The Role of Convection and Nearly Singular Behavior of the 3D Navier-Stokes Equations

The Role of Convection and Nearly Singular Behavior of the 3D Navier-Stokes Equations Thomas Y. Hou Applied and Comput. Mathematics, Caltech PDE Conference in honor of Blake Temple, University of Michigan May 2, 2011

Motivation The question of whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the seven Clay Millennium Problems. u t + (u )u = p + ν u, u = 0, (1) with initial condition u(x, 0) = u 0. Define vorticity ω = u, then ω is governed by ω t + (u )ω = u ω + ν ω. (2) Note u is formally of the same order as ω. Thus the vortex stretching term u ω ω 2. So far, most regularity analysis uses energy estimates and treats the nonlinear terms as a small perturbation to the diffusion term. The global regularity can be obtained only for small data.

A brief review Global existence for small data (Leray, Ladyzhenskaya, Kato, etc). If u 0 L p (p 3) or u 0 L 2 u 0 L 2 is small, then the 3D Navier-Stokes equations have a globally smooth solution. Non-blowup criteria due to G. Prodi 59, J. Serrin 63. A weak solution u of the 3D Navier-Stokes equations is smooth on [0, T ] R 3 provided that u L q t Lp x ([0,T ] R 3 ) < for some p, q satisfying 3 p + 2 q 1 with 3 < p and 2 q <. The critical case of p = 3, q = was proved by L. Escauriaza, G. Seregin, and V. Sverak in 2003. Partial regularity theory (Caffarelli-Kohn-Nirenberg 82, F. Lin 98) For any suitable weak solution of the 3D Navier-Stokes equations on an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero.

Review continud (Beale-Kato-Majda criterion, 1984) u ceases to be classical at T if and only if T 0 ω (t) dt =. Geometry regularity of direction field of ω: Constantin, Fefferman and Majda (1996). Let ω = ω ξ. The solution of 3D Euler is smooth solution up to T if (1) u L (Ω t) is bounded; and (2) t 0 ξ 2 L (Ω τ ) dτ is uniformly bounded for all t < T. For the axisymmetric Navier-Stokes equations, Chen-Strain-Tsai-Yau [07] and Koch-Nadirashvili-Seregin-Sverak [07] recently proved that if u(x, t) C t 1/2 where C is allowed to be large, then the velocity field u is regular at time zero.

The role of convection and 3D Euler Equations Due to the incompressibility condition, the convection term does not contribute to the energy norm of velocity or the L p -norm of ω. ω t + (u )ω = u ω + ν ω. The convection term and the vortex stretching term can be reformulated as a commutator: ω t + (u )ω (ω )u = ν ω. When we consider the two terms together, we preserve the Lagrangian structure of the solution of the 3D Euler equations: ω(x (α, t), t) = X α (α, t)ω 0(α), det( X (α, t)) 1 α where X (α, t) is the flow map: X t = u(x, t), X (α, 0) = α. Convection tends to severely deform and flatten the support of maximum vorticity. Such deformation tends to weaken the nonlinearity of vortex stretching dynamicallty.

Computation of Hou and Li, J. Nonlinear Science, 2006

Dynamic depletion of vortex stretching

Log log plot of maximum vorticity in time

The Stabilizing Effect of Convection Consider the 3D axi-symmetric incompressible Navier-Stokes equations ( ut θ + u r ur θ + u z uz θ = ν 2 1 ) r 2 u θ 1 r ur u θ, (3) ( ωt θ + u r ωr θ + u z ωz θ = ν 2 1 ) r 2 ω θ + 1 ( (u θ ) 2) r + 1 z r ur ω θ, (4) ( 2 1 r 2 ) ψ θ = ω θ, (5) where u θ, ω θ and ψ θ are the angular components of the velocity, vorticity and stream function respectively, and u r = (ψ θ ) z u z = 1 r (rψθ ) r. Note that equations (3)-(5) completely determine the evolution of the 3D axisymmetric Navier-Stokes equations.

