Orbital stability and singularity formation for Vlasov–Poisson systems

Mohammed Lemou; Florian Méhats; Pierre Raphael

doi:10.1016/j.crma.2005.06.018

Mathematical Problems in Mechanics

Orbital stability and singularity formation for Vlasov–Poisson systems

Presented by: Philippe G. Ciarlet

Mohammed Lemou ¹; Florian Méhats ¹; Pierre Raphael ²

¹ MIP (UMR CNRS 5640), université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France
² Département de mathématiques, CNRS et université Paris Sud, bâtiment 425, 91405 Orsay, France

Comptes Rendus. Mathématique, Volume 341 (2005) no. 4, pp. 269-274.

Abstract
Résumé

We study the gravitational Vlasov–Poisson system in dimension $N = 3$ and $N = 4$ and consider the problem of nonlinear stability of steady states solutions within the framework of concentration compactness techniques. In dimension $N = 3$ 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 $N = 4$ where the problem is $L^{1}$ 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.

Nous considérons le système de Vlasov–Poisson gravitationnel en dimensions $N = 3$ et $N = 4$ et replaçons l'étude de la stabilité non linéaire des états stationnaires dans le cadre des techniques de concentration compacité. En dimension $N = 3$ où le problème est sous-critique, nous démontrons la stabilité orbitale dans l'espace d'énergie des polytropes qui sont des solutions stationnaires de type ground state, ce qui améliore pour cette classe les résultats déjà publiés. En dimension $N = 4$ où le problème est $L^{1}$ 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 d'une inégalité de type Gagliardo–Nirenberg. Un fait remarquable est maintenant l'existence d'une symétrie conforme qui nous permet d'écrire des solutions explosives explicites de masse critique. Ceci constitue à notre connaissance le premier résultat de description d'une 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.

Received: 2005-05-17
Accepted: 2005-06-02
Published online: 2005-08-03

DOI: 10.1016/j.crma.2005.06.018

Author's affiliations:

Mohammed Lemou ¹; Florian Méhats ¹; Pierre Raphael ²

¹ MIP (UMR CNRS 5640), université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France
² Département de mathématiques, CNRS et université Paris Sud, bâtiment 425, 91405 Orsay, France

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     author = {Mohammed Lemou and Florian M\'ehats and Pierre Raphael},
     title = {Orbital stability and singularity formation for {Vlasov{\textendash}Poisson} systems},
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     pages = {269--274},
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     language = {en},
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Mohammed Lemou; Florian Méhats; Pierre Raphael. Orbital stability and singularity formation for Vlasov–Poisson systems. Comptes Rendus. Mathématique, Volume 341 (2005) no. 4, pp. 269-274. doi : 10.1016/j.crma.2005.06.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.018/

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