On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations (NSν) with initial data in the scaling invariant Besov space, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}^{-1+\frac3p}_{p,\infty},$$\end{document} here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (ANSν), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}^{-\frac12,\frac12}_4$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}^{-\frac12,\frac12}_4(T).$$\end{document} Then with initial data in the scaling invariant space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}^{-\frac12,\frac12}_4,$$\end{document} we prove the global wellposedness for (ANSν) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (ANSν) with high oscillatory initial data (1.2).