Uniqueness and long time asymptotic for the parabolic-parabolic Keller-Segel equation
Carrapatoso, Kleber; Mischler, Stéphane (2017), Uniqueness and long time asymptotic for the parabolic-parabolic Keller-Segel equation, Communications in Partial Differential Equations, 42, 2, p. 291-345. 10.1080/03605302.2017.1280682
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Article accepté pour publication ou publiéLien vers un document non conservé dans cette base
https://hal.archives-ouvertes.fr/hal-01011361Date
2017Nom de la revue
Communications in Partial Differential EquationsVolume
42Numéro
2Éditeur
Marcel Dekker
Ville d’édition
Paris
Pages
291-345
Identifiant publication
Métadonnées
Afficher la notice complèteRésumé (EN)
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<8π, 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 L4/3 Lebesgue norm similarly as for the 2d vorticity Navier-Stokes equation. (2) We prove a radially symmetric and polynomial weighted L2 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}.Mots-clés
Keller-Segel system; long-time behavior; stability; regularization; uniqueness; self-similar variablesPublications associées
Affichage des éléments liés par titre et auteur.
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Fernandez , Giani Egana; Mischler, Stéphane (2016) Article accepté pour publication ou publié
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Campos Serrano, Juan; Dolbeault, Jean (2014) Article accepté pour publication ou publié
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Fernandez, Javier; Escobedo, Miguel; Dolbeault, Jean; Blanchet, Adrien (2010) Article accepté pour publication ou publié
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Carrapatoso, Kleber; Mischler, Stéphane (2017) Article accepté pour publication ou publié
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Lafleche, Laurent; Salem, Samir (2018) Document de travail / Working paper