Blow-up or No Blow-up? Fluid Dynamic Perspective of the Clay Millennium Problem

Blow-up or No Blow-up? Fluid Dynamic Perspective of the Clay Millennium Problem Thomas Y. Hou Applied and Comput. Mathematics, Caltech Research was supported by NSF Theodore Y. Wu Lecture in Aerospace, GALCIT, Caltech October 7, 2011

A tribute to Prof. Ted Wu I feel extremely honored to give this lecture in honor of Prof. Ted Wu. Ted is a great scientist, a powerful applied mathematician, an excellent teacher/mentor, and a real gentleman. His research covers a wide range of areas, and has made fundamental and highly original contributions in every area that he worked on. I very much admire his passion, his persistence and his drive to achieve scientific excellence. It is people like Prof. Ted Wu who have made Caltech such a special and unique university in the world. He is a role model for many of us. He has always been very generous in sharing his knowledge and wisdom with others, nurturing and supporting several generations of Ph.D. students, postdocs, and junior scientists.

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 that u is related to ω by a Riesz operator K of degree zero: u = K(ω), and we have ω L p u L p C ω L p for 1 < p <. Thus the vortex stretching term u ω is formally of the order 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. Localized non-blow-up criteria, Deng-Hou-Yu, 2005. Denote by L(t) the arclength of a vortex line segment L t around the maximum vorticity. If (i) max Lt ( u ξ + u n ) C U (T t) A with A < 1, (ii) C L (T t) B L(t) C 0 / max Lt ( κ, ξ ) with B < 1 A, then the 3D Euler remains smooth up to T.

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.

Numerical evidence of Euler singularity In 1993 (and 2005), R. Kerr [Phys. Fluids] presented numerical evidence of 3D Euler singularity for two anti-parallel vortex tubes: Pseudo-spectral in x and y, Chebyshev in z direction; Best resolution: 512 256 192; Predicted singularity time T = 18.7, but his numerical solutions became under-resolved after T 1 = 17. ω L (T t) 1 ; u L (T t) 1/2 ; Anisotropic scaling: (T t) T t T t; Vortex lines: relatively straight, ξ (T t) 1/2 ; It is important to emphasize that T T 1 = 1.7 is not asymptotically small. Thus the extrapolated blow-up rate of ω L (T t) 1 is not valid asymptotically from T 1 to T. This makes the claim T T 1 ω dt = questionable. Kerr s computation falls into the critical case of Deng-Hou-Yu s non-blowup criterion with A = B = 1/2.

Computation of Hou and Li, J. Nonlinear Science, 2006 Figure : Two slightly perturbed antiparallel vortex tubes at t=0 and t=6

Figure : The local 3D vortex structures and vortex lines around the maximum vorticity at t = 17 with resolution 1536 1024 3072.

Resolution study for energy spectra: 1024 768 2048 vs 1536 1024 3072. FS : the Fourier smoothing method 10 0 energy spectra comparison. 10 5 dashed:1024x768x2048, 2/3rd dealiasing dash dotted:1024x768x2048, FS solid:1536x1024x3072, FS 10 10 10 15 10 20 10 25 10 30 0 200 400 600 800 1000 1200

Maximum velocity in time 0.55 t [0,19],768 512 1536 t [0,19],1024 768 2048 t [10,19],1536 1024 3072 0.5 0.45 0.4 0.35 0.3 0 2 4 6 8 10 12 14 16 18 Figure : Maximum velocity u in time using different resolutions. With maximum velocity being bounded, the non-blowup criterion of Deng-Hou-Yu applies with A = 0 and B = 1/2, implying no blowup at least up to T = 19.

Dynamic depletion of vortex stretching 35 ξ u ω 30 c 1 ω log( ω ) c 2 ω 2 25 20 15 10 5 0 15 15.5 16 16.5 17 17.5 18 18.5 19 Figure : Study of the vortex stretching term in time, resolution 1536 1024 3072. The fact ξ u ω c 1 ω log ω plus ω = ξ u ω implies ω bounded by doubly exponential. D Dt

Log log plot of maximum vorticity in time 1 0.5 0 0.5 1 10 11 12 13 14 15 16 17 18 19 Figure : The plot of log log ω vs time, resolution 1536 1024 3072.

