Smoothing Effect for the Non-linear Vlasov-poisson-fokker-planck System
نویسنده
چکیده
We study a smoothing effect of a non-linear degenerate parabolic problem, the Vlasov-Poisson-Fokker-Planck system, in three dimensions. We prove that a solution gives rise actually to very smooth macroscopic density and force field for positive time. This is obtained by analyzing the effect of the Fokker-Planck kernel on the force term in the Vlasov equation and using classical convolution inequalities. Résumé Nous étudions l’effet régularisant dans un problème parabolique dégénéré nonlinéaire, le système de Vlasov-Poisson-Fokker-Planck, en trois dimensions. Nous montrons qu’une solution faible conduit en fait à une densité macroscopique et un champ de force très réguliers en temps positif. Ceci est obtenu en analysant l’effet du noyau de Fokker-Planck sur le terme de force dans l’équation de Vlasov et en utilisant des inégalités de convolution. Key-words Fokker-Planck operator – Vlasov-Poisson equation – Convolution inequalities – Weighted estimates – Regularizing estimates – Velocity averages A.M.S. Classification numbers: 35B65 35K55 35K65 82A45
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