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May  2014, 13(3): 1119-1140. doi: 10.3934/cpaa.2014.13.1119

## 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

Received  April 2013 Revised  September 2013 Published  December 2013

In this article, we consider a non-autonomous multi-layer quasi-geostrophic equations of the ocean with a singularly oscillating external force $g^{\epsilon}= g_0(t) + \epsilon^{-\rho} g_1(t/\epsilon)$ depending on a small parameter $\epsilon > 0$ and $\rho \in [0, 1)$ together with the averaged system with the external force $g_0(t),$ formally corresponding to the case $\epsilon = 0.$ Under suitable assumptions on the external force, we prove as in [10] the boundness of the uniform global attractor $\mathcal{A}^{\epsilon}$ as well as the upper semi-continuity of the attractors $\mathcal{A}^{\epsilon}$ of the singular systems to the attractor $\mathcal{A}^0$ of the averaged system as $\epsilon \rightarrow 0^+.$ When the external force is small enough and the viscosity is large enough, the convergence rate is controlled by $K \epsilon^{(1 -\rho)}.$ Let us mention that the non-homogenous boundary conditions (and the non-local constraint) present in the multi-layer quasi-geostrophic model makes the estimates more complicated, [3]. These difficulties are overcome using the new formulation presented in [25].
Citation: T. Tachim Medjo. Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1119-1140. doi: 10.3934/cpaa.2014.13.1119
##### 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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