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

TL;DR

This paper proves the existence and uniqueness of weak solutions to a class of elliptic equations with singular, axially symmetric drifts in three-dimensional Lipschitz domains, relevant to Navier-Stokes theory.

Contribution

It establishes the existence and uniqueness of weak solutions for elliptic equations with singular drifts of a specific form, even when the divergence of the drift is positive.

Findings

Existence of weak solutions for negative alpha

Hölder continuity of solutions

Solutions vanish on the axis of symmetry

Abstract

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 $\Gamma:=\{ ~x\in \mathbb R^3:~|x'|=0~\}$.