Global Axisymmetric Solutions to Three-Dimensional Navier–Stokes System

In this paper, we investigate the global well-posedness of three-dimensional Navier–Stokes system with the axisymmetric initial data and with various types of smallness conditions on the swirl component $u^{\theta}_0$ of the initial velocity field. In particular, we proved that if $u^{\theta}_01_{r\leq M}$ is sufficiently small (compared with ∥u0∥L², $\parallel \omega^{\theta}_0\parallel_{L^2}$, and $\parallel \frac {\omega^{\theta}_0}{r}\parallel_{L^2}$) for M large enough, then (1.1) has a unique global smooth solution. Furthermore, we prove that as long as the local axisymmetric smooth solution u satisfying $\parallel u{\theta}1_{r\leq \delta}\parallel_{L^{\alpha}((0,T);L^{\beta})}\lt+\infty$ for δ>0, $\frac {2}{\alpha}+\frac {3}{\beta}\lt 1$ β>6, or (α,β)=(4,6), and any $T\lt \infty$ u will not blow-up in finite time.