Solutions to Landau equation under Prodi-Serrin's like criteria

Abstract

In this paper, we introduce Prodi-Serrin like criteria for weak solutions under which it is possible to show existence of classical solutions to the spatially homogeneous Landau equation for all classical 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}. The first step consists in establishing instantaneous appearance of $L^{p}$ estimates using the Prodi-Serrin criteria. Then, we introduce a new $L^{p}$ to $L^{\infty}$ framework, using a suitable De Giorgi's argument reminiscent of the one found in \cite{ricardo}, which provides appearance of pointwise bounds for such solutions. Our approach is quantitative and does not require a preliminary knowledge of elaborate tools for nonlinear parabolic equations.