Oscillations and integrability of the vorticity in the 3D NS flows

TL;DR

This paper explores conditions under which the vorticity in 3D Navier-Stokes flows can be controlled to prevent singularities, by analyzing oscillations and integrability properties of the vorticity in relation to blow-up profiles.

Contribution

It introduces a weak vorticity direction condition in a local bmo space that breaks energy-level scaling, suggesting a pathway to regularity in 3D Navier-Stokes flows.

Findings

Weak vorticity direction condition suffices to break energy scaling.

Bounds transform vortex filament scenarios into no-singularity scenarios.

Analysis applies to algebraic blow-up profiles of arbitrary degree.

Abstract

In the studies of the Navier-Stokes (NS) regularity problem, it has become increasingly clear that a more realistic path to improved a priori bounds is to try to break away from the scaling of the energy-level estimates in the realm of the blow-up-type arguments (the solution in view is regular/smooth up to the possible blow-up time) rather than to try to improve regularity of arbitrary Leray's weak solutions. The present article is a contribution in this direction; more precisely, it is shown--in the context of an algebraic/polynomial-type blow-up profile of arbitrary degree--that a very weak condition on the vorticity direction field (membership in a local $bmo$ space weighted with arbitrary many logarithms)--suffices to break the energy-level scaling in the bounds on the vorticity. At the same time, the obtained bounds transform a 3D NS criticality scenario depicted by the macro-scale long vortex filaments into a no-singularity scenario.