof, and then, instead of slogging through the whole thing, I will focus on a simple proof of a technically important and physically interesting lower bound -- how ``vacuum regions'' fill in. This is a joint work with Snelson and Tarfulea.<br />
<br />
<br />
<br />
<br />
'''May 8, 2023'''<br />
<br />
[[TBA|Lei Wu (Lehigh)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in '''VV B223 (special room)''' <br />
<br />
'''Title:'''Ghost Effect from Boltzmann Theory<br />
<br />
'''Abstract:'''It is a classical and fundamental problem to study the hydrodynamic limits of kinetic equations as the Knudsen number \varepsilon\rightarrow 0. In this talk, we focus on the stationary Boltzmann equation with diffuse-reflection boundary condition where the Mach number is O(\varepsilon) but the temperature variance is O(1). We rigorously derived the limiting fluid system, the so-called ghost-effect equations, in which an infinitesimal boundary variance will have macroscopic observable effects. This pure kinetic effect reveals the incompleteness of traditional fluid theory. Our proof relies on newly developed kernel estimates with conservation laws and novel BV estimates for the cutoff boundary layer.<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
<br />
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<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=Research_at_UW-Madison_in_DifferentialEquations&diff=25125Research at UW-Madison in DifferentialEquations2023-09-05T12:20:25Z<p>Htran24: </p>
<hr />
<div>==Seminars of interest==<br />
<br />
The weekly [http://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE & Geometric Analysis seminar] is held on Monday afternoons, 3:30-4:30pm. <br />
<br />
Other seminars that will feature PDE related material are the [http://www.math.wisc.edu/wiki/index.php/Geometry_and_Topology_Seminar Geometry and Topology seminar], the [https://www.math.wisc.edu/wiki/index.php/Analysis_Seminar Analysis seminar], and the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math seminar].<br />
<br />
[https://sites.google.com/view/vapseminar Virtual Analysis and PDE Seminar (VAPS)]: organized jointly by 9 US universities including UW.<br />
<br />
==Faculty==<br />
<br />
[https://sites.google.com/view/albri050/dallas-albritton Dallas Albritton] (University of Minnesota, 2020) Pure and applied aspects of PDE and fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~feldman Mikhail Feldman] (UC Berkeley, 1994) Nonlinear PDE, Calculus of Variations.<br />
<br />
[http://www.math.wisc.edu/~ifrim/Home.html Mihaela Ifrim] (UC Davis, 2012) Nonlinear Dispersive Equations (water-wave equations and related dispersive models), Fluid Mechanics, Elastodynamics, Harmonic Analysis, General Relativity.<br />
<br />
[https://sites.google.com/view/ckim Chanwoo Kim] (Brown, 2011) Applied PDE, Kinetic theory, Fluid dynamics.<br />
<br />
[https://sites.google.com/a/umn.edu/math-laurelohm/home Laurel Ohm] (University of Minnesota, 2020) Analysis of PDE arising in biofluid mechanics.<br />
<br />
[http://www.math.wisc.edu/~hung Hung Vinh Tran] (UC Berkeley, 2012) Nonlinear PDE.<br />
<br />
==Faculty in related areas==<br />
<br />
[http://www.math.wisc.edu/~denissov Sergey Denisov] (Moscow State University, 1999) Analysis, PDE.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li] (UW Madison, 2013) Numerical analysis and scientific computing.<br />
<br />
[http://www.math.wisc.edu/~spagnolie/ Saverio Spagnolie] (Courant Institute, 2008) Fluid dynamics, complex fluids, soft matter, computation.<br />
<br />
[http://www.math.wisc.edu/~stechmann/ Sam Stechmann] (Courant Institute, 2008) Applied math, computational math, fluid dynamics, atmospheric science, climate.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (UT Austin, 1998) Mixing in fluids, optimization of mixing.<br />
<br />
[https://www.nicolasgarciat.com/ Nicolas Garcia Trillos] (CMU, 2015) Optimal transport, calculus of variations, and data analysis.<br />
==Current Postdocs (Van Vleck Assistant Professor) in PDE==<br />
<br />
[https://www.math.wisc.edu/~aai/ Albert Ai]<br />
<br />
[https://sites.google.com/view/tnguyen65 Trinh Nguyen]<br />
<br />
[https://schulzmath.wordpress.com/ Simon Schulz]<br />
<br />
==Recent former Postdocs in PDE==<br />
<br />
[https://blog.nus.edu.sg/yyao/ Yao Yao] (VV assist prof 2012-2015). Current position: Dean’s Chair Associate Professor, National University of Singapore.<br />
<br />
[https://sites.google.com/view/KYUDONGCHOI Kyudong Choi] (VV assist prof 2012-2015). Current position: Associate Professor, UNIST (2015-)<br />
<br />
[https://sites.google.com/view/jessicalin-math/home Jessica Lin] (VV assist prof 2014-2017). Current position: Assistant Professor, McGill University (2017-)<br />
<br />
[https://sites.google.com/site/donghyunlee295/ Donghyun Lee] (VV assist prof 2015-2018). Current position: Associate Professor and Mueunjae Distinguished Professor, Postech (2018-) <br />
<br />
[https://cam.uchicago.edu/people/profile/eric-baer/ Eric Baer] (VV assist prof 2015-2018). Current position: Senior Lecturer, University of Chicago (2019-)<br />
<br />
[https://sites.google.com/site/guoxx097/welcome Xiaoqin Guo] (VV assist prof 2017-2020). Current position: Assistant Professor, University of Cincinnati (2020-).<br />
<br />
[http://www.homepages.ucl.ac.uk/~ucahms0/index.htm Matthew Schrecker] (VV assist prof 2018-2020). Current position: EPSRC Postdoctoral Research Fellow, University College London (2020-).<br />
<br />
[https://sites.google.com/site/dhkwonmath/home?authuser=0 Dohyun Kwon] (VV assist prof 9/2020-1/2023). Current position: Assistant Professor, University of Seoul.<br />
<br />
==Emeriti==<br />
<br />
[http://www.math.wisc.edu/~angenent Sigurd Angenent] (Leiden, 1986) Nonlinear PDE, differential geometry, medical imaging, math biology.<br />
<br />
[http://www.math.wisc.edu/~bolotin Sergey Bolotin] (Moscow State University, 1982) Dynamical Systems, Variational Methods, Celestial Mechanics.<br />
<br />
[http://www.math.wisc.edu/~rabinowi Paul Rabinowitz]<br />
PDE, Calculus of Variations, Dynamical Systems, Nonlinear Analysis.<br />
<br />
[http://www.math.wisc.edu/~robbin Joel Robbin]<br />
Global Analysis, Differential Equations.<br />
<br />
[http://www.math.wisc.edu/~turner Robert Turner]<br />
Partial Differential Equations, Fluid Mechanics, Mathematical Biology.<br />
<br />
==Previous events==<br />
<br />
The 81st Midwest PDE seminar '''[https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]''' was held in Madison on April 21/22 (2018).</div>Htran24https://wiki.math.wisc.edu/index.php?title=Research_at_UW-Madison_in_DifferentialEquations&diff=25118Research at UW-Madison in DifferentialEquations2023-09-04T19:04:33Z<p>Htran24: </p>
<hr />
<div>==Seminars of interest==<br />
<br />
The weekly [http://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE & Geometric Analysis seminar] is held on Monday afternoons, 3:30-4:30pm. <br />
<br />
Other seminars that will feature PDE related material are the [http://www.math.wisc.edu/wiki/index.php/Geometry_and_Topology_Seminar Geometry and Topology seminar], the [https://www.math.wisc.edu/wiki/index.php/Analysis_Seminar Analysis seminar], and the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math seminar].<br />
<br />
[https://sites.google.com/view/vapseminar Virtual Analysis and PDE Seminar (VAPS)]: organized jointly by 9 US universities including UW.<br />
<br />
==Faculty==<br />
<br />
[http://www.math.wisc.edu/~angenent Sigurd Angenent] (Leiden, 1986) Nonlinear PDE, differential geometry, medical imaging, math biology.<br />
<br />
[http://www.math.wisc.edu/~feldman Mikhail Feldman] (UC Berkeley, 1994) Nonlinear PDE, Calculus of Variations.<br />
<br />
[http://www.math.wisc.edu/~ifrim/Home.html Mihaela Ifrim] (UC Davis, 2012) Nonlinear Dispersive Equations (water-wave equations and related dispersive models), Fluid Mechanics, Elastodynamics, Harmonic Analysis, General Relativity.<br />
<br />
[https://sites.google.com/view/ckim Chanwoo Kim] (Brown, 2011) Applied PDE, Kinetic theory, Fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~hung Hung Vinh Tran] (UC Berkeley, 2012) Nonlinear PDE.<br />
<br />
==Faculty in related areas==<br />
<br />
[http://www.math.wisc.edu/~denissov Sergey Denisov] (Moscow State University, 1999) Analysis, PDE.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li] (UW Madison, 2013) Numerical analysis and scientific computing.<br />
<br />
[http://www.math.wisc.edu/~spagnolie/ Saverio Spagnolie] (Courant Institute, 2008) Fluid dynamics, complex fluids, soft matter, computation.<br />
<br />
[http://www.math.wisc.edu/~stechmann/ Sam Stechmann] (Courant Institute, 2008) Applied math, computational math, fluid dynamics, atmospheric science, climate.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (UT Austin, 1998) Mixing in fluids, optimization of mixing.<br />
<br />
[https://www.nicolasgarciat.com/ Nicolas Garcia Trillos] (CMU, 2015) Optimal transport, calculus of variations, and data analysis.<br />
==Current Postdocs (Van Vleck Assistant Professor) in PDE==<br />
<br />
[https://www.math.wisc.edu/~aai/ Albert Ai]<br />
<br />
[https://sites.google.com/view/tnguyen65 Trinh Nguyen]<br />
<br />
[https://schulzmath.wordpress.com/ Simon Schulz]<br />
<br />
==Recent former Postdocs in PDE==<br />
<br />
[https://blog.nus.edu.sg/yyao/ Yao Yao] (VV assist prof 2012-2015). Current position: Dean’s Chair Associate Professor, National University of Singapore.<br />
<br />
[https://sites.google.com/view/jessicalin-math/home Jessica Lin] (VV assist prof 2014-2017). Current position: Assistant Professor, McGill University (2017-)<br />
<br />
[https://sites.google.com/site/donghyunlee295/ Donghyun Lee] (VV assist prof 2015-2018). Current position: Assistant Professor, Postech (2018-) <br />
<br />
[https://cam.uchicago.edu/people/profile/eric-baer/ Eric Baer] (VV assist prof 2015-2018). Current position: Senior Lecturer, University of Chicago (2019-)<br />
<br />
[https://sites.google.com/site/guoxx097/welcome Xiaoqin Guo] (VV assist prof 2017-2020). Current position: Assistant Professor, University of Cincinnati (2020-).<br />
<br />
[http://www.homepages.ucl.ac.uk/~ucahms0/index.htm Matthew Schrecker] (VV assist prof 2018-2020). Current position: EPSRC Postdoctoral Research Fellow, University College London (2020-).<br />
<br />
[https://sites.google.com/site/dhkwonmath/home?authuser=0 Dohyun Kwon] (VV assist prof 9/2020-1/2023). Current position: Assistant Professor, University of Seoul.<br />
<br />
==Emeriti==<br />
<br />
[http://www.math.wisc.edu/~bolotin Sergey Bolotin] (Moscow State University, 1982) Dynamical Systems, Variational Methods, Celestial Mechanics.<br />
<br />
[http://www.math.wisc.edu/~rabinowi Paul Rabinowitz]<br />
PDE, Calculus of Variations, Dynamical Systems, Nonlinear Analysis.<br />
<br />
[http://www.math.wisc.edu/~robbin Joel Robbin]<br />
Global Analysis, Differential Equations.<br />
<br />
[http://www.math.wisc.edu/~turner Robert Turner]<br />
Partial Differential Equations, Fluid Mechanics, Mathematical Biology.<br />
<br />
==Previous events==<br />
<br />
The 81st Midwest PDE seminar '''[https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]''' was held in Madison on April 21/22 (2018).</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia/Fall2023&diff=24919Colloquia/Fall20232023-06-29T14:40:46Z<p>Htran24: </p>
<hr />
<div>'''September 8, 2023, Friday 4pm, [https://www.uwlax.edu/profile/tdas/ Tushar Das] (University of Wisconsin-La Crosse)'''<br />
