Convergence of Galerkin solutions and continuous dependence on data in spectrally-hyperviscous models of 3D turbulent flow |
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Authors: | Joel Avrin Chang Xiao |
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Affiliation: | Department of Mathematics and Statistics, University of North Carolina at Charlotte, United States |
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Abstract: | We obtain results on the convergence of Galerkin solutions and continuous dependence on data for the spectrally-hyperviscous Navier-Stokes equations. Let uN denote the Galerkin approximates to the solution u, and let wN=u−uN. Then our main result uses the decomposition wN=PnwN+QnwN where (for fixed n) Pn is the projection onto the first n eigenspaces of A=−Δ and Qn=I−Pn. For assumptions on n that compare well with those in related previous results, the convergence of ‖QnwN(t)Hβ‖ as N→∞ depends linearly on key parameters (and on negative powers of λn), thus reflective of Kolmogorov-theory predictions that in high wavenumber modes viscous (i.e. linear) effects dominate. Meanwhile ‖PnwN(t)Hβ‖ satisfies a more standard exponential estimate, with positive, but fractional, dependence on λn. Modifications of these estimates demonstrate continuous dependence on data. |
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Keywords: | Spectrally-hyperviscous Navier-Stokes equations Galerkin approximations Spectral decomposition methods Strong convergence |
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