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Comptes Rendus Mathématique
Volume 341, n° 4
pages 269-274 (août 2005)
Doi : 10.1016/j.crma.2005.06.018
Received : 17 May 2005 ;  accepted : 2 June 2005
Orbital stability and singularity formation for Vlasov-Poisson systems
Stabilité orbitale et formation de singularité pour des systèmes de Vlasov-Poisson
 

Mohammed Lemou a , Florian Méhats a , Pierre Raphael b
a MIP (UMR CNRS 5640), université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France 
b Département de mathématiques, CNRS et université Paris Sud, bâtiment 425, 91405 Orsay, France 

Abstract

We study the gravitational Vlasov-Poisson system in dimension   and   and consider the problem of nonlinear stability of steady states solutions within the framework of concentration compactness techniques. In dimension   where the problem is subcritical, we prove the orbital stability in the energy space of the polytropes which are ground state type stationary solutions, which improves the already published results for this class. In dimension   where the problem is   critical, polytropic steady states are obtained following Weinstein [M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87 (1983) 567-576] by minimizing a suitable Gagliardo Nirenberg type inequality. Now a striking feature is the existence of a pseudo-conformal symmetry which allows us to derive explicit critical mass finite time blow up solutions. This is to our knowledge the first result of description of a singularity formation in a Vlasov setting. A general mass concentration phenomenon is eventually obtained for finite time blow up solutions. To cite this article: M. Lemou et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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Résumé

Nous considérons le système de Vlasov-Poisson gravitationnel en dimensions   et   et replaçons lʼétude de la stabilité non linéaire des états stationnaires dans le cadre des techniques de concentration compacité. En dimension   où le problème est sous-critique, nous démontrons la stabilité orbitale dans lʼespace dʼénergie des polytropes qui sont des solutions stationnaires de type ground state , ce qui améliore pour cette classe les résultats déjà publiés. En dimension   où le problème est   critique, les polytropes sont obtenus dans la lignée de Weinstein [M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87 (1983) 567-576] comme minimiseurs dʼune inégalité de type Gagliardo-Nirenberg. Un fait remarquable est maintenant lʼexistence dʼune symétrie conforme qui nous permet dʼécrire des solutions explosives explicites de masse critique. Ceci constitue à notre connaissance le premier résultat de description dʼune formation de singularité dans le cadre des équations cinétiques de type Vlasov. Un résultat général de concentration de masse est enfin obtenu pour les solutions explosives. Pour citer cet article : M. Lemou et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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