Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, October 28, 2014 - 15:05 , Location: Skiles 006 , Albert Fathi , École Normale Supérieure de Lyon, France , Organizer:
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
Tuesday, October 7, 2014 - 15:00 , Location: Skiles 006 , Xuwen Chen , Brown University , Organizer:
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
Tuesday, September 30, 2014 - 15:00 , Location: Skiles 006 , Daniel Spirn , University of Minnesota , Organizer:
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
Tuesday, September 9, 2014 - 15:00 , Location: Skiles 006 , Shengguo Zhu , Georgia Tech , Organizer:
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
Tuesday, August 26, 2014 - 15:00 , Location: Skiles 006 , Junxiong Jia , Georgia Tech , Organizer:
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
Tuesday, April 22, 2014 - 15:05 , Location: Skile 006 , John Hunter , University California, Davis , Organizer:
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
Thursday, April 17, 2014 - 15:05 , Location: Skiles 006 , Zaher Hani , New York University , Organizer:
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
Tuesday, April 15, 2014 - 15:05 , Location: Skiles 006 , Constantine Dafermos , Brown University , Organizer:
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
Tuesday, April 8, 2014 - 15:05 , Location: skiles 006 , Soulèye KANE , UCAD , Organizer:
Series: PDE Seminar
Tuesday, March 11, 2014 - 15:05 , Location: Skiles 006 , Dehua Wang , University of Pittsburgh , Organizer:
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.