  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 $\|\ot(t)/r\|_{L^1}$ and $\|\ut(t)/\sqrt
r\|_{L^2}$ are all $0$ as $t$ tends to infinity. And by using this, we can
refine some decay estimates for the axisymmetric solutions.
