# Difference between revisions of "PDE Geometric Analysis seminar"

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A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. | A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. | ||

+ | |||

+ | ===Gung-Min Gie === | ||

+ | |||

+ | Boundary layer analysis of some incompressible flows | ||

+ | |||

+ | The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes. | ||

===Tau Shean Lim=== | ===Tau Shean Lim=== |

## Revision as of 15:36, 14 October 2016

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

## Contents

### Previous PDE/GA seminars

### Tentative schedule for Spring 2017

# PDE GA Seminar Schedule Fall 2016

date | speaker | title | host(s) |
---|---|---|---|

September 12 | Daniel Spirn (U of Minnesota) | Dipole Trajectories in Bose-Einstein Condensates | Kim |

September 19 | Donghyun Lee (UW-Madison) | The Boltzmann equation with specular boundary condition in convex domains | Feldman |

September 26 | Kevin Zumbrun (Indiana) | A Stable Manifold Theorem for a class of degenerate evolution equations | Kim |

October 3 | Will Feldman (UChicago ) | Liquid Drops on a Rough Surface | Lin & Tran |

October 10 | Ryan Hynd (UPenn) | Extremal functions for Morrey’s inequality in convex domains | Feldman |

October 17 | Gung-Min Gie (Louisville) | Kim | |

October 24 | Tau Shean Lim (UW Madison) | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators | Kim & Tran |

October 31 | Tarek Elgindi ( Princeton) | Propagation of Singularities in Incompressible Fluids | Lee & Kim |

November 7 | Adrian Tudorascu (West Virginia) | Feldman | |

November 14 | Alexis Vasseur ( UT-Austin) | Feldman | |

November 21 | Minh-Binh Tran (UW Madison ) | Quantum Kinetic Problems | Hung Tran |

November 28 | David Kaspar (Brown) | Tran | |

December 5 | Brian Weber (University of Pennsylvania) | TBA | Bing Wang |

December 12 |

# Abstracts

### Daniel Spirn

Dipole Trajectories in Bose-Einstein Condensates

Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.

### Donghyun Lee

The Boltzmann equation with specular reflection boundary condition in convex domains

I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.

### Kevin Zumbrun

TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations

ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class

of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and

$D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the

associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here

pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.

### Will Feldman

Liquid Drops on a Rough Surface

I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness.

The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks.

### Ryan Hynd

Extremal functions for Morrey’s inequality in convex domains

A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold.

### Gung-Min Gie

Boundary layer analysis of some incompressible flows

The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.

### Tau Shean Lim

Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators

We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.