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

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.

This version is essentially the same as the two first versions one, we simply made the presentation clearer, correct a lot of typos and add an appendix for a technical result concerning a La Vallée Poussin-type theorem