\Vert_\infty \leq C(1 + t)^{2(\a+1)}.$$

Thus, we have $C^1$ estimates on the fields without requiring any regularity of the density. Next, we utilize energy estimates to bound the density and its derivatives in $L^2(\bfR^3)$.

\[L5\] Assume $f^0\in L^2(\bfR^3)$. Then, for every $t \in [0,T]$ $${\ensuremath{\left\| f(t) \right\|}}_2 \leq {\ensuremath{\left\|  f^0 \right\|}}_2.$$ If additionally, $v_0^\gamma f^0\in L^2(\bfR^3)$ for some $\gamma > 0$ and the hypotheses of Lemma \[L2\] hold, then $${\ensuremath{\left\| v_0^\gamma f(t) \right\|}}_2 \leq C_T$$ for every $t \in [0,T]$.

\[L6\] Assume the hypotheses of Lemma \[L4\] hold with $v_0^{\gamma+1} f^0, v_0^\gamma \partial_x f^0 \in L^2(\bfR^3)$ for some $\gamma \geq 0$. Then for all $t \in [0,T]$ we have $${\ensuremath{\left\| v_0^\gamma {\partial}_x f(t) \right\|}}_2 \leq C_T.$$

\[L7\] Assume the hypotheses of Lemma \[L4\] hold with $v_0^\a f^0 \in L^\infty(\bfR^3)$ and $v_0^{\b-k/2} {\partial}_x^k f^0\in L^2(\bfR^3)$ for some $\a > 5$, $\b > 2$ and any $k =0,1,2$. Then, for all $t \in [0,T]$ $$\Vert v_0^\gamma \partial_{xx} f(t) \Vert_2 + \sum_{k=0}^2 \left ( \Vert \partial^k E(t) \Vert_2 + \Vert \partial^k B(t) \Vert_2  \right ) \leq C_T$$ for every $\gamma \in [0, c]$, where $c= \min \left \{\frac{\a - 3}{2}, \b-1 \right \}$ and $\partial^k$ is any $t$ or $x$ derivative of order 