Asymptotic analysis of Reaction-Diffusion systems with Entropy dissipation

Abstract: We review some result results, joint with Gregoire Allaire, regarding the homogenization of a class of reaction-diffusion systems with highly oscillating coefficients. Reversible chemical reactions are considered. The analyses of reaction-diffusion systems in the case of species-dependent diffusion coefficients are quite non-trivial. We derive uniform (with respect to the heterogeneous length-scale) $L^p$ estimates on the solution. We use the now well-known two-scale convergence method to derive effective equations which happen to be cross-diffusion models. We review some recent results with regard to well-posedness theory for  reaction-diffusion systems. We shall also present a wide range of open problems in connection to reaction-diffusion systems where the rate functions are given by law of mass action.