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

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