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On Landau damping

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Date
2011
Link to item file
http://hal.archives-ouvertes.fr/hal-00376547/en/
Dewey
Probabilités et mathématiques appliquées
Sujet
galactic dynamics; plasma physics; Vlasov-Poisson equation; Landau damping
Journal issue
Acta Mathematica
Volume
207
Number
1
Publication date
2011
Article pages
29-201
Publisher
Springer
DOI
http://dx.doi.org/10.1007/s11511-011-0068-9
URI
https://basepub.dauphine.fr/handle/123456789/3022
Collections
  • CEREMADE : Publications
Metadata
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Author
Villani, Cédric
Mouhot, Clément
Type
Article accepté pour publication ou publié
Abstract (EN)
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.

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