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Non-autonomous 3D primitive equations with oscillating external force and its global attractor
1. | Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States |
References:
[1] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[2] |
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. |
[3] |
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. |
[4] |
C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267.
doi: 10.4007/annals.2007.166.245. |
[5] |
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. |
[6] |
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. |
[7] |
T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. |
[8] |
V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating coefficients, International Conference on Differential and Functional Differential Equations (Moscow, 1999), Funct. Differ. Equ., 8 (2001), 123-140. |
[9] |
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. |
[10] |
V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces, J. Math. Pures Appl. (9), 90 (2008), 469-491.
doi: 10.1016/j.matpur.2008.07.001. |
[11] |
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. |
[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. |
[14] |
V. V. Chepyzhov, M. I. Vishik and W. L. Wendland, On non-autonomous Sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst., 12 (2005), 27-38. |
[15] |
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. |
[16] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differential Equations, 2 (1995), 307-341. |
[17] |
G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology," John Wiley and Sons, New York, 1980. |
[18] |
A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Recherches en Mathématiques Appliquées, 17, Mason, Paris, 1991. |
[19] |
C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear Anal., 61 (2005), 425-460.
doi: 10.1016/j.na.2004.12.005. |
[20] |
C. Hu, R. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations, Chin. Ann. of Math. Ser. B, 23 (2002), 277-292.
doi: 10.1142/S0252959902000262. |
[21] |
C. Hu, R. Temam and M. Ziane, The primitive equations of the large scale ocean under the small depth hypothesis, Discrete Contin. Dyn. Syst., 9 (2003), 97-131. |
[22] |
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. |
[23] |
A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278. |
[24] |
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. |
[25] |
P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dyn. Continuous Impulsive Systems, 4 (1998), 211-226. |
[26] |
G. M. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. |
[27] |
J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications, Nonlinearity, 5 (1992), 237-288.
doi: 10.1088/0951-7715/5/2/001. |
[28] |
J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.
doi: 10.1088/0951-7715/5/5/002. |
[29] |
J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAO I), Computational Mechanics Advance, 1 (1993), 3-54. |
[30] |
J. L. Lions, R. Temam and S. Wang, Numerical analysis of the coupled atmosphere and ocean models (CAOII), Computational Mechanics Advance, 1 (1993), 55-120. |
[31] |
J. L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAOIII), Math. Pures et Appl., 73 (1995), 105-163. |
[32] |
J. L. Lions, R. Temam and S. Wang, On mathematical problems for the primitive equations of the ocean: The mesoscale midlatitude case. Lakshmikantham's legacy: A tribute on his 75th birthday, Nonlinear Anal. Ser. A. Theory Methods, 40 (2000), 439-482.
doi: 10.1016/S0362-546X(00)85026-9. |
[33] |
S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212. |
[34] |
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. |
[35] |
T. Tachim Medjo, On the uniqueness of $z$-weak solutions of the three-dimensional primitive equations of the ocean, Nonlinear Anal. Real World Appl., 11 (2010), 1413-1421.
doi: 10.1016/j.nonrwa.2009.02.031. |
[36] |
J. Pedlosky, "Geophysical Fluid Dynamics," Second edition, Springer-Verlag, New-York, 1987. |
[37] |
J. P. Peixoto and A. H. Oort, "Physics of Climate," American Institute of Physics, New-York, 1992. |
[38] |
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. |
[39] |
E. Simmonet, T. Tachim Medjo and R. Temam, Barotropic-baroclinic formulation of the primitive equations of the ocean, Applicable Analysis, 82 (2003), 439-456.
doi: 10.1080/0003681031000094591. |
[40] |
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. |
[41] |
R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Vol. 68, Second edition, Appl. Math. Sci., Springer-Verlag, New York, 1988. |
[42] |
R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS-Chelsea Series, AMS, Providence, 2001. |
[43] |
S. Wang, "On Solvability for the Equations of the Large-Scale Atmospheric Motion," Ph.D thesis, Lanzhou University, China, 1988. |
[44] |
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. |
[45] |
W. M. Washington and C. L. Parkinson, "An Introduction to Three-Dimensional Climate Modeling," Oxford University Press, Oxford, 1986. |
show all references
References:
[1] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[2] |
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. |
[3] |
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. |
[4] |
C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267.
