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

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