Smoothness Criteria for Navier-Stokes Equations in terms of regularity along the streamlines

Abstract: This article is devoted to a regularity criteria for the Navier-Stokes equa-tions in terms of regularity along the steam lines. More precisely, we proved that if u is a suitable weak solution for the Navier-Stokes equation on [0, T] × R3 satisfying the condition that |u·∇F ||u|γ 6 A|F |, in which F = div ( u|u |), A is some given constant, and γ is some positive number with 0 < γ < 1 3, then it follows that u is smooth over (0, T) × R3