Multi-species Patlak-Keller-Segel system
Abstract
We study the regularity and large-time behavior of a crowd of species driven by chemo-tactic interactions. What distinguishes the different species is the way they interact with the rest of the crowd: the collective motion is driven by different chemical reactions which end up in a coupled system of parabolic Patlak-Keller-Segel equations. We show that the densities of the different species diffuse to zero provided the chemical interactions between the different species satisfy certain sub-critical condition; the latter is intimately related to a log-Hardy-Littlewood-Sobolev inequality for systems due to Shafrir & Wolansky. Thus for example, when two species interact, one of which has mass less than , then the 2-system stays smooth for all time independent of the total mass of the system, in sharp contrast with the well-known breakdown of one specie with initial mass.
Cite
@article{arxiv.1903.02673,
title = {Multi-species Patlak-Keller-Segel system},
author = {Siming He and Eitan Tadmor},
journal= {arXiv preprint arXiv:1903.02673},
year = {2021}
}