f \right )\\
& = & \int 2 v_1 v_2 \partial_{v_1v_2} f  \ dv_1 \ dv_2 \\
& = & 2 \int f \ dv\end{aligned}$$ after two integrations by parts. Hence, using the divergence structure of the Vlasov equation, we arrive at the local energy identity $$\label{localenergy}
\partial_t e + \partial_x m =2 \int f(t,x,v) \ dv$$ where $$e(t,x) = \int v_0 f(t,x,v) \ dv + \frac{1}{2} \left ( \vert E(t,x) \vert^2 + \vert B(t,x) \vert^2 \right )$$ and $$m(t,x) = \int v_1 f(t,x,v) \ dv + E_2(t,x) B(t,x).$$ Since $ f^0$ has compact support in $x$ with suitable decay in $v$, we find $v_0f^0 \in L^1(\bfR^3)$. We can then integrate (\[localenergy\]) over all space to deduce the global energy identity $$\frac{d}{dt} \int e(t,x) \ dx = 2\iint f^0(x,v) \ dx \ dv$$ whence we find $$\int e(t,x) \ dx  \leq C(1+t)$$ for all $t \in [0,T)$ and $E_1, E_2, B \in L^\infty([0,T]; L^2(\bfR))$.

To derive local estimates, we fix $(t,x)$, integrate (\[localenergy\]) along the backwards cone in space-time $\{ (s,y) \in (0,t) \times \bfR : \vert y - x \vert \leq t - s\}$, and use Green’s Theorem to find $$\begin{aligned}
 \int_0^t \biggl [ (e + m)(s, x + 