Landau damping as $B_{0}\rightarrow0.$

In addition, from the dynamics system of the above order differential system, it is known that $S^{0}_{t,\tau}$ satisfies $$S^{0}_{t_{2},t_{3}}\circ S^{0}_{t_{1},t_{2}}=S^{0}_{t_{1},t_{3}}.$$

To estimate solutions and trajectories of kinetic equations, maybe we have to work on the phase space $\mathbb{T}_{x}^{3}\times\mathbb{R}_{v}^{3}.$ And we also use the following three parameters: $\lambda$(gliding analytic regularity), $\mu$(analytic regularity in $ x $) and $\tau$(time-shift along the free transport semigroup). From $Remark $ 2.2, we know that the linear Vlasov equation has the property of the free transport semigroup. This property is crucial to our analysis. In this paper, one of the cornerstones of our analysis is to compare the solution of the nonlinear case at time $\tau$ with the solution of the linear case.

Now we start to introduce the very important tools in our paper. These are time-shift pure and hybrid analytic norms. They are similar with those in the paper \[23\] written by Mouhot and Villani.

First, we introduce some notations. We denote $\mathbb{T}^{3}=\mathbb{R}^{3}/\mathbb{Z}^{3}.$ For function $f(x'_{\tau},v'_{\tau}),$ we define the Fourier transform $\hat{f}(k,\eta),$ where $(k,\eta)\in\mathbb{Z}^{3}\times\mathbb{R}^{3},$ via $$\hat{f}(k,\eta)=\int_{\mathbb{T}^{3}\times\mathbb{R}^{3}}e^{-2i\pi x'_{\tau}\cdot k}e^{-2i\pi v'_{\tau}\cdot \eta}f(x'_{\tau},v'_{\tau})dx'_{\tau}dv'_{\tau},$$ $$\tilde{f}(\omega,k,\eta)=\int_{\mathbb{R}^{+}}e^{2i\pi t\cdot\omega}\int_{\mathbb{T}^{3}\times\mathbb{R}^{3}}
e^{-2i\pi x'_{\tau}\cdot k}e^{-2i\pi v'_{\tau}\cdot \eta}f(x'_{\tau},v'_{\tau})dx'_{\tau}dv'_{\tau}dt.$$

We also write

$$k=(k_{1},k_{2},k_{3})=(k_{\perp}\cos\varphi,k_{\perp}\sin\varphi,k_{3}),\quad
\eta=(\eta_{1},\eta_{2},\eta_{3})=(\eta_{\perp}\cos\gamma,\eta_{\perp}\sin\gamma,\eta_{3}).$$

Now we define some notations $$\tilde{f}=\tilde{f}(\omega,k,\eta),\quad\hat{f}=\hat{f}(t,k,\eta),\quad\tilde{\rho}=\tilde{\rho}(\omega,k),\quad\hat{\rho}=\hat{\rho}(t,k).$$

(Hybrid analytic 