L^1 averaging lemma for transport equations with Lipschitz force fields

Abstract

The purpose of this note is to extend the $L^1$ averaging lemma of Golse and Saint-Raymond \cite{GolSR} to the case of a kinetic transport equation with a force field $F(x)\in W^{1,\infty }$. To this end, we will prove a local in time mixing property for the transport equation ${\partial }_{t}f+v.{\nabla }_{x}f+F.{\nabla }_{v}f=0$.

15 pages, to be published in Kinetic and Related Models