  We investigate the singularity formation of a 3D model that was recently
proposed by Hou and Lei in [16] for axisymmetric 3D incompressible
Navier-Stokes equations with swirl. The main difference between the 3D model of
Hou and Lei and the reformulated 3D Navier-Stokes equations is that the
convection term is neglected in the 3D model. This model shares many properties
of the 3D incompressible Navier-Stokes equations. One of the main results of
this paper is that we prove rigorously the finite time singularity formation of
the 3D inviscid model for a class of initial boundary value problems with
smooth initial data of finite energy. We also prove the global regularity of
the 3D inviscid model for a class of small smooth initial data.
