On the Stabilizing Effect of Convectionin Three-Dimensional Incompressible FlowsTHOMAS Y. HOUCalifornia Institute of TechnologyANDZHEN LEINortheast Normal UniversityAbstractWe investigate the stabilizing effect of convection in three-dimensional incom-pressible Euler and Navier-Stokes equations. The convection term is the mainsource of nonlinearity for these equations. It is often considered destabilizing al-though it conserves energy due to the incompressibility condition. In this paper,we show that the convection term together with the incompressibility conditionactually has a surprising stabilizing effect. We demonstrate this by construct-ing a new three-dimensional model that is derived for axisymmetric flows withswirl using a set of new variables. This model preserves almost all the proper-ties of the full three-dimensional Euler or Navier-Stokes equations except for theconvection term, which is neglected in our model. If we added the convectionterm back to our model, we would recover the full Navier-Stokes equations. Wewill present numerical evidence that seems to support that the three-dimensionalmodel may develop a potential finite time singularity. We will also analyzethe mechanism that leads to these singular events in the new three-dimensionalmodel and how the convection term in the full Euler and Navier-Stokes equa-tions destroys such a mechanism, thus preventing the singularity from formingin a finite time.c 2008 Wiley Periodicals, Inc.1 IntroductionThe question of whether a solution of the three-dimensional incompressibleNavier-Stokes equations can develop a finite time singularity from smooth initialdata with finite energy is one of the most outstanding open problems in math-ematics [12]. A main difficulty in obtaining the global regularity of the three-dimensional Navier-Stokes equations is due to the presence of the vortex-stretchingterm, which has a formal quadratic nonlinearity in vorticity. So far, most regularityanalyses for the three-dimensional Navier-Stokes equations use energy estimatesand require some kind of smallness assumption on the initial data [7, 23, 26, 33].Due to the incompressibility condition, the convection term does not contribute tothe energy norm of the velocity field or any Lp(1<p1) norm of the vorticityfield. As a r esult, the convection term has been basically ignored in the regularityCommunications on Pure and Applied Mathematics, Vol. LXI, 0001–0064 (PREPRINT)c2008 Wiley Periodicals, Inc.2 T. Y. HOU AND Z. LEIanalysis for the Navier-Stokes equations. Most of the efforts have focused on howto use the diffusion term to control the nonlinear vortex-stretching term withoutmaking use of the convection term explicitly.In this paper, we show that the convection term has a surprising stabilizingeffect in the three-dimensional incompressible Euler and Navier-Stokes equations.It plays an essential role in depleting the vortex-stretching term. We demonstratethis stabilizing effect of convection by constructing a new three-dimensional modelfor axisymmetric flows with swirl. This model is formulated in terms of a set ofnew variables related to the angular velocity, the angular vorticity, and the angularstream function. The only difference between our three-dimensional model andthe reformulated Navier-Stokes equations in terms of these new variables is thatwe neglect the convection term in the model. If we add the convection term backto the model, we will recover the full Navier-Stokes equations. This new three-dimensional model preserves almost all the properties of the full three-dimensionalEuler or Navier-Stokes equations. In particular, the strong solution of the modelsatisfies an energy identity similar to that of the full three-dimensional Navier-Stokes equations. We also prove a nonblowup criterion of Beale-Kato-Majda type[1] as well as a nonblowup criterion of Prodi-Serrin type [29, 31] for the model.In a subsequent paper, we will prove a new partial regularity result for the model[16], which is an analogue of the Caffarelli-Kohn-Nirenberg theory [2] for the fullNavier-Stokes equations.Despite the striking similarity at the theoretical level between our model and theNavier-Stokes equations, the former has a completely different behavior from thefull Navier-Stokes equations. We will present numerical evidence which seems tosupport that the model may develop a potential finite time singularity from smoothinitial data with finite energy. By exploiting the axisymmetric geometry of theproblem, we obtain a very efficient adaptive solver with an optimal complexitythat provides effective local resolutions of order 40963for the viscous model and81923for the inviscid model. With this level of resolution, we obtain an excellentfit for the asymptotic blowup rate of maximum axial vorticity in the inviscid model.If we denote by !´the axial vorticity component along the ´-direction, we findthat k!´k1.t/ C.T  t/1with a logarithmic correction, and the potentialsingularity approaches the symmetry axis (the ´-axis) as t ! T . Moreover, ourpreliminary study seems to suggest that the potential singularity is locally self-similar and isotropic. We caution that the evidence for singularity formation ofthe inviscid model is not yet conclusive with the current level of resolutions. Itrequires higher numerical resolutions than what we have currently used to givemore definitive evidence.We also present numerical evidence which seems to suggest that the viscousmodel may develop a potential finite time singularity. The behavior of the nearlysingular solution is similar to that of the solution of the inviscid model. We find thatthe solution of the viscous model experiences tremendous dynamic growth. Thegrowth rate is much faster than what has been observed for the three-dimensionalSTABILIZING EFFECT OF CONVECTION IN 3D INCOMPRESSIBLE FLOWS 3Navier-Stokes equations. On the other hand, we observe that the solution of theviscous model seems to be dominated by the dynamics of the inviscid model dur-ing the time interval of our computation. In order to determine whether the three-dimensional model actually develops a finite time singularity and to study the localscaling property of the potential singularity, we need to solve the viscous modelmuch closer to the potential singularity time to capture the viscous effect accu-rately. This would require substantially higher numerical resolutions than whatwe have used in the current paper. Depending on the