The Landau equation with moderate soft potentials: An approach using $\epsilon$-Poincaré inequality and Lorentz spaces

Abstract

This document presents an elementary approach using $\epsilon$-Poincaré inequality to prove generation of $L^p$-bounds, $p\in (1,\infty )$, for the homogeneous Landau equation with moderate soft potentials $\gamma \in [-2,0)$. The critical case $\gamma =-2$ uses an interpolation approach in the realm of Lorentz spaces and entropy. Alternatively, a direct approach using the Hardy-Littlewood-Sobolev (HLS) inequality and entropy is also presented. On this basis, the generation of pointwise bounds $p=\infty$ is deduced from a De Giorgi argument.