# American Institute of Mathematical Sciences

March  2022, 27(3): 1345-1377. doi: 10.3934/dcdsb.2021093

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

 1 School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China 2 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain

* Corresponding author

Received  November 2020 Revised  February 2021 Published  March 2022 Early access  March 2021

Fund Project: 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)

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 $H^{2\alpha+s}(\mathbb{T}^2).$

Citation: Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $2$D quasi-geostrophic equations with fractional dissipation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1345-1377. doi: 10.3934/dcdsb.2021093
##### References:
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Dlotko, Global Attractors in Abstract Parabolic Problems,, London Mathematical Society Lecture Note Series, vol. 278, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404. [7] P. Constantin, M. Coti Zelati and V. Vicol, Uniformly attracting limit sets for the critically dissipative SQG equation, Nonlinearity, 29 (2016), 298–318. doi: 10.1088/0951-7715/29/2/298. [8] P. Constantin and C. Foiaş, Navier-Stokes Equations,, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988. [9] P. Constantin, A. Tarfulea and V. Vicol, Long time dynamics of forced critical SQG, Comm. Math. Phys., 335 (2015), 93–141. doi: 10.1007/s00220-014-2129-3. [10] A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511–528. doi: 10.1007/s00220-004-1055-1. [11] M. Coti Zelati and P. Kalita, Smooth attractors for weak solutions of the SQG equation with critical dissipation, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1857–1873. doi: 10.3934/dcdsb.2017110. [12] T. Dlotko, M. B. Kania and C. Sun, Quasi-geostrophic equation in $\mathbb{R}^2$, J. Differential Equations, 259 (2015), 531–561. doi: 10.1016/j.jde.2015.02.022. [13] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equation, Research in Applied Mathematics, vol. 37, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. [14] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713–718. doi: 10.1016/S0764-4442(00)00259-7. [15] R. Farwig and C. Qian, Asymptotic behavior for the quasi-geostrophic equations with fractional dissipation in $\mathbb{R}^2$, J. Differential Equations, 266 (2019), 6525–6579. doi: 10.1016/j.jde.2018.11.009. [16] C. Foiaş and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $2$, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1–34. [17] S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117–127. doi: 10.1090/S0002-9939-05-08340-1. [18] Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267–281. doi: 10.1007/BF00276875. [19] A. Gu, D. Li, B. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$, J. Differential Equations, 264 (2018), 7094–7137. doi: 10.1016/j.jde.2018.02.011. [20] J. K. Hale, Asymptotic Behavior of Dissipative Systems, , Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025. [21] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. [22] N. Ju, The maximum principle and the global attractor for the dissipative $2$D quasi-geostrophic equations, Comm. Math. Phys., 255 (2005), 161–181. doi: 10.1007/s00220-004-1256-7. [23] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891–907. doi: 10.1002/cpa.3160410704. [24] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323–347. doi: 10.2307/2939277. [25] C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co., Amsterdam, 2001. [26] C. J. Niche and M. E. Schonbek, Decay of weak solutions to the $2$D dissipative quasi-geostrophic equation, Comm. Math. Phys., 276 (2007), 93–115. doi: 10.1007/s00220-007-0327-y. [27] C. J. Niche and M. E. Schonbek, Decay characterization of solutions to dissipative equations, J. Lond. Math. Soc., 91 (2015), 573–595. doi: 10.1112/jlms/jdu085. [28] J. Pedlosky, Geophysical Fluid Dynamics, Springer, Berlin, 1982. [29] G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885–982. doi: 10.1016/S1874-575X(02)80038-8. [30] J. C. Robinson, Infinite-Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. [31] G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1–33. [32] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, Princeton University Press, Princeton, N.J., 1970. [33] C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal., 63 (2005), 49–65. doi: 10.1016/j.na.2005.04.034. [34] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, , Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [35] S. Wang, Attractors for the $3$D baroclinic quasi-geostrophic equations of large-scale atmosphere, J. Math. Anal. Appl., 165 (1992), 266–283. doi: 10.1016/0022-247X(92)90078-R. [36] M. Wang and Y. Tang, Long time dynamics of $2$D quasi-geostrophic equations with damping in $L^p$, J. Math. Anal. Appl., 412 (2014), 866–877. doi: 10.1016/j.jmaa.2013.11.019. [37] M. Wang and Y. Tang, On dimension of the global attractor for 2D quasi-geostrophic equations, Nonlinear Anal. Real World Appl., 14 (2013), 1887–1895. doi: 10.1016/j.nonrwa.2012.12.005. [38] J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 2001 (2001), 1–13. [39] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5. [40] C. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367–399. doi: 10.1016/j.jde.2005.06.008.

