Averaging of a 3D Lagrangian averaged Navier-Stokes-\alpha model with oscillating external forces

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July 2011, 10(4): 1281-1305. doi: 10.3934/cpaa.2011.10.1281

Averaging of a 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external forces

1.	Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States

Received June 2010 Revised December 2010 Published April 2011

Abstract
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In this article, we consider a non-autonomous three-dimensional Lagrangian averaged Navier-Stokes-$\alpha$ equations with a singularly oscillating external force depending on a small parameter $\epsilon.$ We prove the existence of the uniform global attractor $A^\epsilon.$ Furthermore, using the method of [18] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of $A^\epsilon$ as $\epsilon$ goes to zero.

Keywords: Averaging, global attractor, oscillating force., Navier-Stokes-$\alpha$ model.

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

Citation: T. Tachim Medjo. Averaging of a 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external forces. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1281-1305. doi: 10.3934/cpaa.2011.10.1281

References:

[1]	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
[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. Google Scholar
[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. doi: DOI:10.1002/cpa.10056. Google Scholar
[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., 166 (2007), 245. doi: DOI:10.4007/annals.2007.166.245. Google Scholar
[5]	T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17. doi: DOI:10.3934/dcdss.2009.2.17. Google Scholar
[6]	T. Caraballo, A. M. Márquez-Durán and J. Real, The asymptotic behavior of a stochastic 3D {LANS-$\alpha$ model},, Appl. Math. Optim., 53 (2006), 141. Google Scholar
[7]	T. Caraballo, A. M. Márquez-Durán and J. Real, Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay,, Discrete Contin. Dyn. Syst., 4 (2006), 559. Google Scholar
[8]	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. Google Scholar
[9]	T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271. doi: DOI:10.1016/j.jde.2004.04.012. Google Scholar
[10]	T. Caraballo, J. Real, and T. Taniguchi, On the existence and uniqueness of solutions to stochastic three-dimensional lagrangian averaged Navier-Stokes equations,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2066), 459. doi: DOI:10.1098/rspa.2005.1574. Google Scholar
[11]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows,, Phys. Rev. Lett., 81 (1998), 5338. Google Scholar
[12]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence,, Physica D, 133 (1999), 49. doi: DOI:10.1016/S0167-2789(99)00098-6. Google Scholar
[13]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes,, Phys. Fluids, 11 (1999), 2343. doi: DOI:10.1063/1.870096. Google Scholar
[14]	S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model,, Physica D, 133 (1999), 66. doi: DOI:10.1016/S0167-2789(99)00099-8. Google Scholar
[15]	V. V. Chepyzhov and M. 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. Google Scholar
[16]	V. V. Chepyzhov and M. M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms,, Sb. Math, 192 (2001), 11. doi: DOI:10.1070/SM2001v192n01ABEH000534. Google Scholar
[17]	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. doi: DOI:10.1016/j.matpur.2008.07.001. Google Scholar
[18]	V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces,, Nonlinearity, 22 (2009), 351. doi: DOI:10.1088/0951-7715/22/2/006. Google Scholar
[19]	V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system,, Discrete Contin. Dyn. Syst., 17 (2007), 481. Google Scholar
[20]	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
[21]	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. doi: DOI:10.1007/s10884-007-9077-y. Google Scholar
[22]	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. Google Scholar
[23]	A. Cheskidov, Turbulent boundary layer equations,, C. R. Acad. Sci. Paris S\'er. I, 334 (2002), 423. doi: DOI:10.1016/S1631-073X(02)02275-6. Google Scholar
