Samll BGK waves and nonlinear Landau damping (higher dimensions)

TL;DR

This paper demonstrates the existence of small BGK waves in higher dimensions for the Vlasov-Poisson system, showing nonlinear Landau damping fails in certain Sobolev spaces and identifying critical regularity for invariant structures near stable equilibria.

Contribution

It extends previous one-dimensional results to higher dimensions, establishing the existence of BGK waves and the critical regularity threshold for invariant structures in the Vlasov-Poisson system.

Findings

Small BGK waves exist near homogeneous equilibria in Sobolev spaces with s<1+(1/p).

Nonlinear Landau damping does not hold in these Sobolev spaces.

Critical regularity for invariant structures is identified at s=3/2.

Abstract

Consider Vlasov-Poisson system with a fixed ion background and periodic condition on the space variables, in any dimension d\geq2. First, we show that for general homogeneous equilibrium and any periodic x-box, within any small neighborhood in the Sobolev space W_{x,v}^{s,p} (p>1,s<1+(1/p)) of the steady distribution function, there exist nontrivial travelling wave solutions (BGK waves) with arbitrary traveling speed. This implies that nonlinear Landau damping is not true in W^{s,p}(s<1+(1/p)) space for any homogeneous equilibria and in any period box. The BGK waves constructed are one dimensional, that is, depending only on one space variable. Higher dimensional BGK waves are shown to not exist. Second, for homogeneous equilibria satisfying Penrose's linear stability condition, we prove that there exist no nontrivial invariant structures in the (1+|v|^{2})^{b}-weighted H_{x,v}^{s} (b>((d-1)/4), s>(3/2)) neighborhood. Since arbitrarilly small BGK waves can also be constructed near any homogeneous equilibria in such weighted H_{x,v}^{s} (s<(3/2)) norm, this shows that s=(3/2) is the critical regularity for the existence of nontrivial invariant structures near stable homogeneous equilibria. These generalize our previous results in the one dimensional case.