Title :

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

Author(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]

Publication date :

2024-08-29

English keyword(s) :

Vlasov-Fokker-Planck
Hypocoercivity
Hypoellipticity
Convolution operator
Coulomb interactions

HAL domain(s) :

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

English abstract : [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 ...
Show more >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.Show less >

Language :

Anglais

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL