A new variational approach to the stability of gravitational systems

Mohammed Lemou; Florian Méhats; Pierre Raphaël

doi:10.1016/j.crma.2009.06.005

Mathematical Problems in Mechanics

A new variational approach to the stability of gravitational systems
[Une nouvelle approche variationnelle pour la stabilité des systèmes gravitationnels]

Présenté par : Philippe G. Ciarlet

Mohammed Lemou ¹ ; Florian Méhats ² ; Pierre Raphaël ³

¹ CNRS and IRMAR, université de Rennes 1, 35042 Rennes, France
² IRMAR, université de Rennes 1, 35042 Rennes, France
³ IMT, université Paul-Sabatier, 31062 Toulouse cedex 9, France

Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 979-984.

Résumé
Abstract

Nous considérons le système de Vlasov–Poisson gravitationnel qui décrit l'état d'un système stellaire soumis à la seule force de gravitation. Une conjecture classique en astrophysique est que les états stationnaires du système qui sont des fonctions strictement décroissantes de leur énergie microscopique sont non-linéairement stables par le flot. Ceci fut démontré dès 1961 par Antonov au niveau linéaire. Depuis, l'application des techniques variationelles de concentration compacité telles qu'introduites par P.-L. Lions in 1984 a permis de prouver la stabilité orbitale de certaines sous-classes de profils de type état fondamental. Dans cette Note, nous proposons une nouvelle approche variationnelle basée sur la minimisation du Hamiltonien sous des contraintes d'équimesurabilité, qui sont préservées par le transport non linéaire. Nous démontrons que tout état stationnaire qui est une fonction décroissante de son énergie microscopique est un minimiseur local, ce qui implique sa stabilité non-linéaire contre des perturbations à symétrie sphérique.

We consider the three-dimensional gravitational Vlasov–Poisson system which describes the mechanical state of a stellar system subject to its own gravity. A well-known conjecture in astrophysics is that the steady state solutions which are nonincreasing functions of their microscopic energy are nonlinearly stable by the flow. This was proved at the linear level by Antonov in 1961. Since then, standard variational techniques based on concentration compactness methods as introduced by P.-L. Lions in 1984 have led to the nonlinear stability of subclasses of stationary solutions of ground state type. In this Note, we propose a new variational approach based on the minimization of the Hamiltonian under equimeasurable constraints, which are conserved by the nonlinear transport flow, and recognize any steady state solution which is a nonincreasing function of its microscopic energy as a local minimizer. The outcome is the proof of its nonlinear stability under radially symmetric perturbations.

Reçu le : 2009-04-19
Accepté le : 2009-05-23
Publié le : 2009-07-02

DOI : 10.1016/j.crma.2009.06.005

Affiliations des auteurs :

Mohammed Lemou ¹ ; Florian Méhats ² ; Pierre Raphaël ³

¹ CNRS and IRMAR, université de Rennes 1, 35042 Rennes, France
² IRMAR, université de Rennes 1, 35042 Rennes, France
³ IMT, université Paul-Sabatier, 31062 Toulouse cedex 9, France

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     title = {A new variational approach to the stability of gravitational systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {979--984},
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     year = {2009},
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     language = {en},
}

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Mohammed Lemou; Florian Méhats; Pierre Raphaël. A new variational approach to the stability of gravitational systems. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 979-984. doi : 10.1016/j.crma.2009.06.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.06.005/

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