A geometric characterization of potential Navier-Stokes singularities

TL;DR

This paper establishes a geometric condition involving vorticity directions that guarantees regularity of solutions to the Navier-Stokes equations, providing new insights into potential singularity formation.

Contribution

It introduces a novel geometric criterion based on vorticity directions within a double cone that ensures solution regularity near potential singularities.

Findings

Vorticity directions confined within a double cone imply regularity.

The method uses control of local vorticity fluxes inspired by classical fluid dynamics laws.

The approach links geometric vorticity properties to solution regularity.

Abstract

For a local suitable weak solution to the Navier-Stokes equations, we prove that if the vorticity vectors belong to a double cone in regions of high vorticity magnitude, then the solution is regular. Roughly speaking this implies that, near a potential singularity, the directions of vorticity cannot avoid any great circle on the unit sphere. Our method, based on the control of local vorticity fluxes, is inspired by the classical Kelvin-Helmholtz law for ideal fluids and the Type I regularity theory for axisymmetric Navier-Stokes solutions.