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

TL;DR

This paper investigates blow-up scenarios in 3D Navier-Stokes equations, showing that certain local vorticity structures imply sub-critical behavior, which could prevent singularity formation.

Contribution

It introduces conditions under which the vorticity blow-up leads to sub-critical solutions, advancing understanding of potential singularity mechanisms in 3D Navier-Stokes flows.

Findings

Time-independent estimates on vorticity norms are derived.

High vorticity regions decay faster than critical scaling predicts.

Solutions become sub-critical under certain local structure conditions.

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.