Parabolic-elliptic Keller-Segel's system

TL;DR

This paper investigates the behavior of the compressible Euler-Poisson system with large damping, proving global existence and rigorously deriving the Keller-Segel model as a diffusive limit in critical Besov spaces.

Contribution

It establishes global existence results for the Euler-Poisson system with large damping and provides a rigorous derivation of the Keller-Segel model as a diffusive limit.

Findings

Global existence for large damping Euler-Poisson system

Explicit convergence rate to Keller-Segel model

Analysis in critical Besov spaces

Abstract

We study on the whole space R d the compressible Euler system with damping coupled to the Poisson equation when the damping coefficient tends towards infinity. We first prove a result of global existence for the Euler-Poisson system in the case where the damping is large enough, then, in a second step, we rigorously justify the passage to the limit to the parabolic-elliptic Keller-Segel after performing a diffusive rescaling, and get an explicit convergence rate. The overall study is carried out in 'critical' Besov spaces, in the spirit of the recent survey [16] by R. Danchin devoted to partially dissipative systems.