A cross-diffusion system derived from a Fokker-Planck equation with partial averaging
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.
Cite
@article{arxiv.1601.05039,
title = {A cross-diffusion system derived from a Fokker-Planck equation with partial averaging},
author = {Ansgar Jüngel and Nicola Zamponi},
journal= {arXiv preprint arXiv:1601.05039},
year = {2017}
}