A geometric characterization of potential Navier-Stokes singularities

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.