Elliptic equations with a singular drift from a weak Morrey space

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 $\mathrm{div}b\le 0$ in $\hat{I}\text{c◯}$. 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 $r=\sqrt{x_1^2+x_2^2}$.