2021/10/16 by Nicolas Fournier, Fournier, Nicolas, Yoan Tardy +1 · 2 citations
Mathematics · #60H10 #60K35 #FOS: Mathematics #Mathematical Biology Tumor Growth #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · doi:10.48550/arxiv.2110.08490
openalex publication_date 2021/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 θ/(Nr), where r stands for the distance between two particles. When the intensity θ of this attraction is greater than 2, this particle system explodes in finite time. We assume that N>3θ and study in details what happens near explosion. There are two slightly different scenarios, depending on the values of N and θ, here is one: at explosion, a cluster consisting of precisely k0 particles emerges, for some deterministic k0≥ 7 depending on N and θ. Just before explosion, there are infinitely many (k0-1)-ary collisions. There are also infinitely many (k0-2)-ary collisions before each (k0-1)-ary collision. And there are infinitely many binary collisions before each (k0-2)-ary collision. Finally, collisions of subsets of 3,…,k0-3 particles never occur. The other scenario is similar except that there are no (k0-2)-ary collisions.