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 L1(R2), the Patlak-Keller-Segel equation is globally well-posed if and only if the total mass M≤8π. Our proof is based on some monotonicity formulas of nonnegative mild solutions.