<br />
(host: Stovall)<br />
<br />
==== September 22, 2023, Friday 4pm, TBA ====<br />
(host: Craciun)<br />
<br />
<br />
'''October 2, 2023, Monday at 4pm [https://www.math.tamu.edu/~titi/ Edriss Titi] (Texas A&M University)'''<br />
<br />
Distinguished lectures<br />
<br />
(host: Smith, Stechmann)<br />
<br />
'''October 13, 2023, Friday at 4pm, (tent. reserved)'''<br />
<br />
(host: Stovall)<br />
<br />
<br />
<br />
'''October 20, 2023, Friday at 4pm, Sara Maloni (UVA)''' <br />
<br />
(hosts: Dymarz, Uyanik, GmMaW)<br />
<br />
<br />
<br />
'''October 25, 2023, Wednesday at 4pm, [https://math.mit.edu/~gigliola/ Gigliola Staffilani] (MIT)''' <br />
<br />
(hosts: Ifrim, Smith)<br />
<br />
'''October 31 (Tuesday at 4pm) and November 1 (Wednesday at 4pm)''' '''[https://www.wisdom.weizmann.ac.il/~dinuri/ Irit Dinur] (The Weizmann Institute of Science)'''<br />
<br />
Distinguished lectures, tentatively reserved.<br />
<br />
(host: Gurevich).</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia/Fall2023&diff=24918Colloquia/Fall20232023-06-29T09:43:11Z<p>Htran24: </p>
<hr />
<div>'''September 8, 2023, Friday 4pm, [https://www.uwlax.edu/profile/tdas/ Tushar Das] (University of Wisconsin-La Crosse)'''<br />
<br />
(host: Stovall)<br />
<br />
==== September 22, 2023, Friday 4pm, TBA ====<br />
(host: Craciun)<br />
<br />
<br />
'''October 2, 2023, Monday at 4pm [https://www.math.tamu.edu/~titi/ Edriss Titi] (Texas A&M University)'''<br />
<br />
Distinguished lectures<br />
<br />
(host: Smith, Stechmann)<br />
<br />
'''October 13, 2023, Friday at 4pm, (tent. reserved)'''<br />
<br />
(host: Stovall)<br />
<br />
<br />
<br />
'''October 20, 2023, Friday at 4pm, Sara Maloni (UVA)''' <br />
<br />
(hosts: Dymarz, Uyanik, GmMaW)<br />
<br />
<br />
<br />
'''October 25, 2023, Wednesday at 4pm, [https://math.mit.edu/~gigliola/ Gigliola Staffilani] (MIT)''' <br />
<br />
(host: Leslie Smith)<br />
<br />
'''October 31 (Tuesday at 4pm) and November 1 (Wednesday at 4pm)''' '''[https://www.wisdom.weizmann.ac.il/~dinuri/ Irit Dinur] (The Weizmann Institute of Science)'''<br />
<br />
Distinguished lectures, tentatively reserved.<br />
<br />
(host: Gurevich).</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia/Fall2023&diff=24917Colloquia/Fall20232023-06-29T09:42:02Z<p>Htran24: </p>
<hr />
<div>'''September 8, 2023, Friday 4pm, [https://www.uwlax.edu/profile/tdas/ Tushar Das] (University of Wisconsin-La Crosse)'''<br />
<br />
(host: Stovall)<br />
<br />
==== September 22, 2023, Friday 4pm, TBA ====<br />
(host: Craciun)<br />
<br />
<br />
'''October 2, 2023, Monday at 4pm [https://www.math.tamu.edu/~titi/ Edriss Titi] (Texas A&M University)'''<br />
<br />
Distinguished lectures<br />
<br />
(host: Smith, Stechmann)<br />
<br />
'''October 13, 2023, Friday at 4pm, (tent. reserved)'''<br />
<br />
(host: Stovall)<br />
<br />
<br />
'''October 20, 2023, Friday at 4pm, Sara Maloni (UVA)''' <br />
<br />
(hosts: Dymarz, Uyanik, GmMaW)<br />
<br />
<br />
'''October 25, 2023, Wednesday at 4pm, [https://math.mit.edu/~gigliola/ Gigliola Staffilani] (MIT)''' <br />
<br />
(host: Leslie Smith)<br />
<br />
'''October 31 (Tuesday at 4pm) and November 1 (Wednesday at 4pm)''' '''[https://www.wisdom.weizmann.ac.il/~dinuri/ Irit Dinur] (The Weizmann Institute of Science)'''<br />
<br />
Distinguished lectures, tentatively reserved.<br />
<br />
(host: Gurevich).</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=24906PDE Geometric Analysis seminar2023-06-18T00:28:36Z<p>Htran24: </p>
<hr />
<div>The seminar's format will be a combination of online and in-person; we will make sure to update you as soon as we have more details available. First talk is tentatively scheduled for September 20th ! <br />
<br />
Some of the seminars will be held online. When that would be the case we would use the following zoom link<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
<br />
<br />
[[Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.]]<br />
<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
=== [[Fall 2023-Spring 2024|Schedule for Fall 2023-Spring 2024]]===<br />
<br />
<br />
<br />
'''September 11, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
<br />
<br />
'''September 18, 2023'''<br />
<br />
Hongjie Dong (Brown). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
'''September 25, 2023'''<br />
<br />
Olga Turanova (MSU). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
<br />
<br />
<br />
'''October 2, 2023 '''<br />
<br />
[[ No seminar| Visitor overlapping]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
<br />
<br />
'''October 9, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
<br />
<br />
'''October 16, 2023 '''<br />
<br />
[[ Ian Tice| Ian Tice]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
<br />
<br />
'''October 23, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''October 30, 2023 '''<br />
<br />
[[ Sung-Jin Oh| Sung-Jin Oh]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
<br />
<br />
'''November 6, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''November 13, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
<br />
<br />
'''November 20, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
<br />
<br />
'''November 27, 2023 '''<br />
<br />
[[ No seminar| First Monday after Thanksgiving]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
<br />
<br />
'''December 4, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
<br />
<br />
'''December 11, 2023 '''<br />
<br />
[[ TBA| TBA]]<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901<br />
<br />
= Spring 2024 =<br />
<br />
<br />
'''January 29, 2024'''<br />
<br />
Minh-Binh Tran (TAMU). Host: Hung Tran<br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901.<br />
<br />
Title:<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
<br />
'''March 20, 2023''' <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
<br />
'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
<br />
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
<br />
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Large amplitude solution of BGK model''<br />
<br />
'''Abstract:''' Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time.<br />
This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.<br />
<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Interior W^{2,p} estimates for complex Monge-Ampere equations'' <br />
<br />
'''Abstract:''' The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] (Berkeley)<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion<br />
<br />
'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''The Boltzmann equation with large data<br />
<br />
'''Abstract:''' The Boltzmann equation is a nonlocal, nonlinear equation arising in gas dynamics for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk I will discuss a recent program to understand a more tractable, related question: what is the largest space in which local well-posedness holds and what quantities prevent blow-up at finite times? This program covers several papers and intertwines with a recent push to understand the regularity theory of kinetic Fokker-Planck-type equations. I will give a broad outline of the proof, and then, instead of slogging through the whole thing, I will focus on a simple proof of a technically important and physically interesting lower bound -- how ``vacuum regions'' fill in. This is a joint work with Snelson and Tarfulea.<br />
<br />
<br />
<br />
<br />
'''May 8, 2023'''<br />
<br />
[[TBA|Lei Wu (Lehigh)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in '''VV B223 (special room)''' <br />
<br />
'''Title:'''Ghost Effect from Boltzmann Theory<br />
<br />
'''Abstract:'''It is a classical and fundamental problem to study the hydrodynamic limits of kinetic equations as the Knudsen number \varepsilon\rightarrow 0. In this talk, we focus on the stationary Boltzmann equation with diffuse-reflection boundary condition where the Mach number is O(\varepsilon) but the temperature variance is O(1). We rigorously derived the limiting fluid system, the so-called ghost-effect equations, in which an infinitesimal boundary variance will have macroscopic observable effects. This pure kinetic effect reveals the incompleteness of traditional fluid theory. Our proof relies on newly developed kernel estimates with conservation laws and novel BV estimates for the cutoff boundary layer.<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
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<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
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<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
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<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia/Fall2023&diff=24824Colloquia/Fall20232023-04-29T13:42:38Z<p>Htran24: </p>
<hr />
<div>'''September 8, 2023, Friday 4pm, [https://www.uwlax.edu/profile/tdas/ Tushar Das] (University of Wisconsin-La Crosse)'''<br />
<br />
(host: Stovall)<br />
<br />
==== September 22, 2023, Friday 4pm, TBA ====<br />
(host: Craciun)<br />
<br />
==== October 2, 2023, Monday at 4pm [https://www.math.tamu.edu/~titi/ Edriss Titi] (Texas A&M University) ====<br />
Distinguished lectures<br />
<br />
(host: Smith, Stechmann)<br />
<br />
'''October 13, 2023, Friday at 4pm, (tent. reserved)'''<br />
<br />
(host: Stovall)<br />
<br />
'''October 31 (Tuesday at 4pm) and November 1 (Wednesday at 4pm)''' '''[https://www.wisdom.weizmann.ac.il/~dinuri/ Irit Dinur] (The Weizmann Institute of Science)'''<br />
<br />
Distinguished lectures, tentatively reserved.<br />
<br />
(host: Gurevich).</div>Htran24https://wiki.math.wisc.edu/index.php?title=Fall_2023-Spring_2024&diff=24792Fall 2023-Spring 20242023-04-18T23:52:52Z<p>Htran24: /* Spring 2024 */</p>
<hr />
<div>=Fall 2023= <br />
<br />
<br />
<br />
<br />
<br />
'''September 11, 2023'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''September 18, 2023'''<br />
<br />
Hongjie Dong (Brown). Host: Hung Tran <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''September 25, 2023'''<br />
<br />
Olga Turanova (MSU). Host: Hung Tran <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''October 16, 2023'''<br />
<br />
[[Ian Tice|Ian Tice]](CMU). Host: Chanwoo Kim <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''October 30, 2023'''<br />
<br />
Sung-Jin Oh(Berkeley). Host: Chanwoo Kim <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''December 11, 2023'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901. <br />
<br />
Title:<br />
<br />
=Spring 2024= <br />
<br />
'''January 29, 2024'''<br />
<br />
Minh-Binh Tran (TAMU). Host: Hung Tran <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901. <br />
<br />
Title:<br />
<br />
<br />
<br />
<br />
'''March 25, 2024'''<br />
<br />
No seminar (Spring recess)<br />
<br />
<br />
<br />
'''April 29, 2024'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 901. <br />
<br />
Title:</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia&diff=24761Colloquia2023-04-10T22:13:55Z<p>Htran24: </p>
<hr />
<div>__NOTOC__<br />
<br />
<br />
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm. </b><br />
<br />
<!--- in Van Vleck B239, '''unless otherwise indicated'''. ---><br />
<br />
<br />
== February 3, 2023, Friday at 4pm [https://sites.google.com/a/uwlax.edu/tdas/ Facundo Mémoli] (Ohio State University) ==<br />
(host: Lyu)<br />
<br />
The Gromov-Hausdorff distance between spheres.<br />
<br />
