Note on the Rate of Vortex Stretching for Axisymmetric Euler Flows Without Swirl

Abstract

In this paper, we investigate Childress's conjecture proposed in [Phys.D 237(14-17):1921-1925, 2008] on the growth rate of the vorticity maximum for axisymmetric swirl-free Euler flows in three and higher dimensions. We consider the setting that the axial vorticity is non-positive in the upper half space and odd in the last coordinate, which corresponds to the flow setup for head-on collision of anti-parallel vortex rings. By introducing the \emph{generalized vertical moment} and proving its monotonicity, we obtain a lower bound for the growth of the vorticity maximum, contingent on the initial decay rate in the $z$-direction. Specifically, for three-dimensional flows with initial vorticity sufficiently fast decay in $z$, we obtain a lower bound of $t^{\frac{1}{2}-}$, thereby improving upon existing results.