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Approximation of attractors, large eddy simulations and multiscale methods

Recent advances in the mathematical theory of the Navier-Stokes equations have produced new insight in the mathematical theory of turbulence. In particular, the study of the attractor for the Navier-Stokes equations produced the first connection between two approaches to turbulence that seemed far apart, namely the conventional approach of Kolmogorov and the dynamical systems theory approach. Similarly the study of the approximation of the attractor in connection with the newly introduced concept of approximate inertial manifolds has produced a new approach to large eddy simulations and the study of the interaction of small and large eddies in turbulent flows. Our aim in this article is to survey and describe some of the new results concerning the functional properties of the Navier-Stokes equations and to discuss their relevance to turbulence.