Asymptotic stability for the Navier-Stokes equations

We prove the asymptotic stability for weak solutions to the 3-D Navier-Stokes equations in the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla u \in L^{1}(0, \infty; \dot{B}^{0}_{\infty,\infty}({\mathbb{R}}^{3})) \cap LLogL(0, \infty; \dot{B}^{0}_{\infty,\infty}({\mathbb{R}}^{3}))$$\end{document} with arbitrary initial and external perturbations. This solves a problem due to Yong Zhou (Proc. Roy. Soc. Edinburgh, 136A (2006), 1099-1109).