for regularity of the initial density in $v$ in order to obtain derivative bounds. Hence, solutions achieve a gain in regularity where $f$ and $\partial_x f$ are smooth in $v$ even for initial data which are not.

\[L8\] Assume the hypotheses of Lemma \[L7\] hold. Then for all $t \in (0,T)$, $$\sum_{k=0}^4 \frac{t^k}{2^k k!} {\ensuremath{\left\| v_0^{(4-k)/2} {\nabla}_v^k f(t) \right\|}}_2^2 + \sum_{k=0}^2 \frac{t^k}{2^k k!} {\ensuremath{\left\| v_0^{(3-k)/2} {\nabla}_v^k \partial_x f(t) \right\|}}_2^2 \leq C_T.$$

The gain in regularity achieved from the momentum argument is generally expected from the diffusive term. Additionally, it is possible that the solution gains regularity in its spatial argument as well, but this feature of the system remains unknown. Precedent exists for this possibility, however, as analogous work of Herau [@Herau] and Villani [@Villani] has determined that this does, in fact, occur for the linear, non-relativistic Fokker-Planck equation, as long as the given potential is sufficiently smooth.

Proofs of Lemmas and Estimates
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We first prove Lemma \[L1\], and this will require an additional result regarding the positivity of solutions to the linear Fokker-Planck equations arising from positive initial data.

Much of our argument is adapted from ideas of Lions [@LionsJL], Tartar [@Tartar], and Degond [@Degond]. Thus, we sketch 