A cross-diffusion system derived from a Fokker-Planck equation with partial averaging

TL;DR

This paper analyzes a novel cross-diffusion system derived from a Fokker-Planck equation with partial averaging, establishing global existence of solutions and exploring long-term behavior under certain conditions.

Contribution

It introduces a new cross-diffusion model with a non-symmetric, non-positive definite diffusion matrix derived from a Fokker-Planck equation, and proves global existence of solutions.

Findings

Global-in-time existence of positive weak solutions

Entropy methods applied to non-symmetric diffusion matrix

Analysis of large-time asymptotics under simplifying assumptions

Abstract

A cross-diffusion system for two compoments with a Laplacian structure is analyzed on the multi-dimensional torus. This system, which was recently suggested by P.-L. Lions, is formally derived from a Fokker-Planck equation for the probability density associated to a multi-dimensional It\={o} process, assuming that the diffusion coefficients depend on partial averages of the probability density with exponential weights. A main feature is that the diffusion matrix of the limiting cross-diffusion system is generally neither symmetric nor positive definite, but its structure allows for the use of entropy methods. The global-in-time existence of positive weak solutions is proved and, under a simplifying assumption, the large-time asymptotics is investigated.