The BKM criterion for the 3D Navier-Stokes equations via two velocity components. (English) Zbl 1196.35153

From the paper: The incompressible fluid motion in the whole space \(\mathbb R^3\) is governed by the Navier-Stokes equations
\[ \begin{cases} \partial_tu+(u\cdot\nabla)u+\nabla\pi=\Delta u,\\\nabla\cdot u=0,\\ u(x,0)=u_0.\end{cases} \]
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
\[ \int^T_0\|\nabla_h\widetilde u(s)\|_{\dot B^0_{\infty,\infty}}\,ds<\infty,\quad \widetilde u=(u_1,u_2,0),\quad \nabla_h\widetilde u=(\partial_1\widetilde u,\partial_2\widetilde u,0). \]


35Q30 Navier-Stokes equations
35B65 Smoothness and regularity of solutions to PDEs
35D30 Weak solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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