Collisions of the supercritical Keller-Segel particle system
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 1/r, where r stands for the distance between two particles. When the intensity of this attraction, which is a parameter of the equation, is greater than 2, this particle system explodes in finite time. We study in details what happens near explosion.
There are two slightly different scenarios, depending on the values of N and the intensity of the attraction here is one: at explosion, a cluster consisting of precisely k particles emerges, for some deterministic k>6 depending on N and the intensity of the attraction. Just before explosion, there are infinitely many k-1-ary collisions. There are also infinitely many k-2-ary collisions before each k-1-ary collision. And there are infinitely many binary collisions before each k-2-ary collision. Finally, collisions of subsets of 3,...,k-3 particles never occur. The other scenario is similar except that there are no k-2-ary collisions.
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