Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 29, pp. 1-13.

Title: Regularity for solutions to the Navier-Stokes equations
       with one velocity component regular
   
Author: Cheng  He (Academia Sinica, Beijing, China)

Abstract: 
 In this paper, we establish a regularity criterion for solutions
 to the Navier-stokes equations, which is only related to one
 component of the velocity field. Let $(u, p)$ be a weak solution
 to the Navier-Stokes equations.
 We show that if any one component of the velocity field $u$,
 for example $u_3$, satisfies either
 $u_3 \in L^\infty({\mathbb{R}}^3\times (0, T))$
 or $\nabla u_3 \in L^p (0, T; L^q({\mathbb{R}}^3))$
 with $1/p + 3/2q = 1/2$ and $q \geq 3$ for some $T > 0$,
 then $u$ is regular on $[0, T]$.
 
 
Submitted November 7, 2001. Published March 17, 2002.
Math Subject Classifications: 35Q30, 76D05.
Key Words: Navier-Stokes equations,  weak solutions, regularity.

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