Citation: |
[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. |