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Acta Mathematica

, Volume 207, Issue 1, pp 29–201 | Cite as

On Landau damping

  • Clément Mouhot
  • Cédric Villani
Article

Abstract

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 non-linear echoes; sharp “deflection” 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 non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.

Keywords

Poisson Equation Vlasov Equation Analytic Norm Exponential Moment Analytic Regularity 
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© Institut Mittag-Leffler 2011

Authors and Affiliations

  1. 1.University of Cambridge, DPMMS, Centre for Mathematical SciencesCambridgeUK
  2. 2.Institut Henri Poincaré & Université de Lyon, Institut Camille Jordan, Université Claude BernardVilleurbanne CedexFrance

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