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Global attractors and determining modes for the 3D Navier-Stokes-Voight equations. (English) Zbl 1178.37112

Summary: The authors investigate the long-term dynamics of the three-dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid. Specifically, upper bounds for the number of determining modes are derived for the 3D Navier-Stokes-Voight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view the authors consider the Navier-Stokes-Voight model as a non-viscous (inviscid) regularization of the three-dimensional Navier-Stokes equations. Furthermore, it is also shown that the weak solutions of the Navier-Stokes-Voight equations converge, in the appropriate norm, to the weak solutions of the inviscid simplified Bardina model, as the viscosity coefficient \(\nu \rightarrow 0\).

MSC:

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35Q35 PDEs in connection with fluid mechanics
35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
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[1] Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975.
[2] Babin, A. V. and Vishik, M. I., Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. · Zbl 0778.58002
[3] Bardina, J., Ferziger, J. H. and Reynolds, W. C., Improved subgrid scale models for large eddy simulation, 13th AIAA Fluid and Plasma Dynamics Conference, 1980, 80–1357.
[4] Berselli, L. C., Iliescu, T. and Layton, W. J., Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation, Springer-Verlag, New York, 2006. · Zbl 1089.76002
[5] Cao, Y. P., Lunasin, E. M. and Titi, E. S., Global well-posedness of the three dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4(4), 2006, 823–848. · Zbl 1127.35034
[6] Çelebi, A. O., Kalantarov, V. K. and Polat, M., Attractors for the generalized Benjamin-Bona-Mahony equation, J. Diff. Eqs., 157(2), 1999, 439–451. · Zbl 0934.35151 · doi:10.1006/jdeq.1999.3634
[7] Chueshov, I. D., Theory of functionals that uniquely determine the asymptotic dynamics of infinitedimensional dissipative systems, Russ. Math. Sur., 53(4), 1998, 731–776. · Zbl 0948.34035 · doi:10.1070/RM1998v053n04ABEH000057
[8] Cockburn, B., Jones D. A. and Titi, E. S., Determining degrees of freedom for nonlinear dissipative equations, CR Acad. Sci. Paris, 321(5), 1995, 563–568. · Zbl 0844.35081
[9] Cockburn, B., Jones D. A. and Titi, E. S., Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems, Math. Comp., 66, 1997, 1073–1087. · Zbl 0866.35091 · doi:10.1090/S0025-5718-97-00850-8
[10] Constantin, P., Doering C. R. and Titi, E. S., Rigorous estimates of small scales in turbulent flows, J. Math. Phys., 37, 1996, 6152–6156. · Zbl 0862.35083 · doi:10.1063/1.531769
[11] Constantin, P. and Foias, C., Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1988. · Zbl 0687.35071
[12] Constantin, P., Foias, C., Manley, O. P., et al, Determining modes and fractal dimension of turbulent flows, J. Fluid Mech., 150, 1985, 427–440. · Zbl 0607.76054 · doi:10.1017/S0022112085000209
[13] Constantin, P., Foias, C., Nicolaenko, B., et al, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Appl. Math. Sci., 70, Springer-Verlag, New York, 1989. · Zbl 0683.58002
[14] Constantin, P., Foias, C. and Temam, R., Attractors representing turbulent flows, Mem. Amer. Math. Soc., 53(314), 1985, 1–67. · Zbl 0567.35070
[15] Foias, C., Manley, O., Rosa, R., et al, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001. · Zbl 0994.35002
[16] Foias, C., Manley, O. P., Temam, R., et al, Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9(1–2), 1983, 157–188. · Zbl 0584.35007 · doi:10.1016/0167-2789(83)90297-X
[17] Foias, C. and Prodi, G., Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39, 1967, 1–34. · Zbl 0176.54103
[18] Foias, C. and Titi, E. S., Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4, 1991, 135–153. · Zbl 0714.34078 · doi:10.1088/0951-7715/4/1/009
[19] Hale, J. K., Asymptotic Behavior of Dissipative Systems, Math. Sur. Monographs, Vol. 25, A. M. S., Providence, RI, 1988. · Zbl 0642.58013
[20] Holst, M. J. and Titi, E. S., Determining projections and functionals for weak solutions of the Navier-Stokes equations, Recent Developments in Optimization Theory and Nonlinear Analysis, Y. Censor and S. Reich (eds.), Contemp. Math., Vol. 204, A. M. S., Providence, RI, 1997, 125–138. · Zbl 0873.35062
[21] Ilyin, A. A., Attractors for Navier-Stokes equations in domains with finite measure, Nonlinear Anal., 27, 1996, 605–616. · Zbl 0859.35090 · doi:10.1016/0362-546X(95)00112-9
