Stability for the gravitational Vlasov-Poisson system in dimension two

Jean Dolbeault (1) , Javier Fernández (1) , Óscar Sánchez (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Departamento de Matemática Aplicada

Nous considérons en dimension deux le système de Vlasov-Poisson gravitationnel. Par des méthodes variationnelles, nous prouvons l'existence de solutions stationnaires d'énergie minimale sous une contrainte de type Casimir. La méthode donne aussi un résultat de stabilité de ces solutions pour le problème d'évolution. We consider the two dimensional gravitational Vlasov-Poisson system. Using variational methods, we prove the existence of stationary solutions of minimal energy under a Casimir type constraint. The method also provides a stability criterion of these solutions for the evolution problem.

Mots clés: Vlasov-Poisson system; stellar dynamics; polytropic gas spheres; gravitation; mass; energy; kinetic energy; potential energy; interpolation; Hardy-Littlewood-Sobolev inequality; optimal constants; symmetric nonincreasing rearrangements; Riesz' theorem; bounded solutions; direct variational methods; minimization; scalings; solutions with compact support; minimizers; Lagrange multiplier; Semilinear elliptic equations; Uniqueness; Dirichlet boundary conditions; dynamical stability

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Javier Fernández, Óscar Sánchez. Stability for the gravitational Vlasov-Poisson system in dimension two. Communications in Partial Differential Equations, 2006, 31, pp.1425-1449. ⟨hal-00004799⟩