Particle approximation of the doubly parabolic Keller-Segel equation in the plane - Archive ouverte HAL Access content directly
Journal Articles Journal of Functional Analysis Year : 2023

Particle approximation of the doubly parabolic Keller-Segel equation in the plane

Abstract

In this work, we study a stochastic system of N particles associated with the parabolicparabolic Keller-Segel system. This particle system is singular and non Markovian in that its drift term depends on the past of the particles. When the sensitivity parameter is sufficiently small, we show that this particle system indeed exists for any N ≥ 2, we show tightness in N of its empirical measure, and that any weak limit point of this empirical measure, as N → ∞, solves some nonlinear martingale problem, which in particular implies that its family of time-marginals solves the parabolic-parabolic Keller-Segel system in some weak sense. The main argument of the proof consists of a Markovianization of the interaction kernel: We show that, in some loose sense, the two-by-two path-dependant interaction can be controlled by a two-by-two Coulomb interaction, as in the parabolic-elliptic case.
Fichier principal
Vignette du fichier
ArxivV2.pdf (407.13 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03855110 , version 1 (30-08-2023)

Licence

Identifiers

Cite

Nicolas Fournier, Milica Tomašević. Particle approximation of the doubly parabolic Keller-Segel equation in the plane. Journal of Functional Analysis, 2023, 285 (7), pp.110064. ⟨10.1016/j.jfa.2023.110064⟩. ⟨hal-03855110⟩
55 View
40 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More