Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent
Well-posedness of abstract distributed-order fractional diffusion equations
March 2014, 13(2): 585-603. doi: 10.3934/cpaa.2014.13.585
Regularity criterion for 3D Navier-Stokes equations in Besov spaces
Daoyuan Fang 1, and Chenyin Qian 2,
Department of Mathematics, Zhejiang University, Hangzhou 310027
Department of Mathematics, Zhejiang University, Hangzhou, 310027, China
Received October 2012 Revised July 2013 Published October 2013
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 $\partial_3u_3$.
Keywords: 3D Navier-Stokes equations, Leray-Hopf weak solution, Regularity criterion..
Mathematics Subject Classification: 35Q30, 76D0.
Citation: Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure & Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585
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Daoyuan Fang Chenyin Qian