Collisions of the supercritical Keller-Segel particle system

TL;DR

This paper analyzes the explosion behavior of a particle system associated with the 2D Keller-Segel equation, revealing complex collision patterns and cluster formations near finite-time blow-up.

Contribution

It provides a detailed description of collision dynamics and cluster formation in the supercritical Keller-Segel particle system near explosion, a novel insight into its singularity behavior.

Findings

Finite-time explosion occurs when attraction exceeds a critical threshold.

A cluster of exactly k_0 particles emerges at explosion, with k_0 ≥ 7.

Multiple levels of collisions (binary, (k_0-1)-ary, etc.) occur infinitely often before explosion.

Abstract

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.