Global well-posedness of a conservative relaxed cross diffusion system

TL;DR

This paper establishes the global existence, regularity, and uniqueness of solutions for a class of nonlinear cross diffusion systems with relaxed conservative structure, applicable in any spatial dimension.

Contribution

It proves the global well-posedness of a broad class of nonlinear cross diffusion systems with minimal regularity assumptions on the nonlinearities.

Findings

Global existence of weak solutions in any dimension

Solutions are regular and unique if nonlinearities are locally Lipschitz

Applicable to systems with continuous and bounded nonlinearities

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 $u_i, i=1,...,I$ represent $I$ density-functions, $\tilde{u}$ is a spatially regularized form of $(u_1,...,u_I)$ 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.