Global well-posedness and blow-up for the 2-D Patlak–Keller–Segel equation

In this paper, we prove that for every nonnegative initial data in $L^1({\mathbb{R}}^2)$, the Patlak–Keller–Segel equation is globally well-posed if and only if the total mass $M\le 8\pi$. Our proof is based on some monotonicity formulas of nonnegative mild solutions.