Mathematics – Analysis of PDEs
Scientific paper
2011-06-22
Mathematics
Analysis of PDEs
23 pages
Scientific paper
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.
Lin Zhiwu
Zeng Chongchun
No associations
LandOfFree
Samll BGK waves and nonlinear Landau damping (higher dimensions) does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Samll BGK waves and nonlinear Landau damping (higher dimensions), we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Samll BGK waves and nonlinear Landau damping (higher dimensions) will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-205290