Localized minimizers of flat rotating gravitational systems

Fernandez, Javier; Dolbeault, Jean

Fernandez, Javier; Dolbeault, Jean (2008), Localized minimizers of flat rotating gravitational systems, Annales de l'Institut Henri Poincaré. Analyse non linéaire, 25, 6, p. 1043-1071. http://dx.doi.org/10.1016/j.anihpc.2007.01.001

Type

Article accepté pour publication ou publié

Lien vers un document non conservé dans cette base

http://hal.archives-ouvertes.fr/hal-00112165/en/

Date

2008

Nom de la revue

Annales de l'Institut Henri Poincaré. Analyse non linéaire

Volume

Numéro

Éditeur

Elsevier Masson SAS.

Pages

1043-1071

Résumé (EN)

We study a two dimensional system in solid rotation at constant angular velocity driven by a self-consistent three dimensional gravitational field. We prove the existence of non symmetric stationary solutions of such a flat system in the rotating frame as long as the angular velocity does not exceed some critical value which depends on the mass. The solutions can be seen as stationary solutions of a kinetic equation with a relaxation-time collision kernel forcing the convergence to the polytropic gas solutions, or as stationary solutions of an extremely simplified drift-diffusion model, which is derived from the kinetic equation by formally taking a diffusion limit. In both cases, the solutions are critical points of a free energy functional, and can be seen as localized minimizers in an appropriate sense. Symmetry breaking occurs for small angular velocities.

Mots-clés

angular velocity; entropy; diffusion limit; drift-diffusion; Hardy-Littlewood-Sobolev inequality; bounded solutions; critical bifurcation parameter; rotation; mass; symmetry breaking; localized minimizers; radial solutions; minimization; solutions with compact support; Stellar dynamics; gravitation