# American Institute of Mathematical Sciences

January  2012, 32(1): 265-291. doi: 10.3934/dcds.2012.32.265

## 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

Received  July 2010 Revised  December 2010 Published  September 2011

In this article, we consider a non-autonomous three-dimensional primitive model of the ocean with a singularly oscillating external force depending on a small parameter $\epsilon.$ We prove the existence of the uniform global attractor $\mathcal{A}^{\epsilon}$ in $V,$ (i.e., with the $H^1-$regularity). Furthermore, using the method of [13] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of $\mathcal{A}^{\epsilon}$ as $\epsilon$ goes to zero.
Citation: T. Tachim Medjo. Non-autonomous 3D primitive equations with oscillating external force and its global attractor. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 265-291. doi: 10.3934/dcds.2012.32.265
##### 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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##### 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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