Reformulation of axisymmetric Navier-Stokes equations In [CPAM 08], Hou and Li introduced the following new variables: u 1 = u θ /r, ω 1 = ω θ /r, ψ 1 = ψ θ /r, (6) and derived the following equivalent system that governs the dynamics of u 1, ω 1 and ψ 1 as follows: t u 1 + u r r u 1 + u z z u 1 = ν ( r 2 + 3 r ) r + z 2 u1 + 2u 1 ψ 1z, ) z, t ω 1 + u r r ω 1 + u z z ω 1 = ν ( r 2 + 3 r ) r + z 2 ω1 + ( u1 2 ( r 2 + 3 r ) r + z 2 ψ1 = ω 1, where u r = rψ 1z, u z = 2ψ 1 + rψ 1r. Liu and Wang [SINUM07] showed that if u is a smooth velocity field, then u θ, ω θ and ψ θ must satisfy: u θ r=0 = ω θ r=0 = ψ θ r=0 = 0. Thus u 1, ψ 1 and ω 1 are well defiend. (7)

Stabilizing effect of convection through an exact 1D model for the 3D Navier-Stokes equations In [Hou-Li, CPAM, 61 (2008), no. 5, 661 697], we derived an exact 1D model along the z-axis for the Navier-Stokes equations: (u 1 ) t + 2ψ 1 (u 1 ) z = ν(u 1 ) zz + 2 (ψ 1 ) z u 1, (8) (ω 1 ) t + 2ψ 1 (ω 1 ) z = ν(ω 1 ) zz + ( u1 2 ) z, (9) (ψ 1 ) zz = ω 1. (10) Let ũ = u 1, ṽ = (ψ 1 ) z, and ψ = ψ 1. The above system becomes (ũ) t + 2 ψ(ũ) z = ν(ũ) zz 2ṽũ, (11) (ṽ) t + 2 ψ(ṽ) z = ν(ṽ) zz + (ũ) 2 (ṽ) 2 + c(t), (12) where ṽ = ( ψ) z, ṽ z = ω, and c(t) is an integration constant to enforce the mean of ṽ equal to zero.

The 1D model is exact! A surprising result is that the above 1D model is exact. Theorem 2. Let u 1, ψ 1 and ω 1 be the solution of the 1D model (8)-(10) and define u θ (r, z, t) = ru 1 (z, t), ω θ (r, z, t) = rω 1 (z, t), ψ θ (r, z, t) = rψ 1 (z, t). Then (u θ (r, z, t), ω θ (r, z, t), ψ θ (r, z, t)) is an exact solution of the 3D Navier-Stokes equations. Theorem 2 tells us that the 1D model (8)-(10) preserves some essential nonlinear structure of the 3D axisymmetric Navier-Stokes equations.

Energy method does not work for the 1D model! A standard energy estimate for the 1D model would give we get 1 d 2 dt 1 d 2 dt 1 0 1 0 ũ 2 dz = 3 ṽ 2 dz = 1 0 1 0 (ũ) 2 ṽdz ν ũ 2 ṽdz 3 1 0 1 0 ũ 2 z dz, (ṽ) 3 dz ν 1 0 ṽ 2 z dz. One can obtain essentially the same result for the corresponding reaction-diffusion model by dropping convection and c(t): (ũ) t = ν(ũ) zz 2ṽũ, (ṽ) t = ν(ṽ) zz + (ũ) 2 (ṽ) 2, (13) which admits finite time blowup solutions. It is not clear how to control the nonlinear vortex stretching like terms by the diffusion terms, unless we assume T 0 ṽ L dt <, t T.

Global Well-Posedness of the full 1D Model Theorem 3. Assume that ũ(z, 0) and ṽ(z, 0) are in C m [0, 1] with m 1 and periodic with period 1. Then the solution (ũ, ṽ) of the 1D model will be in C m [0, 1] for all times and for ν 0. Proof. The key is to obtain a priori pointwise estimate for the Lyapunov function ũ 2 z + ṽ 2 z. Differentiating the ũ and ṽ-equations w.r.t z, we get (ũ z ) t + 2 ψ(ũ z ) z 2ṽũ z = 2ṽũ z 2ũṽ z + ν(ũ z ) zz, (ṽ z ) t + 2 ψ(ṽ z ) z 2ṽṽ z = 2ũũ z 2ṽṽ z + ν(ṽ z ) zz. Note that the convection term contributes to stability by cancelling one of the nonlinear terms on the right hand side. This gives (ũ z ) t + 2 ψ(ũ z ) z = 2ũṽ z + ν(ũ z ) zz, (14) (ṽ z ) t + 2 ψ(ṽ z ) z = 2ũũ z + ν(ṽ z ) zz. (15)

