American Institute of Mathematical Sciences

Advanced
June  2014, 19(4): 1213-1226. doi: 10.3934/dcdsb.2014.19.1213

Pullback attractors for three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation

Bo You 1, , Chengkui Zhong 2, and  Fang Li 3,

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an,710049, China
2. 

Department of Mathematics, Nanjing University, Nanjing 210093
3. 

Department of Mathematics, Nanjing University, Nanjing, 210093, China

Received  June 2013 Revised  January 2014 Published  April 2014

This paper is devoted to the existence of pullback attractors for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation. We first prove the existence of pullback absorbing sets in $H$ and $V$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with (1)-(8), and then we prove the existence of a pullback attractor in $H$ by the Sobolev compactness embedding theorem. Finally, we obtain the existence of a pullback attractor in $V$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with (1)-(8) by verifying the pullback $\mathcal{D}$ condition $(PDC)$.
Keywords: planetary geostrophic viscous equations, Sobolev compactness embedding theory, pullback $\mathcal{D}$ condition $(PDC)$., Pullback attractor.
Mathematics Subject Classification: Primary: 37B55; Secondary: 35Q8.
Citation: Bo You, Chengkui Zhong, Fang Li. Pullback attractors for three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1213-1226. doi: 10.3934/dcdsb.2014.19.1213
References:
[1]

R. A. Adams, Sobolev Spaces,, Academic Press, (1975).   Google Scholar

[2]

C. S. Cao, E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model,, Communications on Pure and Applied Mathematics, 56 (2003), 198.  doi: 10.1002/cpa.10056.  Google Scholar

[3]

C. S. Cao, E. S. Titi and M. Ziane, A "horizontal" hyper-diffusion 3-D thermocline planetary geostrophic model: well-posedness and long-time behavior,, Nonlinearity, 17 (2004), 1749.  doi: 10.1088/0951-7715/17/5/011.  Google Scholar

[4]

C. S. Cao, E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics,, Annals of Mathematics, 166 (2007), 245.  doi: 10.4007/annals.2007.166.245.  Google Scholar

[5]

D. N. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems,, Nonlinear Dynamics and Systems Theory, 2 (2002), 125.   Google Scholar

[6]

T. Caraballo, G. Łukasiewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Analysis, 64 (2006), 484.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[7]

V. Chepyzhov, M. I. Vishik, Attractors for equations of mathematical physics,, volume 49 of American Mathematical Society Colloquium Publications, (2002).   Google Scholar

[8]

B. D. Ewaldy, R. Temam, Maximum principles for the primitive equations of the atmosphere,, Discrete and Continuous Dynamical Systems- A, 7 (2001), 343.  doi: 10.3934/dcds.2001.7.343.  Google Scholar

[9]

P. E. Kloeden, B. Schmalfuß, Non-autonomous systems, cocycle attractors and variable time-step discretization,, Numerical Algorithms, 14 (1997), 141.  doi: 10.1023/A:1019156812251.  Google Scholar

[10]

P. E. Kloeden, D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dynamics of Continuous, 4 (1998), 211.   Google Scholar

[11]

Y. J. Li, C. K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations,, Applied Mathematics and Computation, 190 (2007), 1020.  doi: 10.1016/j.amc.2006.11.187.  Google Scholar

[12]

Y. J. Li, S. Y. Wang and H. Q. Wu, Pullback attractors for non-autonomous reaction-diffusion equations in $L^p,$, Applied Mathematics and Computation, 207 (2009), 373.  doi: 10.1016/j.amc.2008.10.065.  Google Scholar

[13]

J. Pedlosky, The equations for geostrophic motion in the ocean,, Journal of Physical Oceanography, 14 (1984), 448.  doi: 10.1175/1520-0485(1984)014<0448:TEFGMI>2.0.CO;2.  Google Scholar

[14]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987).  doi: 10.1115/1.3157711.  Google Scholar

[15]

N. A. Phillips, Geostrophic motion,, Reviews of Geophysics, 1 (1963), 123.  doi: 10.1029/RG001i002p00123.  Google Scholar

[16]

A. Robinson, H. Stommel, The oceanic thermocline and associated thermohaline circulation,, Tellus, 11 (1959), 295.  doi: 10.1111/j.2153-3490.1959.tb00035.x.  Google Scholar

