Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces

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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

Abstract
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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].

Keywords: global attractor, Averaging, multi-layer quasi-geostrophic, oscillating force.

Mathematics Subject Classification: Primary: 35Q30,35Q35; Secondary: 35Q7.

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.

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 ,, \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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