an $x$-derivative and a $v_0$ moment. For instance, use of this lemma will allow us to conclude the estimates $$\Vert {\nabla}^4_v f(t) \Vert_2^2 \lesssim \Vert v_0^{1/2} {\nabla}^2_v \partial_x f(t) \Vert_2^2 \lesssim \Vert \partial_{xx} f(t) \Vert_2^2 \leq C_T$$ along with the previously obtained bound on the second spatial derivative.

\[v\_deriv\_lemma\] Assume the hypotheses of Lemma \[L7\] hold. Then, for all $t \in (0,T)$, we have $$\begin{aligned}
\frac{d}{dt} {\ensuremath{\left\| v_0^\gamma {\nabla}_v^k f (t) \right\|}}_2^2 & \leq &
  C_T\biggl({\ensuremath{\left\| v_0^\gamma {\nabla}_v^k f (t) \right\|}}_2^2 +
  \sum_{j=1}^{k-1} {\ensuremath{\left\| v_0^{\gamma+j-k} {\nabla}_v^j f (t) \right\|}}_2^2
  + {\ensuremath{\left\| v_0^{\gamma+1/2}{\partial}_x{\nabla}_v^{k-2} f(t) \right\|}}_2^2\Big) \\
  & \ & \   - (1- \eps) {\ensuremath{\left\| v_0^{\gamma -1/2} {\nabla}_v^{k+1} f (t) \right\|}}_2^2 \end{aligned}$$ for every $\gamma \in [0, \b -k/2]$, $k=2, 3, 4$, and $\eps > 0$ sufficiently small. Additionally, we have $$\begin{aligned}
\frac{d}{dt} \Vert v_0^{1/2} {\nabla}_v^2 \partial_x f(t) \Vert_2^2 & \leq & C_T \biggl(\Vert v_0^{1/2} {\nabla}_v^2 \partial_x f(t) \Vert_2^2
+ \Vert v_0 {\nabla}_v \partial_x f(t) \Vert_2^2
+  \sum_{j=1}^2 {\ensuremath{\left\| v_0^{2 - j/2} {\nabla}_v^j f (t) \right\|}}_2^2 \\
& \ &  + \Vert \partial_{xx} f(t) \Vert_2^2  \biggr)- (1- \eps) {\ensuremath{\left\| {\nabla}_v^3 \partial_x f (t) \right\|}}_2^2 \end{aligned}$$ for all $t \in (0,T)$ and $\eps > 0$ sufficiently small.

Our final lemma removes the need 