On the local existence for an active scalar equation in critical regularity setting

Abstract

In this note, we address the local well-posedness for the active scalar equation ${\partial }_t\theta +u\cdot \nabla \theta =0$, where $u=-{\nabla }^{\perp }(-\Delta {)}^{-1+\beta /2}\theta$. The local existence of solutions in the Sobolev class $H^{1+\beta +\epsilon }$, where $\epsilon >0$ and $\beta \in (1,2)$, has been recently addressed in \cite{HKZ}. The critical case $\epsilon =0$ has remained open. Using a different technique, we prove the local well-posedness in the Besov space $B_{2,1}^{1+\beta }$, where $\beta \in (1,2)$. The proof is based on log-Lipschitz estimates for the transport equation.