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

TL;DR

This paper establishes that in two dimensions, solutions to the PDE with an $L^2$ drift and small Hardy norm of divergence exhibit optimal H"older regularity, matching the divergence-free case.

Contribution

It proves that small Hardy norm of divergence ensures optimal H"older regularity for solutions with $L^2$ drift in two dimensions.

Findings

Solutions are in $C^{eta}_{loc}$ for all $eta ext{ in }(0,1)$

Optimal regularity holds under small Hardy norm of divergence

Regularity matches divergence-free case

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 $\alpha\in (0, 1)$.