Local regularity for the space-homogeneous Landau equation with very soft potentials

This paper deals with the space-homogenous Landau equation with very soft potentials, including the Coulomb case. This nonlinear equation is of parabolic type with diffusion matrix given by the convolution product of the solution with the matrix $a_{ij}(z)=\mathrm{\mid }z{\mathrm{\mid }}^{\gamma }(\mathrm{\mid }z{\mathrm{\mid }}^2{\delta }_{ij}-z_i{z}_j)$ for $\gamma \in [-3,-2)$. We derive local truncated entropy estimates and use them to establish two facts. Firstly, we prove that the set of singular points (in time and velocity) for the weak solutions constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has zero ${\mathcal{P}}^{m_{\ast }}$ parabolic Hausdorff measure with $m_{\ast }:=\frac{7}{2}\mathrm{\mid }2+\gamma \mathrm{\mid }$. Secondly, we prove that if such a weak solution is axisymmetric, then it is smooth away from the symmetry axis. In particular, radially symmetric weak solutions are smooth away from the origin.