A double-logarithmic lifespan bound for axisymmetric Euler equations with a small initial swirl

We study the three-dimensional axisymmetric incompressible Euler equations in ${\mathbb{R}}^3$ with a small initial swirl. Fix $s>7\mathrm{/}2$, and let ${\mathit{v}}_0\in H^s({\mathbb{R}}^3)$ be axisymmetric and divergence-free. If $\epsilon =\mathrm{\parallel }\mathrm{\nabla }\times (v_{0,\theta }{\mathit{e}}_{\theta }){\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$ is sufficiently small, then the lifespan $T_{\ast }$ of the corresponding $H^s$ solution satisfies $$T_{\ast }\ge \frac{C_s}{1+\mathrm{\parallel }{\mathit{v}}_0{\mathrm{\parallel }}_{H^s}}{[\log \log ({\epsilon }^{-1})]}^{3\mathrm{/}7}\mathrm{.}$$ Moreover, if ${\mathit{v}}_0$ only belongs to $H^3$, we improve the fourfold iterated-logarithmic lifespan bound by Li and Zhou (Z. Angew. Math. Phys. 75:219 2024) to a threefold iterated-logarithmic bound.