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

Abstract

<p style='text-indent:20px;'>In this paper we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in <inline-formula><tex-math id="M2">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ \alpha>\frac{1}{2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ s>1. $\end{document}</tex-math></inline-formula> We prove the existence of <inline-formula><tex-math id="M5">\begin{document}$ (H^{2\alpha^-+s}(\mathbb{T}^2),H^{2\alpha+s}(\mathbb{T}^2)) $\end{document}</tex-math></inline-formula>-global attractor <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{A}, $\end{document}</tex-math></inline-formula> that is, <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> is compact in <inline-formula><tex-math id="M8">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> and attracts all bounded subsets of <inline-formula><tex-math id="M9">\begin{document}$ H^{2\alpha^-+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> with respect to the norm of <inline-formula><tex-math id="M10">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2). $\end{document}</tex-math></inline-formula> The asymptotic compactness of solutions in <inline-formula><tex-math id="M11">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> is established by using commutator estimates for nonlinear terms, the spectral decomposition of solutions and new estimates of higher order derivatives. Furthermore, we show the existence of the exponential attractor in <inline-formula><tex-math id="M12">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2), $\end{document}</tex-math></inline-formula> whose compactness, boundedness of the fractional dimension and exponential attractiveness for the bounded subset of <inline-formula><tex-math id="M13">\begin{document}$ H^{2\alpha^-+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> are all in the topology of <inline-formula><tex-math id="M14">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2). $\end{document}</tex-math></inline-formula></p>