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

TL;DR

This paper proves local well-posedness for an active scalar equation in the critical regularity setting using Besov spaces and log-Lipschitz estimates, addressing an open problem for the case when epsilon equals zero.

Contribution

It introduces a new technique to establish local existence in the critical Besov space for the active scalar equation, extending previous results.

Findings

Proves local well-posedness in the critical Besov space B^{1+eta}_{2,1}

Addresses the open case of epsilon=0 in the regularity setting

Uses log-Lipschitz estimates for the transport equation

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 $B^{1+\beta}_{2,1}$, where $\beta \in (1,2)$. The proof is based on log-Lipschitz estimates for the transport equation.