On Vorticity Gradient Growth for the Axisymmetric 3D Euler Equations Without Swirl

Abstract

We consider the 3D axisymmetric Euler equations without swirl on some bounded axial symmetric domains. In this setting, well-posedness is well known due to the essentially 2D geometry. The quantity $\omega^\theta/r$ plays the role of vorticity in 2D. First, we prove that the gradient of $\omega^\theta/r$ can grow at most double exponentially with improving a priori bound close to the axis of symmetry. Next, on the unit ball, we show that at the boundary, one can achieve double exponential growth of the gradient of $\omega^\theta/r$.