Acta Mathematica

On Landau damping

Clément Mouhot and Cédric Villani

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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.

Dedication

Dedicated to Vladimir Arnold and Carlo Cercignani.

Note

AMS Subject Classification: 82C99 (85A05, 82D10).

Article information

Source
Acta Math. Volume 207, Number 1 (2011), 29-201.

Dates
Received: 10 December 2009
Revised: 10 July 2011
First available in Project Euclid: 31 January 2017

Permanent link to this document
http://projecteuclid.org/euclid.acta/1485892568

Digital Object Identifier
doi:10.1007/s11511-011-0068-9

Mathematical Reviews number (MathSciNet)
MR2863910

Zentralblatt MATH identifier
1239.82017

Rights
2011 © Institut Mittag-Leffler

Citation

Mouhot, Clément; Villani, Cédric. On Landau damping. Acta Math. 207 (2011), no. 1, 29--201. doi:10.1007/s11511-011-0068-9. http://projecteuclid.org/euclid.acta/1485892568.


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