Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in ${\mathbb{R}}^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $\mathrm{\mid }v(x,t)\mathrm{\mid }\le C_{\ast }\mathrm{\mid }t{\mathrm{\mid }}^{-1\mathrm{/}2}$ or, for some $\text{e}>0$, $|v (x,t)| \le C_* r^{-1+\epsilon} |t|^{-\epsilon /2}$for$-T_0\le t < 0$and$0<C_*<\infty$allowedtobelarge\mathrm{.}Weprovethat$v$ is regular at time zero.