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

Abstract

We deal with a weak solution v to the Navier-Stokes initial value problem in R^3 x(0,T). We denote by ω^+ a spectral projection of ω=\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 ω^+ or, alternatively, only on ω^+_3 (the third component of ω^+) imply regularity of solution v.

16 pages