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Pré-Publication, Document De Travail Année : 2014

Uniqueness and long time asymptotic for the parabolic-parabolic Keller-Segel equation

Résumé

The present paper deals with the parabolic-parabolic Keller-Segel equation in the plane inthe general framework of weak (or ``free energy") solutions associated to an initial datum with finite mass $M< 8\pi$, finite second log-moment and finite entropy. The aim of the paper is twofold:(1) We prove the uniqueness of the ``free energy" solution. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the ``optimal regularity" as well as an estimate of the difference of two possible solutions in the critical $L^{4/3}$ Lebesgue norm similarly as for the $2d$ vorticity Navier-Stokes equation. (2) We prove a radially symmetric and polynomial weighted $L^2$ exponential stability of the self-similar profile in the quasi parabolic-elliptic regime. The proof is based on a perturbation argument which takes advantage of the exponential stability of the self-similar profile for the parabolic-elliptic Keller-Segel equation established in \cite{CamposDolbeault2012,EM}.
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Dates et versions

hal-01011361 , version 1 (23-06-2014)
hal-01011361 , version 2 (01-11-2014)
hal-01011361 , version 3 (22-11-2014)
hal-01011361 , version 4 (21-12-2016)

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Kleber Carrapatoso, Stéphane Mischler. Uniqueness and long time asymptotic for the parabolic-parabolic Keller-Segel equation. 2014. ⟨hal-01011361v3⟩
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