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

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