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

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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

Abstract
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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 and 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, New York, 1975.
[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), 198-233. doi: 10.1002/cpa.10056.
[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), 1749-1776. doi: 10.1088/0951-7715/17/5/011.
[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), 245-267. doi: 10.4007/annals.2007.166.245.
[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), 125-144.
[6]	T. Caraballo, G. Łukasiewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.
[7]	V. Chepyzhov, M. I. Vishik, Attractors for equations of mathematical physics, volume 49 of American Mathematical Society Colloquium Publications, AMS, Providence, RI, 2002.
[8]	B. D. Ewaldy, R. Temam, Maximum principles for the primitive equations of the atmosphere, Discrete and Continuous Dynamical Systems- A, 7 (2001), 343-362. doi: 10.3934/dcds.2001.7.343.
[9]	P. E. Kloeden, B. Schmalfuß, Non-autonomous systems, cocycle attractors and variable time-step discretization, Numerical Algorithms, 14 (1997), 141-152. doi: 10.1023/A:1019156812251.
[10]	P. E. Kloeden, D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1998), 211-226.
[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), 1020-1029. doi: 10.1016/j.amc.2006.11.187.
[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), 373-379. doi: 10.1016/j.amc.2008.10.065.
[13]	J. Pedlosky, The equations for geostrophic motion in the ocean, Journal of Physical Oceanography, 14 Academic Press, (1984) 448-455. doi: 10.1175/1520-0485(1984)014<0448:TEFGMI>2.0.CO;2.
[14]	J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1115/1.3157711.
[15]	N. A. Phillips, Geostrophic motion, Reviews of Geophysics, 1 (1963), 123-176. doi: 10.1029/RG001i002p00123.
[16]	A. Robinson, H. Stommel, The oceanic thermocline and associated thermohaline circulation, Tellus, 11 (1959), 295-308. doi: 10.1111/j.2153-3490.1959.tb00035.x.
[17]	B. Schmalfuß, Attractors for non-autonomous dynamical systems, in Proc. Equadiff 99 (eds. B. Fiedler, K. Gröer and J. Sprekels), Berlin, World Scientific, Singapore, (2000), 684-689.
[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), 147-173. doi: 10.1080/00036819808840682.
[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), 1-14.
[20]	R. M. Samelson, G. K. Vallis, A simple friction and diffusion scheme for planetary geostrophic basin models, Journal of Physical Oceanography, 27 (1997), 186-194. doi: 10.1175/1520-0485(1997)027<0186:ASFADS>2.0.CO;2.
[21]	R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, New York, Springer-Verlag, 1997.
[22]	P. Welander, An advective model of the ocean thermocline, Tellus, 11 (1959), 309-318. doi: 10.1111/j.2153-3490.1959.tb00036.x.
[23]	Y. H. Wang, C. K. Zhong, Pullback $\mathcalD$-attractors for nonautonomous sine-Gordon equations, Nonlinear Analysis, 67 (2007), 2137-2148. doi: 10.1016/j.na.2006.09.019.
[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.
[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), 2635-2651. doi: 10.3934/dcdsb.2012.17.2635.

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References:

[1]	R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[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), 198-233. doi: 10.1002/cpa.10056.
[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), 1749-1776. doi: 10.1088/0951-7715/17/5/011.
[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), 245-267. doi: 10.4007/annals.2007.166.245.
[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), 125-144.
[6]	T. Caraballo, G. Łukasiewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.
[7]	V. Chepyzhov, M. I. Vishik, Attractors for equations of mathematical physics, volume 49 of American Mathematical Society Colloquium Publications, AMS, Providence, RI, 2002.
[8]	B. D. Ewaldy, R. Temam, Maximum principles for the primitive equations of the atmosphere, Discrete and Continuous Dynamical Systems- A, 7 (2001), 343-362. doi: 10.3934/dcds.2001.7.343.
[9]	P. E. Kloeden, B. Schmalfuß, Non-autonomous systems, cocycle attractors and variable time-step discretization, Numerical Algorithms, 14 (1997), 141-152. doi: 10.1023/A:1019156812251.
[10]	P. E. Kloeden, D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1998), 211-226.
[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), 1020-1029. doi: 10.1016/j.amc.2006.11.187.
[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), 373-379. doi: 10.1016/j.amc.2008.10.065.
[13]	J. Pedlosky, The equations for geostrophic motion in the ocean, Journal of Physical Oceanography, 14 Academic Press, (1984) 448-455. doi: 10.1175/1520-0485(1984)014<0448:TEFGMI>2.0.CO;2.
[14]	J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1115/1.3157711.
[15]	N. A. Phillips, Geostrophic motion, Reviews of Geophysics, 1 (1963), 123-176. doi: 10.1029/RG001i002p00123.
[16]	A. Robinson, H. Stommel, The oceanic thermocline and associated thermohaline circulation, Tellus, 11 (1959), 295-308. doi: 10.1111/j.2153-3490.1959.tb00035.x.
[17]	B. Schmalfuß, Attractors for non-autonomous dynamical systems, in Proc. Equadiff 99 (eds. B. Fiedler, K. Gröer and J. Sprekels), Berlin, World Scientific, Singapore, (2000), 684-689.
[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), 147-173. doi: 10.1080/00036819808840682.
[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), 1-14.
[20]	R. M. Samelson, G. K. Vallis, A simple friction and diffusion scheme for planetary geostrophic basin models, Journal of Physical Oceanography, 27 (1997), 186-194. doi: 10.1175/1520-0485(1997)027<0186:ASFADS>2.0.CO;2.
[21]	R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, New York, Springer-Verlag, 1997.
[22]	P. Welander, An advective model of the ocean thermocline, Tellus, 11 (1959), 309-318. doi: 10.1111/j.2153-3490.1959.tb00036.x.
[23]	Y. H. Wang, C. K. Zhong, Pullback $\mathcalD$-attractors for nonautonomous sine-Gordon equations, Nonlinear Analysis, 67 (2007), 2137-2148. doi: 10.1016/j.na.2006.09.019.
[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.
[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), 2635-2651. doi: 10.3934/dcdsb.2012.17.2635.

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