$
  |\widehat{W}(k)|\leq\frac{1}{1+|k|^{\gamma}},\gamma>1.$ Further we assume that, for some constant $\lambda_{0}$ such that $\lambda_{0}-B_{0}>0,$ $$\begin{aligned}
 \sup_{\eta\in\mathbb{R}^{3}}e^{2\pi(\lambda_{0}-B_{0})|\eta|}|\tilde{f}^{0}(\eta)|\leq C^{0},\quad \sum_{n\in\mathbb{N}_{0}^{3}}\frac{(\lambda_{0}-B_{0})^{n}}{n!}\|\nabla^{n}_{v}f^{0}\|_{L_{dv}^{1}}\leq C_{0}<\infty.
 \end{aligned}$$

And we consider the following system,

$$\left\{\begin{array}{l}
\partial_{t}f+v\cdot\nabla_{x}f+\frac{q}{m}(v\times B_{0})\cdot\nabla_{v}f=-\frac{q}{m}(E+v\times B)\cdot\nabla_{v}f,\\
\partial_{t}B=\nabla_{x}\times E,\quad \nabla\cdot B=0,\\
E=W(x)\ast\rho(t,x),\rho(t,x)=\int_{\mathbb{R}^{3}}f(t,x,v)dv,\\
f(0,x,v)=f_{0}(x,v)=f_{0}(x,v_{1},v_{2},v_{z}),f^{0}(v)=f^{0}(v_{1},v_{2},v_{z}),
\end{array}\right.$$

there is $\varepsilon=\varepsilon(\lambda_{0},\mu_{0},\beta,\gamma,\lambda'_{0},\mu'_{0})$ verifying the following property: $f_{0}=f_{0}(x,v)$ is an initial data such that $$\begin{aligned}
\sup_{k\in\mathbb{Z}^{3},\eta\in\mathbb{R}^{3}}e^{2\pi(\lambda_{0}-B_{0})|\eta|}e^{2\pi\mu_{0}|k|}|f^{0}-f_{0}|+\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}
|f^{0}-f_{0}|e^{\beta|v|}dvdx\leq\varepsilon,\end{aligned}$$ where any $\beta>0,\lambda_{0}>\lambda'_{0}>B_{0},\mu_{0}>\mu'_{0}>0.$

In addition, we also assume that the $(\mathbf{PSC})$ holds. Then there exists a unique classical solution $(f(t,x,v),E(t,x),B(t,x))$ to the non-linear Vlasov system (0.12).

Moreover, for any fixed $\eta_{3},k_{3},\forall r\in\mathbb{N},$ as $|t|\rightarrow\infty,$ we have $$\begin{aligned}
         &|\hat{f}(t,k,\eta_{1}+k_{1}\frac{\sin\Omega t}{\Omega},\eta_{2}+k_{2}\frac{\sin\Omega t}{\Omega},\eta_{3})-\hat{f}_{0}(k,\eta)|\leq  e^{-(\lambda'_{0}-B_{0})|\eta_{3}+k_{3}t|},\quad\|\rho(t,\cdot)-\rho_{0}\|_{C^{r}(\mathbb{T}^{3})}
         =O(e^{-2\pi(\lambda'_{0}-B_{0})|t|}),\notag\\
          &\|E(t,\cdot)\|_{C^{r}(\mathbb{T}^{3})}=O(e^{-2\pi(\lambda'_{0}-B_{0})|t|}),\quad \|B(t,\cdot)\|
          _{C^{r}(\mathbb{T}^{3})}=O(e^{-2\pi(\lambda'_{0}-B_{0})|t|}),\notag\\
         \end{aligned}$$ where $\rho_{0}=\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}f_{0}(x,v)dvdx.$

Now we simply analyze the relation among the Vlasov-Poisson equations, the Vlasov-Maxwell equations,and our model. If we assume that both the electric induction and the magnetic field are zero, then Vlasov-Maxwell equations reduce to the Vlasov-Poisson equations; if we only assume that the electric induction is zero, then the Vlasov-Maxwell equations reduce to our case. In other words, our case is the generalized case of the Vlasov-Poisson equations, and provides a new observation from the physical viewpoint to solve the corresponding problem of the Vlasov-Maxwell equations. However, we cannot still solve the Vlasov-Maxwell equations completely only through this new 