March  2011, 10(2): 415-433. doi: 10.3934/cpaa.2011.10.415

A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor

1. 

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

Received  December 2009 Revised  May 2010 Published  December 2010

In this article, we consider a non-autonomous three-dimensional Lagrangian averaged Navier-Stokes-$\alpha$ model with a singulary 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 [15] 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. A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor. Communications on Pure & Applied Analysis, 2011, 10 (2) : 415-433. doi: 10.3934/cpaa.2011.10.415
References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations. Studies in Mathematics and its Applications,", 25, (1992).   Google Scholar

[2]

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

[3]

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.  doi: doi:10.1007/s00245-005-0839-9.  Google Scholar

[4]

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

[5]

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 (2006), 459.   Google Scholar

[6]

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.  doi: doi:10.1103/PhysRevLett.81.5338.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

V. V. Chepyzhov and M. M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms,, Sb. Math, 192 (2001), 11.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

A. Cheskidov, Turbulent boundary layer equations,, C. R. Acad. Sci. Paris S\'er. I, 334 (2002), 423.   Google Scholar

[18]

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

[19]

C. Foais, 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

[20]

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

[21]

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

[22]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories,, Adv. Math., 137 (1998), 1.  doi: doi:10.1006/aima.1998.1721.  Google Scholar

[23]

D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion,, Phys. Rev. Lett., 349 (1998), 4173.  doi: doi:10.1103/PhysRevLett.80.4173.  Google Scholar

[24]

D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation,, J. Phys. Oceanogr, 33 (2003), 2355.  doi: doi:10.1175/1520-0485(2003)033<2355:MMTITB>2.0.CO;2.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

K. Mohseni, Kosovič, S. Shkoller and J. E. Marsden, Numerical simulations of the Lagrangian averaged Navier-Stokes equations for homogeneous turbulence,, Phys. Fluids, 15 (2003), 524.  doi: doi:10.1063/1.1533069.  Google Scholar

[31]

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

[32]

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

[33]

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, (1992).   Google Scholar

[2]

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

[3]

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.  doi: doi:10.1007/s00245-005-0839-9.  Google Scholar

[4]

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

[5]

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 (2006), 459.   Google Scholar

[6]

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.  doi: doi:10.1103/PhysRevLett.81.5338.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

V. V. Chepyzhov and M. M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms,, Sb. Math, 192 (2001), 11.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

A. Cheskidov, Turbulent boundary layer equations,, C. R. Acad. Sci. Paris S\'er. I, 334 (2002), 423.   Google Scholar

[18]

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

[19]

C. Foais, 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

[20]

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

[21]

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

[22]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories,, Adv. Math., 137 (1998), 1.  doi: doi:10.1006/aima.1998.1721.  Google Scholar

[23]

D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion,, Phys. Rev. Lett., 349 (1998), 4173.  doi: doi:10.1103/PhysRevLett.80.4173.  Google Scholar

[24]

D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation,, J. Phys. Oceanogr, 33 (2003), 2355.  doi: doi:10.1175/1520-0485(2003)033<2355:MMTITB>2.0.CO;2.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

K. Mohseni, Kosovič, S. Shkoller and J. E. Marsden, Numerical simulations of the Lagrangian averaged Navier-Stokes equations for homogeneous turbulence,, Phys. Fluids, 15 (2003), 524.  doi: doi:10.1063/1.1533069.  Google Scholar

[31]

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

[32]

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

[33]

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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