The Landau equation with moderate soft potentials: An approach using $\varepsilon$-Poincar\'e inequality and Lorentz spaces

TL;DR

This paper introduces an elementary method using $psilon$-Poincare9 inequality and Lorentz spaces to establish $L^p$ bounds for the homogeneous Landau equation with moderate soft potentials, including the critical case.

Contribution

It provides a novel elementary approach to derive $L^p$ bounds for the Landau equation using $psilon$-Poincare9 inequality and Lorentz spaces, including the critical case $psilon=-2$.

Findings

Established $L^p$ bounds for $p eq\infty$ in the Landau equation with soft potentials.

Derived pointwise bounds $p=psilon=8$ using De Giorgi's method.

Presented an alternative approach via Hardy-Littlewood-Sobolev inequality.

Abstract

This document presents an elementary approach using $\varepsilon$-Poincar\'e 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.