Infinite time aggregation for the critical Patlak-Keller-Segel model in ℝ²

We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space ℝ². Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local-in-time existence for any mass of "free-energy solutions," namely weak solutions with some free-energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free-energy solutions with initial data as before for the critical mass 8π/χ. Actually, we prove that solutions blow up as a delta Dirac at the center of mass when t → ∞ when their second moment is kept constant at any time. Furthermore, all moments larger than 2 blowup as t → ∞ if initially bounded.