Collisions of the supercritical Keller-Segel particle system

Publication date: 16 October 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

h e t a / (N r)

, where

r

stands for the distance between two particles. When the intensity

h e t a

of this attraction is greater than

2

, this particle system explodes in finite time. We assume that

N > 3 h e t a

and study in details what happens near explosion. There are two slightly different scenarios, depending on the values of

N

and

h e t a

, here is one: at explosion, a cluster consisting of precisely

k_{0}

particles emerges, for some deterministic

k_{0} g e q 7

depending on

N

and

h e t a

. 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, d o t s, 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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