A note on a $L^p$ stability estimate for regular Lagrangian flows

TL;DR

This paper extends a known stability estimate for regular Lagrangian flows from the case p=1 to a broader range of p in (1, +∞), providing a more general $L^p$ stability framework.

Contribution

It introduces an $L^p$ stability estimate for regular Lagrangian flows, generalizing previous results limited to p=1, with minor proof modifications.

Findings

Established $L^p$ stability estimate for p in (1, +∞)

Extended the applicability of stability estimates beyond p=1

Provided a proof adaptation for the generalized estimate

Abstract

In this note, we prove a $L^p$ version of the well known stability estimate for regular Lagrangian flows derived by Gianluca Crippa and Camillo De Lellis in \cite{crippa2008estimates}. As far as we know, the only estimate of this kind readily available in the literature is for the case $p=1$. With minor modifications to the proof in \cite{crippa2008estimates}, we show that an analogous estimate holds in $L^p$ norm with $p \in (1, + \infty)$.