Weak convergence of the empirical measure for the Keller-Segel model in both subcritical and critical cases

TL;DR

This paper proves the weak convergence of the empirical measure for the particle system modeling the Keller-Segel equations in both subcritical and critical cases, using a simple two-particle moment argument for the subcritical case.

Contribution

It introduces a straightforward proof of weak convergence for the empirical measure in the Keller-Segel particle system, applicable to general initial conditions and both critical and subcritical regimes.

Findings

Weak convergence of empirical measures established in both regimes

Two-particle moment argument prevents finite-time aggregation in subcritical case

Extension to general initial conditions beyond regular ones

Abstract

We show the weak convergence, up to extraction of a subsequence, of the empirical measure for the Keller-Segel system of particles in both subcritical and critical cases, for general initial conditions. This particle system consists of $N$ planar Brownian motions interacting through a Coulombian attractive force, which is quite singular. In the subcritical case, a stronger result has been established by Bresch-Jabin-Wang \cite{bjw} at the price of two simplifications: the whole space $\rr^2$ is replaced by a torus and the initial condition is assumed to be regular. In the subcritical case, our proof is fairly straightforward: we use a {\it two particles} moment argument, which shows that particles do not aggregate in finite time, uniformly in the number of particles. The critical case requires more work.