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.
© 2014 Springer-Verlag Berlin Heidelberg. Received March 13, 2012. Accepted November 21, 2013. Published online January 9, 2014. Communicated by V. Šveràk. We would like to thank the referee for making several valuable suggestions which help improve the quality of our original manuscript. This work was supported in part by NSF by Grants DMS-0908546 and DMS-1159138. Zhen Lei would like to thank the Applied and Computational Mathematics of Caltech and Prof. Thomas Hou for hosting his visit and for their hospitality during his visit. Zhen Lei was in part supported by NSFC (grant No.11171072), the Foundation for Innovative Research Groups of NSFC (grant No.11121101), FANEDD, Innovation Program of Shanghai Municipal Education Commission (grant No.12ZZ012) and SGST 09DZ2272900. The research of Dr. S. Wang was supported by China 973 Program (Grant no. 2011CB808002), the Grants NSFC 11071009 and CIT&TCD20130312.
Submitted - 1203.2980v2.pdf