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

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 $ρ_λ$, $λ>0$, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional ${\mathcal H}_λ$ coming from the critical fast diffusion equation in $\R^2$. We construct solutions of Patlak-Keller-Segel equation satisfying an entropy-entropy dissipation inequality for ${\mathcal H}_λ$. While the entropy dissipation for ${\mathcal H}_λ$ is strictly positive, it turns out to be a difference of two terms, neither of which need 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 $ρ_λ$. In the present paper, we do not provide any estimate of the rate of convergence, but we discuss how this would result from a stability result for a certain sharp Gagliardo-Nirenberg-Sobolev inequality.

This version of the paper improves on the previous version by removing the small size condition on the value of the second Lyapunov functional of the initial data. The improved methodology makes greater use of techniques from optimal mass transportation, and so the second and third sections have changed places, and the current third section completely rewritten