  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.
