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

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  • 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.
    Mathematics Subject Classification: Primary: 35Q30, 35Q35; Secondary: 35Q72.

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  • [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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