Abstracts for Rivière-Fabes Symposium on Analysis and PDE
April 17-19, 2009
All talks in Vincent Hall 16
Lecture 1 - Interaction of Vortices in Viscous Planar Flows
Abstract 1: It is a well-established fact that the long-time behavior of two-dimensional decaying turbulence is essentially governed by a few basic mechanisms, such as vortex interactions and, especially, vortex merging. The aim of this talk is to describe in a rigorous way the interaction of widely separated viscous vortices. To this end, we consider the inviscid limit of the solution of the two-dimensional incompressible Navier-Stokes equation in the particular case where the initial vorticity is a finite collection of point vortices. Assuming that vortex collisions do not occur, we obtain to leading order a superposition of Lamb-Oseen vortices whose centers evolve according to a viscous regularization of the Helmholtz-Kirchhoff system. Our approach also gives an accurate description of the asymptotic profile of each individual vortex, and this allows to estimate the self-interactions which play a crucial role in the convergence proof.
Lecture 1 - Regularity of Elliptic and Parabolic Equations with Rough Coefficients
Abstract 1: I will present some recent results about the regularity and solvability of elliptic and parabolic equations in divergence and non-divergence forms. The leading coefficients are assumed to be measurable in one or two directions and have vanishing mean oscillation in the orthogonal directions. Applications of the results and extensions to higher order (fully couples) systems will also be discussed. Most part of the talk is based on joint work with Nicolai Krylov and with Doyoon Kim.
Lecture 1 - A Lie Groups Approach to the Analysis of Kolmogorov-Fokker-Planck Equations
Abstract 1:Let L be a Hormander-type operator, sum of squares of vector fields+drift. We show sufficient conditions on the vector fields and on the drift term for the existence of a Lie group structure G such that L is left invarient on G. We also investigate the existence of a global fundamental solution for L, providing results that ensure a suitable left invariance property. We will show several examples of operators to which our results apply: some are new, some appear in recent literature usually quoted as Kolmogorov-Fokker-Planck operators. Our examples arise in several theoretical and applied settings, such as diffusion theory, computer and human vision, phasr noise Fokker-Planck equations, curvature Brownian motion.
Lecture 1 - Landau Damping: Relaxation Without Dissipation
Abstract 1: The Landau damping may be the single most famous and paradoxical phenomenon in classical plasma physics, predicting relaxation without any irreversibility. While it has been treated by various authors at the linear level, its nonlinear version has remained elusive so far. In a joint work with Clement Mouhot, we develop a new theory for Landau damping, for the full (not linearized) model. I will describe some of the physical and mathematical advances uncovered (to our own surprise) in this study.
Lecture 1 - Global Schrodinger Maps in Dimensions d ≥ 2: Small Data in the Critical Sobolev Spaces
Lecture 1 - Recent Results in Fluid Mechanics
Abstract 1: We will present, in this talk, new applications of De Giorgi's methods and blow-up techniques to fluid mechanics problems. Those techniques have been successfully applied to show full regularity of the solutions to the surface quasi-geostrophic equation in the critical case.
We will present, also, a new nonlinear family of spaces allowing to control higher derivatives of solutions to the 3D Navier-Stokes equation. Finally, we will present a regularity result for a reaction-diffusion system which has almost the same supercriticality than the 3D Navier-Stokes equation.