Collisions of the supercritical Keller-Segel particle system - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2021

Collisions of the supercritical Keller-Segel particle system

Résumé

We study a particle system naturally associated to the $2$-dimensional Keller-Segel equation. It consists of $N$ Brownian particles in the plane, interacting through a binary attraction in $\theta/(Nr)$, where $r$ stands for the distance between two particles. When the intensity $\theta$ of this attraction is greater than $2$, this particle system explodes in finite time. We assume that $N>3\theta$ and study in details what happens near explosion. There are two slightly different scenarios, depending on the values of $N$ and $\theta$, here is one: at explosion, a cluster consisting of precisely $k_0$ particles emerges, for some deterministic $k_0\geq 7$ depending on $N$ and $\theta$. Just before explosion, there are infinitely many $(k_0-1)$-ary collisions. There are also infinitely many $(k_0-2)$-ary collisions before each $(k_0-1)$-ary collision. And there are infinitely many binary collisions before each $(k_0-2)$-ary collision. Finally, collisions of subsets of $3,\dots,k_0-3$ particles never occur. The other scenario is similar except that there are no $(k_0-2)$-ary collisions.

Dates et versions

hal-03384516 , version 1 (19-10-2021)

Identifiants

Citer

Nicolas Fournier, Yoan Tardy. Collisions of the supercritical Keller-Segel particle system. 2021. ⟨hal-03384516⟩
22 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More