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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 & 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 ,, \emph{Comput. Math. Math. Phys.}, 37 (1997), 348. [2] A. V. Babin and M. I. Vishik, Attractors of evolution equations. Studies in Mathematics and its Applications, 25,, North-Holland Publishing Co, (1992). [3] C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier-Stokes equations and estimate of its dimension ,, \emph{Adv. Math. Sci. Appl.}, 4 (1994), 465. [4] C. Bernier-Kazantsev and I. D. Chueshov, The finiteness of determining degrees of freedom for the quasi-geostrophic multi-layer ocean model ,, \emph{Nonlinear Anal.}, 42 (2000), 1499. 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 ,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 198. doi: 10.1002/cpa.10056. [6] T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations ,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 17. doi: 10.3934/dcdss.2009.2.17. [7] T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays ,, \emph{R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.}, 459 (2003), 3181. doi: 10.1098/rspa.2003.1166. [8] T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays ,, \emph{J. Differential Equations}, 205 (2004), 271. 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 ,, \emph{J. Math. Pures Appl.}, 90 (2008), 469. 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 ,, \emph{Nonlinearity}, 22 (2009), 351. 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 ,, \emph{Sb. Math}, 192 (2001), 11. 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, (2002). [13] V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor ,, \emph{J. Dynam. Differential Equations}, 19 (2007), 655. 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 ,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 55. 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 ,, \emph{SIAM J. Math. Anal.}, 28 (1997), 516. doi: 10.1137/S0036141095291269. [16] H. Crauel, A. Debussche and F. Flandoli, Random attractors ,, \emph{J. Dyn. Differential Equations}, 2 (1995), 307. doi: 10.1007/BF02219225. [17] A. Haraux, Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées,17,, Mason, (1991). [18] N. Ju, The global attractor for the solutions to the 3D viscous primitive equations ,, \emph{Discrete Contin. Dyn. Syst.}, 17 (2007), 159. doi: 10.3934/dcds.2007.17.159. [19] P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization ,, \emph{Numer. Algorithms}, 14 (1997), 141. doi: 10.1023/A:1019156812251. [20] P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations ,, \emph{Dyn. Continuous Impulsive Systems}, 4 (1998), 211. [21] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications ,, \emph{Nonlinearity}, 5 (1992), 237. [22] J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean ,, \emph{Nonlinearity}, 5 (1992), 1007. [23] S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces ,, \emph{J. Differential Equations}, 230 (2006), 196. 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 ,, \emph{Discrete Contin. Dyn. Syst.}, 13 (2005), 701. doi: 10.3934/dcds.2005.13.701. [25] T. Tachim Medjo, On strong solutions of the multi-layer quasi-geostrophic equations of the ocean ,, \emph{Nonlinear Anal.}, 68 (2008), 3550. 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 ,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 265. doi: 10.3934/dcds.2012.32.265. [27] J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987). [28] P. Peixoto and A. H. Oort, Physics of Climate,, American Institute of Physics, (1992). [29] R. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation ,, \emph{Appl. Anal}, 70 (1998), 147. doi: 10.1080/00036819808840682. [30] H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations ,, \emph{Nonlinearity}, 22 (2009), 667. doi: 10.1088/0951-7715/22/3/008. [31] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68., Appl. Math. Sci., (1988). doi: 10.1007/978-1-4684-0313-8. [32] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS-Chelsea Series, (2001). [33] Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations ,, \emph{Dyn. Syst.}, 23 (2008), 1. doi: 10.1080/14689360701611821.

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##### 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 ,, \emph{Comput. Math. Math. Phys.}, 37 (1997), 348. [2] A. V. Babin and M. I. Vishik, Attractors of evolution equations. Studies in Mathematics and its Applications, 25,, North-Holland Publishing Co, (1992). [3] C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier-Stokes equations and estimate of its dimension ,, \emph{Adv. Math. Sci. Appl.}, 4 (1994), 465. [4] C. Bernier-Kazantsev and I. D. Chueshov, The finiteness of determining degrees of freedom for the quasi-geostrophic multi-layer ocean model ,, \emph{Nonlinear Anal.}, 42 (2000), 1499. 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 ,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 198. doi: 10.1002/cpa.10056. [6] T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations ,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 17. doi: 10.3934/dcdss.2009.2.17. [7] T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays ,, \emph{R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.}, 459 (2003), 3181. doi: 10.1098/rspa.2003.1166. [8] T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays ,, \emph{J. Differential Equations}, 205 (2004), 271. 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 ,, \emph{J. Math. Pures Appl.}, 90 (2008), 469. 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 ,, \emph{Nonlinearity}, 22 (2009), 351. 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 ,, \emph{Sb. Math}, 192 (2001), 11. 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, (2002). [13] V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor ,, \emph{J. Dynam. Differential Equations}, 19 (2007), 655. 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 ,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 55. 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 ,, \emph{SIAM J. Math. Anal.}, 28 (1997), 516. doi: 10.1137/S0036141095291269. [16] H. Crauel, A. Debussche and F. Flandoli, Random attractors ,, \emph{J. Dyn. Differential Equations}, 2 (1995), 307. doi: 10.1007/BF02219225. [17] A. Haraux, Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées,17,, Mason, (1991). [18] N. Ju, The global attractor for the solutions to the 3D viscous primitive equations ,, \emph{Discrete Contin. Dyn. Syst.}, 17 (2007), 159. doi: 10.3934/dcds.2007.17.159. [19] P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization ,, \emph{Numer. Algorithms}, 14 (1997), 141. doi: 10.1023/A:1019156812251. [20] P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations ,, \emph{Dyn. Continuous Impulsive Systems}, 4 (1998), 211. [21] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications ,, \emph{Nonlinearity}, 5 (1992), 237. [22] J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean ,, \emph{Nonlinearity}, 5 (1992), 1007. [23] S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces ,, \emph{J. Differential Equations}, 230 (2006), 196. 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 ,, \emph{Discrete Contin. Dyn. Syst.}, 13 (2005), 701. doi: 10.3934/dcds.2005.13.701. [25] T. Tachim Medjo, On strong solutions of the multi-layer quasi-geostrophic equations of the ocean ,, \emph{Nonlinear Anal.}, 68 (2008), 3550. 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 ,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 265. doi: 10.3934/dcds.2012.32.265. [27] J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987). [28] P. Peixoto and A. H. Oort, Physics of Climate,, American Institute of Physics, (1992). [29] R. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation ,, \emph{Appl. Anal}, 70 (1998), 147. doi: 10.1080/00036819808840682. [30] H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations ,, \emph{Nonlinearity}, 22 (2009), 667. doi: 10.1088/0951-7715/22/3/008. [31] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68., Appl. Math. Sci., (1988). doi: 10.1007/978-1-4684-0313-8. [32] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS-Chelsea Series, (2001). [33] Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations ,, \emph{Dyn. Syst.}, 23 (2008), 1. doi: 10.1080/14689360701611821.
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