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 and 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, North-Holland Publishing Co, Amsterdam, 1992.

[2]

T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17-36. doi: doi:10.3934/dcdss.2009.2.17.

[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), 141-161. doi: doi:10.1007/s00245-005-0839-9.

[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), 559-578.

[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), 459-479.

[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), 5338-5341. doi: doi:10.1103/PhysRevLett.81.5338.

[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), 49-65. doi: doi:10.1016/S0167-2789(99)00098-6.

[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), 2343-2353. doi: doi:10.1063/1.870096.

[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), 66-83. doi: doi:10.1016/S0167-2789(99)00099-8.

[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), 123-140.

[11]

V. V. Chepyzhov and M. M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math, 192 (2001), 11-47.

[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), 469-491. doi: doi:10.1016/j.matpur.2008.07.001.

[13]

V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D navier-stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), 351-370. doi: doi:10.1088/0951-7715/22/2/006.

[14]

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

[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), 655-684. doi: doi:10.1007/s10884-007-9077-y.

[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), 27-38.

[17]

A. Cheskidov, Turbulent boundary layer equations, C. R. Acad. Sci. Paris Sér. I, 334 (2002), 423-427.

[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), 55-66. doi: doi:10.3934/dcdss.2009.2.55.

[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), 1-35. doi: doi:10.1023/A:1012984210582.

[20]

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

[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), 344009, 23pp.

[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), 1-81. doi: doi:10.1006/aima.1998.1721.

[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), 4173-4177. doi: doi:10.1103/PhysRevLett.80.4173.

[24]

D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation, J. Phys. Oceanogr, 33 (2003), 2355-2365. doi: doi:10.1175/1520-0485(2003)033<2355:MMTITB>2.0.CO;2.

[25]

P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms, 14 (1997), 141-152. doi: doi:10.1023/A:1019156812251.

[26]

P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dyn. Continuous Impulsive Systems, 4 (1998), 211-226.

[27]

S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212. doi: doi:10.1016/j.jde.2006.07.009.

[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), 701-719. doi: doi:10.3934/dcds.2005.13.701.

[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), 1449-1468. doi: doi:10.1098/rsta.2001.0852.

[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), 524-544. doi: doi:10.1063/1.1533069.

[31]

H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations, Nonlinearity, 22 (2009), 667-681. doi: doi:10.1088/0951-7715/22/3/008.

[32]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1988.

[33]

Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations, Dyn. Syst., 23 (2008), 1-16. 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, Amsterdam, 1992.

[2]

T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17-36. doi: doi:10.3934/dcdss.2009.2.17.

[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), 141-161. doi: doi:10.1007/s00245-005-0839-9.

[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), 559-578.

[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), 459-479.

[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), 5338-5341. doi: doi:10.1103/PhysRevLett.81.5338.

[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), 49-65. doi: doi:10.1016/S0167-2789(99)00098-6.

[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), 2343-2353. doi: doi:10.1063/1.870096.

[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), 66-83. doi: doi:10.1016/S0167-2789(99)00099-8.

[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), 123-140.

[11]

V. V. Chepyzhov and M. M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math, 192 (2001), 11-47.

[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), 469-491. doi: doi:10.1016/j.matpur.2008.07.001.

[13]

V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D navier-stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), 351-370. doi: doi:10.1088/0951-7715/22/2/006.

[14]

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

[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), 655-684. doi: doi:10.1007/s10884-007-9077-y.

[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), 27-38.

[17]

A. Cheskidov, Turbulent boundary layer equations, C. R. Acad. Sci. Paris Sér. I, 334 (2002), 423-427.

[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), 55-66. doi: doi:10.3934/dcdss.2009.2.55.

[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), 1-35. doi: doi:10.1023/A:1012984210582.

[20]

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

[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), 344009, 23pp.

[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), 1-81. doi: doi:10.1006/aima.1998.1721.

[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), 4173-4177. doi: doi:10.1103/PhysRevLett.80.4173.

[24]

D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation, J. Phys. Oceanogr, 33 (2003), 2355-2365. doi: doi:10.1175/1520-0485(2003)033<2355:MMTITB>2.0.CO;2.

[25]

P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms, 14 (1997), 141-152. doi: doi:10.1023/A:1019156812251.

[26]

P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dyn. Continuous Impulsive Systems, 4 (1998), 211-226.

[27]

S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212. doi: doi:10.1016/j.jde.2006.07.009.

[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), 701-719. doi: doi:10.3934/dcds.2005.13.701.

[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), 1449-1468. doi: doi:10.1098/rsta.2001.0852.

[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), 524-544. doi: doi:10.1063/1.1533069.

[31]

H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations, Nonlinearity, 22 (2009), 667-681. doi: doi:10.1088/0951-7715/22/3/008.

[32]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1988.

[33]

Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations, Dyn. Syst., 23 (2008), 1-16. doi: doi:10.1080/14689360701611821.

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