[22] Ilyin, A. A. and Titi, E. S., Sharp estimates for the number of degrees of freedom for the damped-driven 2-D Navier-Stokes equations, J. Nonlinear Sci., 16(3), 2006, 233–253. · Zbl 1106.35049 · doi:10.1007/s00332-005-0720-7
[23] Jones, D. A. and Titi, E. S., Determining finite volume elements for the 2D Navier-Stokes equations, Phys. D, 60, 1992, 165–174. · Zbl 0778.35084 · doi:10.1016/0167-2789(92)90233-D
[24] Jones, D. A. and Titi, E. S., Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana Univ. Math. J., 42, 1993, 875–887. · Zbl 0796.35128 · doi:10.1512/iumj.1993.42.42039
[25] Kalantarov, V. K., Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. LOMI, 152, 1986, 50–54. · Zbl 0621.35022
[26] Kalantarov, V. K., Global behavior of solutions of nonlinear equations of mathematical physics of classical and non-classical type, Postdoctoral Thesis, St. Petersburg, 1988. · Zbl 0645.35011
[27] Kalantarov, V. K., Levant, B. and Titi, E. S., Gevrey regularity of the global attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19, 2009, 133–152. · Zbl 1177.35152 · doi:10.1007/s00332-008-9029-7
[28] Karazeeva, N. A., Kotsiolis, A. A. and Oskolkov, A. P., Dynamical systems generated by initial-boundary value problems for equations of motion of linear viscoelastic fluids, Proc. Steklov Inst. Math., 3, 1991, 73–108. · Zbl 0731.35084
[29] Khouider, B. and Titi, E. S., An inviscid regularization for the surface quasi-geostrophic equation, Comm. Pure Appl. Math., 61, 2008, 1331–1346. · Zbl 1149.35018 · doi:10.1002/cpa.20218
[30] Henshaw, W. D., Kreiss, H. O. and Yström, J., Numerical experiments on the interaction between the large and small-scale motions of the Navier-Stokes equations, Multiscale Model. Simul., 1, 2003, 119–149. · Zbl 1146.76590 · doi:10.1137/S1540345902406240
[31] Ladyzhenskaya, O. A., On the dynamical system generated by the Navier-Stokes equations, Zap. Nauchn. Sem. LOMI, 27, 1972, 91–114. · Zbl 0301.35077
[32] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York, 1963. · Zbl 0121.42701
[33] Ladyzhenskaya, O. A., Attractors for Semigroups and Evolution Equations, Lezioni Lincee, Cambridge University Press, Cambridge, 1991. · Zbl 0755.47049
[34] Ladyzhenskaya, O. A., Solonnikov, V. A. and Uraltseva, N. N., Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967.
[35] Larios, A. and Titi, E. S., On the high-order global regularity of the three-dimensional inviscid {\(\alpha\)}-regularization of various hydrodynmaic models, preprint. · Zbl 1202.35172
[36] Layton, R. and Lewandowski, R., On a well-posed turbulence model, Discrete Continuous Dyn. Sys. B, 6, 2006, 111–128. · Zbl 1089.76028
[37] Levant, B., Ramos, F. and Titi, E. S., On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Commun. Math. Sci., 7, 2009, in press. · Zbl 1188.35135
[38] Moise, I., Rosa, R. and Wang, X. M., Attractors for non-compact semigroups via energy equations, Non-linearity, 11(5), 1998, 1369–1393. · Zbl 0914.35023
[39] Olson, E. and Titi, E. S., Determining modes for continuous data assimilation in 2D turbulence, J. Stat. Phys., 113(5–6), 2003, 799–840. · Zbl 1137.76402 · doi:10.1023/A:1027312703252
[40] Olson, E. and Titi, E. S., Determining modes and Grashof number in 2D turbulence – A numerical case study, 2007, preprint. · Zbl 1178.76190
[41] Oskolkov, A. P., The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem. LOMI, 38, 1973, 98–136.
[42] Oskolkov, A. P., A certain nonstationary quasilinear system with a small parameter, that regularizes the system of Navier-Stokes equations, Problems of Mathematical Analysis, No. 4: Integral and Differential Operators. Differential Equations, St. Petersburg University, St. Petersburg, 143, 1973, 78–87.
[43] Oskolkov, A. P., On the theory of Voight fluids, Zap. Nauchn. Sem. LOMI, 96, 1980, 233–236. · Zbl 0476.76016
[44] Ramos, F. and Titi, E. S., Invariant measures for the 3D Navier-Stokes-Voigt equations and their Navier-Stokes limit, preprint. · Zbl 1387.35462
[45] Robinson, J., Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
[46] Stanislavova, M., Stefanov, A. and Wang, B. X., Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on \(\mathbb{R}\)3, J. Diff. Eqs., 219(2), 2005, 451–483. · Zbl 1160.35354 · doi:10.1016/j.jde.2005.08.004
[47] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. · Zbl 0871.35001
[48] Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis, Third Revised Edition, North-Holland, Amsterdam, 2001. · Zbl 0981.35001
[49] Wang, B. X. and Yang, W. L., Finite-dimensional behaviour for the Benjamin-Bona-Mahony equation, J. Phys. A, 30(13), 1997, 4877–4885. · Zbl 0924.35139 · doi:10.1088/0305-4470/30/13/035
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