Regularity of a Weak Solution to the Navier-Stokes Equations via One Component of a Spectral Projection of Vorticity

TL;DR

This paper establishes conditions involving a spectral projection of vorticity that guarantee the regularity of weak solutions to the Navier-Stokes equations in three dimensions.

Contribution

It introduces new regularity criteria based on a spectral projection of vorticity, focusing on a single component, which advances understanding of solution smoothness.

Findings

Spectral projection of vorticity can determine solution regularity.

Conditions on the third component of the spectral projection imply regularity.

Provides new criteria for weak solution regularity in Navier-Stokes equations.

Abstract

We deal with a weak solution v to the Navier-Stokes initial value problem in R^3 x(0,T). We denote by \omega^+ a spectral projection of \omega=\curl\, v, defined by means of the spectral resolution of identity associated with the self-adjoint operator \curl. We show that certain conditions imposed on \omega^+ or, alternatively, only on \omega^+_3 (the third component of \omega^+) imply regularity of solution v.