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In this note, we address the local well-posedness for the active scalarequation $\partial_t \theta + u\cdot \nabla \theta =0$, where $u = -\nabla^\perp(-\Delta)^{-1+\beta/2}\theta$. The local existence of solutions inthe 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 localwell-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.

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