Electron. J. Diff. Eqns., Vol. 2002(2002), No. 29, pp. 1-13.

Regularity for solutions to the Navier-Stokes equations with one velocity component regular

Cheng He

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 positive $T$, 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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Cheng He
Department of Mathematics, Faculty of Science
Kobe University
Rokko, Kobe, 657-8501, Japan
e-mail: chenghe@math.kobe-u.ac.jp
On leave from:
Academy of Mathematics and System Sciences,
Academia Sinica, Beijing, 100080, China.

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