Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 11, pp. 1-8.

Title: On the result of He concerning the smoothness of solutions to
       the Navier-Stokes equations

Author: Milan Pokorny (Math. Institute of Charles Univ., Czech Republic)

Abstract: 
 We improve the regularity criterion for the Navier-Stokes equations
 proved  by He [4]. We show that for the Cauchy problem the
 Leray-Hopf weak solution is smooth provided
 $\nabla u_3 \in L^t(0,T;L^s)$, $\frac 2t + \frac 3s \leq \frac 32$.
 
Submitted November 6, 2002. Published February 6, 2003.
Math Subject Classifications: 35Q35, 76D05.
Key Words: Navier-Stokes equations; regularity of systems of PDE's; 
           anisotropic regularity criteria.