Multiplying (14) by 2ũ z and (15) by 2ṽ z, we have (ũz 2 ) t + 2 ψ(ũ z 2 ) z = 4ũũ z ṽ z + 2νũ z (ũ z ) zz, (16) (ṽz 2 ) t + 2 ψ(ṽ z 2 ) z = 4ũũ z ṽ z + 2νṽ z (ṽ z ) zz. (17) Now, we add (16) to (17). Surprisingly, the nonlinear vortex stretching-like terms cancel each other exactly. We get (ũ2 z + ṽz 2 ) + 2 ψ ( ũ 2 t z + ṽz 2 ) = 2ν (ũ z z(ũ z ) zz + ṽ z (ṽ z ) zz ). Moreover we can rewrite the diffusion term in the following form: (ũ2 z + ṽz 2 ) + 2 ψ ( ũ 2 t z + ṽz 2 ) = ν ( ũ 2 z z + ṽz 2 ) 2ν [ (ũ zz zz ) 2 + (ṽ zz ) 2]. Thus, (ũz 2 + ṽz 2 ) satisfies a maximum principle for all ν 0: ũz 2 + ṽz 2 L (ũ 0 ) 2 z + (ṽ 0 ) 2 z L.

Construction of a family of globally smooth solutions Theorem 4. Let φ(r) be a smooth cut-off function and u 1, ω 1 and ψ 1 be the solution of the 1D model. Define u θ (r, z, t) = ru 1 (z, t)φ(r) + ũ(r, z, t), ω θ (r, z, t) = rω 1 (z, t)φ(r) + ω(r, z, t), ψ θ (r, z, t) = rψ 1 (z, t)φ(r) + ψ(r, z, t). Then there exists a family of globally smooth functioons ũ, ω and ψ such that u θ, ω θ and ψ θ are globally smooth solutions of the 3D Navier-Stokes equations with finite energy.

A New 3D Model for NSE, [Hou-Lei, CPAM, 09] Recall the reformulated 3D Navier-Stokes equations: t u 1 + u r r u 1 + u z z u 1 = ν ( r 2 + 3 r ) r + z 2 u1 + 2u 1 ψ 1z, t ω 1 + u r r ω 1 + u z z ω 1 = ν ( r 2 + 3 r ) r + z 2 ω1 + ( ) u1 2 z, ( r 2 + 3 r ) r + z 2 ψ1 = ω 1, (18) where u r = rψ 1z, u z = 2ψ 1 + rψ 1r. Our 3D model is derived by simply dropping the convective term from (18): t u 1 = ν ( r 2 + 3 r ) r + z 2 u1 + 2u 1 ψ 1z, t ω 1 = ν ( r 2 + 3 r ) r + z 2 ω1 + (u1 2) z, ( (19) r 2 + 3 r ) r + z 2 ψ1 = ω 1. Note that (19) is already a closed system, and u 1 = u θ /r characterizes the axial vorticity near r = 0.

Properties of the 3D Model [Hou-Lei, CPAM, 09] This 3D model shares many important properties with the axisymmetric Navier-Stokes equations. First of all, one can define an incompressible velocity field in the model equations (19). u(t, x) = u r (t, r, z)e r + u θ (t, r, z)e θ + u z (t, r, z)e z, (20) u θ = ru 1, u r = rψ 1z, u z = 2ψ 1 + rψ 1r, (21) where x = (x 1, x 2, z), r = x1 2 + x 2 2. It is easy to check that u = r u r + z u z + ur r = 0, (22) which is the same as the Navier-Stokes equations.

Properties of the 3D Model continued Our model enjoys the following properties ([Hou-Lei, CPAM-09]): Theorem 5. Energy identity. The strong solution of (19) satisfies 1 d ( u1 2 + 2 Dψ 1 2) ( Du1 r 3 drdz + ν 2 + 2 D 2 ψ 1 2) r 3 drdz = 0, 2 dt which has been proved to be equivalent to that of the Navier-Stokes equations. Here D is the first order derivative operator defined in R 5. Theorem 6. A non-blowup criterion of Beale-Kato-Majda type. A smooth solution (u 1, ω 1, ψ 1 ) of the model (19) for 0 t < T blows up at time t = T if and only if T where u is defined in (20)-(21). 0 u BMO(R 3 )dt =,

Properties of the 3D Model continued Theorem 7. A non-blowup criterion of Prodi-Serrin type. A weak solution (u 1, ω 1, ψ 1 ) of the model (19) is smooth on [0, T ] R 3 provided that u θ L q t Lp x ([0,T ] R 3 ) < (23) for some p, q satisfying 3 p + 2 q 1 with 3 < p and 2 q <. Theorem 8. An analog of Caffarelli-Kohn-Nirenberg partial regularity result [Hou-Lei, CMP-09]. For any suitable weak solution of the 3D model equations (19) on an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero.

