WIAS Preprint No. 2883, (2021)

On multi-species diffusion with size exclusion


Authors

  • Hopf, Katharina
    ORCID: 0000-0002-6527-2256
  • Burger, Martin

2020 Mathematics Subject Classification

  • 35K59 35K65 35A02 35B35

Keywords

  • Cross diffusion, size exclusion, volume filling, gradient-flow structure, degenerate nonlinear mobility, weak-strong stability, large-data weak solutions, long-time asymptotics, relative entropy, convexity

DOI

10.20347/WIAS.PREPRINT.2883

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.

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