Entropic structure and duality for multiple species cross-diffusion systems

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 L2 control. The heart of our proof is a semi-implicit scheme tailored for cross-diffusion systems firstly defined in Desvillettes et al. (2015) and a (nonlinear Aubin–Lions type) compactness result developed in Moussa (2016) and Andreianov et al. (2015) that turns the (potentially weak) gradient estimates into almost everywhere convergence. We apply our results to models having an entropy relying on the detailed balance condition exhibited by Chen et. al. in Chen et al. (2016).