The Gromov-Hausdorff distance is a fundamental tool in Riemanian geometry (through the topology it generates) and is also utilized in applied geometry and topological data analysis as a metric for expressing the stability of methods which process geometric data (e.g. hierarchical clustering and persistent homology barcodes via the Vietoris-Rips filtration). In fact, distances such as the Gromov-Hausdorff distance or its Optimal Transport variants (i.e. the so-called Gromov-Wasserstein distances) are nowadays often invoked in applications related to data classification.<br />
<br />
Whereas it is often easy to estimate the value of the Gromov-Hausdorff distance between two given metric spaces, its ''precise'' value is rarely easy to determine. Some of the best estimates follow from considerations related to both the stability of persistent homology and to Gromov's filling radius. However, these turn out to be non-sharp.<br />
<br />
In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between pairs of spheres (endowed with their usual geodesic distance). These results involve lower bounds which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and also matching upper bounds which are induced from specialized constructions of (a posteriori optimal) ``correspondences" between spheres.<br />
<br />
== February 24, 2023, Cancelled/available ==<br />
== March 3, 2023, Friday at 4pm [https://faculty.washington.edu/steinerb/ Stefan Steinerberger] (University of Washington) ==<br />
<br />
Title: How curved is a combinatorial graph?<br />
<br />
Abstract: Curvature is one of the fundamental ingredients in differential geometry. People are increasingly interested in whether it is possible to think of combinatorial graphs as behaving like manifolds and a number of different notions of curvature have been proposed. I will introduce some of the existing ideas and then propose a new notion based on a simple and explicit linear system of equations that is easy to compute. This notion satisfies a surprisingly large number of desirable properties -- connections to game theory (especially the von Neumann Minimax Theorem) and potential theory will be sketched; simultaneously, there is a certain "magic" element to all of this that is poorly understood and many open problems remain. I will also sketch some curious related problems that remain mostly open. No prior knowledge of differential geometry (or graphs) is required.<br />
<br />
(hosts: Shaoming Guo, Andreas Seeger)<br />
<br />
== March 8, 2023, Wednesday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
'''''Distinguished lectures'''''<br />
<br />
Title: Surfaces and foliations in hyperbolic 3-manifolds<br />
<br />
Abstract: How does the geometric theory of hyperbolic 3-manifolds interact with the topological theory of foliations within them? Both points of view have seen profound developments over the past 40 years, and yet we have only an incomplete understanding of their overlap. I won't have much to add to this understanding! Instead, I will meander through aspects of both stories, saying a bit about what we know and pointing out some interesting questions.<br />
<br />
(host: Kent)<br />
<br />
== March 10, 2023, Friday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
'''''Distinguished lectures'''''<br />
<br />
Title: End-periodic maps, via fibered 3-manifolds<br />
<br />
Abstract: In the second lecture I will focus on some joint work with Michael Landry and Sam Taylor. Thurston showed how a certain ``spinning<nowiki>''</nowiki> construction in a fibered 3-manifold produces a depth-1 foliation, which is described by an end-periodic map of an infinite genus surface. The dynamical properties of such maps were then studied by Handel-Miller, Cantwell-Conlon-Fenley and others. We show how to reverse this construction, obtaining every end-periodic map from spinning in a fibered manifold. This allows us to recover the dynamical features of the map, and more, directly from the more classical theory of fibered manifolds.<br />
<br />
(host: Kent)<br />
<br />
== March 24, 2023 , Friday at 4pm [https://www.carolynrabbott.com/ Carolyn Abbott] (Brandeis University) ==<br />
'''Title''': Boundaries, boundaries, and more boundaries <br />
<br />
'''Abstract:''' It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I will discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I will describe various relationships between these boundaries and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups. This is joint work with Jason Behrstock and Jacob Russell. <br />
<br />
== March 31, 2023 , Friday at 4pm [http://www.math.toronto.edu/balint/ Bálint Virág] (University of Toronto) ==<br />
'''Title:''' Random plane geometry -- a gentle introduction<br />
<br />
'''Abstract:''' Consider Z^2, and assign a random length of 1 or 2 to each edge based on independent fair coin tosses. The resulting random geometry, first passage percolation, is conjectured to have a scaling limit. Most random plane geometric models (including hidden geometries) should have the same scaling limit. I will explain the basics of the limiting geometry, the "directed landscape", and its relation to traffic jams, tetris, coffee stains and random matrices.<br />
<br />
(host: Valko)<br />
<br />
== April 7, 2023, Friday at 4pm [https://www.mi.fu-berlin.de/math/groups/fluid-dyn/members/rupert_klein.html Rupert Klein] (FU Berlin) ==<br />
<br />
'''''Wasow lecture'''''<br />
<br />
Title: Mathematics: A key to climate research<br />
<br />
Abstract: Mathematics in climate research is often thought to be mainly a provider of techniques for solving, e.g., the atmosphere and ocean flow equations. Three examples elucidate that its role is much broader and deeper:<br />
<br />
1) Climate modelers often employ reduced forms of “the flow equations” for efficiency. Mathematical analysis helps assessing the regimes of validity of such models and defining conditions under which they can be solved robustly.<br />
<br />
2) Climate is defined as “weather statistics”, and climate research investigates its change in time in our “single realization of Earth” with all its complexity. The required reliable notions of time dependent statistics for sparse data in high dimensions, however, remain to be established. Recent mathematical research offers advanced data analysis techniques that could be “game changing” in this respect.<br />
<br />
3) Climate research, economy, and the social sciences are to generate a scientific basis for informed political decision making. Subtle misunderstandings often hamper systematic progress in this area. Mathematical formalization can help structuring discussions and bridging language barriers in interdisciplinary research.<br />
<br />
(hosts: Smith, Stechmann)<br />
<br />
== April 21, 2023, Friday at 4pm [https://sternber.pages.iu.edu/ Peter Sternberg] (Indiana University) ==<br />
<br />
(hosts: Feldman, Tran)<br />
<br />
Title: A family of toy problems modeling liquid crystals exhibiting large disparity in the elastic coefficients.<br />
<br />
Abstract: Certain classes of liquid crystals have been found to strongly favor particular types of deformations over others; for example, the cost of splay may greatly exceed the cost of bend or twist. In a series of studies with Dmitry Golovaty (Akron), Michael Novack (UT Austin) and Raghav Venkatraman (Courant), we explore the implications of assuming various asymptotic regimes for the elastic constants. Through a mixture of formal and rigorous analysis, along with computations, we identify the limiting behavior of minimizers to the associated energies. We find that a variety of singular structures emerge corresponding to jumps in the profile of these limiting minimizers that effectively save on the cost of splay, bend or twist—whichever is assumed to be most expensive.<br />
<br />
<br />
== April 28, 2023, Friday at 4pm [https://nqle.pages.iu.edu/ Nam Q. Le] (Indiana University) ==<br />
Title: Hessian eigenvalues and hyperbolic polynomials<br />
<br />
Abstract: Hessian eigenvalues are natural nonlinear analogues of the classical Dirichlet eigenvalues. The Hessian eigenvalues and their corresponding eigenfunctions are expected to share many analytic and geometric properties (such as uniqueness, stability, max-min principle, global smoothness, Brunn-Minkowski inequality, convergence of numerical schemes, etc) as their Dirichlet counterparts. In this talk, I will discuss these issues and some recent progresses in various geometric settings. I will also explain the unexpected role of hyperbolic polynomials in our analysis. I will not assume any familiarity with these concepts. <br />
<br />
== Future Colloquia ==<br />
<br />
[[Colloquia/Fall2023|Fall 2023]]<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Fall2022|Fall 2022]]<br />
<br />
[[Spring 2022 Colloquiums|Spring 2022]]<br />
<br />
[[Colloquia/Fall2021|Fall 2021]]<br />
<br />
[[Colloquia/Spring2021|Spring 2021]]<br />
<br />
[[Colloquia/Fall2020|Fall 2020]]<br />
<br />
[[Colloquia/Spring2020|Spring 2020]]<br />
<br />
[[Colloquia/Fall2019|Fall 2019]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]<br />
<br />
[[WIMAW]]</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia&diff=24742Colloquia2023-04-05T18:05:49Z<p>Htran24: /* April 21, 2023, Friday at 4pm Peter Sternberg (Indiana University) */</p>
<hr />
<div>__NOTOC__<br />
<br />
<br />
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm. </b><br />
<br />
<!--- in Van Vleck B239, '''unless otherwise indicated'''. ---><br />
<br />
<br />
== February 3, 2023, Friday at 4pm [https://sites.google.com/a/uwlax.edu/tdas/ Facundo Mémoli] (Ohio State University) ==<br />
(host: Lyu)<br />
<br />
The Gromov-Hausdorff distance between spheres.<br />
<br />
The Gromov-Hausdorff distance is a fundamental tool in Riemanian geometry (through the topology it generates) and is also utilized in applied geometry and topological data analysis as a metric for expressing the stability of methods which process geometric data (e.g. hierarchical clustering and persistent homology barcodes via the Vietoris-Rips filtration). In fact, distances such as the Gromov-Hausdorff distance or its Optimal Transport variants (i.e. the so-called Gromov-Wasserstein distances) are nowadays often invoked in applications related to data classification.<br />
<br />
Whereas it is often easy to estimate the value of the Gromov-Hausdorff distance between two given metric spaces, its ''precise'' value is rarely easy to determine. Some of the best estimates follow from considerations related to both the stability of persistent homology and to Gromov's filling radius. However, these turn out to be non-sharp.<br />
<br />
In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between pairs of spheres (endowed with their usual geodesic distance). These results involve lower bounds which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and also matching upper bounds which are induced from specialized constructions of (a posteriori optimal) ``correspondences" between spheres.<br />
<br />
== February 24, 2023, Cancelled/available ==<br />
== March 3, 2023, Friday at 4pm [https://faculty.washington.edu/steinerb/ Stefan Steinerberger] (University of Washington) ==<br />
<br />
Title: How curved is a combinatorial graph?<br />
<br />
