Existence Theory for a Cross-Diffusion System with Independent Drifts: Mixing Dynamics

Abstract: We consider a cross-diffusion system for which the diffusion of each species is governed solely by the aggregate density through a pressure law of logarithmic or fast diffusion type. The model is set over a one dimensional bounded interval, equipped with no-flux boundary conditions, and accommodates for the presence of potential drifts which are allowed to differ across each species. We establish the global existence of solutions without having to assume the total mixing of solutions. As a consequence, we give a full resolution of the PDE systems recently studied by the authors and by Elbar--Santambrogio, by allowing a general class of initial data with finite bounded variation, with no further structural assumptions on their supports.