About the entropic structure of detailed balanced multi-species cross-diffusion equations

TL;DR

This paper establishes a rigorous connection between the entropy structure of multi-species cross-diffusion equations and microscopic Markov processes, offering a new approach for proving global solutions.

Contribution

It introduces a novel method linking entropy structures of cross-diffusion systems with microscopic Markov processes through mean-field and discretisation limits.

Findings

Established a formal link between entropy structures and microscopic processes.

Provided a rigorous proof of the discretisation limits.

Proposed a new strategy for proving global existence of solutions.

Abstract

This paper links at the formal level the entropy structure of a multi-species cross-diffusion system of Shigesada-Kawasaki-Teramoto (SKT) type satisfying the detailed balance condition with the entropy structure of a reversible microscopic many-particle Markov process on a discretised space. The link is established by first performing a mean-field limit to a master equation over discretised space. Then the spatial discretisation limit is performed in a completely rigorous way. This by itself provides a novel strategy for proving global existence of weak solutions to a class of cross-diffusion systems.