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 $$t^{4/3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> 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.