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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 $\Delta u+b\cdot\nabla u=0$have the same optimal H\"older regularity as in the case of divergence-freedrift, that is, $u\in C^{\alpha}_{\text{loc}}$ for all $\alpha\in (0, 1)$.

On optimal H\"older regularity of solutions to the equation $\Delta u+b\cdot\nabla u=0$ in two dimensions