Geodesic
A remark on the regularity for the 3D Navier-Stokes equations in terms of the two components of the velocity
Electronic journal of differential equations, Tome 2009 (2009)
In this note, we study the regularity of Leray-Hopf weak solutions to the Navier-Stokes equation, with the condition

$ \nabla (u_{1},u_{2},0) \in L^{\frac{2}{1-r}}(0,T; \dot{\mathcal{M}}_{2,3/r} (\mathbb{R}^3) , $

where

$ L^{1/3}(\mathbb{R}^3)\subset \dot{X}_r( \mathbb{R}^3) \subset \dot{\mathcal{M}}_{2,3/r}(\mathbb{R}^3), $

the above regularity condition allows us to improve the results obtained by Fan and Gao [6].
Classification : 35Q30, 35K15, 76D05
Keywords: Navier-Stokes equations, regularity criterion, Morrey-campanato spaces 
@article{EJDE_2009__2009__a75,
     author = {Gala,  Sadek},
     title = {A remark on the regularity for the {3D} {Navier-Stokes} equations in terms of the two components of the velocity},
     journal = {Electronic journal of differential equations},
     year = {2009},
     volume = {2009},
     zbl = {1182.35180},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2009__2009__a75/}
}
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Gala,  Sadek. A remark on the regularity for the 3D Navier-Stokes equations in terms of the two components of the velocity. Electronic journal of differential equations, Tome 2009 (2009). http://geodesic.mathdoc.fr/item/EJDE_2009__2009__a75/