On a class of triangular cross-diffusion systems and its fast reaction approximation

Abstract: The purpose of this article is to investigate the emergence of cross-diffusion in the time evolution of two slow-fast species in competition. A class of triangular cross-diffusion system is obtained as the singular limit of a fast reaction-diffusion system. We first prove the convergence of the unique strict solution of the fast reaction-diffusion system towards a (weak, strong) solution of the cross-diffusion system, as the reaction rate $\epsilon^{-1}$ goes to $+\infty$. Furthermore, under the assumption of small cross-diffusion, we obtain a convergence rate as well as the influence of the initial layer, due to initial data, on the convergence rate itself. Both results are obtained through energy functionals that handle the fast reaction terms uniformly in $\epsilon$.