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 with normally elliptic diffusion operators, due to Amann, and global-in-time existence theorems by Lepoutre, Moussa, and co-workers for cross-diffusion systems with an additional Laplace structure. The boundedness-by-entropy method allows 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.

Homogenization results for a coupled system of reaction–diffusion equations

“…The form of the nonlinearities arising at the microscale and this non-standard coupling lead, at the macroscale, to the appearance of additional source or sink terms and of a non-standard diffusion matrix, which is not constant (see Theorem 24 and Corollary 25). Hence, we show that some fast reactions occurring, at the microscale, on the boundaries of S ε (see the second equation in system (2.1)) could lead, at the macroscale, to a cross -diffusion system (see for this terminology the recent papers [37] and [38]).…”