[17]

B. Schmalfuß, Attractors for non-autonomous dynamical systems,, in Proc. Equadiff 99 (eds. B. Fiedler, (2000), 684.   Google Scholar

[18]

R. M. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation,, Applicable Analysis, 70 (1998), 147.  doi: 10.1080/00036819808840682.  Google Scholar

[19]

R. M. Samelson, R. Temam and S. Wang, Remarks on the planetary geostrophic model of gyre scale ocean circulation,, Differential and Integral Equations, 13 (2000), 1.   Google Scholar

[20]

R. M. Samelson, G. K. Vallis, A simple friction and diffusion scheme for planetary geostrophic basin models,, Journal of Physical Oceanography, 27 (1997), 186.  doi: 10.1175/1520-0485(1997)027<0186:ASFADS>2.0.CO;2.  Google Scholar

[21]

R. Temam, Infinite-dimensional dynamical systems in mechanics and physics,, New York, (1997).   Google Scholar

[22]

P. Welander, An advective model of the ocean thermocline,, Tellus, 11 (1959), 309.  doi: 10.1111/j.2153-3490.1959.tb00036.x.  Google Scholar

[23]

Y. H. Wang, C. K. Zhong, Pullback $\mathcalD$-attractors for nonautonomous sine-Gordon equations,, Nonlinear Analysis, 67 (2007), 2137.  doi: 10.1016/j.na.2006.09.019.  Google Scholar

[24]

B. You, C. K. Zhong and F. Li, Regularity of the global attractor for three dimensional planetary geostrophic viscous equations of large-scale ocean circulation,, in preparation., ().   Google Scholar

[25]

L. Yang, M. H. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with a dynamical boundary condition,, Discrete Continuous Dynam. Systems - B, 17 (2012), 2635.  doi: 10.3934/dcdsb.2012.17.2635.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces,, Academic Press, (1975).   Google Scholar

[2]

C. S. Cao, E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model,, Communications on Pure and Applied Mathematics, 56 (2003), 198.  doi: 10.1002/cpa.10056.  Google Scholar

[3]

C. S. Cao, E. S. Titi and M. Ziane, A "horizontal" hyper-diffusion 3-D thermocline planetary geostrophic model: well-posedness and long-time behavior,, Nonlinearity, 17 (2004), 1749.  doi: 10.1088/0951-7715/17/5/011.  Google Scholar

[4]

C. S. Cao, E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics,, Annals of Mathematics, 166 (2007), 245.  doi: 10.4007/annals.2007.166.245.  Google Scholar

[5]

D. N. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems,, Nonlinear Dynamics and Systems Theory, 2 (2002), 125.   Google Scholar

[6]

T. Caraballo, G. Łukasiewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Analysis, 64 (2006), 484.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[7]

V. Chepyzhov, M. I. Vishik, Attractors for equations of mathematical physics,, volume 49 of American Mathematical Society Colloquium Publications, (2002).   Google Scholar

[8]

B. D. Ewaldy, R. Temam, Maximum principles for the primitive equations of the atmosphere,, Discrete and Continuous Dynamical Systems- A, 7 (2001), 343.  doi: 10.3934/dcds.2001.7.343.  Google Scholar

[9]

P. E. Kloeden, B. Schmalfuß, Non-autonomous systems, cocycle attractors and variable time-step discretization,, Numerical Algorithms, 14 (1997), 141.  doi: 10.1023/A:1019156812251.  Google Scholar

[10]

P. E. Kloeden, D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dynamics of Continuous, 4 (1998), 211.   Google Scholar

[11]

Y. J. Li, C. K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations,, Applied Mathematics and Computation, 190 (2007), 1020.  doi: 10.1016/j.amc.2006.11.187.  Google Scholar

[12]

Y. J. Li, S. Y. Wang and H. Q. Wu, Pullback attractors for non-autonomous reaction-diffusion equations in $L^p,$, Applied Mathematics and Computation, 207 (2009), 373.  doi: 10.1016/j.amc.2008.10.065.  Google Scholar

[13]

