# American Institute of Mathematical Sciences

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 and Applied Analysis, 2011, 10 (4) : 1281-1305. doi: 10.3934/cpaa.2011.10.1281
##### 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] 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), 465-489. [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), 198-233. 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), 245-267. 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), 17-36. 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), 141-161. [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), 559-578. [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), 3181-3194. [9] T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. 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), 459-479. 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), 5338-5341. [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), 49-65. 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), 2343-2353. 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), 66-83. 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), 123-140. [16] V. V. Chepyzhov and M. M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math, 192 (2001), 11-47. 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), 469-491. 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), 351-370. 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), 481-500. [20] 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. [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), 655-684. 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), 27-38. [23] A. Cheskidov, Turbulent boundary layer equations, C. R. Acad. Sci. Paris Sér. I, 334 (2002), 423-427. 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), 333-362. 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), 55-66. doi: DOI:10.3934/dcdss.2009.2.55. [26] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differential Equations, 2 (1995), 307-341. 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), 505-519. 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), 1-35. 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), 215-269. 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), L13-L16. 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), 69-78. doi: doi:10.1016/j.physd.2006.06.012. [32] A. Haraux, "Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées," 17, Mason, Paris, 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), 344009(23pp). [34] D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation, J. Phys. Oceanogr., 33 (2003), 2355-2365. [35] N. Ju, The global attractor for the solutions to the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179. 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), 141-152. 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), 211-226. [38] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications, Nonlinearity, 5 (1992), 237-288. [39] J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. [40] 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. [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), 701-719. 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), 1449-1468. 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), 524-544. 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), 147-173. [46] 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. [47] R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1988. [48] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS-Chelsea Series, AMS, Providence, 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), 1703-1736. [50] 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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##### 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] 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), 465-489. [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), 198-233. 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), 245-267. 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), 17-36. 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), 141-161. [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), 559-578. [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), 3181-3194. [9] T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. 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), 459-479. 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), 5338-5341. [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), 49-65. 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), 2343-2353. 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), 66-83. 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), 123-140. [16] V. V. Chepyzhov and M. M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math, 192 (2001), 11-47. 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), 469-491. 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), 351-370. 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), 481-500. [20] 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. [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), 655-684. 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), 27-38. [23] A. Cheskidov, Turbulent boundary layer equations, C. R. Acad. Sci. Paris Sér. I, 334 (2002), 423-427. 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), 333-362. 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), 55-66. doi: DOI:10.3934/dcdss.2009.2.55. [26] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differential Equations, 2 (1995), 307-341. 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), 505-519. 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), 1-35. 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), 215-269. 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), L13-L16. 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), 69-78. doi: doi:10.1016/j.physd.2006.06.012. [32] A. Haraux, "Systèmes dynamiques dissipatifs et applications. Recherches en Mathématiques Appliquées," 17, Mason, Paris, 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), 344009(23pp). [34] D. D. Holm and B. T. Nadiga, Modeling mesocale turbulence in the barotropic double-gyre circulation, J. Phys. Oceanogr., 33 (2003), 2355-2365. [35] N. Ju, The global attractor for the solutions to the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179. 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), 141-152. 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), 211-226. [38] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications, Nonlinearity, 5 (1992), 237-288. [39] J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. [40] 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. [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), 701-719. 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), 1449-1468. 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), 524-544. 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), 147-173. [46] 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. [47] R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1988. [48] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS-Chelsea Series, AMS, Providence, 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), 1703-1736. [50] 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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