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

TL;DR

This paper presents a straightforward proof demonstrating the existence of global weak solutions for the 2D Keller-Segel equation with initial measures below a critical mass, using a two-particles moment approach.

Contribution

It introduces a simple, particle-based proof of non-explosion for measure solutions of the Keller-Segel equation, simplifying previous methods.

Findings

Existence of global weak solutions for initial mass less than 8π.

Proof relies on a two-particles moment computation.

Applicable to initial measures with finite total mass.

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 $f_0(\mathbb{R}^2)< 8 \pi$.