Blow-up scenarios for 3D NSE exhibiting sub-criticality with respect to the scaling of one-dimensional local sparseness

Abstract

It is shown that, if the vorticity magnitude associated with a (presumed singular) three-dimensional incompressible Navier-Stokes flow blows-up in a manner exhibiting certain {\em time dependent local structure}, then {\em time independent} estimates on the $L^1$ norm of $|\omega |\log \sqrt{1+|\omega {|}^2}$ follow. The implication is that the volume of the region of high vorticity decays at a rate of greater order than a rate connected to the critical scaling of one-dimensional local sparseness and, consequently, the solution becomes sub-critical.

final version, to appear in J. Math. Fluid Mech., 13pp