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

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This work was supported by the National Natural Science Foundation of China under grant 41875084, and by FEDER and Ministerio de Ciencia, Innovación y Universidades of Spain (Grant PGC2018-096540-B-I00), and Junta de Andalucía, Spain (Grant US-1254251)

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  • In this paper we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in $ H^{2\alpha+s}(\mathbb{T}^2) $ with $ \alpha>\frac{1}{2} $ and $ s>1. $ We prove the existence of $ (H^{2\alpha^-+s}(\mathbb{T}^2),H^{2\alpha+s}(\mathbb{T}^2)) $-global attractor $ \mathcal{A}, $ that is, $ \mathcal{A} $ is compact in $ H^{2\alpha+s}(\mathbb{T}^2) $ and attracts all bounded subsets of $ H^{2\alpha^-+s}(\mathbb{T}^2) $ with respect to the norm of $ H^{2\alpha+s}(\mathbb{T}^2). $ The asymptotic compactness of solutions in $ H^{2\alpha+s}(\mathbb{T}^2) $ 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 $ H^{2\alpha+s}(\mathbb{T}^2), $ whose compactness, boundedness of the fractional dimension and exponential attractiveness for the bounded subset of $ H^{2\alpha^-+s}(\mathbb{T}^2) $ are all in the topology of $ H^{2\alpha+s}(\mathbb{T}^2). $

    Mathematics Subject Classification: Primary: 35B40, 35B41, 37L30, 35Q86.

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