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

Abstract

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-free drift, that is, $u\in C^{\alpha}_{\text{loc}}$ for all $\alpha\in (0, 1)$.