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 $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with a singular drift of the form $b_0=b-\alpha \frac{x^{\mathrm{\prime }}}{\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^2}$ where $x^{\mathrm{\prime }}=(x_1,x_2,0)$, $\alpha \in \mathbb{R}$ 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 Hölder continuity of a unique weak solution which vanishes on the axis $\mathrm{\Gamma }:=\{x\in {\mathbb{R}}^3:\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }=0\}$.