some constants $\lambda_{0},\alpha_{0},C^{0}>0,\delta_{0}>0;$

-   In addition, $(\mathbf{PSC})$ holds,

    $(\mathbf{PSC}):$ for any component velocity in the $\hat{z}$ direction $v_{3}\in\mathbb{R},$ there exists some positive constant $v_{Te}$ such that if $v_{3}=\frac{\omega}{k_{3}} \textmd{ when } k_{3}\neq0; \textmd{ or } k_{3}=0$ where $\omega,k$ are frequencies of time and space $t,x,$ respectively, then $|v_{3}|\gg v_{Te}.$

Then for any $\lambda'_{0}<\lambda_{0},$ we have $$\begin{aligned}
         &\sup_{t\geq 0}\|\rho(t,\cdot)\|_{\mathcal{F}^{\lambda'_{0}t+\mu_{0}}}\leq C(C^{0},\lambda_{0},\lambda'_{0},\mu)\delta_{0},\notag\\
         &\sup_{t\geq0}\|f-f^{0}\|_{\mathcal{Z}^{\lambda''_{0},\mu;1}_{t}}\leq C(C^{0},\lambda_{0},\lambda'_{0},\mu)\delta_{0}.
         \end{aligned}$$

The revisited proof of Lemma 1.1

$Proof.$ First $\omega\neq0,k_{3}\neq0,$ we have

$$\tilde{\rho}(\omega,k)=\int_{\mathbb{R}^{+}}\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ix\cdot k}f_{0}(x'(0,x,v),v'(0,x,v))dvdxdt$$ $$+\frac{q}{m}\tilde{\rho}(\omega,k)\frac{1}{\omega}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}
e^{2\pi it\omega}e^{-2\pi ik\cdot(v'_{t1}\frac{\sin\Omega t}{t},v'_{t2}\frac{\sin\Omega t}{t},v_{3}t)}
(k_{3}v'_{3})(\widehat{W}_{1}\partial_{v'_{1}}f^{0}-\widehat{W}_{2}\partial_{v'_{2}}f^{0})dvdt$$ $$+\frac{q}{m}\tilde{\rho}(\omega,k)\frac{1}{\omega}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ik\cdot(v'_{t1}\frac{\sin\Omega t}{t},v'_{t2}\frac{\sin\Omega t}{t},v_{3}t)}(k_{1}\widehat{W}_{2}-k_{2}\widehat{W}_{1})\cdot(v_{2}\partial_{v'_{1}}f^{0}-v_{1}\partial_{v'_{2}}f^{0})dvdt$$ $$\begin{aligned}
&-\frac{q}{m}\tilde{\rho}(\omega,k)\frac{1}{\omega}\int_{\mathbb{R}^{+}}
\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ik\cdot(v'_{t1}\frac{\sin\Omega t}{t},v'_{t2}\frac{\sin\Omega t}{t},v_{3}t)}\cdot(v_{1}\widehat{W}_{1}+v_{2}\widehat{W}_{2})k_{3}\partial_{v'_{3}}f^{0}
dvdt\notag\\
&-\frac{q}{m}\tilde{\rho}(\omega,k)\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ik\cdot(v'_{t1}\frac{\sin\Omega t}{t},v'_{t2}\frac{\sin\Omega t}{t},v_{3}t)}(\widehat{W}_{1}\partial_{v'_{1}}f^{0}+\widehat{W}_{2}\partial_{v'_{2}}f^{0})dvdt.\notag\\\end{aligned}$$

