  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 $d \geq 3$.
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.
