Abstracts for Rivière-Fabes Symposium on Analysis and PDE
April 17-19, 2015
All talks in Vincent Hall 16
Lecture 1 - Transport theory: from functional inequalities to random matrices
Abstract 1: The optimal transport problem consists in finding the cheapest way to transport a distribution of mass from one place to another. Apart from its applications to economics, optimal transport theory is an efficient tool to construct change of variables between probability densities, and this fact can be applied for instance to prove stability of minimizers of several geometric/functional inequalities.
More recently, motivated by problems arising in random matrix theory, people have tried to apply these methods in very large dimensions. However the regularity of optimal maps seem to play an important role in this context, and unfortunately one cannot hope in general to obtain regularity estimates that are uniform with respect to the dimension. Based on these considerations, it seems hopeless to apply optimal transport theory in this context. Still, ideas coming from optimal transport can be used to construct approximate transport maps (i.e., maps which send a density onto another up to a small error) which enjoy regularity estimates that are uniform in the dimension, and such maps can then be used to show universality results for the distribution of eigenvalues in random matrices.
The aim of these lectures is to give a self-contained presentation of all these results.
Lecture 1 - Inviscid damping and enhanced dissipation in 2D Euler and Navier-Stokes.
Abstract:I will review a few aspects of the inviscid damping in 2D Euler near the 2D Couette flow (Joint with J. Bedrossian) and then will talk about Enhanced dissipation and inviscid damping in the Navier-Stokes equations near the 2D Couette flow and study the stability in the inviscid limit.
The main goal is the study of the long time, inviscid limit of the 2D Navier-Stokes equations near the periodic Couette flow, and in particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin's 1887 linear analysis. (joint with J. Bedrossian and V. Vicol).
Lecture 1 - Analytic and algebraic related structures on the families of convex sets and log-concave functions. (Based on Joint works with Shiri Artstein and Liran Rotem)
Lecture 2 - Elementary operator equations and classical constructions in analysis. (Based on joint works with Hermann Koenig)
Abstract: The main goal of the talks is to show how some classical constructions in Geometry and Analysis appear (and in a unique way) from elementary and very simplest properties. For example, the polarity relation and support functions are very important and well known constructions in Convex Geometry, but what would be their functional version, say, in the class of log-concave functions? And yes, they are uniquely defined also for this class, as well as for many other classes of functions.
Another example: How can one identify the "square root of a convex body"? Yes, it is possible (however, "the square of a convex body" does not exist in general).
In the first talk we will mostly deal with Geometric results of this nature. We also construct summation operation on the class of log-concave functions which polarizes the Lebegue Integral and introduces the notion of a mixed integral parallelly to mixed volumes for convex bodies. We will show some inequalities coming from Convex geometry, but which are presented already on the level of log-concave (or even more generally quasi-concave) functions.
In the preperation to the second talk we will characterize the Fourier transform (on the Schwartz class in R^n) as, essentially, the only map which transforms the product to the convolution.
The talks will be (obviously) accessible to graduate students.
Lecture 1 - From molecular dynamics to kinetic theory and fluid mechanics
Abstract 1: In his sixth problem, Hilbert asked for an axiomatization of gas dynamics, and he suggested to use the Boltzmann equation as an intermediate description between the (microscopic) atomic dynamics and (macroscopic) fluid models. The main difficulty to achieve this program is to prove the asymptotic decorrelation between the local microscopic interactions (referred to as propagation of chaos) on a time scale much larger than the mean free time. This is indeed the key property to observe some relaxation towards local thermodynamic equilibrium.
This control of the collision process can be obtained in fluctuation regimes [1, 2]. In , we have established a long time convergence result to the linearized Boltzmann equation, and eventually derived the acoustic and incompressible Stokes-Fourier equations in dimension 2. The proof relies crucially on symmetry arguments, combined with a suitable pruning procedure to discard super exponential collision trees.
Keywords: system of particles, low density, Boltzmann-Grad limit, kinetic equation, fluid models
 T. Bodineau, I. Gallagher, L. Saint-Raymond. The Brownian motion as the limit of a deterministic system of hard-spheres, to appear in Invent. Math. (2015).
 T. Bodineau, I. Gallagher, L. Saint-Raymond. From hard spheres to the linearized Boltzmann equation: an L2 analysis of the Boltzmann-Grad limit, in preparation.