Periodic homogenization of Hamilton-Jacobi equations with defects: cell problems in the non-convex setting.

Speaker: 

Hung Tran

Institution: 

University of Chicago

Time: 

Thursday, May 22, 2014 - 2:00am to 3:00am

Host: 

Location: 

440R

 

We study the effect of defects in the periodic homogenization of
Hamilton-Jacobi equations with non convex Hamiltonians. More precisely, we
handle the question about existence of sublinear solutions of the cell
problems.

On Multi-Dimensional Compressible Navier-Stokes Systems with Large Oscillations (joint with Applied and Computational Math seminar)

Speaker: 

Zhouping Xin

Institution: 

The Chinese University of Hongkong

Time: 

Wednesday, April 30, 2014 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

 In this talk, I will discuss recent results on the
 large time well-posedness of classical solutions to the
 multi-dimensional compressible Navier-Stokes system with possible
 large oscillations and vacuum.
 The focus will be on finite-time blow-up of classical solutions for
 the 3-D full compressible Navier-Stokes system, and the global
 existence of classical solutions to the isentropic compressible
 Navier-Stokes system in both 2-D and 3-D in the presence of vacuum
 and possible large oscillations.  New estimates on the fast decay
 of the pressure in the presence of vacuum will be presented,  which
 are crucial for the well-posedness theory in 2-dimensional case.

On the size of the nodal sets of solutions of elliptic and parabolic PDEs

Speaker: 

Igor Kucavica

Institution: 

University of Southern California

Time: 

Tuesday, March 4, 2014 - 3:00pm

Location: 

RH 306

In this talk we will present various results on the size of the nodal (zero) set for solutions of partial differential equations of elliptic and parabolic type. In particular, we will establish a sharp upper bound for the (n-1)-dimensional Hausdorff measure of the nodal sets of the eigenfunctions of regular analytic elliptic problems. We will also show certain more recent results concerning semilinear equations (e.g.  Navier-Stokes equations) and equations with non-analytic coefficients.

Regularity for almost minimizers with free boundary

Speaker: 

Tatiana Toro

Institution: 

University of Washington at Seattle

Time: 

Tuesday, February 25, 2014 - 3:00pm

Host: 

Location: 

306RH

 

In recent work with Guy David we introduce the notion of almost
minimizer for a series of functionals previously studied by Alt-Caffarelli
and Alt-Caffarelli-Friedman.

We prove regularity results for these almost minimizers and explore the
structure of the corresponding free boundary. A key ingredient in the study
of the 2-phase problem is the existence of almost monotone quantities. The
goal of this talk is to present these results in a self-contained manner,
emphasizing both the similarities and differences between minimizers and
almost minimizers.

Shock Reflection, von Neumann conjectures, and free boundary problems

Speaker: 

Mikhail Feldman

Institution: 

University of Wisconsin-Madison

Time: 

Tuesday, May 6, 2014 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle, and discuss some techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed. The talk is based on joint work with Gui-Qiang Chen.

Viscosity solutions of Hamilton-Jacobi equations in metric spaces

Speaker: 

Andrzej Swiech

Institution: 

Georgia Institute of Technology

Time: 

Tuesday, January 14, 2014 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

 

The theory of Hamilton-Jacobi equations in Hilbert and some
Banach spaces is relatively well developed. Much less is known about equations in spaces of measures, and more general metric spaces. We will present a notion of metric viscosity solution which applies to a class of Hamilton-Jacobi equations in geodesic metric spaces and gives well posedness for such equations. We will also discuss other approaches to Hamilton-Jacobi equations in metric spaces, in particular in the Wasserstein space, and discuss some applications of such equations.

 

The limit as p tends to infinity of a free boundary problem for p-Laplacian

Speaker: 

Peiyong Wang

Institution: 

Wayne state university

Time: 

Tuesday, May 21, 2013 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

 

I will introduce the free boundary problem for the p-Laplacian with
emphasis on the free boundary condition. Then any uniform sub-
sequential limit is proved to solve the free boundary problem for
the infinity Laplacian.

 

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