Abstract

We discuss the analysis of a cross-diffusion PDE system for mixtures of hard spheres, which was derived by Bruna and Chapman (J Chem Phys 137, 2012) from a stochastic system of interacting Brownian particles using the methods of matched asymptotics. 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. 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. To gain further insights into the dynamics of asymptotic gradient flows, we study the system in the special case of two specific species – namely diffusing (Brownian) and immobile (obstacle) particles. In this case the cross-diffusion system reduces to a single nonlinear nonlinear Fokker—Planck equation, which again has no full gradient flow structure. However it can be interpreted as an asymptotic gradient flow for different entropy and mobility pairs. We discuss several possible such pairs and present global in time existence results as well as study the long time behavior of the corresponding full gradient flow equation. Furthermore we illustrate the dynamics of the different equations with numerical simulations.

This is joint work with M. Bruna (Oxford), M. Burger (Münster) and H. Ranetbauer (Vienna)