Elliptic equations with a singular drift from a weak Morrey space

TL;DR

This paper establishes the existence and uniqueness of weak solutions for elliptic equations with singular drifts in weak Morrey spaces, including cases with axisymmetric singularities.

Contribution

It introduces a novel approach to handle elliptic equations with drifts in weak Morrey spaces, covering singularities along an axis.

Findings

Proved existence of weak solutions under singular drift conditions.

Established uniqueness of solutions in the specified setting.

Extended the theory to include drifts with axisymmetric singularities.

Abstract

In this paper we prove the existence and uniqueness of weak solutions to the Dirichlet problem for an elliptic equation with a drift $b$ satisfying $\operatorname{div} b\le 0$ in $\Omega$. We assume $b$ belongs to some weak Morrey class which includes in the 3D case, in particular, drifts having a singularity along the axis $x_3$ with the asymptotic $b(x)\sim c/r$, where $r=\sqrt{x_1^2+x_2^2}$.