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

Lin Yang 1, , Yejuan Wang 1,, and  Tomás Caraballo 2,

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 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). $

Keywords: Global attractor, fractional dissipation, quasi-geostrophic equations, asymptotic compactness, exponential attractor.
Mathematics Subject Classification: Primary: 35B40, 35B41, 37L30, 35Q86.
Citation: Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021093
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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.  Google Scholar

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[34]

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[35]

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[36]

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[37]

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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.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, vol. 65, Academic Press, New York-London, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[8] P. Constantin and C. Foiaş, Navier-Stokes Equations,, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, Berlin, 1982. Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[31]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1–33. Google Scholar

[32] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, Princeton University Press, Princeton, N.J., 1970.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 2001 (2001), 1–13.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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