IUMJ

Title: Small BGK waves and nonlinear Landau damping (higher dimensions)

Authors: Zhiwu Lin and Chongchun Zeng

Issue: Volume 61 (2012), Issue 5, 1711-1735

Abstract:

Consider the Vlasov-Poisson system with a fixed ion background and periodic boundary conditions on the space variables, in dimension $d=2,3$. 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 traveling 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, we prove that there exist no nontrivial invariant structures in some neighborhood of stable homogeneous equilibria, in the $(1+|v|^2)^b$-weighted $H_{x,v}^s$ ($b>(d-1)/4$, $s>\frac{3}{2}$) space. Since arbitrarily small BGK waves can also be constructed near any homogeneous equilibria in such weighted $H_{x,v}^s$ ($s<\frac{3}{2}$) norm, this shows that $s=\frac{3}{2}$ 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.