On the optimal rate of vortex stretching for axisymmetric Euler flows without swirl

TL;DR

This paper establishes an upper bound of t^{4/3} for the growth of maximum vorticity in axisymmetric Euler flows without swirl, confirming a long-standing conjecture and linking vorticity growth to conserved quantities.

Contribution

It provides a rigorous proof of the vorticity growth bound in axisymmetric Euler flows without swirl, supporting prior conjectures and numerical findings.

Findings

Proves the t^{4/3} upper bound for vorticity growth.

Links velocity maximum to kinetic energy and conserved vorticity quantities.

Confirms conjecture by Childress and numerical results from previous studies.

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, 2008] and supported by numerical computations from Childress--Gilbert--Valiant [J. Fluid Mech. 2016]. The key is to estimate the velocity maximum by the kinetic energy together with conserved quantities involving the vorticity.