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Article Dans Une Revue Comptes Rendus. Mathématique Année : 2009

A new variational approach to the stability of gravitational systems

Résumé

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 1983 have led to the nonlinear stability of subclasses of stationary solutions of ground state type. In this paper, 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 anisotropic steady state solution which is a decreasing function of its microscopic energy as a local minimizer. The outcome is the proof of its nonlinear stability of under radially symmetric perturbations.

Dates et versions

hal-00378457 , version 1 (24-04-2009)

Identifiants

Citer

Mohammed Lemou, Florian Méhats, Pierre Raphaël. A new variational approach to the stability of gravitational systems. Comptes Rendus. Mathématique, 2009, 347 (15-16), pp.979-984. ⟨10.1016/j.crma.2009.06.005⟩. ⟨hal-00378457⟩
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