Interacting particle approximation of cross-diffusion systems

Abstract

We prove the existence of weak solutions of a class of multi-species cross-diffusion systems as well as the propagation of chaos result by means of nonlocal approximation of the nonlinear diffusion terms, coupling methods and compactness arguments. We also prove the uniqueness under further structural assumption on the mobilities by combining the uniqueness argument for viscous porous medium equations and linear Fokker-Planck equations. We show that these equations capture the macroscopic behavior of stochastic interacting particle systems if the localisation parameter is chosen logarithmically with respect to the number of particles.