On vortex stretching for anti-parallel axisymmetric flows

We consider axisymmetric incompressible inviscid flows without swirl in $\mathbb{R}^3$, under the assumption that the axial vorticity is non-positive in the upper half space and odd in the last coordinate. This flow setup corresponds to the head-on collision of anti-parallel vortex rings. We prove infinite growth of the vorticity impulse on the upper half space. As direct applications, we achieve $\limsup_{t\to\infty} |\omega(t,\cdot)|_{L^\infty} =\infty$ for certain compactly supported data $\omega_0\in C^{1,\alpha}(\mathbb{R}^3)$ with $0<\alpha<3/17$ and $|\omega(t,\cdot)|_{L^p} \gtrsim t^{1/15-}$ with any $2\leq p\leq \infty$ for patch type data with smooth boundary and profile.