Local conditional regularity for the Landau equation with Coulomb potential

This paper studies the regularity of Villani solutions of the space homogeneous Landau equation with Coulomb interaction in dimension 3. Specifically, we prove that any such solution belonging to the Lebesgue space \begin{document}$ L_{t}^{\infty}L_{v}^{q} $\end{document} with \begin{document}$ q>3 $\end{document} in an open cylinder \begin{document}$ (0,S)\times B $\end{document}, where \begin{document}$ B $\end{document} is an open ball of \begin{document}$ \mathbb{R}^{3} $\end{document}, must have Hölder continuous second order derivatives in the velocity variables, and first order derivative in the time variable locally in any compact subset of that cylinder.