  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 $\omega^\theta/r$.
