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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 3 is governed by the Navier-Stokes equations

t u+(u·)u+π=Δu,·u=0,u(x,0)=u 0 ·

Here represents the gradient ( 1 , 2 , 3 ), u 0 is a given initial velocity, u=(u 1 ,u 2 ,u 3 ) and π 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·)u= i=1 3 u i i u k (k=1,2,3),·u= i=1 3 i u 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

0 T h u ˜(s) B ˙ , 0 ds<,u ˜=(u 1 ,u 2 ,0), h u ˜=( 1 u ˜, 2 u ˜,0)·

35Q30Stokes and Navier-Stokes equations
35B65Smoothness and regularity of solutions of PDE
35D30Weak solutions of PDE
76D05Navier-Stokes equations (fluid dynamics)
76D03Existence, uniqueness, and regularity theory