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Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (Large time behavior and steady states)

Dolbeault, Jean (1999), Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (Large time behavior and steady states), Journal de mathématiques pures et appliquées, 78, 2, p. 121-157. http://dx.doi.org/10.1016/S0021-7824(01)80006-4

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Type
Article accepté pour publication ou publié
Date
1999
Nom de la revue
Journal de mathématiques pures et appliquées
Volume
78
Numéro
2
Éditeur
Elsevier
Pages
121-157
Identifiant publication
Métadonnées
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Auteur(s)
Dolbeault, Jean cc
Résumé (EN)
This paper is devoted to the characterization of external electrostatic potentials for which the Vlasov-Poisson-Fokker-Planck system satisfies one of the following properties: (i) the system admits stationary solutions, (ii) any solution to the evolution problem converges to a stationary solution, or, equivalently, no mass vanishes for large times, (iii) the free energy is bounded from below, We give conditions under which these different notions of confinement are equivalent.
Mots-clés
Kinetic equations; Fokker-Planck equation; Vlasov equation; Poisson equation; Poisson-Boltzmann-Emden equation; Confinement; Stationary solutions; Long time asymptotics; Steady states; Renormalized solutions; A priori estimates; Entropy; Free energy; Configurational free energy; Jensen's inequality; Marcinkiewicz spaces; Semilinear elliptic equations