Global Well-Posedness of a Conservative Relaxed Cross Diffusion System

Abstract

We prove global existence in time of solutions to relaxed conservative cross diffusion systems governed by nonlinear operators of the form $u_i\to \partial_tu_i-\Delta(a_i(\tilde{u})u_i)$, where the $u_i, i=1,\ldots,I$, represent I density functions, $\tilde{u}$ is a spatially regularized form of $(u_1,\ldots,u_I)$, and the nonlinearities $a_i$ are merely assumed to be continuous and bounded from below. Existence of global weak solutions is obtained in any space dimension. Solutions are proved to be regular and unique when the $a_i$ are locally Lipschitz continuous.