Trend to equilibrium for systems with small cross-diffusion

Abstract

This paper presents new analytical results for a class of nonlinear parabolic systems of partial different equations with small cross-diffusion which have been proposed to describe the 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 equilibration result obtained for the general system and we derive 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.1991 Mathematics Subject Classification. 35B40, 35B45, 35K51, 65N08.

Global existence for a class of viscous systems of conservation laws

“…The following theorem, combined with uniform-in-time estimates, allows to show global existence of classical solutions for a relatively wide class of cross-diffusion systems (see, for example, [2,3] and references therein). Remark 4 (Notation).…”

“…In some cases it is possible to extend many of the results we present to d > 1 (see e.g. [2,3]), however, in general, one can not guarantee existence of global, classical solutions of strongly coupled systems (see, for example, [13,16]). The system we consider is complemented by mixed, time-dependent boundary conditions which are imposed at the extreme points of the domain.…”

A Degenerate Cross-Diffusion System as the Inviscid Limit of a Nonlocal Tissue Growth Model

“…( 1) and nonlinear diffusion similar to Eq. (2). In a regime of low volume fraction and extremely localised repulsion, they showed that the probability density of a representative particle evolves according to a linear diffusion equation with a nonlinear correction of the form as in Eq.…”

“…Unlike other multi-species cross-diffusion systems that incorporate size exclusion effects that were recently introduced and considered, see [1,2,5,15,21,38,46], System (3) assumes a rather prominent place. This is due to the high symmetry in the velocity-pressure relation which renders the system non-strictly parabolic and is only convex rather than strictly convex.…”

Active Crowds

“…for some parametric energy functional E. For example, (30) exhibits a GF structure if the red and blue particles have the same size and diffusivity, but lacks it for differently sized particles (a variation of the model not discussed here). The closeness of AGF to GF can be used to study for example its stationary solutions and the behaviour of solution close to equilibrium, see [2,3,15].…”