Abstract: Curvature is one of the fundamental ingredients in differential geometry. People are increasingly interested in whether it is possible to think of combinatorial graphs as behaving like manifolds and a number of different notions of curvature have been proposed. I will introduce some of the existing ideas and then propose a new notion based on a simple and explicit linear system of equations that is easy to compute. This notion satisfies a surprisingly large number of desirable properties -- connections to game theory (especially the von Neumann Minimax Theorem) and potential theory will be sketched; simultaneously, there is a certain "magic" element to all of this that is poorly understood and many open problems remain. I will also sketch some curious related problems that remain mostly open. No prior knowledge of differential geometry (or graphs) is required.<br />
<br />
(hosts: Shaoming Guo, Andreas Seeger)<br />
<br />
== March 8, 2023, Wednesday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
'''''Distinguished lectures'''''<br />
<br />
Title: Surfaces and foliations in hyperbolic 3-manifolds<br />
<br />
Abstract: How does the geometric theory of hyperbolic 3-manifolds interact with the topological theory of foliations within them? Both points of view have seen profound developments over the past 40 years, and yet we have only an incomplete understanding of their overlap. I won't have much to add to this understanding! Instead, I will meander through aspects of both stories, saying a bit about what we know and pointing out some interesting questions.<br />
<br />
(host: Kent)<br />
<br />
== March 10, 2023, Friday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
'''''Distinguished lectures'''''<br />
<br />
Title: End-periodic maps, via fibered 3-manifolds<br />
<br />
Abstract: In the second lecture I will focus on some joint work with Michael Landry and Sam Taylor. Thurston showed how a certain ``spinning<nowiki>''</nowiki> construction in a fibered 3-manifold produces a depth-1 foliation, which is described by an end-periodic map of an infinite genus surface. The dynamical properties of such maps were then studied by Handel-Miller, Cantwell-Conlon-Fenley and others. We show how to reverse this construction, obtaining every end-periodic map from spinning in a fibered manifold. This allows us to recover the dynamical features of the map, and more, directly from the more classical theory of fibered manifolds.<br />
<br />
(host: Kent)<br />
<br />
== March 24, 2023 , Friday at 4pm [https://www.carolynrabbott.com/ Carolyn Abbott] (Brandeis University) ==<br />
'''Title''': Boundaries, boundaries, and more boundaries <br />
<br />
'''Abstract:''' It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I will discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I will describe various relationships between these boundaries and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups. This is joint work with Jason Behrstock and Jacob Russell. <br />
<br />
== March 31, 2023 , Friday at 4pm [http://www.math.toronto.edu/balint/ Bálint Virág] (University of Toronto) ==<br />
'''Title:''' Random plane geometry -- a gentle introduction<br />
<br />
'''Abstract:''' Consider Z^2, and assign a random length of 1 or 2 to each edge based on independent fair coin tosses. The resulting random geometry, first passage percolation, is conjectured to have a scaling limit. Most random plane geometric models (including hidden geometries) should have the same scaling limit. I will explain the basics of the limiting geometry, the "directed landscape", and its relation to traffic jams, tetris, coffee stains and random matrices.<br />
<br />
(host: Valko)<br />
<br />
== April 7, 2023, Friday at 4pm [https://www.mi.fu-berlin.de/math/groups/fluid-dyn/members/rupert_klein.html Rupert Klein] (FU Berlin) ==<br />
<br />
'''''Wasow lecture'''''<br />
<br />
Title: Mathematics: A key to climate research<br />
<br />
Abstract: Mathematics in climate research is often thought to be mainly a provider of techniques for solving, e.g., the atmosphere and ocean flow equations. Three examples elucidate that its role is much broader and deeper:<br />
<br />
1) Climate modelers often employ reduced forms of “the flow equations” for efficiency. Mathematical analysis helps assessing the regimes of validity of such models and defining conditions under which they can be solved robustly.<br />
<br />
2) Climate is defined as “weather statistics”, and climate research investigates its change in time in our “single realization of Earth” with all its complexity. The required reliable notions of time dependent statistics for sparse data in high dimensions, however, remain to be established. Recent mathematical research offers advanced data analysis techniques that could be “game changing” in this respect.<br />
<br />
3) Climate research, economy, and the social sciences are to generate a scientific basis for informed political decision making. Subtle misunderstandings often hamper systematic progress in this area. Mathematical formalization can help structuring discussions and bridging language barriers in interdisciplinary research.<br />
<br />
(hosts: Smith, Stechmann)<br />
<br />
== April 21, 2023, Friday at 4pm [https://sternber.pages.iu.edu/ Peter Sternberg] (Indiana University) ==<br />
<br />
(hosts: Feldman, Tran)<br />
<br />
Title: A family of toy problems modeling liquid crystals exhibiting large disparity in the elastic coefficients.<br />
<br />
Abstract: Certain classes of liquid crystals have been found to strongly favor particular types of deformations over others; for example, the cost of splay may greatly exceed the cost of bend or twist. In a series of studies with Dmitry Golovaty (Akron), Michael Novack (UT Austin) and Raghav Venkatraman (Courant), we explore the implications of assuming various asymptotic regimes for the elastic constants. Through a mixture of formal and rigorous analysis, along with computations, we identify the limiting behavior of minimizers to the associated energies. We find that a variety of singular structures emerge corresponding to jumps in the profile of these limiting minimizers that effectively save on the cost of splay, bend or twist—whichever is assumed to be most expensive.<br />
<br />
<br />
== April 28, 2023, Friday at 4pm [https://nqle.pages.iu.edu/ Nam Q. Le] (Indiana University) ==<br />
<br />
== Future Colloquia ==<br />
<br />
[[Colloquia/Fall2023|Fall 2023]]<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Fall2022|Fall 2022]]<br />
<br />
[[Spring 2022 Colloquiums|Spring 2022]]<br />
<br />
[[Colloquia/Fall2021|Fall 2021]]<br />
<br />
[[Colloquia/Spring2021|Spring 2021]]<br />
<br />
[[Colloquia/Fall2020|Fall 2020]]<br />
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[[Colloquia/Spring2020|Spring 2020]]<br />
<br />
[[Colloquia/Fall2019|Fall 2019]]<br />
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[[Colloquia/Spring2019|Spring 2019]]<br />
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[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
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[[Colloquia/Spring2016|Spring 2016]]<br />
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[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]<br />
<br />
[[WIMAW]]</div>Htran24https://wiki.math.wisc.edu/index.php?title=Research_at_UW-Madison_in_DifferentialEquations&diff=24685Research at UW-Madison in DifferentialEquations2023-03-22T18:15:45Z<p>Htran24: </p>
<hr />
<div>==Seminars of interest==<br />
<br />
The weekly [http://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE & Geometric Analysis seminar] is held on Monday afternoons, 3:30-4:30pm. <br />
<br />
Other seminars that will feature PDE related material are the [http://www.math.wisc.edu/wiki/index.php/Geometry_and_Topology_Seminar Geometry and Topology seminar], the [https://www.math.wisc.edu/wiki/index.php/Analysis_Seminar Analysis seminar], and the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math seminar].<br />
<br />
[https://sites.uci.edu/pdeonlineseminar/ Virtual Analysis and PDE Seminar (VAPS)]: organized jointly by 9 US universities including UW.<br />
<br />
==Faculty==<br />
<br />
[http://www.math.wisc.edu/~angenent Sigurd Angenent] (Leiden, 1986) Nonlinear PDE, differential geometry, medical imaging, math biology.<br />
<br />
[http://www.math.wisc.edu/~feldman Mikhail Feldman] (UC Berkeley, 1994) Nonlinear PDE, Calculus of Variations.<br />
<br />
[http://www.math.wisc.edu/~ifrim/Home.html Mihaela Ifrim] (UC Davis, 2012) Nonlinear Dispersive Equations (water-wave equations and related dispersive models), Fluid Mechanics, Elastodynamics, Harmonic Analysis, General Relativity.<br />
<br />
[https://sites.google.com/view/ckim Chanwoo Kim] (Brown, 2011) Applied PDE, Kinetic theory, Fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~hung Hung Vinh Tran] (UC Berkeley, 2012) Nonlinear PDE.<br />
<br />
==Faculty in related areas==<br />
<br />
[http://www.math.wisc.edu/~denissov Sergey Denisov] (Moscow State University, 1999) Analysis, PDE.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li] (UW Madison, 2013) Numerical analysis and scientific computing.<br />
<br />
[http://www.math.wisc.edu/~spagnolie/ Saverio Spagnolie] (Courant Institute, 2008) Fluid dynamics, complex fluids, soft matter, computation.<br />
<br />
[http://www.math.wisc.edu/~stechmann/ Sam Stechmann] (Courant Institute, 2008) Applied math, computational math, fluid dynamics, atmospheric science, climate.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault] (UT Austin, 1998) Mixing in fluids, optimization of mixing.<br />
<br />
[https://www.nicolasgarciat.com/ Nicolas Garcia Trillos] (CMU, 2015) Optimal transport, calculus of variations, and data analysis.<br />
==Current Postdocs (Van Vleck Assistant Professor) in PDE==<br />
<br />
[https://www.math.wisc.edu/~aai/ Albert Ai]<br />
<br />
[https://sites.google.com/view/tnguyen65 Trinh Nguyen]<br />
<br />
[https://schulzmath.wordpress.com/ Simon Schulz]<br />
<br />
==Recent former Postdocs in PDE==<br />
<br />
[https://blog.nus.edu.sg/yyao/ Yao Yao] (VV assist prof 2012-2015). Current position: Dean’s Chair Associate Professor, National University of Singapore.<br />
<br />
[https://sites.google.com/view/jessicalin-math/home Jessica Lin] (VV assist prof 2014-2017). Current position: Assistant Professor, McGill University (2017-)<br />
<br />
[https://sites.google.com/site/donghyunlee295/ Donghyun Lee] (VV assist prof 2015-2018). Current position: Assistant Professor, Postech (2018-) <br />
<br />
[https://cam.uchicago.edu/people/profile/eric-baer/ Eric Baer] (VV assist prof 2015-2018). Current position: Senior Lecturer, University of Chicago (2019-)<br />
<br />
[https://sites.google.com/site/guoxx097/welcome Xiaoqin Guo] (VV assist prof 2017-2020). Current position: Assistant Professor, University of Cincinnati (2020-).<br />
<br />
[http://www.homepages.ucl.ac.uk/~ucahms0/index.htm Matthew Schrecker] (VV assist prof 2018-2020). Current position: EPSRC Postdoctoral Research Fellow, University College London (2020-).<br />
<br />
[https://sites.google.com/site/dhkwonmath/home?authuser=0 Dohyun Kwon] (VV assist prof 9/2020-1/2023). Current position: Assistant Professor, University of Seoul.<br />
<br />
==Emeriti==<br />
<br />
[http://www.math.wisc.edu/~bolotin Sergey Bolotin] (Moscow State University, 1982) Dynamical Systems, Variational Methods, Celestial Mechanics.<br />
<br />
[http://www.math.wisc.edu/~rabinowi Paul Rabinowitz]<br />
PDE, Calculus of Variations, Dynamical Systems, Nonlinear Analysis.<br />
<br />
[http://www.math.wisc.edu/~robbin Joel Robbin]<br />
Global Analysis, Differential Equations.<br />
<br />
[http://www.math.wisc.edu/~turner Robert Turner]<br />
