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This paper studies the regularity of Villani solutions of the spacehomogeneous Landau equation with Coulomb interaction in dimension 3.Specifically, we prove that any such solution belonging to the Lebesgue spaceL_{t}^{\infty}L_{v}^{q} with q>3 in an open cylinder (0,S)\times B, where B isan open ball of \mathbb{R}^{3}, must have Holder continuous second orderderivatives in the velocity variables, and first order derivative in the timevariable locally in any compact subset of that cylinder.

Local conditional regularity for the Landau equation with Coulomb potential