Potential singularity formation of the 3D model It is interesting to study the invicsid model. t u 1 = 2u 1 ψ 1z, t ω 1 = (u1 2 ) z, ( r 2 + 3 r ) r + z 2 ψ1 = ω 1. (24) If we let v = log(u1 2 ), then we can further reduce the 3D model to the following nonlocal nonlinear wave equation: v tt = 4 ( ( ) 1 e v ) zz, (25) where = ( 2 r + 3 r r + 2 z ), and e v r 3 drdz C 0. Note that ( ) 1 is a positive smoothing operator. This is a nonlinear nonlocal wave equation along the z-direction.

Numerical evidence for a potential finite time singularity Innitial condition we consider in our numerical computations is given by u 1 (z, r, 0) = (1 + sin(4πz))(r 2 1) 20 (r 2 1.2) 30, 0 r 1, ψ 1 (z, r, 0) = 0, ω 1 (z, r, 0) = 0. A second order finite difference discretization is used in space, and the classical fourth order Runge-Kutta method is used to discretize in time. We use the following coordinate transformation along the r-direction to achieve the adaptivity: r = f (α) α 0.9 sin(πα)/π. We use an effective resolution up to 4096 3 for the 3D problem.

A 3D view of u 1 at t = 0.02.

A 3D view of u 1 at t = 0.021.

Asymptotic blowup rate: u 1 T = 0.02109 and C = 8.20348. C (T t), with

Stabilizing effect of convection. To study the stabilizing effect of convection, we add the convection term back to the 3D model and solve the Navier-Stokes equations using the solution of the 3D model at t = 0.02 as the initial condition.

Contours of initial data for u 1.

Contours of u 1 at t = 0.021, solution of full NSE.

Contours of u 1 at t = 0.0235, solution of full NSE.

Recent theortical progress for the 3D model Theorem 9. [Hou-Shi-Wang, 09]. Consider the 3D inviscid model { u t = 2uψ z, ω t = (u 2 ) z, ( ) x 2 + y 2 + z 2 (26) ψ = ω, 0 x, y, z 1, with boundary condition ψ = 0 at x = 0, 1, y = 0, 1, z = 1, and (α ψ z + ψ) z=0 = 0 for some 0 < α < 1. If the initial conditions, u 0 and ψ 0, are smooth, satisfying u 0 = 0 at z = 0, 1, and log(u 0 )φ(x, y, z)dxdydz 0, (ψ 0 ) z φ(x, y, z)dxdydz > 0, [0,1] 3 [0,1] 3 where φ(x, y, z) = sin(x) sin(y) cosh(α(1 z)), then the 3D inviscid model must develop a finite time singularity. Moreover, if the ω-equation is viscous and ω satisfies the same boundary condition as ψ, then the 3D model with partial viscosity must develop a finite time singularity.

Global regularity for a class of initial-boundary data Let v = ψ z, we can rewrite the 3D model as follows: { u t = 2uv v t = (u 2 ) zz, (x, z) Ω = [0, δ] 3. (27) The initial and boundary conditions are given as follows: v Ω = 4, v(x, y, z, 0) = v 0 (x, y, z), u(x, y, z, 0) = u 0 (x, y, z). (28) Theorem 10. Assume u 2 0, v 0 H m (Ω) for m 2 and v 0 4 for x Ω, then the solution of the 3D model remains smooth for all times as long as the following holds max (4C m, 1) ( v 0 H m + C m u 2 0 H m) δ < 1, (29) where C m is a Sobolev interpolation constant.

Recent progress: potential blowup of full NSE Surprisingly, a recent study of the full 3D axisymmetric NS and Euler equations reveals that the convection terms are not always stabilizing, as suggested by our previous work. Indeed, the solution grows rapidly with a strong, apparently quadratic nonlinearity which seems to be reenforced by the convection terms. The dynamic evolution of u 1 and w 1 on the (r, z)-plane can be visualized in the following contour plots.

