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On Landau damping. (English) Zbl 1239.82017
Acta Math. 207, No. 1, 29-201 (2011); correction ibid. 207, No. 2, 391 (2011).
Going beyond the linearised study has been a long-standing problem in the theory of Landau damping. In the present paper, the non-linear version of Landau damping is studied for arbitrarily large times for attractive and repulsive interactions of any regularity. This leads the authors to a distinctive mathematical theory of Landau damping with its own functional spaces and inequalities.
The article provides first an introduction to Landau damping, including historical comments and a review of the existing literature. Then, the authors’ theorem on non-linear Landau damping is stated. The main ingredients of the new theory, which are the Fourier transform to quantify analytic regularity and to implement phase mixing; the introduction of a time-shift parameter to keep memory of the initial time; flexible analytic norms behaving well with respect to composition; a control of the deflection of trajectories induced by the force field; new functional inequalities of bilinear type; a new analysis of the time response aimed at controlling self-induced plasma echoes; and a Newtonian iteration scheme, are pointed out. Further, after a complete treatment of linear Landau damping, the spaces of analytic functions are defined which are needed in the following.
Next, exponential Landau damping for the linearised Vlasov equation is established 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 further analysis involves new families of analytic norms having up to five parameters. After their presentation, a number of functional inequalities (i.e., gradient inequality, inversion inequality, Sobolev corrections, individual mode estimates) which are crucial in the subsequent analysis, are derived. As a first application of the new approach, linear Landau damping is revisited.
Then, four types of new estimates are established – deflection estimates (in hybrid norm), short-term and long-term regularity extortion, and echo control. This is the physically new material of the article. The short-term considerations lead to a twist on the popular view on Landau damping according to which the wave gives energy to the particles that it interacts with. Here, the wave gains regularity from the background, and regularity is converted into decay. Using the estimates a theorem of the authors about the growth control via integral inequalities is derived and proven.
Furthermore, the Newton algorithm is adapted to the setting of the non-linear Vlasov equation. Some iterative estimations (local-in-time, global-in-time) are performed during the course of the scheme. In particular, a technical refinement is presented allowing to handle Coulomb-Newton interactions. Using these estimates the main theorem of the authors on non-linear Landau damping is proven. Limiting cases are included. As a side result, the stability of homogeneous equilibria of the nonlinear Vlasov equation is established under sharp assumptions. The authors point out the strong analogy with KAM (Kolmogorov-Arnold-Mozer) theory. Further, an extension to non-analytic Gevrey distribution functions is presented. Some counterexamples and asymptotic expansions are studied. The paper is concluded with some general comments on the physical implication of Landau damping.
A short summary of the results of the study and methods of proofs can be found in the authors’ expository paper [“Landau damping”, J. Math. Phys. 51, No. 1, Paper No. 015204, 7 p. (2010; Zbl 1247.82081)].

##### MSC:
 82D10 Statistical mechanical studies of plasmas 35Q83 Vlasov equations
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##### References:
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