Partial Differential Equations, Fluid Mechanics, Mathematical Biology.<br />
<br />
==Previous events==<br />
<br />
The 81st Midwest PDE seminar '''[https://sites.google.com/view/81stmidwestpdeseminar/home Midwest PDE seminar]''' was held in Madison on April 21/22 (2018).</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=24670PDE Geometric Analysis seminar2023-03-20T15:46:19Z<p>Htran24: /* Spring 2023 */</p>
<hr />
<div>The seminar's format will be a combination of online and in-person; we will make sure to update you as soon as we have more details available. First talk is tentatively scheduled for September 20th ! <br />
<br />
Some of the seminars will be held online. When that would be the case we would use the following zoom link<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
<br />
<br />
[[Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.]]<br />
<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
=== [[Fall 2023-Spring 2024|Schedule for Fall 2023-Spring 2024]]===<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
<br />
'''March 20, 2023''' <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Lagrangian solutions to the Porous Media Equation (and friends)<br />
<br />
'''Abstract:''' Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.<br />
<br />
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? <br />
<br />
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] <br />
<br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''May 9, 2023'''<br />
<br />
Lei Wu (Lehigh). Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
<br />
<br />
<br />
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<br />
<br />
<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia&diff=24665Colloquia2023-03-19T17:03:50Z<p>Htran24: /* March 24, 2023 , Friday at 4pm Carolyn Abbott (Brandeis University) */</p>
<hr />
<div>__NOTOC__<br />
<br />
<br />
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm. </b><br />
<br />
<!--- in Van Vleck B239, '''unless otherwise indicated'''. ---><br />
<br />
<br />
== February 3, 2023, Friday at 4pm [https://sites.google.com/a/uwlax.edu/tdas/ Facundo Mémoli] (Ohio State University) ==<br />
(host: Lyu)<br />
<br />
The Gromov-Hausdorff distance between spheres.<br />
<br />
The Gromov-Hausdorff distance is a fundamental tool in Riemanian geometry (through the topology it generates) and is also utilized in applied geometry and topological data analysis as a metric for expressing the stability of methods which process geometric data (e.g. hierarchical clustering and persistent homology barcodes via the Vietoris-Rips filtration). In fact, distances such as the Gromov-Hausdorff distance or its Optimal Transport variants (i.e. the so-called Gromov-Wasserstein distances) are nowadays often invoked in applications related to data classification.<br />
<br />
Whereas it is often easy to estimate the value of the Gromov-Hausdorff distance between two given metric spaces, its ''precise'' value is rarely easy to determine. Some of the best estimates follow from considerations related to both the stability of persistent homology and to Gromov's filling radius. However, these turn out to be non-sharp.<br />
<br />
In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between pairs of spheres (endowed with their usual geodesic distance). These results involve lower bounds which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and also matching upper bounds which are induced from specialized constructions of (a posteriori optimal) ``correspondences" between spheres.<br />
<br />
== February 24, 2023, Cancelled/available ==<br />
== March 3, 2023, Friday at 4pm [https://faculty.washington.edu/steinerb/ Stefan Steinerberger] (University of Washington) ==<br />
<br />
Title: How curved is a combinatorial graph?<br />
<br />
Abstract: Curvature is one of the fundamental ingredients in differential geometry. People are increasingly interested in whether it is possible to think of combinatorial graphs as behaving like manifolds and a number of different notions of curvature have been proposed. I will introduce some of the existing ideas and then propose a new notion based on a simple and explicit linear system of equations that is easy to compute. This notion satisfies a surprisingly large number of desirable properties -- connections to game theory (especially the von Neumann Minimax Theorem) and potential theory will be sketched; simultaneously, there is a certain "magic" element to all of this that is poorly understood and many open problems remain. I will also sketch some curious related problems that remain mostly open. No prior knowledge of differential geometry (or graphs) is required.<br />
<br />
(hosts: Shaoming Guo, Andreas Seeger)<br />
<br />
== March 8, 2023, Wednesday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
'''''Distinguished lectures'''''<br />
<br />
Title: Surfaces and foliations in hyperbolic 3-manifolds<br />
<br />
Abstract: How does the geometric theory of hyperbolic 3-manifolds interact with the topological theory of foliations within them? Both points of view have seen profound developments over the past 40 years, and yet we have only an incomplete understanding of their overlap. I won't have much to add to this understanding! Instead, I will meander through aspects of both stories, saying a bit about what we know and pointing out some interesting questions.<br />
<br />
(host: Kent)<br />
<br />
== March 10, 2023, Friday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
'''''Distinguished lectures'''''<br />
<br />
Title: End-periodic maps, via fibered 3-manifolds<br />
<br />
Abstract: In the second lecture I will focus on some joint work with Michael Landry and Sam Taylor. Thurston showed how a certain ``spinning<nowiki>''</nowiki> construction in a fibered 3-manifold produces a depth-1 foliation, which is described by an end-periodic map of an infinite genus surface. The dynamical properties of such maps were then studied by Handel-Miller, Cantwell-Conlon-Fenley and others. We show how to reverse this construction, obtaining every end-periodic map from spinning in a fibered manifold. This allows us to recover the dynamical features of the map, and more, directly from the more classical theory of fibered manifolds.<br />
<br />
(host: Kent)<br />
<br />
== March 24, 2023 , Friday at 4pm [https://www.carolynrabbott.com/ Carolyn Abbott] (Brandeis University) ==<br />
'''Title''': Boundaries, boundaries, and more boundaries <br />
<br />
'''Abstract:''' It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I will discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I will describe various relationships between these boundaries and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups. This is joint work with Jason Behrstock and Jacob Russell. <br />
<br />
== March 31, 2023 , Friday at 4pm [http://www.math.toronto.edu/balint/ Bálint Virág] (University of Toronto) ==<br />
(host: Benedek Valko)<br />
<br />
== April 7, 2023, Friday at 4pm [https://www.mi.fu-berlin.de/math/groups/fluid-dyn/members/rupert_klein.html Rupert Klein] (FU Berlin) ==<br />
<br />
Wasow lecture<br />
<br />
(hosts: Smith, Stechmann)<br />
<br />
== April 21, 2023, Friday at 4pm [https://sternber.pages.iu.edu/ Peter Sternberg] (Indiana University) ==<br />
<br />
(hosts: Feldman, Tran)<br />
<br />
<br />
== April 28, 2023, Friday at 4pm [https://nqle.pages.iu.edu/ Nam Q. Le] (Indiana University) ==<br />
<br />
== Future Colloquia ==<br />
<br />
[[Colloquia/Fall2023|Fall 2023]]<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Fall2022|Fall 2022]]<br />
<br />
[[Spring 2022 Colloquiums|Spring 2022]]<br />
<br />
[[Colloquia/Fall2021|Fall 2021]]<br />
<br />
[[Colloquia/Spring2021|Spring 2021]]<br />
<br />
[[Colloquia/Fall2020|Fall 2020]]<br />
<br />
[[Colloquia/Spring2020|Spring 2020]]<br />
<br />
[[Colloquia/Fall2019|Fall 2019]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]<br />
<br />
[[WIMAW]]</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia&diff=24646Colloquia2023-03-14T00:04:46Z<p>Htran24: </p>
<hr />
<div>__NOTOC__<br />
<br />
<br />
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm. </b><br />
<br />
<!--- in Van Vleck B239, '''unless otherwise indicated'''. ---><br />
<br />
<br />
== February 3, 2023, Friday at 4pm [https://sites.google.com/a/uwlax.edu/tdas/ Facundo Mémoli] (Ohio State University) ==<br />
(host: Lyu)<br />
<br />
The Gromov-Hausdorff distance between spheres.<br />
<br />
The Gromov-Hausdorff distance is a fundamental tool in Riemanian geometry (through the topology it generates) and is also utilized in applied geometry and topological data analysis as a metric for expressing the stability of methods which process geometric data (e.g. hierarchical clustering and persistent homology barcodes via the Vietoris-Rips filtration). In fact, distances such as the Gromov-Hausdorff distance or its Optimal Transport variants (i.e. the so-called Gromov-Wasserstein distances) are nowadays often invoked in applications related to data classification.<br />
<br />
Whereas it is often easy to estimate the value of the Gromov-Hausdorff distance between two given metric spaces, its ''precise'' value is rarely easy to determine. Some of the best estimates follow from considerations related to both the stability of persistent homology and to Gromov's filling radius. However, these turn out to be non-sharp.<br />
<br />
In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between pairs of spheres (endowed with their usual geodesic distance). These results involve lower bounds which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and also matching upper bounds which are induced from specialized constructions of (a posteriori optimal) ``correspondences" between spheres.<br />
<br />
== February 24, 2023, Cancelled/available ==<br />
== March 3, 2023, Friday at 4pm [https://faculty.washington.edu/steinerb/ Stefan Steinerberger] (University of Washington) ==<br />
<br />
Title: How curved is a combinatorial graph?<br />
<br />
Abstract: Curvature is one of the fundamental ingredients in differential geometry. People are increasingly interested in whether it is possible to think of combinatorial graphs as behaving like manifolds and a number of different notions of curvature have been proposed. I will introduce some of the existing ideas and then propose a new notion based on a simple and explicit linear system of equations that is easy to compute. This notion satisfies a surprisingly large number of desirable properties -- connections to game theory (especially the von Neumann Minimax Theorem) and potential theory will be sketched; simultaneously, there is a certain "magic" element to all of this that is poorly understood and many open problems remain. I will also sketch some curious related problems that remain mostly open. No prior knowledge of differential geometry (or graphs) is required.<br />
<br />
(hosts: Shaoming Guo, Andreas Seeger)<br />
<br />
== March 8, 2023, Wednesday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
'''''Distinguished lectures'''''<br />
<br />
Title: Surfaces and foliations in hyperbolic 3-manifolds<br />
<br />
Abstract: How does the geometric theory of hyperbolic 3-manifolds interact with the topological theory of foliations within them? Both points of view have seen profound developments over the past 40 years, and yet we have only an incomplete understanding of their overlap. I won't have much to add to this understanding! Instead, I will meander through aspects of both stories, saying a bit about what we know and pointing out some interesting questions.<br />
