Sobolev regularity in \[5\] and so on. They \[6\] also proved Landau damping for the collisionless Vlasov equation with a class of $L^{1}$ interaction potentials on $\mathbb{R}_{x}^{3}\times\mathbb{R}_{v}^{3}$ for localized disturbances of infinite, inhomogeneous background. Also, there are counterexamples that can be found in \[2,10,13\], showing that there is in general no exponential decay without analyticity and confining. Bedrossian stated one of these counterexamples by proving that the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations on $\mathbb{T}_{x}\times\mathbb{R}_{v}$ cannot, in general, be extended to Sobolev spaces by constructing a sequence of homogeneous background distributions and arbitrarily small perturbations in $H^{s}$ which deviates arbitrary far from free transport for a long time. Lin and Zeng \[10\] also showed that there exist nontrivial traveling wave solutions to the Vlasov equation in Sobolev space $W^{s.p}_{x,v}$ $(p>1,s<1+\frac{1}{p})$ with arbitrary traveling speed. This implies that nonlinear Landau dampping is not true in $W^{s.p}_{x,v}$ $(p>1,s<1+\frac{1}{p})$ space for any homogeneous equilibria and in any period box. In addition, Deng and Masmoudi \[13\] showed the instability of the Couette flow in low Gevrey spaces. In recent years, there are also lots of results on the stability in other setting such as those in 