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

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 ${\partial }_3{u}_3$.

arXiv admin note: text overlap with arXiv:1005.4463 by other authors

References in corpus (1)

• On the regularity of weak solutions of the 3D Navier-Stokes equations in $B_{\infty ,\infty }^{-1}$

Cited by in corpus (1)

• Some new regularity criteria for the 3D Navier-Stokes Equations