Scalar elliptic equations with a singular drift

TL;DR

This paper studies the existence and properties of weak solutions to a scalar elliptic PDE with a singular drift term in a bounded domain, focusing on the impact of the singularity and the divergence-free condition.

Contribution

It introduces a framework for analyzing elliptic equations with singular drift terms involving a parameter and establishes weak solvability results.

Findings

Weak solutions exist under certain conditions.

The singular drift affects solution regularity.

Parameter ontrols the strength of the singularity.

Abstract

We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation $-\Delta u + b^{(\alpha)}\cdot \nabla u= f$ in a bounded domain $\Omega\subset {\mathbb R^2}$ containing the origin, where $f \in W^{-1}_q(\Omega) $ with $q>2$ and $b^{(\alpha)}:=b-\alpha \frac{x}{|x|^2}$, $b$ is a divergence-free vector field and $\alpha\in {\mathbb R}$ is a parameter.