as the improved Newton iteration.

Indeed, there are many papers that contribute to Landau damping. Here we only list some results. Bedrossian, Masmoudi, and Mouhot \[4\] provided a new, simple and short proof of nonlinear Landau damping on $\mathbb{T}^{d}$ in only electric field case that nearly obtains the “ critical " Gevrey-$\frac{1}{s}$ regularity predicted in \[23\]. Although their proofs have lots of the same ingredients as the proof in \[23\] from a physical point of view, at a mathematical level, the two proofs are quite different, they “mod out" by the characteristics of free transport and work in the coordinates $z=x-vt$ with $(t,x,v)\rightarrow(t,z,v).$ The evolution equation (0.1) $ (B=0)$ becomes $$\partial_{t}f+E(t,z+vt)\cdot (\nabla_{v}-t\nabla_{z})f+E(t,z+vt)
 \cdot\nabla_{v}f^{0}=0.$$ From this formula, it is easy to see the phase mixing mechanism. And this coordinate shift is related to the notion of “gliding regularity" used in \[23\]. One of the main ingredients of their proof is to split nonlinear terms into the transport structure term and “reaction" term in \[23\] by using paradifferential calculus . Bedrossian, Masmoudi \[3\] also used this method to prove the inviscid damping and asymptotic stability of 2-D Euler equations and later also proved the stability threshold for the 3D Couette flow in 