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

Abstract

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

15 pages, proof of Theorem 1.1 is updated