Geodesic
Regularity for solutions to the Navier-Stokes equations with one velocity component regular
Electronic journal of differential equations, Tome 2002 (2002)
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 $[0, T]$.
Classification : 35Q30, 76D05
Keywords: Navier-Stokes equations, weak solutions, regularity 
@article{EJDE_2002__2002__a96,
     author = {He,  Cheng},
     title = {Regularity for solutions to the {Navier-Stokes} equations with one velocity component regular},
     journal = {Electronic journal of differential equations},
     year = {2002},
     volume = {2002},
     zbl = {0993.35072},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a96/}
}
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He,  Cheng. Regularity for solutions to the Navier-Stokes equations with one velocity component regular. Electronic journal of differential equations, Tome 2002 (2002). http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a96/