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

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 $p\in (1,+\mathrm{\infty })$.