Long-time asymptotics of axisymmetric Navier–Stokes equations in critical spaces

We prove that any global strong solution to axisymmetric Navier–Stokes equations must eventually become small. In particular, the limits of ‖ωθ(t)/r‖L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \omega ^\theta (t)/r\Vert _{L^1}$$\end{document} and ‖uθ(t)/r‖L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert u^\theta (t)/\sqrt{r}\Vert _{L^2}$$\end{document} are all 0 as t tends to infinity. Then by using these, we can refine a series of decay estimates. In particular, for the global axisymmetric solutions we know till now, namely the axisymmetric without swirl or with small swirl ones, these decay estimates hold. But our result here do not require any smallness conditions beforehand, thus more general.