On Landau damping

By Clément Mouhot, Cédric Villani. 2009

688 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/25362af74db860dd7d7925ba2141d6f443e7d3de

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Citations in the corpus

Modeling the propagation characteristics of chorus using CRRES suprathermal electron fluxes (2007, methodology): “as has been done previously [Bell et al., 2002], then combining (1)–(5) gives: 219”
Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow (2014, background): “[14, 50, 16, 3, 45, 4, 9] and the references therein for some discussion about the connections between inviscid damping and Landau damping and the more general concept of phase mixing.”
On the stability threshold for the 3D Couette flow in Sobolev regularity (2015, methodology): “6 for a more in-depth discussion of the relationship between this work and previous related infinite regularity results in [38, 10, 9, 11, 54, 7, 8].”
Linear inviscid damping and enhanced dissipation for the Kolmogorov flow (2017, background): “The first part of this paper is devoted to studying the linear inviscid damping for the linearized Euler equations around the Kolmogorov flow (− cos y, 0), which take in the vorticity formulation: { ∂tω + Lω = 0, ω|t=0 = ω0(x, y),(1.2) where ω = ∂xV 2 − ∂yV 1 is the vorticity and L = u(y)∂x + u′′(y)”
Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations (2013, methodology): “The gap between the linear and nonlinear theory of Landau damping was only bridged recently by the ground-breaking work of Mouhot and Villani, who showed that the phase-mixing indeed persists in the nonlinear Vlasov equations for small perturbations [68] (see also [18, 44]).”
Linear Inviscid Damping for a Class of Monotone Shear Flow in Sobolev Spaces (2015, background): “Motivated by [18], Bedrossian and Masmoudi also proved the nonlinear inviscid damping of twodimensional Euler equations around Couette flow in an infinite channel for the perturbation in the Gevrey class.”
Small Data Solutions of the Vlasov-Poisson System and the Vector Field Method (2015, background): “Finally, let us mention the celebrated work [16] on Landau damping concerning the stability of stationnary solutions to (1)-(3) with periodic initial data.”
A review of the mean field limits for Vlasov equations (2014, background): “This famous Landau damping, first surmised in [113], was eventually proved in [129].”
Losses in plasmonics: from mitigating energy dissipation to embracing loss-enabled functionalities (2017, background): “This in turn reduces the maximum local electric field intensity that can be achieved on the nanoparticle surface [38].”
Factorization for non-symmetric operators and exponential H-theorem (2010, result): “al L1 case at the price of assuming more initial regularity on the velocity variable. It shares some similarity with the results in [20]. The proof makes use of the \gliding norms" introduced in [81]. Proof of Lemma 4.17. Introducing the dierential operator (4.47) D t:= tr x+ r v; we observe th”
Quantum Hydrodynamics (2013, methodology): “A related approach to derive the hydrodynamics equations is based on the Dawson (classical) multistream model [68] which is extended to the quantum case [69] by considering a statistical mixture of K pure states representing K “streams” of particles each characterized by the same velocity.”
Asymptotic Stability of Equilibria for Screened Vlasov–Poisson Systems via Pointwise Dispersive Estimates (2019, methodology): “Though dispersion on the whole space is used at some crucial points in order to close the estimates in finite regularity, the approach is much more related to the one of [4] for the torus than that of Bardos-Degond (it is actually dispersive properties of the free transport in the frequency space th”
Landau Damping: Paraproducts and Gevrey Regularity (2013, background): “See [30,31,67,84] for more discussion about what can happen without confinement.”
Well-posedness for the Prandtl system without analyticity or monotonicity (2013, background): “Hence, stability in the Gevrey context has more physical insight that stability in the analytic context: see [21, 2] for similar issues.”