On the global well-posedness of a class of Boussinesq–Navier–Stokes systems

In this paper we consider the following 2D Boussinesq–Navier–Stokes systems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\begin{array}{lll}\partial_t u + u \cdot \nabla u + \nabla p = - \nu |D|^\alpha u + \theta e_2\\ \quad\quad \partial_t \theta+u\cdot\nabla \theta = - \kappa|D|^\beta \theta \\ \quad\quad\quad\quad\quad{\rm div} u = 0\end{array}}$$\end{document}with ν > 0, κ > 0 and 0 < β < α < 1. When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{6-\sqrt{6}}{4}({\doteq}0.888) < \alpha < 1, 1-\alpha < \beta \leq f(\alpha)}$$\end{document}, where f(α) < 1 is an explicit function as a technical bound, we prove the global well-posedness results for the rough initial data.