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Zbl 1196.35153
Dong, Bo-Qing; Zhang, Zhifei
The BKM criterion for the 3D Navier-Stokes equations via two velocity components. (English)
[J] Nonlinear Anal., Real World Appl. 11, No. 4, 2415-2421 (2010). ISSN 1468-1218

From the paper:: The incompressible fluid motion in the whole space $\Bbb R^3$ is governed by the Navier-Stokes equations $$\cases \partial_tu+(u\cdot\nabla)u+\nabla\pi=\Delta u,\\\nabla\cdot u=0,\\ u(x,0)=u_0.\endcases$$ Here $\nabla$ represents the gradient $(\partial_1,\partial_2,\partial_3)$, $u_0$ is a given initial velocity, $u=(u_1,u_2,u_3)$ and $\pi$ denote the unknown velocity vector Field and scalar pressure field of the fluid motion, respectively. Here and in what follows, we use the notations for a vector function $u$, $$(u\cdot\nabla)u=\sum^3_{i=1}u_i\partial_i u_k\quad (k=1,2,3),\quad \nabla\cdot u=\sum^3_{i=1}\partial_iu_i.$$ In the study of the regularity criterion of Leray-Hopf weak solutions to the 3D Navier-Stokes equations, the Beale-Kato-Majda type criterion is obtained in terms of the horizontal derivatives of the two velocity components $$\int^T_0\|\nabla_h\widetilde u(s)\|_{\dot B^0_{\infty,\infty}}\,ds<\infty,\quad \widetilde u=(u_1,u_2,0),\quad \nabla_h\widetilde u=(\partial_1\widetilde u,\partial_2\widetilde u,0).$$

MSC 2000:: *35Q30 Stokes and Navier-Stokes equations
35B65 Smoothness of solutions of PDE
35D30
76D05 Navier-Stokes equations (fluid dynamics)
76D03 Existence, uniqueness, and regularity theory

Keywords: Navier-Stokes equations; Beale-Kato-Majda criterion

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