A priori estimates for solutions to Landau equation under Prodi-Serrin like criteria

In this paper, we introduce Prodi-Serrin like criteria which enable to provide a priori estimates for the solutions to the spatially homogeneous Landau equation for all classical soft potentials and dimensions $d\ge 3$. The physical case of Coulomb interaction in dimension $d=3$ is included in our analysis, which generalizes the work of \cite{silvestre}. Our approach is quantitative and does not require a preliminary knowledge of elaborate tools for nonlinear parabolic equations.