On optimal Hölder regularity of solutions to the equation $Δu+b\cdot\nabla u=0$ in two dimensions

We show that for an $L^2$ drift $b$ in two dimensions, if the Hardy norm of $\text{div }b$ is small, then the weak solutions to $Δu+b\cdot\nabla u=0$ have the same optimal Hölder regularity as in the case of divergence-free drift, that is, $u\in C^α_{\text{loc}}$ for all $α\in (0, 1)$.

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