Global regularity for some axisymmetric Euler flows in ${\mathbb{R}}^d$

We consider axisymmetric Euler flows without swirl in ${\mathbb{R}}^d$ with $d\ge 4$, for which the global regularity of smooth solutions is an open problem. When $d=4$, we obtain global regularity under the assumption that the initial vorticity satisfies some decay at infinity and is vanishing at the axis. Assuming further that the initial vorticity is of one sign guarantees global regularity for $d\le 7$.