Landau Damping: Paraproducts and Gevrey Regularity

Abstract

We give a new, simpler, but also and most importantly more general and robust, proof of nonlinear Landau damping on T d in Gevrey− 1 s regularity (s > 1/3) 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.

Landau damping in relativistic plasmas

“…The author also thanks Markus Kunze for many enlightening conversations and for pointing out the paper by Bedrossian, Masmoudi, and Mouhot [1] on which this paper is heavily dependent.…”

“…There are a number of important identities regarding the terms appearing in these norms which are identified in [1]. We collect them in the following two lemmas.…”

“…On of the principal reasons for using · as opposed to | · | is the following elementary estimate from [1]: Lemma 4.3 Let λ(t) be decreasing. Then for all m ∈ N 3 and all σ ∈ R, there is a constant C = C(|m|, λ(0), σ, ν) so that…”

“…The following useful inequalities are established in [1] (and so we refer to this reference for all proofs). The first set are a variant of Young's Inequality (Lemma 3.1 in [1]) while the second set are proven by the paraproduct decomposition above (Lemma 3.3 in [1]).…”

“…Instead of the original analysis of Mouhot and Villani [16], we choose to follow the analysis of Landau damping for VP on the torus given by Bedrossian, Masmoudi, and Mouhot in [1]. Our results are largely identical.…”