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