J. Pedlosky, The equations for geostrophic motion in the ocean,, Journal of Physical Oceanography, 14 (1984), 448.  doi: 10.1175/1520-0485(1984)014<0448:TEFGMI>2.0.CO;2.  Google Scholar

[14]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987).  doi: 10.1115/1.3157711.  Google Scholar

[15]

N. A. Phillips, Geostrophic motion,, Reviews of Geophysics, 1 (1963), 123.  doi: 10.1029/RG001i002p00123.  Google Scholar

[16]

A. Robinson, H. Stommel, The oceanic thermocline and associated thermohaline circulation,, Tellus, 11 (1959), 295.  doi: 10.1111/j.2153-3490.1959.tb00035.x.  Google Scholar

[17]

B. Schmalfuß, Attractors for non-autonomous dynamical systems,, in Proc. Equadiff 99 (eds. B. Fiedler, (2000), 684.   Google Scholar

[18]

R. M. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation,, Applicable Analysis, 70 (1998), 147.  doi: 10.1080/00036819808840682.  Google Scholar

[19]

R. M. Samelson, R. Temam and S. Wang, Remarks on the planetary geostrophic model of gyre scale ocean circulation,, Differential and Integral Equations, 13 (2000), 1.   Google Scholar

[20]

R. M. Samelson, G. K. Vallis, A simple friction and diffusion scheme for planetary geostrophic basin models,, Journal of Physical Oceanography, 27 (1997), 186.  doi: 10.1175/1520-0485(1997)027<0186:ASFADS>2.0.CO;2.  Google Scholar

[21]

R. Temam, Infinite-dimensional dynamical systems in mechanics and physics,, New York, (1997).   Google Scholar

[22]

P. Welander, An advective model of the ocean thermocline,, Tellus, 11 (1959), 309.  doi: 10.1111/j.2153-3490.1959.tb00036.x.  Google Scholar

[23]

Y. H. Wang, C. K. Zhong, Pullback $\mathcalD$-attractors for nonautonomous sine-Gordon equations,, Nonlinear Analysis, 67 (2007), 2137.  doi: 10.1016/j.na.2006.09.019.  Google Scholar

[24]

B. You, C. K. Zhong and F. Li, Regularity of the global attractor for three dimensional planetary geostrophic viscous equations of large-scale ocean circulation,, in preparation., ().   Google Scholar

[25]

L. Yang, M. H. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with a dynamical boundary condition,, Discrete Continuous Dynam. Systems - B, 17 (2012), 2635.  doi: 10.3934/dcdsb.2012.17.2635.  Google Scholar

[1]

Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058
[2]

Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991
[3]

Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060
[4]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[5]

Ning Ju. The global attractor for the solutions to the 3D viscous primitive equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 159-179. doi: 10.3934/dcds.2007.17.159
[6]

Ning Ju. The finite dimensional global attractor for the 3D viscous Primitive Equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7001-7020. doi: 10.3934/dcds.2016104
[7]

Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143
[8]

Yangrong Li, Jinyan Yin. A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1203-1223. doi: 10.3934/dcdsb.2016.21.1203
[9]

Julia García-Luengo, Pedro Marín-Rubio, José Real. Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1603-1621. doi: 10.3934/cpaa.2015.14.1603
[10]

Julia García-Luengo, Pedro Marín-Rubio, José Real, James C. Robinson. Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203
[11]

Pedro Marín-Rubio, José Real. Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 989-1006. doi: 10.3934/dcds.2010.26.989
[12]

Grzegorz Łukaszewicz. Pullback attractors and statistical solutions for 2-D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 643-659. doi: 10.3934/dcdsb.2008.9.643
[13]

Julia García-Luengo, Pedro Marín-Rubio. Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2127-2146. doi: 10.3934/cpaa.2020094
[14]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195
[15]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[16]

Shulin Wang, Yangrong Li. Probabilistic continuity of a pullback random attractor in time-sample. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020028
[17]

Bo You, Chunxiang Zhao. Approximation of stationary statistical properties of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020057
[18]

Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653
[19]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[20]

Tomás Caraballo, José Real, I. D. Chueshov. Pullback attractors for stochastic heat equations in materials with memory. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 525-539. doi: 10.3934/dcdsb.2008.9.525

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences