characteristics in order to derive estimates. However, the appearance of the Fokker-Planck term changes the structure of the operator in (\[RVMFP\]), and the values of the distribution function are not conserved along such curves. Hence, the following lemma (similar to that of [@Degond]) will be utilized to estimate the particle distribution in such situations.

\[L1\] Let $g \in L^1((0,T), L^\infty(\bfR^3))$, $F \in W^{1,\infty}((0,T) \times \bfR^3; \bfR^2)$, and $h_0 \in L^\infty(\bfR^3) \cap L^2(\bfR^3)$ be given with $D \in C(\bfR^2; \bfR^{2 \times 2})$ positive semi-definite. Assume $h(t,x,v)$ is a weak solution of $$\label{LinVlasov}
\left \{ \begin{gathered} 
\partial_t  h + \hat{v}_1 \partial_x h + F(t,x,v) \cdot \nabla_v  h - \nabla_v \cdot \left ( D \nabla_v h \right)  =  g(t,x,v)\\
  h(0,x,v) =   h_0(x,v).
\end{gathered} \right.$$ Then, for every $t \in [0,T]$ $${\ensuremath{\left\| h(t) \right\|}}_\infty \leq {\ensuremath{\left\| h_0 \right\|}}_\infty + \int_0^t {\ensuremath{\left\| g(s) \right\|}}_\infty \ ds.$$

Next, we state a lemma that will allow us to control the fields and moments of the particle distribution.

\[L2\] Assume $v_0^\a  f^0 \in L^\infty(\bfR^3)$ for some $\a > 3$, $ f^0$ possesses compact support in $x$, and $E_2^0, B^0 \in H^1(\bfR)$. Then, for any $t \in [0,T]$, $x\in\mathbb{R}$, we have $$\label{cone}
\begin{gathered}
\int_0^t \left ( \int (v_0 \pm 