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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

T. Tachim Medjo 1,

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].
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.   Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of evolution equations. Studies in Mathematics and its Applications, 25,, North-Holland Publishing Co, (1992).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics. American Mathematical Society Colloquium Publications, 49,, American Mathematical Society, (2002).   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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H. Crauel, A. Debussche and F. Flandoli, Random attractors ,, \emph{J. Dyn. Differential Equations}, 2 (1995), 307.  doi: 10.1007/BF02219225.  Google Scholar

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A. Haraux, Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées,17,, Mason, (1991).   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[20]

P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations ,, \emph{Dyn. Continuous Impulsive Systems}, 4 (1998), 211.   Google Scholar

[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.   Google Scholar

[22]

J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean ,, \emph{Nonlinearity}, 5 (1992), 1007.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987).   Google Scholar

[28]

P. Peixoto and A. H. Oort, Physics of Climate,, American Institute of Physics, (1992).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS-Chelsea Series, (2001).   Google Scholar

[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.  Google Scholar

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.   Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of evolution equations. Studies in Mathematics and its Applications, 25,, North-Holland Publishing Co, (1992).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics. American Mathematical Society Colloquium Publications, 49,, American Mathematical Society, (2002).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

H. Crauel, A. Debussche and F. Flandoli, Random attractors ,, \emph{J. Dyn. Differential Equations}, 2 (1995), 307.  doi: 10.1007/BF02219225.  Google Scholar

[17]

A. Haraux, Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées,17,, Mason, (1991).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations ,, \emph{Dyn. Continuous Impulsive Systems}, 4 (1998), 211.   Google Scholar

[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.   Google Scholar

[22]

J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean ,, \emph{Nonlinearity}, 5 (1992), 1007.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer-Verlag, (1987).   Google Scholar

[28]

P. Peixoto and A. H. Oort, Physics of Climate,, American Institute of Physics, (1992).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS-Chelsea Series, (2001).   Google Scholar

[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.  Google Scholar

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