Smallest scale estimates for the Navier-Stokes equations for incompressible fluids

We consider solutions of the Navier-Stokes equations for incompressible fluids in two and three space dimensions. We obtain improved estimates, in the limit of vanishing viscosity, for the Fourier coefficients. The coefficients decay exponentially fast for wave numbers larger than the square root of the maximum of the velocity gradients divided by the square root of the viscosity. This defines the minimum scale, the size of the smallest feature in the flow.The work of Kreiss was supported in part by National Science Foundation under Grant DMS-8312264 and Office of Naval Research under Contract N-00014-83-K-0422.