Stable solutions and nonstandard diffusions: PDE questions arising in Mathematical Physics

Introduction

The concept of diffusion is ubiquitous in the physical sciences. From the mathematical point of view, its study started in the early 19th century with the development of PDE theory, and has many connections to Physics, Probability, Geometry, and Functional Analysis. This project aims to answer several outstanding questions related to the mathematics of diffusion.

Project Structure

The proposal is divided into two blocks:

1. Stable Solutions to Reaction-Diffusion PDE

   • The first block corresponds to the study of stable solutions to reaction-diffusion PDE, and more precisely the classification of global stable solutions in the physical space (i.e., in 3D) for a general class of problems including:
     • The Allen-Cahn equation
     • The Alt-Phillips equation
     • The thin Alt-Caffarelli equations
   • We will also investigate the same question for complex-valued solutions in 2D, which arises in the construction of travelling waves for the Gross-Pitaevskii equation.

2. Nonstandard Diffusions

   • The second block corresponds to nonstandard diffusions. In particular, we will study:
     • The Boltzmann equation (a fundamental model in statistical mechanics)
     • Nonlocal diffusions (deeply related to Lévy processes and "anomalous diffusions")
     • The porous medium equation (a classical nonlinear PDE that arises in various physical models in which diffusion is "slow").

Motivation

The highly ambitious goals of the project are motivated by some recent results obtained by the PI in these areas.