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Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces
1. | Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States |
References:
[1] |
V. I. Agoshkov and V. M. Ipatova, Solvability of the altimeter data assimilation problem in the quasi-geostrophic multi-layer model of ocean circulation , Comput. Math. Math. Phys., 37 (1997), 348-358. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of evolution equations. Studies in Mathematics and its Applications, 25, North-Holland Publishing Co, Amsterdam, 1992. |
[3] |
C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier-Stokes equations and estimate of its dimension , Adv. Math. Sci. Appl., 4 (1994), 465-489. |
[4] |
C. Bernier-Kazantsev and I. D. Chueshov, The finiteness of determining degrees of freedom for the quasi-geostrophic multi-layer ocean model , Nonlinear Anal., 42 (2000), 1499-1512.
doi: 10.1016/S0362-546X(99)00188-1. |
[5] |
C. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model , Comm. Pure Appl. Math., 56 (2003), 198-233.
doi: 10.1002/cpa.10056. |
[6] |
T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations , Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17-36.
doi: 10.3934/dcdss.2009.2.17. |
[7] |
T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays , R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194.
doi: 10.1098/rspa.2003.1166. |
[8] |
T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays , J. Differential Equations, 205 (2004), 271-297.
doi: 10.1016/j.jde.2004.04.012. |
[9] |
V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces , J. Math. Pures Appl., 90 (2008), 469-491.
doi: 10.1016/j.matpur.2008.07.001. |
[10] |
V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces , Nonlinearity, 22 (2009), 351-370.
doi: 10.1088/0951-7715/22/2/006. |
[11] |
V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms , Sb. Math, 192 (2001), 11-47.
doi: 10.1070/SM2001v192n01ABEH000534. |
[12] |
V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics. American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. |
[13] |
V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor , J. Dynam. Differential Equations, 19 (2007), 655-684.
doi: 10.1007/s10884-007-9077-y. |
[14] |
A. Cheskidov and S. Lu, The existence and the structure of uniform global attractors for nonautonomous reaction-diffusion systems without uniqueness , Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 55-66.
doi: 10.3934/dcdss.2009.2.55. |
[15] |
T. Colin, The cauchy problem and the continuous limit for the multilayer model in geophysical fluid dynamics , SIAM J. Math. Anal., 28 (1997), 516-529.
doi: 10.1137/S0036141095291269. |
[16] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors , J. Dyn. Differential Equations, 2 (1995), 307-341.
doi: 10.1007/BF02219225. |
[17] |
A. Haraux, Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées,17, Mason, Paris, 1991. |
[18] |
N. Ju, The global attractor for the solutions to the 3D viscous primitive equations , Discrete Contin. Dyn. Syst., 17 (2007), 159-179.
doi: 10.3934/dcds.2007.17.159. |
[19] |
P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization , Numer. Algorithms, 14 (1997), 141-152.
doi: 10.1023/A:1019156812251. |
[20] |
P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations , Dyn. Continuous Impulsive Systems, 4 (1998), 211-226. |
[21] |
J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications , Nonlinearity, 5 (1992), 237-288. |
[22] |
J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean , Nonlinearity, 5 (1992), 1007-1053. |
[23] |
S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces , J. Differential Equations, 230 (2006), 196-212.
doi: 10.1016/j.jde.2006.07.009. |
[24] |
S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces , Discrete Contin. Dyn. Syst., 13 (2005), 701-719.
doi: 10.3934/dcds.2005.13.701. |
[25] |
T. Tachim Medjo, On strong solutions of the multi-layer quasi-geostrophic equations of the ocean , Nonlinear Anal., 68 (2008), 3550-3564.
doi: 10.1016/j.na.2007.03.046. |
[26] |
T. Tachim Medjo, Non-autonomous 3D primitive equations with oscillating external force and its global attractor , Discrete Contin. Dyn. Syst., 32 (2012), 265-291.
doi: 10.3934/dcds.2012.32.265. |
[27] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New-York, second edition, 1987. |
[28] |
P. Peixoto and A. H. Oort, Physics of Climate, American Institute of Physics, New-York, 1992. |
[29] |
R. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation , Appl. Anal, 70 (1998), 147-173.
doi: 10.1080/00036819808840682. |
[30] |
H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations , Nonlinearity, 22 (2009), 667-681.
doi: 10.1088/0951-7715/22/3/008. |
[31] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[32] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS-Chelsea Series, AMS, Providence, 2001. |
[33] |
Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations , Dyn. Syst., 23 (2008), 1-16.
