Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model

Adrien Blanchet, Eric A. Carlen, José A. Carrillo

Research output: Contribution to journalArticleResearchpeer-review

87 Citations (Scopus)

Abstract

We investigate the long time behavior of the critical mass Patlak-Keller-Segel equation. This equation has a one parameter family of steady-state solutions ρ{variant} λ, λ>0, with thick tails whose second moment is unbounded. We show that these steady-state solutions are stable, and find basins of attraction for them using an entropy functional Hλ coming from the critical fast diffusion equation in R2. We construct solutions of Patlak-Keller-Segel equation satisfying an entropy-entropy dissipation inequality for Hλ. While the entropy dissipation for Hλ is strictly positive, it turns out to be a difference of two terms, neither of which needs to be small when the dissipation is small. We introduce a strategy of controlled concentration to deal with this issue, and then use the regularity obtained from the entropy-entropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards ρ{variant} λ. © 2011 Elsevier Inc.
Original languageEnglish
Pages (from-to)2142-2230
JournalJournal of Functional Analysis
Volume262
Issue number5
DOIs
Publication statusPublished - 1 Mar 2012

Keywords

  • Basins of attraction
  • Critical mass
  • Gradient flows with respect to transport distances
  • Keller-Segel model

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