Dynamics of Scaled Norms of Vorticity for the Three-dimensional Navier-Stokes and Euler Equations

A series of numerical experiments is suggested for the three-dimensional Navier-Stokes and Euler equations on a periodic domain based on a set of L2m-norms of vorticity Ωm for m ≥ 1. These are scaled to form the dimensionless sequence Dm=(ϖ0-1 Ωm)αm where ϖ0 is a constant frequency and αm = 2m/(4m-3). A numerically testable Navier-Stokes regularity criterion comes from comparing the relative magnitudes of Dm and Dm+1 while another is furnished by imposing a critical lower bound on ʃ0t Dmdτ The behaviour of the Dm is also important in the Euler case in suggesting a method by which possible singular behaviour might also be tested.