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##### References:
 [1] R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, vol. 65, Academic Press, New York-London, 1975. [2] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992. [3] L. C. Berselli, Vanishing viscosity limit and long-time behavior for $2$D quasi-geostrophic equations, Indiana Univ. Math. J., 51 (2002), 905–930. doi: 10.1512/iumj.2002.51.2075. [4] V. Berti and S. Gatti, Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations, Quart. Appl. Math., 64 (2006), 617–639. doi: 10.1090/S0033-569X-06-01044-9. [5] J. W. Cholewa and T. Dlotko, Bi-spaces global attractors in abstract parabolic equations, Evolution equations, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 60 (2003), 13–26. [6] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems,, London Mathematical Society Lecture Note Series, vol. 278, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404. [7] P. Constantin, M. Coti Zelati and V. Vicol, Uniformly attracting limit sets for the critically dissipative SQG equation, Nonlinearity, 29 (2016), 298–318. doi: 10.1088/0951-7715/29/2/298. [8] P. Constantin and C. Foiaş, Navier-Stokes Equations,, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988. [9] P. Constantin, A. Tarfulea and V. Vicol, Long time dynamics of forced critical SQG, Comm. Math. Phys., 335 (2015), 93–141. doi: 10.1007/s00220-014-2129-3. [10] A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511–528. doi: 10.1007/s00220-004-1055-1. [11] M. Coti Zelati and P. Kalita, Smooth attractors for weak solutions of the SQG equation with critical dissipation, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1857–1873. doi: 10.3934/dcdsb.2017110. [12] T. Dlotko, M. B. Kania and C. Sun, Quasi-geostrophic equation in $\mathbb{R}^2$, J. Differential Equations, 259 (2015), 531–561. doi: 10.1016/j.jde.2015.02.022. [13] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equation, Research in Applied Mathematics, vol. 37, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. [14] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713–718. doi: 10.1016/S0764-4442(00)00259-7. [15] R. Farwig and C. Qian, Asymptotic behavior for the quasi-geostrophic equations with fractional dissipation in $\mathbb{R}^2$, J. Differential Equations, 266 (2019), 6525–6579. doi: 10.1016/j.jde.2018.11.009. [16] C. Foiaş and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $2$, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1–34. [17] S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117–127. doi: 10.1090/S0002-9939-05-08340-1. [18] Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267–281. doi: 10.1007/BF00276875. [19] A. Gu, D. Li, B. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$, J. Differential Equations, 264 (2018), 7094–7137. doi: 10.1016/j.jde.2018.02.011. [20] J. K. Hale, Asymptotic Behavior of Dissipative Systems, , Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025. [21] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. [22] N. Ju, The maximum principle and the global attractor for the dissipative $2$D quasi-geostrophic equations, Comm. Math. Phys., 255 (2005), 161–181. doi: 10.1007/s00220-004-1256-7. [23] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891–907. doi: 10.1002/cpa.3160410704. [24] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323–347. doi: 10.2307/2939277. [25] C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co., Amsterdam, 2001. [26] C. J. Niche and M. E. Schonbek, Decay of weak solutions to the $2$D dissipative quasi-geostrophic equation, Comm. Math. Phys., 276 (2007), 93–115. doi: 10.1007/s00220-007-0327-y. [27] C. J. Niche and M. E. Schonbek, Decay characterization of solutions to dissipative equations, J. Lond. Math. Soc., 91 (2015), 573–595. doi: 10.1112/jlms/jdu085. [28] J. Pedlosky, Geophysical Fluid Dynamics, Springer, Berlin, 1982. [29] G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885–982. doi: 10.1016/S1874-575X(02)80038-8. [30] J. C. Robinson, Infinite-Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. [31] G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1–33. [32] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, Princeton University Press, Princeton, N.J., 1970. [33] C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal., 63 (2005), 49–65. doi: 10.1016/j.na.2005.04.034. [34] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, , Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [35] S. Wang, Attractors for the $3$D baroclinic quasi-geostrophic equations of large-scale atmosphere, J. Math. Anal. Appl., 165 (1992), 266–283. doi: 10.1016/0022-247X(92)90078-R. [36] M. Wang and Y. Tang, Long time dynamics of $2$D quasi-geostrophic equations with damping in $L^p$, J. Math. Anal. Appl., 412 (2014), 866–877. doi: 10.1016/j.jmaa.2013.11.019. [37] M. Wang and Y. Tang, On dimension of the global attractor for 2D quasi-geostrophic equations, Nonlinear Anal. Real World Appl., 14 (2013), 1887–1895. doi: 10.1016/j.nonrwa.2012.12.005. [38] J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 2001 (2001), 1–13. [39] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5. [40] C. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367–399. doi: 10.1016/j.jde.2005.06.008.
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