On the Optimal Rate of Vortex Stretching for Axisymmetric Euler Flows Without Swirl

Abstract

For axisymmetric flows without swirl and compactly supported initial vorticity, we prove the upper bound of \(t^{4/3}\) for the growth of the vorticity maximum, which was conjectured by Childress (Phys. D 237(14-17):1921-1925, 2008) and supported by numerical computations from Childress–Gilbert–Valiant (J. Fluid Mech. 805:1-30, 2016). The key is to estimate the velocity maximum by the kinetic energy together with conserved quantities involving the vorticity.