Length Scales and the Navier-Stokes Equations
Despite the lack of a regularity proof for the 3d Navier-Stokes equations, it is an interesting and important question whether it can be shown that large fluctuations or excursions away from temporal and spatial averages can occur. If it can be demonstrated that the Navier-Stokes equations allow such fluctuations away from averages, then these must have narrow spatial and temporal bandwidths and the width of these will give information about the smallest scale in the flow. Any numerical scheme must ‘resolve’ these spikes to get an accurate representation of the flow. The question of the smallest length scale is the topic of this paper.
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