On the lifespan of axisymmetric incompressible Euler equations with a small initial swirl

We show the lifespan of a strong solution to the incompressible 3D axially symmetric Euler system in \(\mathbb {R}^3\) ${\mathbb{R}}^3$ can be arbitrarily large if the initial swirl is small enough. A precise lower bound of the lifespan, depending on the size of \(\Vert \nabla \times (v_{0,\theta }\,\varvec{e}_{\varvec{\theta }})\Vert _{L^\infty }\) $\Vert \mathrm{\nabla }\times (v_{0,\theta }{\mathit{e}}_{\mathit{\theta }}){\Vert }_{L^{\infty }}$ , is given. This extends the conclusion of Danchin (Proc Am Math Soc 141:1979–1993, 2012), in which the domain must be distant from the axis of symmetry. This may help us to have a better understanding of the nonlinear framework and the blowup mechanism inherent in the 3D axisymmetric Euler equations.