On partial type I solutions to the Axially symmetric Navier-Stokes equations

Let $v= v_{r}e_{r} + v_{\th}e_{\th} + v_{3}e_{3}$ be a Leray-Hopf solution to the axially symmetric Navier-Stokes equations (ASNS). We call it a partial type I solution if $v_r(x, t) \ge -C/\sqrt{T-t}$ for some constant $C>0$ and $(x, t) \in \mathbf{R}^3 \times [0, T)$. In this paper, it is proven that such solution does not blow up at time $T$ under the extra mild assumption that $|v_\theta(x, 0)| |x'|$ is bounded. This extends a well known result by two groups of people who proved the no blowup conclusion under the full type I condition: $|v(x, t)| \le C/\sqrt{T-t}$. The result also confirms the physical intuition that potential blow ups for ASNS are caused by super-critical inward radial velocity.