Anisotropic Boltzmann-Gibbs dynamics of strongly magnetized Vlasov-Fokker-Planck equations

Maxime Herda (1, 2) , Luis Miguel Miguel Rodrigues (3)

1 RAPSODI - Reliable numerical approximations of dissipative systems
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 IRMAR - Institut de Recherche Mathématique de Rennes

We consider various sets of Vlasov-Fokker-Planck equations modeling the dynamics of charged particles in a plasma under the effect of a strong magnetic field. For each of them in a regime where the strength of the magnetic field is effectively stronger than that of collisions we first formally derive asymptotically reduced models. In this regime, strong anisotropic phenomena occur; while equilibrium along magnetic field lines is asymptotically reached our asymptotic models capture a non trivial dynamics in the perpendicular directions. We do check that in any case the obtained asymptotic model defines a well-posed dynamical system and when self consistent electric fields are neglected we provide a rigorous mathematical justification of the formally derived systems. In this last step we provide a complete control on solutions by developing anisotropic hypocoercive estimates.

Mots clés: plasma physics; Gibbs equilibrium; Maxwell-Boltzmann distribution; Fokker-Planck operator; Vlasov equation; disparate mass; hypocoercivity; strong magnetic field; gyrokinetic theory

Domaines: Equations aux dérivées partielles [math.AP]; Physique des plasmas [physics.plasm-ph]

Maxime Herda, Luis Miguel Miguel Rodrigues. Anisotropic Boltzmann-Gibbs dynamics of strongly magnetized Vlasov-Fokker-Planck equations. Kinetic and Related Models , 2019, 12 (3), pp.593-636. ⟨10.3934/krm.2019024⟩. ⟨hal-01382854v2⟩