Scalar elliptic equations with a singular drift

ABSTRACT

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

Keywords:

• Elliptic equations
• singular drift
• weak solutions

AMS SUBJECT CLASSIFICATION:

• 35J15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of the second author was supported by RFBR [grant number 20-01-00397].