Small BGK waves and nonlinear Landau damping (higher dimensions) (Q2866927)

The authors study the Vlasov-Poisson system with a fixed ion background and periodic boundary conditions on the space variables in dimensions \(d=2,3\). Firstly, it is shown 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. These BGK waves are one dimensional; the proofs use ODE bifurcations and the dominated convergence theorem; three cases are studied. Higher dimensional BGK waves are shown not to exist. Secondly, it is proved that no nontrivial invariant structures exist 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. The proof uses Fourier transform, principal value and Hardy inequalities. 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 the authors previous results in the one-dimensional case.