Hardy's inequality and (almost) the Landau equation

TL;DR

This paper proves an $L^ olinebreak^\infty$ estimate for the isotropic homogeneous Landau equation with very soft potentials, using Hardy and Poincaré inequalities combined with Sobolev and De Giorgi-Nash-Moser theory.

Contribution

It introduces a novel approach linking Hardy inequalities to $L^ olinebreak^\infty$ bounds for the Landau equation in the soft potential regime.

Findings

Established $L^ olinebreak^\infty$ bounds for solutions

Connected Hardy inequalities to propagation of $L^p$ norms

Applied Sobolev and De Giorgi-Nash-Moser techniques

Abstract

In this manuscript we establish an $L^\infty$ estimate for the isotropic analogue of the homogeneous Landau equation. This is done for values of the interaction exponent $\gamma$ in (a part of) the range of very soft potentials. The main observation in our proof is that the classical weighted Hardy inequality leads to a weighted Poincar\'e inequality, which in turn implies the propagation of some $L^p$ norms of solutions. From here, the $L^\infty$ estimate follows from certain weighted Sobolev inequalities and De Giorgi-Nash-Moser theory.