Global existence analysis of cross-diffusion population systems for multiple species

TL;DR

This paper proves the global existence of weak solutions for reaction-cross-diffusion systems modeling multiple competing species, extending classical models and using advanced entropy methods under certain conditions.

Contribution

It introduces a refined entropy approach and a new approximation scheme to establish global solutions for complex multi-species cross-diffusion models.

Findings

Global existence of solutions is proved under detailed balance or weak cross-diffusion conditions.

The model extends classical two-species population models to multiple species.

Entropy decreases over time under detailed balance, ensuring stability.

Abstract

The existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species is proved. The equations can be derived from an on-lattice random-walk model with general transition rates. In the case of linear transition rates, it extends the two-species population model of Shigesada, Kawasaki, and Teramoto. The equations are considered in a bounded domain with homogeneous Neumann boundary conditions. The existence proof is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed balance or weak cross-diffusion condition. The detailed balance condition is related to the symmetry of the mobility matrix, which mirrors Onsager's principle in thermodynamics. Under detailed balance (and without reaction), the entropy is nonincreasing in time, but counter-examples show that the entropy may increase initially if detailed balance does not hold.