Continuous maximal regularity on manifolds with singularities and applications to geometric flows

Speaker: 

Yuanzhen Shao

Institution: 

Vanderbilt University

Time: 

Tuesday, September 30, 2014 - 3:00pm

Location: 

RH 306

In this talk, we study continuous maximal regularity theory for a class of degenerate or singular differential operators on manifolds with singularities. Based on this theory, we show local existence and uniqueness of solutions for several nonlinear geometric flows and diffusion equations on non-compact, or even incomplete, manifolds, including the Yamabe flow and parabolic p-Laplacian equations. In addition, we also establish regularity properties of solutions by means of a technique consisting of continuous maximal regularity theory, a parameter-dependent diffeomorphism and the implicit function theorem.

Magnetohydrodynamic fluids with zero magnetic diffusivity

Speaker: 

Xianpeng Hu

Institution: 

Courant Institute

Time: 

Tuesday, May 20, 2014 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

 

Understanding the incompressible/compressible fluid is a fundamental, but
challenging, project not only in numerical analysis, but also in
theoretical analysis, especially when extra effects, such as the elastic
deformation or the magnetic field, interact with the flow. In this talk,
the incompressible fluid and its associated flow map will be reviewed first.
The main object of this talk devotes to a recent work in understanding
incompressible/compressible magnetohydrodynamic fluids with zero magnetic
diffusivity (which is equivalent to infinite conductivity). This is a
joint work with Fanghua Lin.

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.

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