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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

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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

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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

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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

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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

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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

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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

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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

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