Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off

Abstract

We show that in the setting of the spatially homogeneous Boltzmann equation without cut-off, the entropy dissipation associated to a function $f\in L^1({\mathbb{R}}^N)$ yields a control of $\sqrt{f}$ in Sobolev norms as soon as $f$ is locally bounded below. Under this additional assumption of lower bound our result is an improvement of a recent estimate given by P.L. Lions and is optimal in a certain sense.