  We consider axisymmetric Euler flows without swirl in $\mathbb{R}^{d}$ with
$d\geq 4$, 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 $d\leq 7$.
