Compactness and Sobolev-Poincaré Inequalities for Solutions of Kinetic Equations

Abstract

In this paper, we prove a regularity result on the velocity averages of the solution of a kinetic equation whose data have a Sobolev regularity \vspace{-0.05cm} $W^{s,p}$, $0<s<\frac{1}{p}$, $p\in [1,+\infty )$, in the velocity variable. Namely, the velocity averages have the same Sobolev regularity in the time-space variable.