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We show that any unique global solution (here we do not require any smallnesscondition beforehand) to 3-D axisymmetric Navier-Stokes equations in somescaling invariant spaces must eventually become a small solution. Inparticular, 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 byusing this, we can refine some decay estimates for the axisymmetric solutions.

Long-time asymptotics of $3$-D axisymmetric Navier-Stokes equations in critical spaces