Geodesic
A regularity criterion for the Navier-Stokes equations in terms of the horizontal derivatives of the two velocity components
Electronic journal of differential equations, Tome 2011 (2011)
In this article, we consider the regularity for weak solutions to the Navier-Stokes equations in

$ \nabla _h\widetilde{u}\in L^{2/(2-r)}(0,T;\dot{\mathcal{M}}_{2,3/r} (\mathbb{R}^3)),\quad \hbox{for }01, $

then the weak solution is actually strong, where $\dot{\mathcal{M}} _{2,3/r}$ is the critical Morrey-Campanato space and $\widetilde{u} =(u_1,u_2,0), \nabla_h\widetilde{u}=(\partial _1u_1,\partial _2u_2,0)$.
Classification : 35Q30, 76F65
Keywords: Navier-Stokes equations, Leray-Hopf weak solutions, regularity criterion 
@article{EJDE_2011__2011__a56,
     author = {Chen,  Wenying and Gala,  Sadek},
     title = {A regularity criterion for the {Navier-Stokes} equations in terms of the horizontal derivatives of the two velocity components},
     journal = {Electronic journal of differential equations},
     year = {2011},
     volume = {2011},
     zbl = {1220.35116},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a56/}
}
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Chen,  Wenying; Gala,  Sadek. A regularity criterion for the Navier-Stokes equations in terms of the horizontal derivatives of the two velocity components. Electronic journal of differential equations, Tome 2011 (2011). http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a56/