Quantitative control of solutions to the axisymmetric Navier-Stokes equations in terms of the weak $L^3$ norm

We are concerned with strong axisymmetric solutions to the $3$D incompressible Navier-Stokes equations. We show that if the weak $L^3$ norm of a strong solution $u$ on the time interval $[0,T]$ is bounded by $A\gg 1$ then for each $k\ge 0$ there exists $C_k>1$ such that $\mathrm{\parallel }{D}^{k}u(t){\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^3)}\le t^{-(1+k)\mathrm{/}2}\exp \exp A^{C_k}$ for all $t\in (0,T]$.