<br />
(host: Kent)<br />
<br />
== March 10, 2023, Friday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
'''''Distinguished lectures'''''<br />
<br />
Title: End-periodic maps, via fibered 3-manifolds<br />
<br />
Abstract: In the second lecture I will focus on some joint work with Michael Landry and Sam Taylor. Thurston showed how a certain ``spinning<nowiki>''</nowiki> construction in a fibered 3-manifold produces a depth-1 foliation, which is described by an end-periodic map of an infinite genus surface. The dynamical properties of such maps were then studied by Handel-Miller, Cantwell-Conlon-Fenley and others. We show how to reverse this construction, obtaining every end-periodic map from spinning in a fibered manifold. This allows us to recover the dynamical features of the map, and more, directly from the more classical theory of fibered manifolds.<br />
<br />
(host: Kent)<br />
<br />
== March 24, 2023 , Friday at 4pm [https://www.carolynrabbott.com/ Carolyn Abbott] (Brandeis University) ==<br />
<br />
(host: Dymarz, Uyanik, WIMAW)<br />
<br />
== March 31, 2023 , Friday at 4pm [http://www.math.toronto.edu/balint/ Bálint Virág] (University of Toronto) ==<br />
(host: Benedek Valko)<br />
<br />
== April 7, 2023, Friday at 4pm [https://www.mi.fu-berlin.de/math/groups/fluid-dyn/members/rupert_klein.html Rupert Klein] (FU Berlin) ==<br />
<br />
Wasow lecture<br />
<br />
(hosts: Smith, Stechmann)<br />
<br />
== April 21, 2023, Friday at 4pm [https://sternber.pages.iu.edu/ Peter Sternberg] (Indiana University) ==<br />
<br />
(hosts: Feldman, Tran)<br />
<br />
<br />
== April 28, 2023, Friday at 4pm [https://nqle.pages.iu.edu/ Nam Q. Le] (Indiana University) ==<br />
<br />
== Future Colloquia ==<br />
<br />
[[Colloquia/Fall2023|Fall 2023]]<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Fall2022|Fall 2022]]<br />
<br />
[[Spring 2022 Colloquiums|Spring 2022]]<br />
<br />
[[Colloquia/Fall2021|Fall 2021]]<br />
<br />
[[Colloquia/Spring2021|Spring 2021]]<br />
<br />
[[Colloquia/Fall2020|Fall 2020]]<br />
<br />
[[Colloquia/Spring2020|Spring 2020]]<br />
<br />
[[Colloquia/Fall2019|Fall 2019]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]<br />
<br />
[[WIMAW]]</div>Htran24https://wiki.math.wisc.edu/index.php?title=Fall_2023-Spring_2024&diff=24564Fall 2023-Spring 20242023-02-28T23:50:12Z<p>Htran24: </p>
<hr />
<div>=Fall 2023= <br />
<br />
<br />
<br />
<br />
<br />
'''September 11, 2023'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
'''September 18, 2023'''<br />
<br />
Hongjie Dong (Brown). Host: Hung Tran <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
'''September 25, 2023'''<br />
<br />
Olga Turanova (MSU). Host: Hung Tran <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''October 16, 2023'''<br />
<br />
[[Ian Tice|Ian Tice]](CMU). Host: Chanwoo Kim <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''October 30, 2023'''<br />
<br />
Sung-Jin Oh(Berkeley). Host: Chanwoo Kim <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''December 11, 2023'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
=Spring 2024= <br />
<br />
'''January 29, 2024'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''March 25, 2024'''<br />
<br />
No seminar (Spring recess)<br />
<br />
<br />
<br />
'''April 29, 2024'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:</div>Htran24https://wiki.math.wisc.edu/index.php?title=Fall_2023-Spring_2024&diff=24563Fall 2023-Spring 20242023-02-28T23:49:55Z<p>Htran24: </p>
<hr />
<div>=Fall 2023= <br />
<br />
<br />
<br />
<br />
<br />
'''September 11, 2023'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
'''September 25, 2023'''<br />
<br />
Hongjie Dong (Brown). Host: Hung Tran <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
'''September 25, 2023'''<br />
<br />
Olga Turanova (MSU). Host: Hung Tran <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''October 16, 2023'''<br />
<br />
[[Ian Tice|Ian Tice]](CMU). Host: Chanwoo Kim <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''October 30, 2023'''<br />
<br />
Sung-Jin Oh(Berkeley). Host: Chanwoo Kim <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''December 11, 2023'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
=Spring 2024= <br />
<br />
'''January 29, 2024'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
<br />
'''March 25, 2024'''<br />
<br />
No seminar (Spring recess)<br />
<br />
<br />
<br />
'''April 29, 2024'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia/Fall2023&diff=24552Colloquia/Fall20232023-02-27T17:37:56Z<p>Htran24: Created page with "==October 2, 2023, Monday at 4pm [https://www.math.tamu.edu/~titi/ Edriss Titi] (Texas A&M University)== Distinguished lectures (host: Smith, Stechmann)"</p>
<hr />
<div>==October 2, 2023, Monday at 4pm [https://www.math.tamu.edu/~titi/ Edriss Titi] (Texas A&M University)==<br />
Distinguished lectures<br />
<br />
(host: Smith, Stechmann)</div>Htran24https://wiki.math.wisc.edu/index.php?title=Fall_2023-Spring_2024&diff=24543Fall 2023-Spring 20242023-02-24T15:37:43Z<p>Htran24: /* Fall 2023 */</p>
<hr />
<div>=Fall 2023= <br />
<br />
<br />
<br />
<br />
'''September 11, 2023'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
'''September 25, 2023'''<br />
<br />
Olga Turanova (MSU). Host: Hung Tran <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
'''October 16, 2023'''<br />
<br />
[[Ian Tice|Ian Tice]](CMU). Host: Chanwoo Kim <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
'''October 30, 2023'''<br />
<br />
Sung-Jin Oh(Berkeley). Host: Chanwoo Kim <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
'''December 11, 2023'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
=Spring 2024= <br />
<br />
'''January 29, 2024'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:<br />
<br />
<br />
'''March 25, 2024'''<br />
<br />
No seminar (Spring recess)<br />
<br />
<br />
'''April 29, 2024'''<br />
<br />
[[|]] (). Host: <br />
<br />
Format: in-person. Time: 3:30-4:30PM, VV 911. <br />
<br />
Title:</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=24542PDE Geometric Analysis seminar2023-02-24T02:55:08Z<p>Htran24: /* PDE GA Seminar Schedule Fall 2022-Spring 2023 */</p>
<hr />
<div>The seminar's format will be a combination of online and in-person; we will make sure to update you as soon as we have more details available. First talk is tentatively scheduled for September 20th ! <br />
<br />
Some of the seminars will be held online. When that would be the case we would use the following zoom link<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
<br />
<br />
[[Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.]]<br />
<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
=== [[Fall 2023-Spring 2024|Schedule for Fall 2023-Spring 2024]]===<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[No seminar]] <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
[[Trinh Tien Nguyen]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] UC Berkeley<br />
<br />
Format: Format: In person in Room VV901, Time: 3:30-4:30PM. <br />
<br />
'''Title: ''' WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION <br />
<br />
'''Abstract: ''' We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[Yuxi Han]] (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales<br />
<br />
'''Abstract:''' In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
'''March 20, 2023'''<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
ANTOINE REMOND-TIEDREZ. <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] <br />
<br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
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<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
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<br />
<br />
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<br />
<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=24423PDE Geometric Analysis seminar2023-02-08T01:01:42Z<p>Htran24: </p>
<hr />
<div>The seminar's format will be a combination of online and in-person; we will make sure to update you as soon as we have more details available. First talk is tentatively scheduled for September 20th ! <br />
<br />
Some of the seminars will be held online. When that would be the case we would use the following zoom link<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
<br />
<br />
[[Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.]]<br />
<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
===[[Fall 2022-Spring 2023 | Schedule for Fall 2022-Spring 2023]]===<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
<br />
'''Title:'''<br />
<br />
'''Abstract:''' <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
Trinh Tien Nguyen (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' The inviscid limit of Navier-Stokes for domains with curved boundaries<br />
<br />
'''Abstract:''' Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
Yuxi Han (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus <br />
<br />
'''Abstract:''' This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
'''March 20, 2023'''<br />
<br />
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim <br />
<br />
Format: In-person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
Zhihan Wang (Princeton). Host: Sigurd Angenent <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:''' ''Translating mean curvature flow with simple end.''<br />
<br />
'''Abstract:''' Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
<br />
ANTOINE REMOND-TIEDREZ. <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] <br />
<br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA|Chris Henderson (Arizona)]]. Host: Chanwoo Kim<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=24318PDE Geometric Analysis seminar2023-01-30T19:54:42Z<p>Htran24: </p>
<hr />
<div>The seminar's format will be a combination of online and in-person; we will make sure to update you as soon as we have more details available. First talk is tentatively scheduled for September 20th ! <br />
<br />
Some of the seminars will be held online. When that would be the case we would use the following zoom link<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
<br />
<br />
[[Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.]]<br />
<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
===[[Fall 2022-Spring 2023 | Schedule for Fall 2022-Spring 2023]]===<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:''' <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
Trinh Tien Nguyen (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
Yuxi Han (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
Jinwoo Jang (Postech), Host: Chanwoo Kim <br />
<br />
Format: In-person , Time: 3:30-4:30PM. <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
'''March 20, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[Ben Pineau| Ben Pineau]] <br />
<br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
<br />
'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
<br />
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
<br />
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
<br />
<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
<br />
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=24229PDE Geometric Analysis seminar2023-01-22T14:46:16Z<p>Htran24: </p>
<hr />
<div>The seminar's format will be a combination of online and in-person; we will make sure to update you as soon as we have more details available. First talk is tentatively scheduled for September 20th ! <br />
