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Stochastic averaging principle for dynamical systems with fractional Brownian motion
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 |
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. |
show all references
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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