doi: 10.4007/annals.2007.166.245. |
[5] |
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. |
[6] |
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. |
[7] |
T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. |
[8] |
V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating coefficients, International Conference on Differential and Functional Differential Equations (Moscow, 1999), Funct. Differ. Equ., 8 (2001), 123-140. |
[9] |
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. |
[10] |
V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces, J. Math. Pures Appl. (9), 90 (2008), 469-491.
doi: 10.1016/j.matpur.2008.07.001. |
[11] |
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. |
[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. |
[14] |
V. V. Chepyzhov, M. I. Vishik and W. L. Wendland, On non-autonomous Sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst., 12 (2005), 27-38. |
[15] |
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. |
[16] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differential Equations, 2 (1995), 307-341. |
[17] |
G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology," John Wiley and Sons, New York, 1980. |
[18] |
A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Recherches en Mathématiques Appliquées, 17, Mason, Paris, 1991. |
[19] |
C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear Anal., 61 (2005), 425-460.
doi: 10.1016/j.na.2004.12.005. |
[20] |
C. Hu, R. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations, Chin. Ann. of Math. Ser. B, 23 (2002), 277-292.
doi: 10.1142/S0252959902000262. |
[21] |
C. Hu, R. Temam and M. Ziane, The primitive equations of the large scale ocean under the small depth hypothesis, Discrete Contin. Dyn. Syst., 9 (2003), 97-131. |
[22] |
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. |
[23] |
A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278. |
[24] |
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. |
[25] |
P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dyn. Continuous Impulsive Systems, 4 (1998), 211-226. |
[26] |
G. M. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. |
[27] |
J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications, Nonlinearity, 5 (1992), 237-288.
doi: 10.1088/0951-7715/5/2/001. |
[28] |
J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.
doi: 10.1088/0951-7715/5/5/002. |
[29] |
J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAO I), Computational Mechanics Advance, 1 (1993), 3-54. |
[30] |
J. L. Lions, R. Temam and S. Wang, Numerical analysis of the coupled atmosphere and ocean models (CAOII), Computational Mechanics Advance, 1 (1993), 55-120. |
[31] |
J. L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAOIII), Math. Pures et Appl., 73 (1995), 105-163. |
[32] |
J. L. Lions, R. Temam and S. Wang, On mathematical problems for the primitive equations of the ocean: The mesoscale midlatitude case. Lakshmikantham's legacy: A tribute on his 75th birthday, Nonlinear Anal. Ser. A. Theory Methods, 40 (2000), 439-482.
doi: 10.1016/S0362-546X(00)85026-9. |
[33] |
S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212. |
[34] |
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. |
[35] |
T. Tachim Medjo, On the uniqueness of $z$-weak solutions of the three-dimensional primitive equations of the ocean, Nonlinear Anal. Real World Appl., 11 (2010), 1413-1421.
doi: 10.1016/j.nonrwa.2009.02.031. |
[36] |
J. Pedlosky, "Geophysical Fluid Dynamics," Second edition, Springer-Verlag, New-York, 1987. |
[37] |
J. P. Peixoto and A. H. Oort, "Physics of Climate," American Institute of Physics, New-York, 1992. |
[38] |
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. |
[39] |
E. Simmonet, T. Tachim Medjo and R. Temam, Barotropic-baroclinic formulation of the primitive equations of the ocean, Applicable Analysis, 82 (2003), 439-456.
doi: 10.1080/0003681031000094591. |
[40] |
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. |
[41] |
R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Vol. 68, Second edition, Appl. Math. Sci., Springer-Verlag, New York, 1988. |
[42] |
R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS-Chelsea Series, AMS, Providence, 2001. |
[43] |
S. Wang, "On Solvability for the Equations of the Large-Scale Atmospheric Motion," Ph.D thesis, Lanzhou University, China, 1988. |
[44] |
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. |
[45] |
W. M. Washington and C. L. Parkinson, "An Introduction to Three-Dimensional Climate Modeling," Oxford University Press, Oxford, 1986. |
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