## Asymptotic Galerkin convergence and dynamical system results for the 3-D spectrally-hyperviscous Navier-Stokes equations on bounded domains.(English)Zbl 1467.35234

Summary: The spectrally-hyperviscous Navier-Stokes equations (SHNSE) represent a subgrid-scale model of turbulence for which previous studies were limited to periodic-box domains. Then in our work [“The 3-D spectrally-hyperviscous Navier-Stokes equations on bounded domains with zero boundary conditions”, Preprint, arXiv:1908.11005] the SHNSE was adapted to general bounded domains with zero boundary conditions. Here we extend to this new setting the convergence and dynamical-system results in our works [J. Dyn. Differ. Equations 20, No. 2, 479–518 (2008; Zbl 1143.35348); with C. Xiao, J. Differ. Equations 247, No. 10, 2778–2798 (2009; Zbl 1181.35166)], obtaining clear and straightforward Galerkin-convergence estimates, and in the case of decaying turbulence new convergence results featuring asymptotic decay rates in time. In extending the attractor-dimension results in [2008, loc. cit.] our new degrees-of-freedom estimates stay strictly within the Landau-Lifshitz estimates [L. D. Landau and E. M. Lifshitz, Fluid mechanics. Reading: Addison-Wesley. (1959)] for most computationally-relevant parameter values and exhibit a reduction in the number of degrees of freedom in calculations. The foundational properties of our bounded-domain setting also allow us to adapt the quadratic-form machinery of R. Temam [in: Nonlinear partial differential equations and their applications. College de France Seminar Vol. VII, Paris 1983–84, Res. Notes Math. 122, 272–292 (1985; Zbl 0572.35083); “Infinite-dimensional dynamical systems in fluid mechanics”, Proc. Symp. Pure Math. 431–445 (1986; doi:10.1090/pspum/045.2/843630)] to carry over the main inertial-manifold results of the author [J. Dyn. Differ. Equations 20, No. 2, 479–518 (2008; Zbl 1143.35348)].

### MSC:

 35Q30 Navier-Stokes equations 35A35 Theoretical approximation in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B41 Attractors 35B42 Inertial manifolds 76F02 Fundamentals of turbulence 93D20 Asymptotic stability in control theory

### Citations:

Zbl 1143.35348; Zbl 1181.35166; Zbl 0572.35083
Full Text:

