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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
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