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Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $\Bbb R^3$. (English) Zbl 1091.35064
The authors give conditions on vorticity guaranteeing the smoothness of weak solutions to the Navier-Stokes system in $\Bbb R^3$. Since the pioneering work of Beirao da Veiga, this procedure has been used several times in the work of Kozono, Ogawa, Taniuchi. In a paper of Chae and Choe the conditions of Serrin type are imposed only on two components of vorticity in $L_q(L_r)$, while a result of Kozono and Yatsu requires conditions on two components of vorticity in $L_q$(BMO) and $r = \infty$. In this paper, an analogous result is proved in homogeneous Besov spaces $\dot B^0_{r,\sigma}$. The main result reads as follows: Let $T > 0$. Suppose $u(t,x)$ be a weak Leray-Hopf solution to the Navier-Stokes system on $\Bbb R^3 \times (0,T)$ with solenoidal initial value $u_0 \in H^1(\Bbb R^3)$. Set $w = \operatorname{curl} u = [w_1, w_2,w_3], \widetilde{w} = [w_1, w_2, 0] $ and assume that $$ \int_0^T \| \widetilde{w}\|^q_{\dot B^0_{r,\sigma}} \,dt < \infty. $$ Then $u$ is regular provided $\frac{2}{q} + \frac{3}{r} = 2$; $\frac{3}{2} < r \leq \infty$, $\sigma \leq \frac{2r}{3}$.

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