<br />
Some of the seminars will be held online. When that would be the case we would use the following zoom link<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
<br />
<br />
[[Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.]]<br />
<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
===[[Fall 2022-Spring 2023 | Schedule for Fall 2022-Spring 2023]]===<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==<br />
<br />
=Spring 2023= <br />
<br />
<br />
<br />
'''January 30, 2023 '''<br />
<br />
[[Jingwen Chen|Jingwen Chen]] (U Chicago)<br />
<br />
Time: 3:30 PM -4:30 PM, in person in VV901 <br />
<br />
Title: Mean curvature flows in the sphere via phase transitions.<br />
<br />
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory. <br />
<br />
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile). <br />
<br />
<br />
<br />
'''February 6, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:''' <br />
<br />
<br />
<br />
'''February 13, 2023'''<br />
<br />
Trinh Tien Nguyen (UW Madison) <br />
<br />
Format: In person, Time: 3:30-4:30PM. <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
<br />
'''February 20, 2023'''<br />
<br />
[[Ovidiu Avadanei| Ovidiu Avadanei]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''February 27, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 6, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''March 13, 2023'''<br />
<br />
[[Spring Recess: No Seminar]] <br />
<br />
<br />
<br />
<br />
'''March 20, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''March 27, 2023'''<br />
<br />
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.<br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''April 3, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''April 10, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''April 17, 2023'''<br />
<br />
[[Jingrui Cheng]] (Stony Brook). Host: Misha Feldman.<br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
'''April 24, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:''' <br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
'''May 1, 2023'''<br />
<br />
[[TBA| TBA]] <br />
<br />
Format: , Time: <br />
<br />
'''Title:'''<br />
<br />
'''Abstract:'''<br />
<br />
<br />
<br />
<br />
=Fall 2022= <br />
<br />
<br />
<br />
'''September 12, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
<br />
<br />
<br />
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar<br />
<br />
[[Andrej Zlatos]] (UCSD). Host: Hung Tran. <br />
<br />
Format: in-person. Time: 4-5PM, VV B139. <br />
<br />
Title: Homogenization in front propagation models<br />
<br />
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.<br />
<br />
<br />
<br />
<br />
'''September 26, 2022 ''' <br />
<br />
[[Haotian Wu]] (The University of Sydney, Australia). Host: Sigurd Angenent.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
<u>Title:</u> ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''<br />
<br />
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).<br />
<br />
<br />
<br />
<br />
'''October 3, 2022'''<br />
<br />
[[No Seminar]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
'''October 10, 2022'''<br />
<br />
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.<br />
<br />
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).<br />
<br />
Speaker: Sasha Kiselev (Duke)<br />
<br />
Title: The flow of polynomial roots under differentiation<br />
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work<br />
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.<br />
<br />
<br />
<br />
'''October 17, 2022'''<br />
<br />
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.<br />
<br />
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures. <br />
<br />
<br />
<br />
<br />
'''October 24, 2022'''<br />
<br />
[[No seminar.]]<br />
<br />
Format: , Time: <br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
<br />
<br />
<br />
<br />
'''October 31, 2022 '''<br />
<br />
[[Yuan Gao]] (Purdue). Host: Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures<br />
<br />
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.<br />
<br />
<br />
<br />
'''November 7, 2022 '''<br />
<br />
[[Beomjun Choi]] (Postech)<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Liouville theorem for surfaces translating by powers of Gauss curvature<br />
<br />
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.<br />
<br />
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.<br />
<br />
<br />
<br />
<br />
'''November 14, 2022 '''<br />
<br />
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.<br />
<br />
Format: in person, Time: 3:30pm-4:30pm VV 901 <br />
<br />
Title: Sticky Particles with Sticky Boundary<br />
<br />
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.<br />
<br />
<br />
<br />
<br />
<br />
'''November 21, 2022 '''<br />
<br />
[[Jason Murphy]] (Missouri S&T)<br />
<br />
Format: online <br />
<br />
Join Zoom Meeting<br />
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09<br />
Meeting ID: 948 7748 3456<br />
Passcode: 303105<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Sharp scattering results for the 3d cubic NLS<br />
<br />
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''November 28, 2022 '''<br />
<br />
[[No Seminar]]- Thanksgiving <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
'''December 5, 2022 '''<br />
<br />
[[James Rowan]] (UC Berkeley)<br />
<br />
Time: 3:30 PM -4:30 PM <br />
<br />
Title: Solitary waves for infinite depth gravity water waves with constant vorticity<br />
<br />
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.<br />
<br />
<br />
<br />
<br />
<br />
'''December 12, 2022 '''<br />
<br />
[[Calum Rickard ]] UC Davis<br />
<br />
Format: in-person in room VV901 <br />
<br />
Time: 3:00 PM -4:00 PM <br />
<br />
Title: An infinite class of shocks for compressible Euler<br />
<br />
Abstract: <br />
We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==<br />
<br />
=Spring 2022=<br />
<br />
<br />
<br />
<br />
'''January 31th, 2022.'''<br />
<br />
[[No Seminar]]<br />
<br />
<br />
'''February 7th, 2022.'''<br />
<br />
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM <br />
<br />
Title: Estimates and regularity for the k-Yamabe equation <br />
<br />
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.<br />
<br />
<br />
<br />
<br />
'''February 14th, 2022.'''<br />
<br />
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM <br />
<br />
Title: MCF after the Velázquez&mdash;Stolarski example.<br />
<br />
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.<br />
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. <br />
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.<br />
<br />
<br />
<br />
'''February 21th, 2022.'''<br />
<br />
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time: 11:00 AM<br />
<br />
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations<br />
<br />
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck). <br />
<br />
<br />
<br />
<br />
'''February 28th, 2021.'''<br />
<br />
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Time:3:30PM-4:30PM<br />
<br />
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC<br />
<br />
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.<br />
<br />
<br />
'''March 7th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
'''March 14th, 2022.'''<br />
<br />
[[Spring recess - No Seminar]]; <br />
<br />
<br />
<br />
<br />
'''March 21th, 2022.'''<br />
<br />
[[ No Seminar]];<br />
<br />
<br />
<br />
'''March 28th, 2022.'''<br />
<br />
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Determinants, commuting flows, and recent progress on<br />
completely integrable systems<br />
<br />
Abstract: I will survey a number of recent developments in the theory<br />
of completely integrable nonlinear dispersive PDE. These include a<br />
priori bounds, the orbital stability of multisoliton solutions,<br />
well-posedness at optimal regularity, and the existence of dynamics<br />
for Gibbs distributed initial data. I will describe the basic objects<br />
that tie together these disparate results, as well as the diverse<br />
ideas required for each problem.<br />
<br />
<br />
'''April 4th, 2022.'''<br />
<br />
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM <br />
<br />
Title: General-relativistic viscous fluids.<br />
<br />
<br />
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has<br />
<br />
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.<br />
<br />
<br />
<br />
'''April 11th, 2022.'''<br />
<br />
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM<br />
<br />
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09<br />
<br />
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]<br />
Passcode: 180680<br />
<br />
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations<br />
<br />
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo<br />
<br />
<br />
<br />
'''April 18th, 2022.'''<br />
<br />
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.<br />
<br />
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.<br />
<br />
<br />
<br />
'''April 25th, 2022.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
<br />
'''May 2nd, 2022.'''<br />
<br />
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM<br />
<br />
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness<br />
<br />
Abstract: In this talk, I will present some recent results obtained in<br />
collaboration with V. Patel (Oxford) in connection with a system<br />
describing a heat-conducting incompressible fluid. I will introduce the<br />
notion of a dissipative weak solution of the system and highlight the<br />
connections and differences to the existing approaches in the<br />
literature. One of the advantages of the proposed approach is that the<br />
solution satisfies a weak-strong uniqueness principle, which guarantees<br />
that the weak solution will coincide with the strong solution, as long<br />
as the latter exists; moreover, the solutions are constructed via a<br />
finite element approximation, leading (almost, not quite) to the first<br />
convergence result for the full system including viscous dissipation.<br />
<br />
<br />
<br />
=Fall 2021=<br />
<br />
'''September 20th, 2021.'''<br />
<br />
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
<br />
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts<br />
<br />
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).<br />
<br />
<br />
<br />
'''September 27th, 2021.'''<br />
<br />
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Volume-preserving crystalline and anisotropic mean curvature flow<br />
<br />
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).<br />
<br />
<br />
'''October 4th, 2021.'''<br />
<br />
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries<br />
<br />
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.<br />
<br />
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).<br />
<br />
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.<br />
<br />
<br />
'''October 11th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''October 18th, 2021.'''<br />
<br />
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM<br />
<br />
Title: Well-posedness of logarithmic spiral vortex sheets.<br />
<br />
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.<br />
<br />
<br />
'''October 25th, 2021.'''<br />
<br />
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm<br />
