On weighted regularity criteria for the axisymmetric Navier–Stokes equations

In this paper, we consider the axisymmetric Navier–Stokes equations with swirl. By invoking the magic identity of Miao and Zheng, the symmetry properties of Riesz transforms and the Hardy–Sobolev inequality, we establish regularity criterion involving rdωθ   with −1≤d<0−1≤d<0. This improves and extends the results of [3,21].