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Parabolic-elliptic Keller–Segel system

Valentin Lemarié

Vol. 6 (2024), No. 3, 503–542
Abstract

We study on the whole space 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 by R. Danchin devoted to partially dissipative systems.

Keywords
critical regularity, relaxation limit, partially dissipative, Euler–Poisson equations, Keller–Segel equations
Mathematical Subject Classification
Primary: 35A01
Milestones
Received: 18 July 2023
Revised: 24 January 2024
Accepted: 3 March 2024
Published: 30 September 2024
Authors
Valentin Lemarié
Laboratoire d’Analyse et de Mathématiques Appliquées
LAMA — Gustave Eiffel
Saint-Maur-des-Fossés
France