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Series: PDE Seminar

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.
Let D be a compact smooth domain with boundary $\partial D$ contained in
$T^N \times ]0,+\infty[$ . We show that if U is differentiable at each point of $\partial D$, then
this is also the case on the interior of D.
There are several variants of this result in different settings.
To make the result accessible to the layman, we will explain the method
on the function distance to a closed subset of an Euclidean space. This
example contains all the ideas of the general case.

Series: PDE Seminar

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
study the corresponding BBGKY hierarchy which contains a diverging
coefficient as the strength of the confining potential tends to
infinity. We prove that the limiting structure of the density matrices
counterbalances this diverging coefficient. We establish the
convergence of the BBGKY sequence and hence the propagation of chaos
for the focusing quantum many-body system. We derive rigorously the 1D
focusing cubic NLS as the mean-field limit of this 3D focusing quantum
many-body dynamic and obtain the exact 3D to 1D coupling constant.

Series: PDE Seminar

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.

Series: PDE Seminar

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.

Series: PDE Seminar

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 find that the high frequency part is also important for us to get the time decay.

Series: PDE Seminar

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
equations. The nonlinear evolution of nondispersive surface waves, such
as Rayleigh waves or quasi-static SPPs, is described by nonlocal,
quasi-linear, singular integro-differential equations, and we will
discuss some of the properties of these waves, including the formation
of singularities on the boundary.

Series: PDE Seminar

We consider the cubic nonlinear Schr\"odinger equation posed on the
product spaces \R\times \T^d. We prove the existence of global solutions
exhibiting infinite growth of high Sobolev norms. This is a
manifestation of the "direct energy cascade" phenomenon, in which the
energy of the system escapes from low frequency concentration zones to
arbitrarily high frequency ones (small scales). One main ingredient in
the proof is a precise description of the asymptotic dynamics of the
cubic NLS equation when 1\leq d \leq 4. More precisely, we prove
modified scattering to the resonant dynamics in the following sense:
Solutions to the cubic NLS equation converge (as time goes to infinity)
to solutions of the corresponding resonant system (aka first Birkhoff
normal form). This is joint work with Benoit Pausader (Princeton),
Nikolay Tzvetkov (Cergy-Pontoise), and Nicola Visciglia (Pisa).

Series: PDE Seminar

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.

Series: PDE Seminar

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

Series: PDE Seminar

Some mixed-type PDE problems for transonic flow and isometric
embedding will be discussed. Recent results on the solutions to the
hyperbolic-elliptic mixed-type equations and related systems of PDEs will
be presented.