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Consider axisymmetric strong solutions of the incompressible Navier-Stokesequations in $\R^3$ with non-trivial swirl. Such solutions are not known to beglobally defined, but it is shown in \cite{MR673830} that they could only blowup on the axis of symmetry.  Let $z$ denote the axis of symmetry and $r$ measure the distance to thez-axis. Suppose the solution satisfies the pointwise scale invariant bound $|v(x,t)| \le C_*{(r^2 -t)^{-1/2}} $ for $-T_0\le t < 0$ and $0<C_*<\infty$allowed to be large, we then prove that $v$ is regular at time zero.Comment: 25 page

Lower Bound on the Blow-up Rate of the Axisymmetric Navier–Stokes Equations