A simple proof of non-explosion for measure solutions of the Keller-Segel equation

Abstract

We give a simple proof, relying on a {\it two-particles} moment computation, that there exists a global weak solution to the $2$-dimensional parabolic-elliptic Keller-Segel equation when starting from any initial measure $f_0$ such that $f_0({\mathbb{R}}^2)<8\pi$.

A stronger result is already known. However our proof seems new and is very short. In this version, we modified the introduction and removed the tedious critical case to emphasize the simplicity of the proof