A Liouville type theorem for axially symmetric $D$-solutions to steady Navier-Stokes Equations

We study axially symmetric $D$-solutions of three dimensional steady Navier-Stokes equations. We prove that if the velocity $u$ decays like $\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^{-(\frac{2}{3}{)}^{+}}$ uniformly for $z$, or the vorticity $\omega$ decays like $\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^{-(\frac{5}{3}{)}^{+}}$ uniformly for $z$, then $u$ vanishes. Here $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }$ denotes the distance to the axis.