  It is shown--within a mathematical framework based on the suitably defined
scale of sparseness of the super-level sets of the positive and negative parts
of the vorticity components, and in the context of a blow-up-type
argument--that the ever-resisting `scaling gap', i.e., the scaling distance
between a regularity criterion and a corresponding a priori bound (shortly, a
measure of the super-criticality of the 3D Navier-Stokes regularity problem),
can be reduced by an algebraic factor; since (independent) fundamental works of
Ladyzhenskaya, Prodi and Serrin as well as Kato and Fujita in 1960s, all the
reductions have been logarithmic in nature, regardless of the functional set up
utilized. More precisely, it is shown that it is possible to obtain an a priori
bound that is algebraically better than the energy-level bound, while keeping
the corresponding regularity criterion at the same level as all the classical
regularity criteria. The mathematics presented was inspired by morphology of
the regions of intense vorticity/velocity gradients observed in computational
simulations of turbulent flows, as well as by the physics of turbulent cascades
and turbulent dissipation.

Analysis of PDEs