The Landau equation with moderate soft potentials: An approach using $\varepsilon$-Poincaré inequality and Lorentz spaces
This document presents an elementary approach using $\varepsilon$-Poincaré inequality to prove generation of $L^{p}$-bounds, $p\in(1,\infty)$, for the homogeneous Landau equation with moderate soft potentials $γ\in[-2,0)$. The critical case $γ=-2$ 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.
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