Length scales in solutions of the Navier-Stokes equations

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Abstract
A set of ladder inequalities for the 2d and 3d forced Navier-Stokes equations on a periodic domain (0, L)d is developed, leading to a natural definition of a set of length scales. The authors discuss what happens to these scales if intermittent fluctuations in the vorticity field occur, and they consider how these scales compare to those derived from the attractor dimension and the number of determining modes. Their methods are based on estimates of ratios of norms which appear to play a natural role and which make many of the calculations comparatively easy. In 3d they cannot preclude length scales which are significantly shorter than the Kolmogorov length. In 2d their estimate for a length scale l turns out to be (l/L)-21/2 where G is the Grashof number. This estimate of l is shorter than that derived from the attractor dimension. The reason for this is discussed in detail.