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

TL;DR

This paper proves the global existence of solutions for a one-dimensional cross-diffusion system with independent drifts, accommodating general initial data and different potential drifts for each species, advancing understanding of mixing dynamics.

Contribution

It establishes the existence of solutions without assuming total mixing, extending previous models to more general initial conditions and drift configurations.

Findings

Proved global existence of solutions for the system.

Allowed for initial data with finite bounded variation.

Handled different potential drifts for each species.

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.