Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations

Source abstract

We consider the regularity criteria for a weak solution \documentclass[12pt]{minimal}\begin{document}$\bm {u}=(u_1,u_2,u_3)$\end{document}u=(u1,u2,u3) to the Navier-Stokes equations in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3. Denoting by \documentclass[12pt]{minimal}\begin{document}$\bm {\omega }=(\omega _1,\omega _2,\omega _3)=\mbox{curl }\bm {u}$\end{document}ω=(ω1,ω2,ω3)=curlu the vorticity, we then prove that \documentclass[12pt]{minimal}\begin{document}$\bm {u}$\end{document}u is smooth, provided u3, ω3; or ∂3u3, ω3 are in some Serrin-type integrability classes.