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

Abstract

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).

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Correspondence to Joel Avrin.

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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). https://doi.org/10.1007/s40879-019-00370-3

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Keywords

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

Mathematics Subject Classification

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