Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation

In this paper we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in \begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document} with \begin{document}$ \alpha>\frac{1}{2} $\end{document} and \begin{document}$ s>1. $\end{document} We prove the existence of \begin{document}$ (H^{2\alpha^-+s}(\mathbb{T}^2),H^{2\alpha+s}(\mathbb{T}^2)) $\end{document} -global attractor \begin{document}$ \mathcal{A}, $\end{document} that is, \begin{document}$ \mathcal{A} $\end{document} is compact in \begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document} and attracts all bounded subsets of \begin{document}$ H^{2\alpha^-+s}(\mathbb{T}^2) $\end{document} with respect to the norm of \begin{document}$ H^{2\alpha+s}(\mathbb{T}^2). $\end{document} The asymptotic compactness of solutions in \begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document} is established by using commutator estimates for nonlinear terms, the spectral decomposition of solutions and new estimates of higher order derivatives. Furthermore, we show the existence of the exponential attractor in \begin{document}$ H^{2\alpha+s}(\mathbb{T}^2), $\end{document} whose compactness, boundedness of the fractional dimension and exponential attractiveness for the bounded subset of \begin{document}$ H^{2\alpha^-+s}(\mathbb{T}^2) $\end{document} are all in the topology of \begin{document}$ H^{2\alpha+s}(\mathbb{T}^2). $\end{document}