Regularity criterion for  3D Navier-Stokes equations in  Besov spaces

Daoyuan Fang; Chenyin Qian

doi:10.3934/cpaa.2014.13.585

Article Contents

2014, Volume 13, Issue 2: 585-603. Doi: 10.3934/cpaa.2014.13.585

This issue Previous Article Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent Next Article Well-posedness of abstract distributed-order fractional diffusion equations

Regularity criterion for 3D Navier-Stokes equations in Besov spaces

Daoyuan Fang^1, and
Chenyin Qian^2,

1.
Department of Mathematics, Zhejiang University, Hangzhou 310027
2.
Department of Mathematics, Zhejiang University, Hangzhou, 310027

Received: September 30, 2012

Revised: June 30, 2013

Published: February 2014

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Abstract

Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $\nabla_{h}{u}$ (or $ \nabla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; \dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$, where $\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $\mathbb{R}^3$. Finally, we also get a further regularity criterion, when give the sufficient condition on $\partial_3u_3$.

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Mathematics Subject Classification: 35Q30, 76D05.

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