On the Partial Regularity of a 3D Model of the Navier-Stokes Equations
We study the partial regularity of a 3D model of the incompressible Navier-Stokes equations which was recently introduced by the authors in . This model is derived for axisymmetric flows with swirl using a set of new variables. It preserves almost all the properties of the full 3D Euler or Navier-Stokes equations except for the convection term which is neglected in the model. If we add the convection term back to our model, we would recover the full Navier-Stokes equations. In , we presented numerical evidence which seems to support that the 3D model develops finite time singularities while the corresponding solution of the 3D Navier-Stokes equations remains smooth. This suggests that the convection term play an essential role in stabilizing the nonlinear vortex stretching term. In this paper, we prove that for any suitable weak solution of the 3D model in an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero. The partial regularity result of this paper is an analogue of the Caffarelli-Kohn-Nirenberg theory for the 3D Navier-Stokes equations.
© 2009 Springer. Received: 10 March 2008 Accepted: 5 September 2008 Published online: 20 November 2008. Communicated by H.-T. Yau. The work was done when Zhen Lei was a postdoctoral scholar at California Institute of Technology. The work was in part supported by NSF under the FRG Grant DMS-0353838, ITR Grant ACI-0204932 and DMS-0713670. Zhen Lei's work was also supported by NSFC under grant 10801029 and Postdoctoral Science Foundation of China under grant 20070410160. The authors would like to thank Prof. Congming Li for valuable comments and suggestions.