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Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in ${ℝ}^{3}$. (English) Zbl 1091.35064

The authors give conditions on vorticity guaranteeing the smoothness of weak solutions to the Navier-Stokes system in ${ℝ}^{3}$. 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 ${L}_{q}\left({L}_{r}\right)$, 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 ${\stackrel{˙}{B}}_{r,\sigma }^{0}$.

The main result reads as follows: Let $T>0$. Suppose $u\left(t,x\right)$ be a weak Leray-Hopf solution to the Navier-Stokes system on ${ℝ}^{3}×\left(0,T\right)$ with solenoidal initial value ${u}_{0}\in {H}^{1}\left({ℝ}^{3}\right)$. Set $w=curlu=\left[{w}_{1},{w}_{2},{w}_{3}\right],\stackrel{˜}{w}=\left[{w}_{1},{w}_{2},0\right]$ and assume that

${\int }_{0}^{T}{\parallel \stackrel{˜}{w}\parallel }_{{\stackrel{˙}{B}}_{r,\sigma }^{0}}^{q}\phantom{\rule{0.166667em}{0ex}}dt<\infty ·$

Then $u$ is regular provided $\frac{2}{q}+\frac{3}{r}=2$; $\frac{3}{2}, $\sigma \le \frac{2r}{3}$.

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