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

TL;DR

This paper establishes new criteria similar to Prodi-Serrin conditions that provide a priori estimates for solutions to the Landau equation, including Coulomb interactions, in higher dimensions.

Contribution

It introduces a novel set of criteria for the Landau equation that generalize previous results and do not rely on complex nonlinear parabolic tools.

Findings

Provides a priori estimates for all classical soft potentials

Includes Coulomb interaction in three dimensions

Generalizes previous work by Silvestre

Abstract

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.