at the level of kinetic description based on the Vlasov equations from the mathematical view point. First, we analyze the Vlasov-Maxwell equations from perspective of both equilibrium and stability theories.

Now we give a detailed description of the Vlasov-Maxwell equations. We denote the particles distribution function by $f=f(t,x,v),$ and the electric and magnetic fields by $E(t,x)$ and $B(t,x),$ respectively. Then the Vlasov equation says $$\begin{aligned}
 \partial_{t}f+v\cdot\nabla_{x}f+(E+v\times B)\cdot\nabla_{v}f=0.
 \end{aligned}$$ The electric and magnetic fields $E(t,x)$ and $B(t,x)$ in Eq.$(0.1)$ are determined from Maxwell’s equations: $$\begin{aligned}
 &\nabla\cdot E(t,x)=\int_{\mathbb{R}^{3}}f(t,x,v)dv,\quad \nabla\times B(t,x)=\int_{\mathbb{R}^{3}}vf(t,x,v)dv+\frac{\partial E(t,x)}{\partial t},\notag\\
 &\frac{\partial B(t,x)}{\partial t}=-\nabla\times E(t,x),\quad \nabla\cdot B(t,x)=0.
 \end{aligned}$$ Note that Eq.$(0.1)$ is nonlinear since $E(t,x)$ and $B(t,x)$ are determined in terms of $f(t,x,v)$ from Maxwell’s equations $(0.2).$

An equilibrium analysis of Eq.$(0.1)$ and Eqs.$(0.2)$ proceeds by setting $\frac{\partial}{\partial t}=0$ and looking for stationary solutions, $f^{0}(x,v),E^{0}(x),B^{0}(x),$ that satisfy the equations $$\begin{aligned}
 & v\cdot\nabla_{x}f^{0}(x,v)+(E^{0}+v\times B^{0})\cdot\nabla_{v}f^{0}(x,v)=0,\quad \nabla\cdot B^{0}(x)=0,\notag\\
 &\nabla\times B^{0}(x)=\int_{\mathbb{R}^{3}}vf^{0}(x,v)dv,\quad \nabla\cdot E^{0}(x)=\int_{\mathbb{R}^{3}}f^{0}(x,v)dv.\notag\\
 \end{aligned}$$ An analysis of Eq.(0.3) reduces to a determination of the particle constants of the motion in the equilibrium fields $E^{0}(x)$ and $B^{0}(x).$ In this paper, we assume that $E^{0}(x)=0,$ namely, $\int_{\mathbb{R}^{3}}f^{0}(x,v)dv=0.$ This implies that there are no deviations from charge neutrality in equilibrium, $B^{0}(x)$ is produced by external current sources as 