<br />
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature<br />
<br />
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.<br />
<br />
<br />
'''November 1th, 2021.'''<br />
<br />
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM<br />
<br />
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity<br />
<br />
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.<br />
<br />
<br />
<br />
'''November 8th, 2021.'''<br />
<br />
[[ Albert Ai]] (UW Madison); <br />
<br />
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation<br />
<br />
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.<br />
<br />
<br />
<br />
'''November 15th, 2021.'''<br />
<br />
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]] <br />
<br />
[[Please observe the time change! ]]<br />
<br />
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP<br />
<br />
<br />
Title: Global wellposedness for the energy-critical Zakharov system below the ground state<br />
<br />
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.<br />
<br />
<br />
<br />
'''November 22th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''November 29th, 2021.'''<br />
<br />
[[No seminar]]<br />
<br />
<br />
'''December 6th, 2021.'''<br />
<br />
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.<br />
<br />
Title: Quantitative homogenization of Hamilton-Jacobi equations<br />
<br />
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).<br />
<br />
<br />
'''December 13th, 2021.'''<br />
<br />
<br />
[[No seminar ]]<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==<br />
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. <br />
<br />
'''Week 1 (9/1/2020-9/5/2020)'''<br />
<br />
1. Paul Rabinowitz - The calculus of variations and phase transition problems.<br />
https://www.youtube.com/watch?v=vs3rd8RPosA<br />
<br />
2. Frank Merle - On the implosion of a three dimensional compressible fluid.<br />
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be <br />
<br />
'''Week 2 (9/6/2020-9/12/2020)'''<br />
<br />
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.<br />
https://www.youtube.com/watch?v=4ndtUh38AU0<br />
<br />
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI<br />
<br />
<br />
<br />
'''Week 3 (9/13/2020-9/19/2020)'''<br />
<br />
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ<br />
<br />
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE<br />
<br />
<br />
<br />
'''Week 4 (9/20/2020-9/26/2020)'''<br />
<br />
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be<br />
<br />
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM<br />
<br />
<br />
<br />
'''Week 5 (9/27/2020-10/03/2020)'''<br />
<br />
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo<br />
<br />
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c<br />
<br />
<br />
'''Week 6 (10/04/2020-10/10/2020)'''<br />
<br />
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E<br />
<br />
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing<br />
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html<br />
<br />
<br />
'''Week 7 (10/11/2020-10/17/2020)'''<br />
<br />
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s<br />
<br />
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg<br />
<br />
<br />
'''Week 8 (10/18/2020-10/24/2020)'''<br />
<br />
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg<br />
<br />
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ<br />
<br />
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.<br />
<br />
<br />
'''Week 9 (10/25/2020-10/31/2020)'''<br />
<br />
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE<br />
<br />
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764<br />
<br />
<br />
<br />
'''Week 10 (11/1/2020-11/7/2020)'''<br />
<br />
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be<br />
<br />
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html<br />
<br />
<br />
<br />
'''Week 11 (11/8/2020-11/14/2020)'''<br />
<br />
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc<br />
<br />
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0<br />
<br />
<br />
'''Week 12 (11/15/2020-11/21/2020)'''<br />
<br />
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY<br />
<br />
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk<br />
<br />
<br />
'''Week 13 (11/22/2020-11/28/2020)'''<br />
<br />
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be<br />
<br />
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8<br />
<br />
'''Week 14 (11/29/2020-12/5/2020)'''<br />
<br />
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, <br />
https://youtu.be/xfAKGc0IEUw<br />
<br />
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc<br />
<br />
<br />
<br />
'''Week 15 (12/6/2020-12/12/2020)'''<br />
<br />
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be<br />
<br />
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU<br />
<br />
<br />
'''Spring 2021'''<br />
<br />
'''Week 1 (1/31/2021- 2/6/2021)'''<br />
<br />
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be<br />
<br />
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84<br />
<br />
<br />
'''Week 2 ( 2/7/2021- 2/13/2021)'''<br />
<br />
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek<br />
<br />
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE<br />
<br />
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.<br />
<br />
'''Week 3 ( 2/14/2021- 2/20/2021)'''<br />
<br />
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s<br />
<br />
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg<br />
<br />
<br />
<br />
'''Week 4 ( 2/21/2021- 2/27/2021)'''<br />
<br />
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309<br />
<br />
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68<br />
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'''Week 5 ( 2/28/2021- 3/6/2021)'''<br />
<br />
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317<br />
<br />
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k<br />
<br />
'''Week 6 (3/7/2021-3/13/2021)'''<br />
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1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html<br />
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2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c<br />
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<br />
'''Week 7 (3/14/2021-3/20/2021)'''<br />
<br />
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM<br />
<br />
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html<br />
<br />
<br />
<br />
'''Week 8 (3/21/2021- 3/27/2021)'''<br />
<br />
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs<br />
<br />
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A<br />
<br />
<br />
<br />
'''Week 9 (3/28/2021- 4/3/2021)'''<br />
<br />
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM<br />
<br />
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9<br />
<br />
<br />
'''Week 10 (4/4/2021- 4/10/2021)'''<br />
<br />
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235<br />
<br />
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8<br />
<br />
'''Week 11(4/11/2021- 4/17/2021)'''<br />
<br />
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo<br />
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2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html<br />
<br />
'''Week 12(4/18/2021- 4/24/2021)'''<br />
<br />
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0<br />
<br />
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo<br />
<br />
'''Week 13(4/25/2021- 5/1/2021)'''<br />
<br />
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI <br />
<br />
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html<br />
<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
=== ===<br />
<br />
Title: <br />
<br />
Abstract:</div>Htran24https://wiki.math.wisc.edu/index.php?title=Colloquia/Spring2023&diff=24203Colloquia/Spring20232023-01-19T07:34:46Z<p>Htran24: </p>
<hr />
<div>__NOTOC__<br />
<br />
<br />
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm. </b><br />
<br />
<!--- in Van Vleck B239, '''unless otherwise indicated'''. ---><br />
<br />
<br />
== February 3, 2023, Friday at 4pm [https://sites.google.com/a/uwlax.edu/tdas/ Facundo Mémoli] (Ohio State University) ==<br />
(host: Lyu)<br />
<br />
The Gromov-Hausdorff distance between spheres.<br />
<br />
The Gromov-Hausdorff distance is a fundamental tool in Riemanian geometry (through the topology it generates) and is also utilized in applied geometry and topological data analysis as a metric for expressing the stability of methods which process geometric data (e.g. hierarchical clustering and persistent homology barcodes via the Vietoris-Rips filtration). In fact, distances such as the Gromov-Hausdorff distance or its Optimal Transport variants (i.e. the so-called Gromov-Wasserstein distances) are nowadays often invoked in applications related to data classification.<br />
<br />
Whereas it is often easy to estimate the value of the Gromov-Hausdorff distance between two given metric spaces, its ''precise'' value is rarely easy to determine. Some of the best estimates follow from considerations related to both the stability of persistent homology and to Gromov's filling radius. However, these turn out to be non-sharp.<br />
<br />
In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between pairs of spheres (endowed with their usual geodesic distance). These results involve lower bounds which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and also matching upper bounds which are induced from specialized constructions of (a posteriori optimal) ``correspondences" between spheres.<br />
<br />
== February 24, 2023, Friday at 4pm [https://sites.google.com/a/uwlax.edu/tdas/ Tushar Das] (University of Wisconsin - La Crosse) ==<br />
(hosts: Burkart, Stovall)<br />
<br />
== March 3, 2023, Friday at 4pm [https://faculty.washington.edu/steinerb/ Stefan Steinerberger] (University of Washington) ==<br />
<br />
(hosts: Shaoming Guo, Andreas Seeger)<br />
<br />
== March 8, 2023, Wednesday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
Distinguished lectures<br />
<br />
(host: Kent)<br />
<br />
== March 10, 2023, Friday at 4pm [https://math.yale.edu/people/yair-minsky Yair Minsky] (Yale University) ==<br />
<br />
Distinguished lectures<br />
<br />
(host: Kent)<br />
<br />
== March 24, 2023 , Friday at 4pm [https://www.carolynrabbott.com/ Carolyn Abbott] (Brandeis University) ==<br />
<br />
(host: Dymarz, Uyanik, WIMAW)<br />
<br />
== March 31, 2023 , Friday at 4pm [http://www.math.toronto.edu/balint/ Bálint Virág] (University of Toronto) ==<br />
(host: Benedek Valko)<br />
<br />
== April 7, 2023, Friday at 4pm [https://www.mi.fu-berlin.de/math/groups/fluid-dyn/members/rupert_klein.html Rupert Klein] (FU Berlin) ==<br />
<br />
Wasow lecture<br />
<br />
(hosts: Smith, Stechmann)<br />
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== April 21, 2023, Friday at 4pm [https://sternber.pages.iu.edu/ Peter Sternberg] (Indiana University) ==<br />
<br />
(hosts: Feldman, Tran)<br />
<br />
<br />
<br />
== Past Colloquia ==<br />
[[Colloquia/Fall2022|Fall 2022]]<br />
<br />
[[Colloquia/Spring2022|Spring 2022]]<br />
<br />
[[Colloquia/Fall2021|Fall 2021]]<br />
<br />
[[Colloquia/Spring2021|Spring 2021]]<br />
<br />
[[Colloquia/Fall2020|Fall 2020]]<br />
<br />
[[Colloquia/Spring2020|Spring 2020]]<br />
<br />
[[Colloquia/Fall2019|Fall 2019]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]<br />
<br />
[[WIMAW]]</div>Htran24