Abstracts for Rivière-Fabes Symposium on Analysis and PDE
April 25-27, 2014
All talks in Vincent Hall 16
Lecture 1 - Boundary value problems on conformal compact Einstein manifolds
Abstract 1: Given a class of conformally compact Einstein manifolds with boundary, we are interested to study the boundary behavior of their compactified metrics. In the 3+1 case, I will report on some joint work with Yuxin Ge and Paul Yang on a compactness result in such setting, which is a study of some 4th order elliptic system with some matching 3rd order boundary conditions. In the general n+1 case, . I will report on some recent joint work with Jeffrey Case in which we study the positivity of a class of non-local conformal covariant operators, which are fractional GJMS operators on the boundary defined via scattering theory on the interior, and which includes the Dirichlet-Neumann operator as a special case.
Lecture 1 - On the long-term dynamics of solutions of certain fluid models
Abstract 1: I will present several recent theorems on the long-term behavior of solutions of certain equations describing fluid dynamics. The main models to be discussed are (1) two-fluid interface models in 2 dimensions, and (2) the Euler--Maxwell two-fluid system in 3 dimensions. The main questions we are considering concern the long-term dynamics of solutions, i.e. global existence and formation of singularities.
I will present joint work with F. Pusateri, C. Fefferman, V. Lie, B. Pausader, and Y. Guo.
Lecture 1 - Blow-up for mass critical KdV and universality
Lecture 2 - On soliton resolution for the radial critical wave equation
Abstract 1: We give a complete description of the dynamical behavior (including blow-up) of solutions with initial data close to ground state.
Abstract 2: In this joint work with Duyckaerts and Kenig we describe the asymptotic behavior of global solutions in a decoupled sum of soliton and a radiation.
Lecture 1 - Decay of correlations for classical and quantum hyperbolic systems
Abstract 1: The talk will based on joint work with Stéphane Nonnenmacher on the distribution of decay rates for systems with thin hyperbolic trapped sets: either filamentary, or smooth and normally hyperbolic. The results will be illustrated by recent numerical and experimental data (Borthwick, Barkhofen et al) and by the case of constant negative curvature (Dyatlov—Faure—Guillarmou).