Let $$\begin{aligned}
&\tilde{\mathcal{L}}(\omega,k)=\frac{q}{m}\frac{1}{\omega}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ik\cdot(v'_{t1}\frac{\sin\Omega t}{t},v'_{t2}\frac{\sin\Omega t}{t},v_{3}t)}(k_{3}v_{3})(\widehat{W}_{1}\partial_{v'_{1}}f^{0}-\widehat{W}_{2}\partial_{v'_{2}}f^{0})dvdt\notag\\
&-\frac{q}{m}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ik\cdot(v'_{t1}\frac{\sin\Omega t}{t},v'_{t2}\frac{\sin\Omega t}{t},v_{3}t)}
(\widehat{W}_{1}\partial_{v'_{1}}f^{0}+\widehat{W}_{2}\partial_{v'_{2}}f^{0})\cdot dvdt\notag\\
&-\frac{q}{m}\frac{1}{\omega}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ik\cdot(v'_{t1}\frac{\sin\Omega t}{t},v'_{t2}\frac{\sin\Omega t}{t},v_{3}t)}(v_{1}\widehat{W}_{1}+v_{2}\widehat{W}_{2})k_{3}\partial_{v'_{3}}f^{0}
dvdt\notag\\
&+\frac{q}{m}\frac{1}{\omega}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{-2\pi ik\cdot(v'_{t1}\frac{\sin\Omega t}{t},v'_{t2}\frac{\sin\Omega t}{t},v_{3}t)}(k_{1}\widehat{W}_{2}-k_{2}\widehat{W}_{1})(v_{2}\partial_{v'_{1}}f^{0}
-v_{1}\partial_{v'_{2}}f^{0})dvdt,\notag\\\end{aligned}$$ hence $$\begin{aligned}
\tilde{\rho}(\omega,k)=\tilde{A}(\omega,k)+\tilde{\rho}(\omega,k)\tilde{\mathcal{L}}(\omega,k).\end{aligned}$$ Then taking the inverse Fourier transform in time $ t, $ we get $\hat{\rho}(t,k)=\hat{A}(t,k)+\hat{\rho}(t,k)\ast\hat{\mathcal{L}}(t,k),$ and

$$\begin{aligned}
&e^{2\pi\lambda'_{0} |k\cdot(\frac{\sin\Omega t}{t},\frac{\sin\Omega t}{t},t)|}e^{\mu|k|}\hat{\rho}(t,k)=e^{2\pi\lambda'_{0} |k\cdot(\frac{\sin\Omega t}{t},\frac{\sin\Omega t}{t},t)|}e^{\mu|k|}\hat{A}(t,k)+e^{2\pi\lambda'_{0} |k\cdot(\frac{\sin\Omega t}{t},\frac{\sin\Omega t}{t},t)|}e^{\mu|k|}\hat{\rho}(t,k)\ast\hat{\mathcal{L}}(t,k).\end{aligned}$$

Let $\Phi(t,k)=e^{2\pi\lambda'_{0} |k\cdot(\frac{\sin\Omega t}{t},\frac{\sin\Omega t}{t},t)|}e^{\mu|k|}\hat{\rho}(t,k),$ $\mathcal{A}(t,k)=e^{2\pi\lambda'_{0} |k\cdot(\frac{\sin\Omega t}{t},\frac{\sin\Omega t}{t},t)|}e^{\mu|k|}\hat{A}(t,k),$ $\mathcal{K}^{0}(t,k)=e^{2\pi\lambda'_{0} |k\cdot(\frac{\sin\Omega t}{t},\frac{\sin\Omega t}{t},t)|}$ $\cdot\hat{\mathcal{L}}(t,k),$ then from (3.4), we have $\widetilde{\Phi}(\omega,k)=\widetilde{\mathcal{A}}(\omega,k)+\widetilde{\Phi}(\omega,k)\widetilde{\mathcal{K}}^{0}(\omega,k).$

Then $$\begin{aligned}
&\|\Phi(t,k)\|_{L^{2}(dt)}
=\|\widetilde{\Phi}(\omega,k)\|_{L^{2}}\leq\|\widetilde{\mathcal{A}}(\omega,k)\|_{L^{2}}+\|\widetilde{\Phi}(\omega,k)\|_{L^{2}}
\|\widetilde{\mathcal{K}}^{0}(\omega,k)\|_{L^{\infty}}\notag\\
&\leq\|e^{2\pi\lambda'_{0} |k\cdot(\frac{\sin\Omega t}{t},\frac{\sin\Omega t}{t},t)|}e^{\mu|k|}\hat{A}(t,k)\|_{L^{2}}+\|e^{2\pi\lambda'_{0} |k\cdot(\frac{\sin\Omega t}{t},\frac{\sin\Omega t}{t},t)|}e^{\mu|k|}\hat{\rho}(t,k)\|_{L^{2}}
\|\widetilde{\mathcal{K}}^{0}(\omega,k)\|_{L^{\infty}}.\notag\\\end{aligned}$$

Next we have to estimate $\|\widetilde{\mathcal{K}}^{0}(\omega,k)\|_{L^{\infty}}.$

Indeed,

$$\|\widetilde{\mathcal{K}}^{0}(\omega,k)\|_{L^{\infty}}
\leq\sup_{\omega}\frac{q}{m}\bigg[\bigg|\frac{1}{\omega}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}e^{2\pi it\omega}e^{2\pi\lambda'_{0} 