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

Abstract

<p style='text-indent:20px;'>We give a simple proof, relying on a <i>two-particles</i> moment computation, that there exists a global weak solution to the <inline-formula><tex-math id="M1">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-dimensional parabolic-elliptic Keller-Segel equation when starting from any initial measure <inline-formula><tex-math id="M2">\begin{document}$ f_0 $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M3">\begin{document}$ f_0( {\mathbb{R}}^2)< 8 \pi $\end{document}</tex-math></inline-formula>.</p>