2011/06/22 by Zhiwu Lin, Lin, Zhiwu, Chongchun Zeng +1
Mathematics · Physics and Astronomy · #35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Plasma Physics (physics.plasm-ph) #Spectral Theory in Mathematical Physics #math.AP #msc:35 #physics.plasm-ph
paper · pdf · doi:10.48550/arxiv.1106.4368
23 pages
arxiv created 2011/06/22 · openalex publication_date 2011/06/22 · arxiv updated 2011/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider Vlasov-Poisson system with a fixed ion background and periodic condition on the space variables, in any dimension d≥2. First, we show that for general homogeneous equilibrium and any periodic x-box, within any small neighborhood in the Sobolev space Wx,vs,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 Ws,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 Hx,vs (b>((d-1)/4), s>(3/2)) neighborhood. Since arbitrarilly small BGK waves can also be constructed near any homogeneous equilibria in such weighted Hx,vs (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.