  In this paper, we will discuss the axisymetric, swirl-free Euler equation in
four and higher dimensions. We will show that in four and higher dimensions the
axisymetric, swirl-free Euler equation has properties which could allow
finite-time singularity formation of a form that is excluded in three
dimensions. We will also consider a model equation that is obtained by taking
the infinite-dimensional limit of the vorticity equation in this setup. This
model exhibits finite-time blowup of a Burgers shock type. The blowup result
for the infinite dimensional model equation heavily suggests that smooth
solutions of the Euler equation exhibit finite-time blowup in sufficiently high
dimensions.
