Cross-Diffusion Systems with Entropy Structure

Abstract

Some results on cross-diffusion systems with entropy structure are reviewed.The focus is on local-in-time existence results for general systems withnormally elliptic diffusion operators, due to Amann, and global-in-time existencetheorems by Lepoutre, Moussa, and co-workers for cross-diffusion systems withan additional Laplace structure. The boundedness-by-entropy methodallows for global bounded weak solutions to certain diffusion systems.Furthermore, a partial result on the uniqueness of weak solutions is recalled,and some open problems are presented.