Growth estimates for axisymmetric Euler equations without swirl

We consider the axisymmetric Euler equations in $\mathbb{R}^3$ without swirl, and establish several upper and lower bounds for the growth of solutions. On the one hand, we obtain an upper bound $t^2$ for the radial moment $\int_{\mathbb{R}^3} r\omega^\theta dx$, which is the conjectured optimal rate by Childress (Phys. D 237(14-17):1921-1925, 2008). On the other hand, for all initial data satisfying certain symmetry and sign conditions, we prove that the radial moment grows at least like $t/\log t$ as time goes to infinity, and $\|\omega(\cdot,t)\|_{L^p(\mathbb{R}^3)}$ exhibits at least $t^{1/4}$ growth in the limsup sense for all $1\leq p\leq \infty$. To the best of our knowledge, this is the first result to establish power-law $L^p$-norm growth for smooth, compactly supported initial vorticity in $\mathbb{R}^3$. For these initial data, we also show that nearly all vorticity must eventually escape to $r\to\infty$ in the time-integral sense.