Cross-diffusion systems with excluded volume effects and asymptotic gradient flows

TL;DR

This paper analyzes a cross-diffusion PDE system modeling mixtures of hard spheres with excluded volume effects, focusing on stability, existence, and asymptotic behavior, and introduces the concept of an asymptotic gradient flow.

Contribution

It introduces the concept of an asymptotic gradient flow structure for general cases and applies it to study the behavior near equilibrium in cross-diffusion systems.

Findings

Gradient flow structure exists in symmetric cases.

Global existence and local stability are established for symmetric systems.

Numerical simulations illustrate model behavior.

Abstract

In this paper we discuss the analysis of a cross-diffusion PDE system for a mixture of hard spheres, which was derived by Bruna and Chapman from a stochastic system of interacting Brownian particles using the method of matched asymptotic expansions. The resulting cross-diffusion system is valid in the limit of small volume fraction of particles. While the system has a gradient flow structure in the symmetric case of all particles having the same size and diffusivity, this is not valid in general. We discuss local stability and global existence for the symmetric case using the gradient flow structure and entropy variable techniques. For the general case, we introduce the concept of an asymptotic gradient flow structure and show how it can be used to study the behavior close to equilibrium. Finally we illustrate the behavior of the model with various numerical simulations.