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

AbstractWe 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