doi: 10.1080/14689360701611821. |
show all references
References:
[1] |
V. I. Agoshkov and V. M. Ipatova, Solvability of the altimeter data assimilation problem in the quasi-geostrophic multi-layer model of ocean circulation , Comput. Math. Math. Phys., 37 (1997), 348-358. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of evolution equations. Studies in Mathematics and its Applications, 25, North-Holland Publishing Co, Amsterdam, 1992. |
[3] |
C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier-Stokes equations and estimate of its dimension , Adv. Math. Sci. Appl., 4 (1994), 465-489. |
[4] |
C. Bernier-Kazantsev and I. D. Chueshov, The finiteness of determining degrees of freedom for the quasi-geostrophic multi-layer ocean model , Nonlinear Anal., 42 (2000), 1499-1512.
doi: 10.1016/S0362-546X(99)00188-1. |
[5] |
C. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model , Comm. Pure Appl. Math., 56 (2003), 198-233.
doi: 10.1002/cpa.10056. |
[6] |
T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations , Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17-36.
doi: 10.3934/dcdss.2009.2.17. |
[7] |
T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays , R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194.
doi: 10.1098/rspa.2003.1166. |
[8] |
T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays , J. Differential Equations, 205 (2004), 271-297.
doi: 10.1016/j.jde.2004.04.012. |
[9] |
V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces , J. Math. Pures Appl., 90 (2008), 469-491.
doi: 10.1016/j.matpur.2008.07.001. |
[10] |
V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces , Nonlinearity, 22 (2009), 351-370.
doi: 10.1088/0951-7715/22/2/006. |
[11] |
V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms , Sb. Math, 192 (2001), 11-47.
doi: 10.1070/SM2001v192n01ABEH000534. |
[12] |
V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics. American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. |
[13] |
V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor , J. Dynam. Differential Equations, 19 (2007), 655-684.
doi: 10.1007/s10884-007-9077-y. |
[14] |
A. Cheskidov and S. Lu, The existence and the structure of uniform global attractors for nonautonomous reaction-diffusion systems without uniqueness , Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 55-66.
doi: 10.3934/dcdss.2009.2.55. |
[15] |
T. Colin, The cauchy problem and the continuous limit for the multilayer model in geophysical fluid dynamics , SIAM J. Math. Anal., 28 (1997), 516-529.
doi: 10.1137/S0036141095291269. |
[16] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors , J. Dyn. Differential Equations, 2 (1995), 307-341.
doi: 10.1007/BF02219225. |
[17] |
A. Haraux, Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées,17, Mason, Paris, 1991. |
[18] |
N. Ju, The global attractor for the solutions to the 3D viscous primitive equations , Discrete Contin. Dyn. Syst., 17 (2007), 159-179.
doi: 10.3934/dcds.2007.17.159. |
[19] |
P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization , Numer. Algorithms, 14 (1997), 141-152.
doi: 10.1023/A:1019156812251. |
[20] |
P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations , Dyn. Continuous Impulsive Systems, 4 (1998), 211-226. |
[21] |
J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications , Nonlinearity, 5 (1992), 237-288. |
[22] |
J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean , Nonlinearity, 5 (1992), 1007-1053. |
[23] |
S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces , J. Differential Equations, 230 (2006), 196-212.
doi: 10.1016/j.jde.2006.07.009. |
[24] |
S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces , Discrete Contin. Dyn. Syst., 13 (2005), 701-719.
doi: 10.3934/dcds.2005.13.701. |
[25] |
T. Tachim Medjo, On strong solutions of the multi-layer quasi-geostrophic equations of the ocean , Nonlinear Anal., 68 (2008), 3550-3564.
doi: 10.1016/j.na.2007.03.046. |
[26] |
T. Tachim Medjo, Non-autonomous 3D primitive equations with oscillating external force and its global attractor , Discrete Contin. Dyn. Syst., 32 (2012), 265-291.
doi: 10.3934/dcds.2012.32.265. |
[27] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New-York, second edition, 1987. |
[28] |
P. Peixoto and A. H. Oort, Physics of Climate, American Institute of Physics, New-York, 1992. |
[29] |
R. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation , Appl. Anal, 70 (1998), 147-173.
doi: 10.1080/00036819808840682. |
[30] |
H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations , Nonlinearity, 22 (2009), 667-681.
doi: 10.1088/0951-7715/22/3/008. |
[31] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[32] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS-Chelsea Series, AMS, Providence, 2001. |
[33] |
Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations , Dyn. Syst., 23 (2008), 1-16.
doi: 10.1080/14689360701611821. |
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