Collisions of the supercritical Keller–Segel particle system
Nicolas Fournier, Yoan Tardy
Source 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 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.
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