Trend to equilibrium for systems with small cross-diffusion

¹ Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100 L’Aquila, Italy
² University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
³ Sorbonne Université, Laboratoire Jacques-Louis Lions, Paris 75005, France
⁴ Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry, CV 47AL, UK
⁵ Radon Institute for Computational and Applied Mathematics, Altenbergerstr. 69, 4040 Linz, Austria

^* Corresponding author: luca.alasio@gssi.it

Received: 19 June 2019
Accepted: 4 February 2020

Abstract

This paper presents new analytical results for a class of nonlinear parabolic systems of partial different equations with small cross-diffusion which describe the macroscopic dynamics of a variety of large systems of interacting particles. Under suitable assumptions, we prove existence of classical solutions and we show exponential convergence in time to the stationary state. Furthermore, we consider the special case of one mobile and one immobile species, for which the system reduces to a nonlinear equation of Fokker–Planck type. In this framework, we improve the convergence result obtained for the general system and we derive sharper L^∞-bounds for the solutions in two spatial dimensions. We conclude by illustrating the behaviour of solutions with numerical experiments in one and two spatial dimensions.