Entropic structure and duality for multiple species cross-diffusion systems

TL;DR

This paper establishes the existence of global weak solutions for complex multi-species cross-diffusion systems using entropy and duality estimates, with applications to models satisfying detailed balance conditions.

Contribution

It introduces a novel semi-implicit scheme and combines entropy and duality methods to prove existence results for a broad class of cross-diffusion systems.

Findings

Existence of global weak solutions for multi-species cross-diffusion systems.

Application of entropy and duality estimates to control solutions.

Extension to models with entropy based on detailed balance condition.

Abstract

This paper deals with the existence of global weak solutions for a wide class of (multiple species) cross-diffusions systems. The existence is based on two different ingredients: an entropy estimate giving some gradient control and a duality estimate that gives naturally L^2 control. The proof relies on a semi-implicit scheme tailored for cross-diffusion systems firstly defined by the two authors and collaborators. These results are applied to models having an entropy relying on the detailed balance condition recently exhibited by Chen et. al.