  We show that any unique global solution (here we do not require any smallness
condition beforehand) to 3-D axisymmetric Navier-Stokes equations in some
scaling invariant spaces must eventually become a small solution. In
particular, we show that the limits of $\|\omega^\theta(t)/r\|_{L^1}$ and
$\|u^\theta(t)/\sqrt r\|_{L^2}$ are all $0$ as $t$ tends to infinity. And by
using this, we can refine some decay estimates for the axisymmetric solutions.
