Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data

Abstract

Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data Annales de l'I. H. P., section C, tome 2, n o 2 (1985), p. 101-118 <http://www.numdam.org/item?id=AIHPC_1985__2_2_101_0> © Gauthier-Villars, 1985, tous droits réservés. L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/locate/anihpc) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data C. BARDOS (1) and P. DEGOND (2)

Streamline diffusion methods for the Vlasov-Poisson equation

“…For a global existence theorem for n = 3 with small initial data we refer to [3], We start by splitting (0.2) in two parts.…”

“…For uniqueness results we refer to [6], [22] and [3] in one, two and three dimensions respectively. Remark 1.1 : For the case n = 1 assuming a periodicity on JC, the Poisson équation (1.1) becomes…”

“…These methods do not need any restriction on the size of the initial data, which have only to be smooth enough. Global in time solutions for the three dimensional problem are given by Bardos and Degond [3], who consider small initial data and use the dispersive effect of the linearized équation to dérive the existence of a global unique solution. A recent survey of a diffusion process approach to existence of a unique solution is given by Wollman [24].…”

Analytic solutions to a strongly nonlinear Vlasov equation

“…Results showing the existence of strong or classical solutions to the Vlasov-Poisson system are now well known, even in dimension 3; among many important contributions, we refer to [1] for small initial data, to [9], [11] and [12] for long time existence with any initial data, and to [2] for existence of analytic solutions. For Vlasov-Maxwell, existence of classical solutions is known in dimension 1 or 2 for long time, see [7] or [8].…”

“…Parmi de nombreuses contributions importantes, nous renvoyons plus spécifiquementà [1] pour des données initiales petites,à [9], [11], [12] pour des données quelconques en temps grand età [2] pour le cas analytique. Dans le cas plus singulier de Vlasov-Maxwell, l'existence de solutions classiques est connue en temps grand en dimension 1 ou 2 (cf [7] ou [8]).…”