Landau Damping: Paraproducts and Gevrey Regularity

By Jacob Bedrossian, Nader Masmoudi, Clément Mouhot. Annals of PDE, 2013

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We give a new, simpler, but also and most importantly more general and robust, proof of nonlinear Landau damping on Td\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^d$$\end{document} in Gevrey-1s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\frac{1}{s}$$\end{document} regularity (s>1/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s > 1/3$$\end{document}) which matches the regularity requirement predicted by the formal analysis of Mouhot and Villani [67]. Our proof combines in a novel way ideas from the original proof of Landau damping Mouhot and Villani [67] and the proof of inviscid damping in 2D Euler Bedrossian and Masmoudi [10]. As in Bedrossian and Masmoudi [10], we use paraproduct decompositions and controlled regularity loss along time to replace the Newton iteration scheme of Mouhot and Villani [67]. We perform time-response estimates adapted from Mouhot and Villani [67] to control the plasma echoes and couple them to energy estimates on the distribution function in the style of the work Bedrossian and Masmoudi [10]. We believe the work is an important step forward in developing a systematic theory of phase mixing in infinite dimensional Hamiltonian systems.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7a4be34153757d786e5160ebe5103eeb861d3f68

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Citations in the corpus

On the stability threshold for the 3D Couette flow in Sobolev regularity (2015, methodology): “1, we need to use methods which differ significantly from those used in the infinite regularity works [38, 10, 9, 11, 54, 7, 8].”
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”
Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow (2014, background): “The term ‘inviscid damping’ appeared after it was noticed to be the hydrodynamic analogue of Landau damping [39, 49, 18, 32, 45, 10] in the kinetic description of plasmas which are sufficiently collisionless (the analogue of inviscid) to be well-described by the Vlasov equations.”
Linear Inviscid Damping for a Class of Monotone Shear Flow in Sobolev Spaces (2015, background): “ is a very diﬃcult problem to extend linear damping to nonlinear damping. Mouhot and Villani [18] made a breakthrough and proved nonlinear Landau damping for the perturbation in Gevrey class(see also [4]). Motivated by [18], Bedrossian and Masmoudi also proved the nonlinear inviscid damping of 2D Eu”