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 $L^{2m}$-norms of vorticity ${\mathrm{\Omega }}_m$ for $m\ge 1$. These are scaled to form the dimensionless sequence $D_{m}= (\varpi_{0}^{-1}\Omega_{m})^{\alpha_{m}}$where$\varpi_{0}$ is a constant frequency and ${\alpha }_m=2m\mathrm{/}(4m-3)$. A numerically testable Navier-Stokes regularity criterion comes from comparing the relative magnitudes of $D_m$ and $D_{m+1}$ while another is furnished by imposing a critical lower bound on ${\int }_0^t{D}_{m}d\tau$. The behaviour of the $D_m$ is also important in the Euler case in suggesting a method by which possible singular behaviour might also be tested.