Averaging lemmas with a force term in the transport equation

TL;DR

This paper develops new averaging lemmas for the transport equation with a force term, improving regularity results by considering the force explicitly, using local variable changes and stationary phase methods under specific non-degeneracy conditions.

Contribution

It introduces novel averaging lemmas that incorporate the force term directly, enhancing regularity estimates compared to previous results.

Findings

Improved regularity in L^2 space for transport equations with force.

Extension of results to L^p spaces for constant force, with 1<p≤2.

Characterization of optimal non-degeneracy conditions for the lemmas.

Abstract

We obtain several averaging lemmas for transport operator with a force term. These lemmas improve the regularity yet known by not considering the force term as part of an arbitrary right-hand side. Two methods are used: local variable changes or stationary phase. These new results are subjected to two non degeneracy assumptions. We characterize the optimal conditions of these assumptions to compare the obtained regularities according to the space and velocity variables. Our results are mainly in $L^2$, and for constant force, in $L^p$ for $1<p \leq 2$.