### References:

 [1] Avrin, J., Singular initial data and uniform global bounds for the hyperviscous Navier-Stokes equation with periodic boundary conditions, J. Differential Equations, 190, 1, 330-351 (2003) · Zbl 1048.35053 [2] Avrin, J., The asymptotic finite-dimensional character of a spectrally-hyperviscous model of 3D turbulent flow, J. Dynam. Differential Equations, 20, 2, 479-518 (2008) · Zbl 1143.35348 [3] Avrin, J., Exponential asymptotic stability of a class of dynamical systems with applications to models of turbulent flow in two and three dimensions, Proc. Roy. Soc. Edinburgh Sect. A, 142, 2, 225-238 (2012) · Zbl 1236.35012 [4] Avrin, J., High-order Galerkin convergence and boundary characteristics of the 3-D Navier-Stokes equations on intervals of regularity, J. Differential Equations, 257, 7, 2404-2417 (2014) · Zbl 1308.35157 [5] Avrin, J.: The 3-D spectrally-hyperviscous Navier-Stokes equations on bounded domains with zero boundary conditions (2019). arXiv:1908.11005 (submitted) [6] Avrin, J.; Xiao, C., Convergence of Galerkin solutions and continuous dependence on data in spectrally-hyperviscous models of 3D turbulent flow, J. Differential Equations, 247, 10, 2778-2798 (2009) · Zbl 1181.35166 [7] Avrin, J.; Xiao, C., Convergence results for a class of spectrally hyperviscous models of 3-D turbulent flow, J. Math. Anal. Appl., 409, 2, 742-751 (2014) · Zbl 1306.35080 [8] Basdevant, C.; Legras, B.; Sadourny, R.; Béland, M., A study of barotropic model flows: intermittency, waves and predictability, J. Atmos. Sci., 38, 11, 2305-2326 (1981) [9] Borue, V.; Orszag, SA, Numerical study of three-dimensional Kolmogorov flow at high Reynolds numbers, J. Fluid Mech., 306, 293-323 (1996) · Zbl 0857.76034 [10] Borue, V.; Orszag, SA, Local energy flux and subgrid-scale statistics in three-dimensional turbulence, J. Fluid Mech., 366, 1-31 (1998) · Zbl 0924.76035 [11] Cerutti, S.; Meneveau, C.; Knio, OM, Spectral and hyper-eddy viscosity in high-Reynolds-number turbulence, J. Fluid Mech., 421, 307-338 (2000) · Zbl 0958.76507 [12] Cheskidov, A., Global attractors of evolutionary systems, J. Dynam. Differential Equations, 21, 2, 249-268 (2009) · Zbl 1176.35035 [13] Cheskidov, A.; Foias, C., On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231, 2, 714-754 (2006) · Zbl 1113.35140 [14] Chollet, J-P; Lesieur, M., Parametrization of small scales of three-dimensional isotropic turbulence utilizing spectral closures, J. Atmos. Sci., 38, 12, 2747-2757 (1981) [15] Constantin, P.; Foias, C., Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractor for the 2D Navier-Stokes equations, Comm. Pure Appl. Math., 38, 1, 1-27 (1985) · Zbl 0582.35092 [16] Constantin, P.; Foias, C., Navier-Stokes Equations. Chicago Lectures in Mathematics (1988), Chicago: University of Chicago Press, Chicago · Zbl 0687.35071 [17] Constantin, P.; Foias, C.; Manley, OP; Temam, R., Determining modes and fractal dimension of turbulent flows, J. Fluid Mech., 150, 427-440 (1985) · Zbl 0607.76054 [18] Constantin, P.; Foias, C.; Manley, OP; Temam, R., On the dimension of the attractors in two-dimensional turbulence, Phys. D, 30, 3, 284-296 (1988) · Zbl 0658.58030 [19] Foias, C.; Holm, DD; Titi, ES, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152-153, 505-519 (2001) · Zbl 1037.76022 [20] Foias, C.; Holm, DD; Titi, ES, The three dimensional viscous Camassa-Holm equations and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14, 1, 1-35 (2002) · Zbl 0995.35051 [21] Foias, C.; Manley, OP; Temam, R.; Trève, YM, Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9, 1-2, 157-188 (1983) · Zbl 0584.35007 [22] Foias, C.; Manley, O.; Rosa, R.; Temam, R., Navier-Stokes Equations and Turbulence. Encyclopedia of Mathematics and Its Applications (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 0994.35002 [23] Foias, C.; Sell, GR; Temam, R., Variétés inertielles des équations différentielles dissipatives, C. R. Acad. Sci. Paris Ser. I Math., 301, 5, 139-141 (1985) · Zbl 0591.35062 [24] Foias, C.; Sell, GR; Temam, R., Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73, 2, 309-353 (1988) · Zbl 0643.58004 [25] Giga, Y.; Miyakawa, T., Solutions in $$L_r$$ of the Navier-Stokes initial-value problem, Arch. Ration. Mech. Anal., 89, 3, 267-281 (1985) · Zbl 0587.35078 [26] Guermond, J-L; Prudhomme, S., Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows, M2AN