Contour plots of u 1 at t = 0.001 Figure: Contour plot of u 1 at t=0.001.

Contour plots of u 1 at t = 0.01 Figure: Contour plot of u 1 at t=0.01.

Contour plots of u 1 at t = 0.02 Figure: Contour plot of u 1 at t=0.02.

Contour plots of u 1 at t = 0.03 Figure: Contour plot of u 1 at t=0.03.

Contour plots of u 1 at t = 0.036 Figure: Contour plot of u 1 at t=0.036.

Contour plots of u 1 at t = 0.043 Figure: Contour plot of u 1 at t=0.043.

Contour plots of w 1 at t = 0.001 Figure: Contour plot of w 1 at t=0.001.

Contour plots of w 1 at t = 0.01 Figure: Contour plot of w 1 at t=0.01.

Contour plots of w1 at t = 0.02 Figure: Contour plot of w1 at t=0.02. T. Y. Hou, Applied Mathematics, Caltech On Global Regularity of 3D NSE

Contour plots of w1 at t = 0.03 Figure: Contour plot of w1 at t=0.03. T. Y. Hou, Applied Mathematics, Caltech On Global Regularity of 3D NSE

Contour plots of w 1 at t = 0.036 Figure: Contour plot of w 1 at t=0.036.

Contour plots of w 1 at t = 0.043 Figure: Contour plot of w 1 at t=0.043.

Velocity field and streamlines at t = 0.043 The solution can also be visualized in 3D. Figure: Velocity field (u r, u θ, u z) and streamlines at t = 0.043.

Vorticity iso-surface at t = 0.043 Figure: Iso-surface of vorticity ω (with ω 1 max ω ) at t = 0.043. 3 Note the nontrivial geometry of the iso-surface.

Vortex lines at t = 0.043 Figure: Vortex lines at t = 0.043. Note the high curvature of the vortex lines near the maximum vorticity region.

Log log plot of the maximum of u 1 and w 1 The following figure shows the rapid, super-double-exponential growth of the maximum of u 1 and w 1 in time. 1.9 log(log(max u1 )) as a function of t 1.85 1.8 log(log(max u1 )) 1.75 1.7 1.65 1.6 1.55 1.5 0.02 0.025 0.03 0.035 0.04 0.045 t Figure: Log log plot of maximum u 1 as a function of time.

Log log plot of the maximum of u 1 and w 1 (Cont d) 2.6 log(log(max w1 )) as a function of t 2.55 2.5 log(log(max w1 )) 2.45 2.4 2.35 2.3 2.25 2.2 0.02 0.025 0.03 0.035 0.04 0.045 t Figure: Log log plot of maximum w 1 as a function of time. To the best of our knowledge, this type of strong nonlinearity has not been observed before.

The quadratic nonlinearity Carefully note that the nonlinear u 1 -stretching term, 2φ 1, is comparable with u 1 at the maximum of u 1. This quadratic nonlinearity is the driving force behind the rapid growth of u 1 and w 1. 0.6 0.55 0.5 Nonlinear stretching of u1 a function of t 2 phi1 z max u1 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.02 0.025 0.03 0.035 0.04 0.045 t Figure: Ratio of the u 1-stretching term, 2φ 1, to u 1 at the maximum of u 1.

Understanding the strong nonlinearity To understand how this quadratic nonlinearity arises, we plot the cross sections of the solution along the r- and z-coordinate lines that pass through the maximum of u 1. 1600 1400 1200 T=0.043, r cross section through max u1, Nr=2048 u1 phi1 z w1 (rescaled) phi1 (rescaled) 1000 800 600 400 200 0 200 0 0.02 0.04 0.06 0.08 0.1 r Figure: Cross section of the solution at t = 0.043 along the r-line that passes through the maximum of u 1.

Understanding the strong nonlinearity (Cont d) 1600 1400 1200 1000 800 600 400 200 0 200 T=0.043, z cross section through max u1, Nz=8192 u1 phi1 z w1 (rescaled) phi1 (rescaled) 400 0 0.02 0.04 0.06 0.08 0.1 0.12 z Figure: Cross section of the solution at t = 0.043 along the z-line that passes through the maximum of u 1.

