A logarithmically improved regularity criterion for the supercritical quasi-geostrophic equations in Besov space

In this paper, we consider the logarithmically improved regularity criterion for the supercritical quasi-geostrophic equation in Besov space B˙∞,∞−r(ℝ2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot B_{\infty ,\infty }^{ - r}\left( {{\mathbb{R}^2}} \right)$$\end{document}. The result shows that if θ is a weak solutions satisfies ∫0T∥∇θ(⋅,s)∥B˙∞,∞−rαα−r1+ln(e+∥∇⊥θ(⋅,s)∥L2r)!ds<∞forsome0<r<αand0<α<1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_0^T {\frac{{\left\| {\nabla \theta ( \cdot ,s)} \right\|_{\dot B_{\infty ,\infty }^{ - r} }^{\tfrac{\alpha } {{\alpha - r}}} }} {{1 + \ln \left( {e + \left\| {\nabla ^ \bot \theta ( \cdot ,s)} \right\|_{L^{\tfrac{2} {r}} } } \right)!}}ds < \infty for some 0 < r < \alpha and 0 < \alpha < 1,}$$\end{document} then θ is regular at t = T. In view of the embedding L2r⊂M2rp⊂B˙∞,∞−r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^{\frac{2}{r}}} \subset M_{\frac{2}{r}}^p \subset \dot B_{\infty ,\infty }^{ - r}$$\end{document} with 2≤p<2r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \leqslant p < \frac{2}{r}$$\end{document} and 0 ≤ r < 1, we see that our result extends the results due to [20] and [31].