Recently Raugel and Sell obtained global existence results for the Navier Stokes equation requiring that certain products involving the size of the data and the thinness of the domain be small. Thus the initial and forcing data could actually be quite large if the domain was thin enough. These results were obtained for periodic, and a case of homogeneous mixed periodic-Dirichlet, boundary conditions. We develop integral-equation techniques that allow us to obtain similar results in the case of purely homogeneous-Dirichlet boundary conditions. Our results are fairly simple to state and hold in a general setting, whereby we replace the role of the thinness of the domain by the reciprocal of the first eigenvalue of the Laplacian. We show further utulity of the integral-equation techniques by bootstrapping global H 1 -bounds, whenever available in 2-d or 3-d, into higher-order global bounds with slightly smoother forcing functions than those assumed by Guillope, but otherwise more general in that L p -integrable singularities in time are allowed. AcademicPress, Inc.

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We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in which hyperviscous terms are added spectrally to the standard incompressible Navier-Stokes equations (NSE). Let P m be the projection onto the first m eigenspaces of A = − , let µ and α be positive constants with α 3/2, and let Q m = I − P m , then we add to the NSE operators µA ϕ in a general family such that A ϕ Q m A α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers past a cutoff λ m 0 where m 0 m, so that for large enough m 0 the inertial-range wavenumbers see only standard NSE viscosity. We first obtain estimates on the Hausdorff and fractal dimensions of the attractor A (respectively dim H A and dim F A). For a constant K α on the order of unity we show ifrepresents characteristic macroscopic length, and l is the Kolmogorov length scale, i.e. l = (ν 3 / ) where is Kolmogorov's mean rate of dissipation of energy in turbulent flow. All bracketed constants and K α are dimensionless and scale-invariant. The estimate grows in m due to the term λ m /λ 1 but at a rate lower than m 3/5 , and the estimate grows in µ as the relative size of ν to µ. The exponent on l 0 /l is significantly less than the Landau-Lifschitz predicted value of 3. If we impose the condition λ m (1/l ) 2 , the estimates become K α [l 0 /l ] 3 for µ ν and As a corollary, for most of the cases of the operators A ϕ in the distinguished-class case that we expect will be typically used in practice we also obtain an M, now of dimension m 0 for m 0 large enough, though under conditions requiring generally larger m 0 than the m in the special class. In both cases, for large enough m (respectively m 0 ), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics, and in particular trajectories on M are controlled by essentially NSE dynamics.KEY WORDS: 3D turbulent flow models; attractor dimension; inertial manifolds; degrees of freedom.

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