On the smallest scale for the incompressible Navier-Stokes equations

We prove that, for solutions to the two- and three-dimensional incompressible Navier-Stokes equations, the minimum scale is inversely proportional to the square root of the Reynolds number based on the kinematic viscosity and the maximum of the velocity gradients. The bounds on the velocity gradients can be obtained for two-dimensional flows, but have to be assumed in three dimensions. Numerical results in two dimensions are given which illustrate and substantiate the features of the proof. Implications of the minimum scale result, to the decay rate of the energy spectrum are discussed.

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