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 R³ with a singular drift of the form b₀= b- x'|x'|² where x'= (x₁, x₂, 0), 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 <0 the divergence of such drifts is positive which potentially can ruin the uniqueness of solutions. Nevertheless, for <0 we prove existence and H\"older continuity of a unique weak solution which vanishes on the axis: =\ ~x R³: ~|x'|=0~\.