PDE Geometric Analysis seminar

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

Seminar Schedule Fall 2015

date	speaker	title	host(s)
September 7 (Labor Day)
September 14 (special room: B115)	Hung Tran (Madison)	Some inverse problems in periodic homogenization of Hamilton--Jacobi equations
September 21 (special room: B115)	Eric Baer (Madison)	Optimal function spaces for continuity of the Hessian determinant as a distribution
September 28	Donghyun Lee (Madison)	FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT
October 5	Hyung-Ju Hwang (Postech & Brown Univ)	The Fokker-Planck equation in bounded domains	Kim
October 12	Binh Tran (Madison)	TBA
October 19	Bob Jensen (Loyola University Chicago)	TBA	Tran
October 26	Luis Silvestre (Chicago)	TBA	Kim
November 2	Connor Mooney (UT Austin)	TBA	Lin
November 9	Javier Gomez-Serrano (Princeton)	TBA	Zlatos
November 16	Yifeng Yu (UC Irvine)	TBA	Tran
November 23	Nam Le (Indiana)	TBA	Tran
November 30	Qin Li (Madison)	TBA
December 7	Lu Wang (Madison)	TBA
December 14	Christophe Lacave (Paris 7)	TBA	Zlatos

Abstract

Hung Tran

Some inverse problems in periodic homogenization of Hamilton--Jacobi equations.

Abstract: We look at the effective Hamiltonian $\overline{H}$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\overline{H}$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. I will discuss some interesting cases in both convex and nonconvex settings. Joint work with Songting Luo and Yifeng Yu.

Eric Baer

Optimal function spaces for continuity of the Hessian determinant as a distribution.

Abstract: In this talk we describe a new class of optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$, obtained in collaboration with D. Jerison. Inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space $B(2-2/N,N)$ of fractional order, and that all continuity results in this scale of Besov spaces are consequences of this result. A key ingredient in the argument is the characterization of $B(2-2/N,N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ (of codimension 2). The most elaborate part of the analysis is the construction of a counterexample to continuity in $B(2-2/N,p)$ with $p>N$. Tools involved in this step include the choice of suitable ``atoms" having a tensor product structure and Hessian determinant of uniform sign, formation of lacunary series of rescaled atoms, and delicate estimates of terms in the resulting multilinear expressions.

Donghyun Lee

FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT.

Abstract : Free-boundary problems of incompressible fluids have been studied for several decades. In the viscous case, it is basically solved by Stokes regularity. However, the inviscid case problem is generally much harder, because the problem is purely hyperbolic. In this talk, we approach the problem via vanishing viscosity limit, which is a central problem of fluid mechanics. To correct boundary layer behavior, conormal Sobolev space will be introduced. In the spirit of the recent work by N.Masmoudi and F.Rousset (2012, non-surface tension), we will see how to get local regularity of incompressible free-boundary Euler, taking surface tension into account. This is joint work with Tarek Elgindi. If possible, we also talk about applying the similar technique to the free-boundary MHD(Magnetohydrodynamics). Especially, we will see that strong zero initial boundary condition is still valid for this coupled PDE. For the general boundary condition (for perfect conductor), however, the problem is still open.