Math. Model. Numer. Anal., 37, 6, 893-908 (2003) · Zbl 1070.76035 [27] Guermond, J-L; Oden, JT; Prudhomme, S., Mathematical perspectives on large-eddy simulation models for turbulent flows, J. Math. Fluid Mech., 6, 2, 194-248 (2004) · Zbl 1094.76030 [28] Holm, DD; Mardsen, JE; Ratiu, TS, Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137, 1, 1-81 (1998) · Zbl 0951.37020 [29] Jones, D.; Titi, ES, On the number of determining nodes for the 2D Navier-Stokes equations, J. Math. Anal. Appl., 168, 1, 72-88 (1992) · Zbl 0773.35050 [30] Jones, D.; Titi, ES, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana Univ. Math. J., 42, 3, 875-887 (1993) · Zbl 0796.35128 [31] Kalantarov, VK; Titi, ES, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30, 6, 697-714 (2009) · Zbl 1178.37112 [32] Karamanos, G-S; Karniadakis, GE, A spectral vanishing viscosity method for large-eddy simulations, J. Comput. Phys., 163, 1, 22-50 (2000) · Zbl 0984.76036 [33] Kevlahan, NK-R; Farge, M., Vorticity filaments in two-dimensional turbulence: creation, stability and effect, J. Fluid Mech., 346, 49-76 (1997) · Zbl 0910.76026 [34] Kirby, RM; Sherwin, SJ, Stabilisation of spectral/$$hp$$ element methods through spectral vanishing viscosity: application to fluid mechanics modelling, Comput. Methods Appl. Mech. Engrg., 195, 23-24, 3128-3144 (2006) · Zbl 1115.76060 [35] Kolmogorov, A., The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, C. R. (Doklady) Acad. Sci. URSS (N.S.), 30, 301-305 (1941) · JFM 67.0850.06 [36] Kostianko, A., Inertial manifolds for the 3D modified-Leray-$$\alpha$$ model with periodic boundary conditions, J. Dynam. Differential Equations, 30, 1, 1-24 (2018) · Zbl 1390.35023 [37] Kraichnan, RH, Eddy viscosity in two and three dimensions, J. Atmos. Sci., 33, 8, 1521-1536 (1976) [38] Labovsky, A.; Layton, W., Magnetohydrodynamic flows: Boussinesq conjecture, J. Math. Anal. Appl., 434, 2, 1665-1675 (2016) · Zbl 1329.76388 [39] Landau, LD; Lifshitz, EM, Fluid Mechanics (1959), Reading: Addison-Wesley, Reading [40] Layton, W.: The 1877 Boussinesq assumption: turbulent flows are dissipative on the mean flow. Technical report, University of Pittsburgh (2014) [41] Lions, J-L, Quelques résultats d’existence dans des équations aux dérivées partielles non linéaires, Bull. Soc. Math. France, 87, 245-273 (1959) · Zbl 0147.07902 [42] Liu, J-G; Liu, J.; Pego, R., Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate, Comm. Pure Appl. Math., 60, 10, 1443-1487 (2007) · Zbl 1131.35058 [43] Marsden, JE; Shkoller, S., Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$$\alpha )$$ equations on bounded domains, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359, 1784, 1449-1468 (2001) · Zbl 1006.35074 [44] Minguez, M.; Pasquetti, R.; Serre, E., Spectral vanishing viscosity stabilized LES of the Ahmed body turbulent wake, Comm. Comput. Phys., 5, 2-4, 635-648 (2009) · Zbl 1364.76070 [45] Olson, E.; Titi, ES, Determining modes for continuous data assimilation in 2D turbulence, J. Stat. Phys., 113, 5-6, 799-840 (2003) · Zbl 1137.76402 [46] Tadmor, E., Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal., 26, 1, 30-44 (1989) · Zbl 0667.65079 [47] Tadmor, E.; Baines, MJ; Morton, KW, Super-viscosity and spectral approximations of nonlinear conservation laws, Numerical Methods for Fluid Dynamics, 69-81 (1993), New York: Oxford University Press, New York [48] Temam, R.: Attractors for Navier-Stokes equations. In: Brézis, H., Lions, J.-L. (eds.) Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol. VII. Research Notes in Mathematics, vol. 122, pp. 272-292. Pitman, Boston (1985) · Zbl 0572.35083 [49] Temam, R.: Infinite-dimensional dynamical systems in fluid mechanics. In: Browder, F.E. (ed.) Nonlinear Functional Analysis and its Applications, Part 2. Proceedings of Symposia in Pure Mathematics, vol. 45.2, pp. 431-445. American Mathematical Society, Providence (1986) · Zbl 0598.35095 [50] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences (1997), New York: Springer, New York · Zbl 0871.35001 [51] Younsi, A.: Effect of hyperviscosity on the Navier-Stokes turbulence. Electron. J. Differential Equations 2010, # 110 (2010) · Zbl 1402.76052 [52] Yu, Y., The existence of solution for viscous Camassa-Holm equations on bounded domain in five dimensions, J. Math. Anal. Appl., 429, 2, 849-872 (2015) · Zbl 1316.35084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.