On Landau damping

Clément Mouhot (1, 2) , Cédric Villani (3)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 DAMTP - Department of Applied Mathematics and Theoretical Physics
3 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of nonlinear echoes; sharp scattering estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the nonlinear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications.

Mots clés: Landau damping; plasma physics; galactic dynamics; Vlasov-Poisson equation

Domaines: Equations aux dérivées partielles [math.AP]; Physique des plasmas [physics.plasm-ph]; Mécanique statistique [cond-mat.stat-mech]; Astrophysique galactique [astro-ph.GA]; Astrophysique galactique [astro-ph.GA]

Clément Mouhot, Cédric Villani. On Landau damping. Acta Mathematica, 2011, 207 (1), pp.29-201. ⟨10.1007/s11511-011-0068-9⟩. ⟨hal-00376547v5⟩