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

Abstract: 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.