[24]	A. Cheskidov, Boundary layer for the Navier-Stokes-alpha model of fluid turbulence,, Arch. Ration. Mech. Anal., 3 (2004), 333. doi: DOI: 10.1007/s00205-004-0305-x. Google Scholar
[25]	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. doi: DOI:10.3934/dcdss.2009.2.55. Google Scholar
[26]	H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Differential Equations, 2 (1995), 307. doi: DOI: 10.1007/BF02219225. Google Scholar
[27]	C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence,, Physica D, 152 (2001), 505. doi: DOI:10.1016/S0167-2789(01)00191-9. Google Scholar
[28]	C. Foias, D. D. Holm and E. S. Titi, The three-dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes and turbulence theory,, J. Dynam. Diff. Equat., 14 (2002), 1. doi: DOI: 10.1023/A:1012984210582. Google Scholar
[29]	D. D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion,, Physica D, 133 (1999), 215. doi: doi:10.1016/S0167-2789(99)00093-7. Google Scholar
[30]	B. J. Geurts and D. D. Holm, Regularization modeling for large-eddy simulation,, Phys. Fluids, 15 (2003). doi: DOI:10.1063/1.1529180. Google Scholar
[31]	J. D. Gibbon and D. D. Holm, Length-scale estimates for the LANS-$\alpha$ equations in terms of the Reynolds number,, Phys. D, 220 (2006), 69. doi: doi:10.1016/j.physd.2006.06.012. Google Scholar
[32]	A. Haraux, "Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées," 17,, Mason, (1991). Google Scholar
[33]	M. W. Hecht, D. D. Holm, M. R. Petersen and B. A. Wingate, The LANS-$\alpha$ and Leray turbulence parameterizations in primitive equation ocean modeling,, J. Phy. A: Math. Theor., 41 (2008). Google Scholar
[34]	D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation,, J. Phys. Oceanogr., 33 (2003), 2355. Google Scholar
[35]	N. Ju, The global attractor for the solutions to the 3D viscous primitive equations,, Discrete Contin. Dyn. Syst., 17 (2007), 159. doi: 10.3934/dcds.2007.17.159. Google Scholar
[36]	P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Numer. Algorithms, 14 (1997), 141. doi: DOI: 10.1023/A:1019156812251. Google Scholar
[37]	P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dyn. Continuous Impulsive Systems, 4 (1998), 211. Google Scholar
[38]	J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications,, Nonlinearity, 5 (1992), 237. Google Scholar
[39]	J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean,, Nonlinearity, 5 (1992), 1007. Google Scholar
[40]	S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces,, J. Differential Equations, 230 (2006), 196. doi: doi:10.1016/j.jde.2006.07.009. Google Scholar
[41]	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. doi: doi:10.3934/dcds.2005.13.701. Google Scholar
[42]	J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on bounded domains,, Phil. Trans. R. Soc. Lond. A, 359 (2001), 1449. doi: DOI:10.1098/rsta.2001.0852. Google Scholar
[43]	T. Tachim Medjo, A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor,, Accepted in Communications on Pure and Applied Analysis., (). Google Scholar
[44]	K. Mohseni, Kosovič, S. Shkoller and J. E. Marsden, Numerical simulations of the Lagrangian averaged Navier-Stokes equations for homogeneous isotropic turbulence,, Phys. Fluids, 15 (2003), 524. doi: DOI:10.1063/1.1533069. Google Scholar
[45]	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. Google Scholar
[46]	H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations,, Nonlinearity, 22 (2009), 667. doi: doi: 10.1088/0951-7715/22/3/008. Google Scholar
[47]	R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," volume 68,, Appl. Math. Sci., (1988). Google Scholar
[48]	R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", AMS-Chelsea Series, (2001). Google Scholar
[49]	M. I. Vishik, E. S. Titi, and V. V. Chepyzhov, On the convergence of trajectory attractors of the three-dimensional Navier-Stokes-$\alpha$ model as $\alpha $ approaches 0,, Sb. Math., 198 (2007), 1703. Google Scholar
[50]	Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations,, Dyn. Syst., 23 (2008), 1. doi: DOI: 10.1080/14689360701611821. Google Scholar

