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Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $$\mathbb R^3$$. (English) Zbl 1091.35064
The authors give conditions on vorticity guaranteeing the smoothness of weak solutions to the Navier-Stokes system in $$\mathbb R^3$$. 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 $$\mathbb R^3 \times (0,T)$$ with solenoidal initial value $$u_0 \in H^1(\mathbb R^3)$$. 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:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 35B65 Smoothness and regularity of solutions to PDEs
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