On a cross-diffusion model for multiple species with nonlocal interaction and size exclusion

TL;DR

This paper analyzes a nonlinear PDE model for two interacting species with size exclusion and attraction, establishing global existence, exploring phase separation, and examining coarsening dynamics through analytical and numerical methods.

Contribution

It introduces a global existence result for a cross-diffusion system with nonlocal interactions and investigates phase separation and coarsening phenomena.

Findings

Global existence of solutions established

Phase separation effects analyzed

Coarsening dynamics observed numerically

Abstract

The aim of this paper is to study a PDE model for two diffusing species interacting by local size exclusion and global attraction. This leads to a nonlinear degenerate cross-diffusion system, for which we provide a global existence result. The analysis is motivated by the formulation of the system as a formal gradient flow for an appropriate energy functional consisting of entropic terms as well as quadratic nonlocal terms. Key ingredients are entropy dissipation methods as well as the recently developed boundedness by entropy principle. Moreover, we investigate phase separation effects inherent in the cross-diffusion model by an analytical and numerical study of minimizers of the energy functional and their asymptotics to a previously studied case as the diffusivity tends to zero. Finally we briefly discuss coarsening dynamics in the system, which can be observed in numerical results and is motivated by rewriting the PDEs as a system of nonlocal Cahn-Hilliard equations.