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. © 2011 Institut Mittag-Leffler.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: This project started from an unlikely conjunction of discussions of the authors withvarious people, most notably Yan Guo, Dong Li, Freddy Bouchet and Etienne Ghys.We also got crucial inspiration from the books  and  by James Binney and ScottTremaine; and  by Serge Alinhac and Patrick Gerard. Warm thanks to Julien Barre,Jean Dolbeault, Thierry Gallay, Stephen Gustafson, Gregory Hammett, Donald Lynden-Bell, Michael Sigal, Eric Sere and especially Michael Kiessling for useful exchanges andreferences; and to Francis Filbet and Irene Gamba for providing numerical simulations.We are also grateful to Patrick Bernard, Freddy Bouchet, Emanuele Caglioti, YvesElskens, Yan Guo, Zhiwu Lin, Michael Loss, Peter Markowich, Govind Menon, YannOllivier, Mario Pulvirenti, Jeff Rauch, Igor Rodnianski, Peter Smereka, Yoshio Sone,Tom Spencer, and the team of the Princeton Plasma Physics Laboratory for further con-structive discussions about our results. Finally, we acknowledge the generous hospitalityof several institutions: Brown University, where the first author was introduced to Lan-dau damping by Yan Guo in early 2005; the Institute for Advanced Study in Princeton,who offered the second author a serene atmosphere of work and concentration during thebest part of the preparation of this work; Cambridge University, who provided repeatedhospitality to the first author thanks to the Award No. KUK-I1-007-43, funded by theKing Abdullah University of Science and Technology; and the University of Michigan,where conversations with Jeff Rauch and others triggered a significant improvement ofour results.Our deep thanks go to the referees for their careful examination of the manuscript.We dedicate this paper to two great scientists who passed away during the elaboration ofour work. The first one is Carlo Cercignani, one of the leaders of kinetic theory, author ofseveral masterful treatises on the Boltzmann equation, and a long-time personal friend ofthe second author. The other one is Vladimir Arnold, a mathematician of extraordinaryinsight and influence; in this paper we shall uncover a tight link between Landau dampingand the theory of perturbation of completely integrable Hamiltonian systems, to whichArnold has made major contributions.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.