Asymptotic Galerkin convergence and dynamical system results for the 3-D spectrally-hyperviscous Navier–Stokes equations on bounded domains


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 Avrin (The 3-D spectrally-hyperviscous Navier–Stokes equations on bounded domains with zero boundary conditions. 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 Avrin (J Dyn Differ Equ 20(2):479–518, 2008) and Avrin and Xiao (J Differ Equ 247(10):2778–2798, 2009), 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 Avrin (2008) our new degrees-of-freedom estimates stay strictly within the Landau–Lifschitz estimates (Landau and Lifshitz in Fluid mechanics, Addison-Wesley, Reading, 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 Temam (in: Brézis, Lions (eds) Nonlinear partial differential equations and their applications, Pitman, Boston, 1985; Browder (ed) Nonlinear functional analysis and its applications, American Mathematical Society, Providence, 1986) to carry over the main inertial-manifold results of Avrin (2008).

This is a preview of subscription content, access via your institution.


  1. 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)

    MathSciNet  MATH  Google Scholar 

  2. 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)

    MathSciNet  MATH  Google Scholar 

  3. 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)

    MathSciNet  MATH  Google Scholar 

  4. 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)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Avrin, J.: The 3-D spectrally-hyperviscous Navier–Stokes equations on bounded domains with zero boundary conditions (2019). arXiv:1908.11005 (submitted)

  6. 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)

    MathSciNet  MATH  Google Scholar 

  7. 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)

    MathSciNet  MATH  Google Scholar 

  8. 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)

    Google Scholar 

  9. 9.

    Borue, V., Orszag, S.A.: Numerical study of three-dimensional Kolmogorov flow at high Reynolds numbers. J. Fluid Mech. 306, 293–323 (1996)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Borue, V., Orszag, S.A.: Local energy flux and subgrid-scale statistics in three-dimensional turbulence. J. Fluid Mech. 366, 1–31 (1998)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Cerutti, S., Meneveau, C., Knio, O.M.: Spectral and hyper-eddy viscosity in high-Reynolds-number turbulence. J. Fluid Mech. 421, 307–338 (2000)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Cheskidov, A.: Global attractors of evolutionary systems. J. Dynam. Differential Equations 21(2), 249–268 (2009)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Cheskidov, A., Foias, C.: On global attractors of the 3D Navier–Stokes equations. J. Differential Equations 231(2), 714–754 (2006)

    MathSciNet  MATH  Google Scholar 

  14. 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)

    Google Scholar 

  15. 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)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Constantin, P., Foias, C.: Navier–Stokes Equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1988)

    Google Scholar 

  17. 17.

    Constantin, P., Foias, C., Manley, O.P., Temam, R.: Determining modes and fractal dimension of turbulent flows. J. Fluid Mech. 150, 427–440 (1985)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Constantin, P., Foias, C., Manley, O.P., Temam, R.: On the dimension of the attractors in two-dimensional turbulence. Phys. D 30(3), 284–296 (1988)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Foias, C., Holm, D.D., Titi, E.S.: The Navier–Stokes-alpha model of fluid turbulence. Phys. D 152–153, 505–519 (2001)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Foias, C., Holm, D.D., Titi, E.S.: 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)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Foias, C., Manley, O.P., Temam, R., Trève, Y.M.: Asymptotic analysis of the Navier–Stokes equations. Phys. D 9(1–2), 157–188 (1983)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Foias, C., Manley, O., Rosa, R., Temam, R.: Navier–Stokes Equations and Turbulence. Encyclopedia of Mathematics and Its Applications, vol. 83. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  23. 23.

    Foias, C., Sell, G.R., Temam, R.: Variétés inertielles des équations différentielles dissipatives. C. R. Acad. Sci. Paris Ser. I Math. 301(5), 139–141 (1985)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Foias, C., Sell, G.R., Temam, R.: Inertial manifolds for nonlinear evolutionary equations. J. Differential Equations 73(2), 309–353 (1988)

    MathSciNet  MATH  Google Scholar 

  25. 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)

    MATH  Google Scholar 

  26. 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)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Guermond, J.-L., Oden, J.T., Prudhomme, S.: Mathematical perspectives on large-eddy simulation models for turbulent flows. J. Math. Fluid Mech. 6(2), 194–248 (2004)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Holm, D.D., Mardsen, J.E., Ratiu, T.S.: Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137(1), 1–81 (1998)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Jones, D., Titi, E.S.: On the number of determining nodes for the 2D Navier–Stokes equations. J. Math. Anal. Appl. 168(1), 72–88 (1992)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Jones, D., 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(3), 875–887 (1993)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Kalantarov, V.K., Titi, E.S.: Global attractors and determining modes for the 3D Navier–Stokes–Voight equations. Chin. Ann. Math. Ser. B 30(6), 697–714 (2009)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Karamanos, G.-S., Karniadakis, G.E.: A spectral vanishing viscosity method for large-eddy simulations. J. Comput. Phys. 163(1), 22–50 (2000)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Kevlahan, N.K.-R., Farge, M.: Vorticity filaments in two-dimensional turbulence: creation, stability and effect. J. Fluid Mech. 346, 49–76 (1997)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Kirby, R.M., Sherwin, S.J.: 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)

    MathSciNet  MATH  Google Scholar 

  35. 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)

    MathSciNet  Google Scholar 

  36. 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)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Kraichnan, R.H.: Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33(8), 1521–1536 (1976)

    Google Scholar 

  38. 38.

    Labovsky, A., Layton, W.: Magnetohydrodynamic flows: Boussinesq conjecture. J. Math. Anal. Appl. 434(2), 1665–1675 (2016)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Addison-Wesley, Reading (1959)

    Google Scholar 

  40. 40.

    Layton, W.: The 1877 Boussinesq assumption: turbulent flows are dissipative on the mean flow. Technical report, University of Pittsburgh (2014)

  41. 41.

    Lions, J.-L.: Quelques résultats d’existence dans des équations aux dérivées partielles non linéaires. Bull. Soc. Math. France 87, 245–273 (1959)

    MathSciNet  MATH  Google Scholar 

  42. 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)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Marsden, J.E., 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)

    MathSciNet  MATH  Google Scholar 

  44. 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)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Olson, E., Titi, E.S.: Determining modes for continuous data assimilation in 2D turbulence. J. Stat. Phys. 113(5–6), 799–840 (2003)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Tadmor, E.: Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal. 26(1), 30–44 (1989)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Tadmor, E.: Super-viscosity and spectral approximations of nonlinear conservation laws. In: Baines, M.J., Morton, K.W. (eds.) Numerical Methods for Fluid Dynamics, vol. 4, pp. 69–81. Oxford University Press, New York (1993)

    Google Scholar 

  48. 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)

  49. 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)

  50. 50.

    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68, 2nd edn. Springer, New York (1997)

    Google Scholar 

  51. 51.

    Younsi, A.: Effect of hyperviscosity on the Navier-Stokes turbulence. Electron. J. Differential Equations 2010, # 110 (2010)

  52. 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)

    MathSciNet  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Joel Avrin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Avrin, J. Asymptotic Galerkin convergence and dynamical system results for the 3-D spectrally-hyperviscous Navier–Stokes equations on bounded domains. European Journal of Mathematics 6, 1342–1374 (2020).

Download citation


  • Spectral hyperviscosity
  • Zero boundary conditions
  • Galerkin convergence
  • Stability
  • Degrees of freedom
  • Inertial manifolds

Mathematics Subject Classification

  • 35A35
  • 35B40
  • 35B41
  • 35B42
  • 35Q35
  • 76F02
  • 93D20