# Collisions of the supercritical Keller-Segel particle system

Nicolas Fournier,
Yoan Tardy

January 2021

### 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 detail 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 of precisely k_0 particles emerges, for some k_0 \ge 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 collisions. And there are infinitely many binary collisions before each (k_0-2)-ary collision. Finally, collisions of subset of 3,…,k_0-3 particles never occur. The other scenario is similar except that there are no (k_0-2)-ary collisions.