On multi-species diffusion with size exclusion

TL;DR

This paper analyzes a classical multi-species diffusion model with size exclusion, establishing existence, long-term behavior, and stability results using a gradient-flow approach, including cases with vanishing coefficients.

Contribution

It provides the first systematic study of weak solutions, long-time asymptotics, and a weak-strong stability estimate for the model, including novel existence results for partial interaction cases.

Findings

Existence of weak solutions established.

Long-time asymptotic behavior characterized.

Weak-strong stability estimate proven.

Abstract

We revisit a classical continuum model for the diffusion of multiple species with size-exclusion constraint, which leads to a degenerate nonlinear cross-diffusion system. The purpose of this article is twofold: first, it aims at a systematic study of the question of existence of weak solutions and their long-time asymptotic behaviour. Second, it provides a weak-strong stability estimate for a wide range of coefficients, which had been missing so far. In order to achieve the results mentioned above, we exploit the formal gradient-flow structure of the model with respect to a logarithmic entropy, which leads to best estimates in the full-interaction case, where all cross-diffusion coefficients are non-zero. Those are crucial to obtain the minimal Sobolev regularity needed for a weak-strong stability result. For meaningful cases when some of the coefficients vanish, we provide a novel existence result based on approximation by the full-interaction case.