  We investigate the large time behavior of an axisymmetric model for the 3D
Euler equations. In \cite{HL09}, Hou and Lei proposed a 3D model for the
axisymmetric incompressible Euler and Navier-Stokes equations with swirl. This
model shares many properties of the 3D incompressible Euler and Navier-Stokes
equations. The main difference between the 3D model of Hou and Lei and the
reformulated 3D Euler and Navier-Stokes equations is that the convection term
is neglected in the 3D model. In \cite{HSW09}, the authors proved that the 3D
inviscid model can develop a finite time singularity starting from smooth
initial data on a rectangular domain. A global well-posedness result was also
proved for a class of smooth initial data under some smallness condition. The
analysis in \cite{HSW09} does not apply to the case when the domain is
axisymmetric and unbounded in the radial direction. In this paper, we prove
that the 3D inviscid model with an appropriate Neumann-Robin boundary condition
will develop a finite time singularity starting from smooth initial data in an
axisymmetric domain. Moreover, we prove that the 3D inviscid model has globally
smooth solutions for a class of large smooth initial data with some appropriate
boundary condition.
