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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Mathematical Society Colloquium Publications, 49,, American Mathematical Society, (2002).

[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.

[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.

[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.

[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.

[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.

[26]

H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Differential Equations, 2 (1995), 307. doi: DOI: 10.1007/BF02219225.

[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.

[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.

[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.

[30]

B. J. Geurts and D. D. Holm, Regularization modeling for large-eddy simulation,, Phys. Fluids, 15 (2003). doi: DOI:10.1063/1.1529180.

[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.

[32]

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

[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).

[34]

D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation,, J. Phys. Oceanogr., 33 (2003), 2355.

[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.

[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.

[37]

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

[38]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications,, Nonlinearity, 5 (1992), 237.

[39]

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

[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.

[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.

[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.

[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., ().

[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.

[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.

[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.

[47]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," volume 68,, Appl. Math. Sci., (1988).

[48]

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

[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.

[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.

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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Mathematical Society Colloquium Publications, 49,, American Mathematical Society, (2002).

[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.

[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.

[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.

[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.

[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.

[26]

H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Differential Equations, 2 (1995), 307. doi: DOI: 10.1007/BF02219225.

[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.

[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.

[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.

[30]

B. J. Geurts and D. D. Holm, Regularization modeling for large-eddy simulation,, Phys. Fluids, 15 (2003). doi: DOI:10.1063/1.1529180.

[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.

[32]

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

[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).

[34]

D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation,, J. Phys. Oceanogr., 33 (2003), 2355.

[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.

[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.

[37]

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

[38]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications,, Nonlinearity, 5 (1992), 237.

[39]

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

[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.

[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.

[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.

[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., ().

[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.

[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.

[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.

[47]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," volume 68,, Appl. Math. Sci., (1988).

[48]

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

[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.

[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.

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