Georgia Institute of Technology College of Sciences

In this lecture, we will explain a new method to show that regularity on the boundary of a domain implies regularity in the inside for PDE's of the Hamilton-Jacobi type. The method can be applied in different settings. One of these settings concerns continuous viscosity solutions $U : T^N\times [0,+\infty[ \rightarrow R$ of the evolutionary equation $\partial_t U(x, t) + H(x, \partial_x U(x, t) ) = 0,$ where $T^N = R^N / Z^N$, and $H: T^N \times R^N$ is a Tonelli Hamiltonian, i.e. H(x, p) is $C^2$, strictly convex superlinear in p.

Vortices arise in many problems in condensed matter physics, including superconductivity, superfluids, and Bose-Einstein condensates. I will discuss some results on the behavior of two of these systems when there are asymptotically large numbers of vortices. The methods involve suitable renormalization of the energies both at the vortex cores and at infinity, along with a renormalization of the vortex density function.

We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is focusing and matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We

In this talk, firstly, we study the local and global well-posedness for full Navier-Stokes equations with temperature dependent coefficients in the framework of Besov space. We generalized R. Danchin's results for constant transport coefficients to obtain the local and global well-posedness for the initial with low regularity in Besov space framework. Secondly, we give a time decay rate results of the global solution in the Besov space framework which is not investigated before. Due to the low regularity assumption,

We identify sufficient conditions on initial data to ensure the existence of a unique strong solution to the Cauchy problem for the Compressible Navier-Stokes equations with degenerate viscosities and vacuum (such as viscous Saint-Venants model in $\mathbb{R}^2$). This is a recent work joint with Yachun Li and Ronghua Pan.

Surface waves are waves that propagate along a boundary or interface, with energy that is localized near the surface. Physical examples are water waves on the free surface of a fluid, Rayleigh waves on an elastic half-space, and surface plasmon polaritons (SPPs) on a metal-dielectric interface. We will describe some of the history of surface waves and explain a general Hamiltonian framework for their analysis. The weakly nonlinear evolution of dispersive surface waves is described by well-known PDEs like the KdV or nonlinear Schrodinger

The Abstract can be found at http://people.math.gatech.edu/~gchen73/PDEseminar_abstract.pdf

ABSTRACT: The lecture will outline a research program which aims at establishing the existence and long time behavior of BV solutions for hyperbolic systems of balance laws, in one space dimension, with partially dissipative source, manifesting relaxation. Systems with such structure are ubiquitous in classical physics.

The short wave-long wave interactions for viscous compressibleheat conductive fluids is modeled, following Dias & Frid (2011), by a Benney-type system coupling Navier-Stokes equations with a nonlinear Schrodingerequation along particle paths. We study the global existence of smooth solutions to the Cauchy problem in R^3 when the initial data are small smooth perturbations of an equilibrium state. This is a joint work with Ronghua Panand Weizhe Zhang.

In a comparison theorem, one compares the solution of a given PDE to a solution of a second PDE where the data are "rearranged." In this talk, we begin by discussing some of the classical comparison results, starting with Talenti's Theorem. We then discuss Neumann comparison results, including a conjecture of Kawohl, and end with some new results in balls and shells involving cap symmetrization.