Convergence of Galerkin solutions and continuous dependence on data in spectrally-hyperviscous models of 3D turbulent flow

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:247

Article

Avrin, Joel¹  Xiao, Chang¹ 

[1] Univ N Carolina, Dept Math & Stat, Charlotte, NC 28223 USA

关键词: Spectrally-hyperviscous Navier-Stokes equations;    Galerkin approximations;    Spectral decomposition methods;    Strong convergence;   

DOI  :  10.1016/j.jde.2009.08.012

来源: Elsevier

【 摘 要 】

We obtain results on the convergence of Galerkin solutions and continuous dependence on data for the spectrally-hyperviscous Navier-Stokes equations. Let u(N) denote the Galerkin approximates to the solution u, and let W-N = u - u(N). Then our main result uses the decomposition W-N = PnWN + Q(n)W(N) where (for fixed n) P-n is the projection onto the first n eigenspaces of A = -Delta and Q(n) = I - P-n. For assumptions on n that compare well with those in related previous results. the convergence of parallel to Q(n)W(n)(t)parallel to(H beta) as N -> infinity depends linearly on key parameters (and on negative powers of lambda(n)). thus reflective of Kolmogorov-theory predictions that in high wavenumber modes viscous (i.e. linear) effects dominate. Meanwhile parallel to PnWN(t)parallel to(H beta) satisfies a more standard exponential estimate. with positive, but fractional, dependence on lambda(n). Modifications of these estimates demonstrate continuous dependence on data. (C) 2009 Published by Elsevier Inc.

【 授权许可】

Free