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

TL;DR

This paper extends the $L^1$ averaging lemma to kinetic transport equations with Lipschitz continuous force fields, establishing a local in time mixing property that broadens the lemma's applicability.

Contribution

It introduces an extension of the $L^1$ averaging lemma to cases with Lipschitz force fields, providing new mixing estimates for such transport equations.

Findings

Proves a local in time mixing property for transport equations with Lipschitz force fields.

Extends the $L^1$ averaging lemma to a broader class of kinetic equations.

Provides mathematical tools for analyzing kinetic equations with non-smooth 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 $\partial_t f + v.\nabla_x f + F.\nabla_v f =0$.