$k$.

Next, we derive dissipative inequalities for lower-order derivatives of the density. Ultimately, these will be used to prove the gain in regularity achieved by Lemma \[L8\].

\[dissipative\] Assume the hypotheses of Lemma \[L7\] hold. Then, for all $t \in (0,T)$, we have the following $$\frac{d}{dt} \Vert v_0^2 f(t) \Vert_2^2 \leq  C_T \Vert v_0^2 f(t) \Vert_2^2 - \Vert v_0^{3/2} {\nabla}_v f(t) \Vert_2^2$$ $$\frac{d}{dt} \Vert v_0^{3/2} {\nabla}_v f(t) \Vert_2^2 \leq C_T\left ( \Vert v_0^{3/2} {\nabla}_v f(t) \Vert_2^2 + \Vert v_0 \partial_x f(t) \Vert_2^2 \right )- \Vert v_0 {\nabla}^2_v f(t) \Vert_2^2$$ $$\frac{d}{dt} \Vert v_0^{3/2} \partial_x f(t) \Vert_2^2 \leq C_T \left (\Vert v_0^{3/2} \partial_x f(t) \Vert_2^2 + \Vert v_0^2 f(t) \Vert_2^2 \right ) - (1-\eps) \Vert v_0 {\nabla}_v \partial_x f(t) \Vert_2^2$$ $$\begin{aligned}
\frac{d}{dt} \Vert v_0 {\nabla}_v \partial_x f(t) \Vert_2^2 & \leq & C_T \left ( \Vert v_0 {\nabla}_v \partial_x f(t) \Vert_2^2 + \Vert \partial_{xx} f(t) \Vert_2^2 +  \Vert v_0^{3/2} {\nabla}_v f(t) \Vert_2^2  \right ) \\
& \ & \ - (1-\eps) \Vert v_0^{1/2} {\nabla}^2_v \partial_x f(t) \Vert_2^2.\end{aligned}$$

The next lemma contains dissipative inequalities for higher-order derivatives of the density. In particular, it will allow us to trade $v$-derivatives of the density for those which are two orders less with the associated penalty of 