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).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 
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     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},
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     volume = {2009},
     year = {2009},
     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/