# American Institute of Mathematical Sciences

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

 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 $(PDC)$.
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:
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##### 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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