A regularity criterion for axially symmetric solutions to the Navier–Stokes equations

We study the axially-symmetric solutions to the Navier–Stokes equations. Assume that the radial component of velocity $(v_r)$ belongs either to $L_\infty(0,T;L_3(\Omega_0))$ or to $v_r/r$ to $L_\infty(0,T;L_{3/2}(\Omega_0))$, where $\Omega_0$ is some neighbourhood of the axis of symmetry. Assume additionally that there exist subdomains $\Omega_k$, $k=1,\dots,N$, such that $\Omega_0\subset\bigcup^N_{k=1}\Omega_k$ and assume that there exist constants $\alpha_1,\alpha_2$ such that either $\big\|v_r\big\|_{L_\infty(0,T;L_3(\Omega_k))}\le\alpha_1$ or $\big\|\frac{v_r}r\Big\|_{L_\infty(0,T;L_{3/2}(\Omega_k))}\le\alpha_2$ for $k=1,\dots,N$. Then the weak solution becomes strong ($v\in W_2^{2,1}(\Omega\times(0,T))$, $\nabla p\in L_2(\Omega\times(0,T))$). Bibl. 28 titles.