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

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.
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:
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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

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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

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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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