On the existence and uniqueness of weak solutions to elliptic equations with a singular drift

In this paper we study the Dirichlet problem for a scalar elliptic equation in a bounded Lipschitz domain $\Omega \subset \mathbb R^3$ with a singular drift of the form $b_0= b-\alpha \frac {x'}{|x'|^2}$ where $x'=(x_1,x_2,0)$, $\alpha \in \mathbb R$ is a parameter and $b$ is a divergence free vector field having essentially the same regularity as the potential part of the drift. Such drifts naturally arise in the theory of axially symmetric solutions to the Navier-Stokes equations. For $\alpha <0$ the divergence of such drifts is positive which potentially can ruin the uniqueness of solutions. Nevertheless, for $\alpha<0$ we prove existence and H\"older continuity of a unique weak solution which vanishes on the axis $\Gamma:=\{ ~x\in \mathbb R^3:~|x'|=0~\}$.