On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations

In (Comm Pure Appl Math 62(4):502-564, 2009), Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier-Stokes equations with swirl. This model shares a number of properties of the 3D incompressible Euler and Navier-Stokes equations. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin or Dirichlet-Robin boundary condition will develop a finite time singularity in an axisymmetric domain. We also provide numerical confirmation for our finite time blowup results. We further demonstrate that the energy of the blowup solution is bounded up to the singularity time, and the blowup mechanism for the mixed Dirichlet-Robin boundary condition is essentially the same as that for the energy conserving homogeneous Dirichlet boundary condition. Finally, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition. Both the analysis and the results we obtain here improve the previous work in a rectangular domain by Hou et al. (Adv Math 230:607-641, 2012) in several respects.