In this paper, we consider the logarithmically improved regularity criterion for the supercritical quasi-geostrophic equation in Besov space
\(\dot B_{\infty ,\infty }^{ - r}\left( {{\mathbb{R}^2}} \right)\). The result shows that if
θ is a weak solutions satisfies
$$\int_0^T {\frac{{\left\| {\nabla \theta ( \cdot ,s)} \right\|_{\dot B_{\infty ,\infty }^{ - r} }^{\tfrac{\alpha }{{\alpha - r}}} }}{{1 + \ln \left( {e + \left\| {\nabla ^ \bot \theta ( \cdot ,s)} \right\|_{L^{\tfrac{2}{r}} } } \right)!}}ds < \infty for some 0 < r < \alpha and 0 < \alpha < 1,}$$
then
θ is regular at
t =
T. In view of the embedding
\({L^{\frac{2}{r}}} \subset M_{\frac{2}{r}}^p \subset \dot B_{\infty ,\infty }^{ - r}\) with
\(2 \leqslant p < \frac{2}{r}\) and 0 ≤
r < 1, we see that our result extends the results due to 20] and 31].