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 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.