Titre :

Well-posedness and long-time behavior for self-consistent Vlasov-Fokker-Planck equations with general potentials

Auteur(s) :

Gervais, Pierre [Auteur]
Reliable numerical approximations of dissipative systems [RAPSODI]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Herda, Maxime [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Reliable numerical approximations of dissipative systems [RAPSODI]

Date de publication :

2024-08-29

Mot(s)-clé(s) en anglais :

Vlasov-Fokker-Planck
Hypocoercivity
Hypoellipticity
Convolution operator
Coulomb interactions

Discipline(s) HAL :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

Résumé en anglais : [en]

We study the well-posedness, steady states and long time behavior of solutions to Vlasov-Fokker-Planck equation with external confinement potential and nonlinear self-consistent interactions. Our analysis introduces newly ...
Lire la suite >We study the well-posedness, steady states and long time behavior of solutions to Vlasov-Fokker-Planck equation with external confinement potential and nonlinear self-consistent interactions. Our analysis introduces newly characterized conditions on the interaction kernel that ensure the local asymptotic stability of the unique steady state. Compared to previous works on this topic, our results allow for large, singular and non-symmetric interactions. As a corollary of our main results, we show exponential decay of solutions to the Vlasov-Poisson-Fokker-Planck equation in dimension 3, for low regularity initial data. In the repulsive case, the result holds in strongly nonlinear regimes (i.e. for arbitrarily small Debye length). Our techniques rely on the design of new Lyapunov functionals based on hypocoercivity and hypoellipticity theories. We use norms which include part of the interaction kernel, and carefully mix "macroscopic quantities based"-hypocoercivity with "commutators based"-hypocoercivity.Lire moins >

Langue :

Anglais

Projet ANR :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL