On Landau damping

2009/04/30 by Clément Mouhot, Cédric Villani · 459 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Classical mechanics #Computer science #Coulomb #Exponential function #Gas Dynamics and Kinetic Theory #Landau damping #Mathematical analysis #Mathematics #Nonlinear system #Physics #Plasma #Quantum mechanics #Quantum, superfluid, helium dynamics #Stability (learning theory) #Statistical physics #Vlasov equation #astro-ph.GA #cond-mat.stat-mech #math.AP #msc:35B35 #msc:35Q60 #msc:62E20 #msc:70F15 #msc:82C99 #msc:82D10 #msc:85A05 #physics.plasm-ph

paper · pdf · doi:10.1007/s11511-011-0068-9

published in Acta Mathematica 207(1), 29-201 (Mittag-Leffler Institute) · News: (1) the main result now covers Coulomb and Newton potentials, and (2) some classes of Gevrey data; (3) as a corollary this implies new results of stability of homogeneous nonmonotone equilibria for the gravitational Vlasov-Poisson equation

arxiv created 2009/12/06 · openalex publication_date 2011/01/01 · arxiv updated 2012/02/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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.

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