On the stabilizing effect of convection in three-dimensional incompressible flows
We investigate the stabilizing effect of convection in three-dimensional incompressible Euler and Navier-Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. In this paper, we show that the convection term together with the incompressibility condition actually has a surprising stabilizing effect. We demonstrate this by constructing a new three-dimensional model that is derived for axisymmetric flows with swirl using a set of new variables. This model preserves almost all the properties of the full three-dimensional Euler or Navier-Stokes equations except for the convection term, which is neglected in our model. If we added the convection term back to our model, we would recover the full Navier-Stokes equations. We will present numerical evidence that seems to support that the three-dimensional model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new three-dimensional model and how the convection term in the full Euler and Navier-Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time.
© 2008 Wiley Periodicals, Inc. Received November 2007. Revised March 2008. This work was done when Zhen Lei was visiting Applied & Computational Mathematics at the California Institute of Technology. He would like to thank the institute for its hospitality. The authors were in part supported by the National Science Foundation under grants FRG DMS-0353838, ITR ACI-0204932, and DMS-0713670. We would like to thank Prof. Hector Ceniceros, Congming Li, and the two anonymous referees for their valuable comments and suggestions on the original manuscript.