Collisions of the supercritical Keller–Segel particle system

  • Nicolas Fournier

    Sorbonne Université, Paris, France
  • Yoan Tardy

    École Polytechnique, Palaiseau, France
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Abstract

We study a particle system naturally associated to the -dimensional Keller–Segel equation. It consists of Brownian particles in the plane, interacting through a binary attraction in , where stands for the distance between two particles. When the intensity of this attraction is greater than , this particle system explodes in finite time. We assume that and study in detail what happens near explosion. There are two slightly different scenarios, depending on the values of and , here is one: at explosion, a cluster consisting of precisely particles emerges, for some deterministic depending on and . Just before explosion, there are infinitely many -ary collisions. There are also infinitely many -ary collisions before each -ary collision. And there are infinitely many binary collisions before each -ary collision. Finally, collisions of subsets of particles never occur. The other scenario is similar except that there are no -ary collisions.

Cite this article

Nicolas Fournier, Yoan Tardy, Collisions of the supercritical Keller–Segel particle system. J. Eur. Math. Soc. 27 (2025), no. 10, pp. 4083–4158

DOI 10.4171/JEMS/1450