Understanding the strong nonlinearity (Cont d) Carefully note from the above figures how u 1 and φ 1,z are closely aligned, both in r- and in z-directions. More interestingly, since the radial velocity u r = rφ 1,z is negative near the maximum of u 1, the r-convection term u r u 1,r creates an attracting force that not only fuels the growth of u 1 but also pushes the peak of u 1 towards the axis r = 0. And it is well known that the solution of the axisymmetric NSE can only blow up at r = 0, if there is a blowup at all.

Approaching the axis r = 0 To see how fast the solution is approaching r = 0, we plot the location (r 1, z 1 ) of the maximum of u 1 as a function of time. 3.5 x 10 3 r1 2 as a function of t, where r1 is the r coordinate of maxloc u1 3 2.5 2 r1 2 1.5 1 0.5 0 0.02 0.025 0.03 0.035 0.04 0.045 t Figure: Plot of r 2 1 as a function of time, where (r 1, z 1) is the point at which u 1 attains its maximum.

Approaching the axis r = 0 (Cont d) 3.5 x 10 3 z1 2 as a function of t, where z1 is the z coordinate of maxloc u1 3 2.5 2 z1 2 1.5 1 0.5 0 0.02 0.025 0.03 0.035 0.04 0.045 t Figure: Plot of z 2 1 as a function of time, where (r 1, z 1) is the point at which u 1 attains its maximum.

Asymptotic blowup rate Assuming that the solution blows up at a finite time T with a blowup rate O(T t) α, we can obtain a crude estimate of T and α using a line fitting procedure. 9 x 10 3 1/max u1 as a function of t 8 7 6 1/max u1 5 4 3 2 1 0 0.02 0.025 0.03 0.035 0.04 0.045 t Figure: Plot of 1/ max u 1 as a function of time.

Asymptotic blowup rate (Cont d) 1/max w1 1/2 as a function of t 0.01 0.009 0.008 1/max w1 1/2 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0.02 0.025 0.03 0.035 0.04 0.045 t Figure: Plot of 1/ max w 1 1/2 as a function of time.

Asymptotic blowup rate (Cont d) 0.4 1/max phi1 2 as a function of t 0.35 0.3 0.25 1/phi1 2 0.2 0.15 0.1 0.05 0 0.02 0.025 0.03 0.035 0.04 0.045 t Figure: Plot of 1/ max φ 1 2 as a function of time. The estimated singularity time is T 0.0447.

Concluding remarks This preliminary study shows that the solution of the axisymmetric NSE may develop a finite time singularity, with the blowup rate u 1 = O(T t) 1, w 1 = O(T t) 2, φ 1 = O(T t) 1/2, if the nonlinear alignment between u 1 and φ 1,z is maintained for a sufficiently long period of time. We are carrying out more careful numerical studies to confirm whether a finite-time blowup can occur for NSE (and also for Euler).

Concluding Remarks Our study shows that convection could play an important role in the dynamic depletion of the nonlinear vortex stretching. By neglecting the convection term, we contruct a new 3D model which shares almost all properties of the Navier-Stokes equations, but could develop finite time singularities. This seems to suggest that one should take advantage of the stabilizing effect of convection in an essential way in our global regularity analysis of the 3D NSE. The energy estimates seem too crude to capture the stabilizing effect of convection. A more localized analytic method may be required to study the global regualrity or blowup of 3D NSE.

References J. Deng, T. Y. Hou, and X. Yu, Geometric Properties and the non-blow-up of the Three-Dimensional Euler Equation, Comm. PDEs, 30:1 (2005), 225-243. T. Y. Hou and R. Li, Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations, J. Nonlinear Science, 16 (2006), 639-664. T. Y. Hou and C. Li, Dynamic Stability of the 3D Axisymmetric Navier-Stokes Equations with Swirl, CPAM, 61 (2008), 661-697. T. Y. Hou and Z. Lei, On the Stabilizing Effect of Convection in 3D Incompressible Flows, CPAM, 62, pp. 501-564, 2009. T. Y. Hou, Z. Shi, and S. Wang, On Singularity Formation of a 3D Model for Incompressible Navier-Stokes Equations, 2009, arxiv:0912.1316v1 [math.ap]. T. Y. Hou, Blow-up or no blowup? A unified computational and analytical approach to 3D incompressible Euler and Navier-Stokes equations, Acta Numerica, pp. 277-346, 2009.