show all references

References:

[1]	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
[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. Google Scholar
[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. doi: DOI:10.1002/cpa.10056. Google Scholar
[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., 166 (2007), 245. doi: DOI:10.4007/annals.2007.166.245. Google Scholar
[5]	T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17. doi: DOI:10.3934/dcdss.2009.2.17. Google Scholar
[6]	T. Caraballo, A. M. Márquez-Durán and J. Real, The asymptotic behavior of a stochastic 3D {LANS-$\alpha$ model},, Appl. Math. Optim., 53 (2006), 141. Google Scholar
[7]	T. Caraballo, A. M. Márquez-Durán and J. Real, Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay,, Discrete Contin. Dyn. Syst., 4 (2006), 559. Google Scholar
[8]	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. Google Scholar
[9]	T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271. doi: DOI:10.1016/j.jde.2004.04.012. Google Scholar
[10]	T. Caraballo, J. Real, and T. Taniguchi, On the existence and uniqueness of solutions to stochastic three-dimensional lagrangian averaged Navier-Stokes equations,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2066), 459. doi: DOI:10.1098/rspa.2005.1574. Google Scholar
[11]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows,, Phys. Rev. Lett., 81 (1998), 5338. Google Scholar
[12]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence,, Physica D, 133 (1999), 49. doi: DOI:10.1016/S0167-2789(99)00098-6. Google Scholar
[13]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes,, Phys. Fluids, 11 (1999), 2343. doi: DOI:10.1063/1.870096. Google Scholar
[14]	S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model,, Physica D, 133 (1999), 66. doi: DOI:10.1016/S0167-2789(99)00099-8. Google Scholar
[15]	V. V. Chepyzhov and M. 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. Google Scholar
[16]	V. V. Chepyzhov and M. M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms,, Sb. Math, 192 (2001), 11. doi: DOI:10.1070/SM2001v192n01ABEH000534. Google Scholar
[17]	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. doi: DOI:10.1016/j.matpur.2008.07.001. Google Scholar
[18]	V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces,, Nonlinearity, 22 (2009), 351. doi: DOI:10.1088/0951-7715/22/2/006. Google Scholar
[19]	V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system,, Discrete Contin. Dyn. Syst., 17 (2007), 481. Google Scholar
[20]	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
[21]	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. doi: DOI:10.1007/s10884-007-9077-y. Google Scholar
[22]	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. Google Scholar
[23]	A. Cheskidov, Turbulent boundary layer equations,, C. R. Acad. Sci. Paris S\'er. I, 334 (2002), 423. doi: DOI:10.1016/S1631-073X(02)02275-6. Google Scholar
[24]	A. Cheskidov, Boundary layer for the Navier-Stokes-alpha model of fluid turbulence,, Arch. Ration. Mech. Anal., 3 (2004), 333. doi: DOI: 10.1007/s00205-004-0305-x. Google Scholar
[25]	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. doi: DOI:10.3934/dcdss.2009.2.55. Google Scholar
[26]	H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Differential Equations, 2 (1995), 307. doi: DOI: 10.1007/BF02219225. Google Scholar
[27]	C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence,, Physica D, 152 (2001), 505. doi: DOI:10.1016/S0167-2789(01)00191-9. Google Scholar
[28]	C. Foias, D. D. Holm and E. S. Titi, The three-dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes and turbulence theory,, J. Dynam. Diff. Equat., 14 (2002), 1. doi: DOI: 10.1023/A:1012984210582. Google Scholar
[29]	D. D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion,, Physica D, 133 (1999), 215. doi: doi:10.1016/S0167-2789(99)00093-7. Google Scholar
[30]	B. J. Geurts and D. D. Holm, Regularization modeling for large-eddy simulation,, Phys. Fluids, 15 (2003). doi: DOI:10.1063/1.1529180. Google Scholar
[31]	J. D. Gibbon and D. D. Holm, Length-scale estimates for the LANS-$\alpha$ equations in terms of the Reynolds number,, Phys. D, 220 (2006), 69. doi: doi:10.1016/j.physd.2006.06.012. Google Scholar
[32]	A. Haraux, "Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées," 17,, Mason, (1991). Google Scholar
[33]	M. W. Hecht, D. D. Holm, M. R. Petersen and B. A. Wingate, The LANS-$\alpha$ and Leray turbulence parameterizations in primitive equation ocean modeling,, J. Phy. A: Math. Theor., 41 (2008). Google Scholar
[34]	D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation,, J. Phys. Oceanogr., 33 (2003), 2355. Google Scholar
[35]	N. Ju, The global attractor for the solutions to the 3D viscous primitive equations,, Discrete Contin. Dyn. Syst., 17 (2007), 159. doi: 10.3934/dcds.2007.17.159. Google Scholar
[36]	P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Numer. Algorithms, 14 (1997), 141. doi: DOI: 10.1023/A:1019156812251. Google Scholar
[37]	P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dyn. Continuous Impulsive Systems, 4 (1998), 211. Google Scholar
[38]	J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications,, Nonlinearity, 5 (1992), 237. Google Scholar
[39]	J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean,, Nonlinearity, 5 (1992), 1007. Google Scholar
[40]	S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces,, J. Differential Equations, 230 (2006), 196. doi: doi:10.1016/j.jde.2006.07.009. Google Scholar
[41]	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. doi: doi:10.3934/dcds.2005.13.701. Google Scholar
[42]	J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on bounded domains,, Phil. Trans. R. Soc. Lond. A, 359 (2001), 1449. doi: DOI:10.1098/rsta.2001.0852. Google Scholar
[43]	T. Tachim Medjo, A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor,, Accepted in Communications on Pure and Applied Analysis., (). Google Scholar
[44]	K. Mohseni, Kosovič, S. Shkoller and J. E. Marsden, Numerical simulations of the Lagrangian averaged Navier-Stokes equations for homogeneous isotropic turbulence,, Phys. Fluids, 15 (2003), 524. doi: DOI:10.1063/1.1533069. Google Scholar
[45]	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. Google Scholar
[46]	H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations,, Nonlinearity, 22 (2009), 667. doi: doi: 10.1088/0951-7715/22/3/008. Google Scholar
[47]	R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," volume 68,, Appl. Math. Sci., (1988). Google Scholar
[48]	R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", AMS-Chelsea Series, (2001). Google Scholar
[49]	M. I. Vishik, E. S. Titi, and V. V. Chepyzhov, On the convergence of trajectory attractors of the three-dimensional Navier-Stokes-$\alpha$ model as $\alpha $ approaches 0,, Sb. Math., 198 (2007), 1703. Google Scholar
[50]	Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations,, Dyn. Syst., 23 (2008), 1. doi: DOI: 10.1080/14689360701611821. Google Scholar

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