Geometry, Control and Genericity for Partial Differential Equations

Introduction

Many physics models are described by waves or more generally dispersive equations (Schrödinger equations) with propagation in a non-homogeneous and bounded medium. Toy models (mostly in flat backgrounds) have been developed by mathematicians. However, many questions remain open even on these simplified models in the presence of inhomogeneities and boundaries.

Pathological Behaviours

In particular, the works of mathematicians in the last decade have allowed the exhibition of some pathological behaviours which appear to be quite unstable. A first point in this proposal will be to expand the understanding of the influence of the geometry (inhomogeneities of the media, boundaries) on the behaviour of solutions to dispersive PDEs.

Stability of Behaviours

When these behaviours appear to be unstable, a natural question is whether they are actually rare. The last years have seen the emergence of a new point of view on these questions: random data Cauchy theories. The idea behind this is that for random initial data, the solution’s behaviours are better than expected (deterministically).

Project Goals

The second point of this project is precisely to go further in this direction. After identifying these pathological behaviours, is it possible to show that for almost all initial data and almost all geometries, they do not happen?

Understanding how to combine the powerful techniques from micro-local and harmonic analysis with a probabilistic approach in this context should allow for a much better understanding of these physically relevant models.

Summary of Objectives

Summarising, the purpose of my project is to develop tools and give answers to the following questions in the context of dispersive PDEs (and to some extent fluid mechanics):

1. Can we understand the influence of the geometric background (and boundaries) on concentration properties and the behaviour of solutions to dispersive evolution PDEs?
2. Can we define generic behaviours for solutions to waves and fluids PDEs?
3. Can we show that some very pathological behaviours (which do happen) are actually very rare?