Axisymmetric flows in the exterior of a cylinder

We study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $\Pi=\{x=(x_{h}, x_3)\ |\ |x_{h} |>1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(\Pi)$ satisfying the decay condition of the swirl component $ru^{\theta}_{0}\in L^{\infty}(\Pi)$.
Comment: Final version. The title is changed from "Axisymmetric flows in an exterior domain"