Regularity condition of solutions to the quasi-geostrophic equations in Besov spaces with negative indices

With a Hölder type inequality in Besov spaces, we show that every strong solution on θ(t, x) on (0, T) of the dissipative quasi-geostrophic equations can be continued beyond T provided that ▿⊥θ(t,x) ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L^{\frac{{2\gamma }} {{\gamma - 2\delta }}} $$\end{document} ((0, T); \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \dot B_{\infty ,\infty }^{{{ - \delta - \gamma } \mathord{\left/ {\vphantom {{ - \delta - \gamma } 2}} \right. \kern-\nulldelimiterspace} 2}} $$\end{document} (ℝ2)) for 0 < δ < \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\gamma } {2} $$\end{document}.