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Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ 2
Author(s) -
Blanchet Adrien,
Carrillo José A.,
Masmoudi Nader
Publication year - 2008
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20225
Subject(s) - bounded function , mathematics , moment (physics) , entropy (arrow of time) , center of mass (relativistic) , euclidean space , constant (computer programming) , euclidean geometry , energy (signal processing) , integrable system , mathematical analysis , physics , classical mechanics , geometry , statistics , quantum mechanics , energy–momentum relation , computer science , programming language
We analyze the two‐dimensional parabolic‐elliptic Patlak‐Keller‐Segel model in the whole Euclidean space ℝ 2 . 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. © 2007 Wiley Periodicals, Inc.