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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