We　investigate　the　singularity　formation　of　a　3D　model　that　was　recently　proposed　by　Hou　and　Lei　(2009)　in　[15]　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.　(C)　2012　Elsevier　Inc.　All　rights　reserved.