Log log plot of peak vorticity in time from Kerr s 93 paper 1 Double logarithm of peak vorticity in time from Kerr 93 paper 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 14 14.5 15 15.5 16 16.5 17 17.5 time

The 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 (??)-(??) 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)

Study of dynamic stability through an exact 1D model 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 1. Let u 1, ψ 1 and ω 1 be the solution of the 1D model (??)-(??) 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 1 tells us that the 1D model (??)-(??) preserves some essential nonlinear structure of the 3D axisymmetric Navier-Stokes equations.

Global Well-Posedness of the full 1D Model Theorem 2. 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. Surprisingly, nonlinear terms cancel each other and we obtain (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, (ũ 2 z + ṽ 2 z ) satisfies a maximum principle for all ν 0: ũ 2 z + ṽ 2 z L (ũ 0 ) 2 z + (ṽ 0 ) 2 z L.

Construction of a family of globally smooth solutions Theorem 3. 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, (13) where u r = rψ 1z, u z = 2ψ 1 + rψ 1r. Our 3D model is derived by simply dropping the convective term from (??): 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, ( (14) r 2 + 3 r ) r + z 2 ψ1 = ω 1. Note that (??) 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 (??). u(t, x) = u r (t, r, z)e r + u θ (t, r, z)e θ + u z (t, r, z)e z, (15) u θ = ru 1, u r = rψ 1z, u z = 2ψ 1 + rψ 1r, (16) 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, (17) 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 4. Energy identity. The smooth solution of (??) 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 5. A non-blowup criterion of Beale-Kato-Majda type. A smooth solution (u 1, ω 1, ψ 1 ) of the model (??) for 0 t < T blows up at time t = T if and only if T where u is defined in (??)-(??). 0 u BMO(R 3 )dt =,

Properties of the 3D Model continued Theorem 6. A non-blowup criterion of Prodi-Serrin type. A weak solution (u 1, ω 1, ψ 1 ) of the model (??) is smooth on [0, T ] R 3 provided that u θ L q t Lp x ([0,T ] R 3 ) < (18) for some p, q satisfying 3 p + 2 q 1 with 3 < p and 2 q <. Theorem 7. An analog of Caffarelli-Kohn-Nirenberg partial regularity result [Hou-Lei, CMP-09]. For any suitable weak solution of the 3D model equations (??) on an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero.

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.

(T t) Asymptotic blowup fit: u 1 1 C, with limiting values T = 0.021083 and C = 8.1901. 1.5 x 10 4 The inverse of u 1 in time, N=1024, 2048, 3072, 4096, ν=0.001 N=1024 N=2048 N=3072 N=4096 Asymptotic fit 1 u 1 1 0.5 0 0.02 0.0201 0.0202 0.0203 0.0204 0.0205 0.0206 0.0207 0.0208 0.0209 0.021 Time

Asymptotic blowup rate: u 1 T = 0.02109 and C = 8.20348. C (T t), with 10 x u 104 1 (blue) vs C/(T t) (red) in time, T=0.02109, C=8.20348, N=4096, ν=0.001 u, N=4096 1 9 Fitted solution 8 7 6 u 1 5 4 3 2 1 0 0.018 0.0185 0.019 0.0195 0.02 0.0205 0.021 Time

Local alignment of u 1 and ψ 1z at t = 0.02. Recall (u 1 ) t = 2u 1 ψ 1z + ν u 1, (ω 1 ) t = (u 2 1 ) z + ν ω 1. 8000 u 1 vs (ψ 1 ) z at r=0, t=0.02, N z =4096, N r =400, t=2.5 10 7, ν=0.001 u 1 6000 (ψ 1 ) z 4000 2000 0 2000 4000 0.25 0.3 0.35 0.4 0.45 0.5 z

Local alignment of u 1 and ψ 1z at t = 0.021. 10 x u 1 vs (ψ 1 ) z at r=0, t=0.021, N z =4096, N r =400, t=2.5 10 7, ν=0.001 104 u 1 8 (ψ 1 ) z 6 4 2 0 2 4 0.25 0.3 0.35 0.4 0.45 0.5 z

The effect of convection. To study the 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. 8000 Maximum u 1 in time, N z =2048, N r =1024, t=2.5 10 7, ν=0.001, full NSE 7000 6000 5000 u1 4000 3000 2000 1000 0.02 0.0205 0.021 0.0215 0.022 0.0225 0.023 0.0235 Time

Contours of initial data for u 1. Contour of u 1 at t=0.02, N z =4096, N r =400, t=2.5 10 7, ν=0.001, 3D model 0.2 7000 0.18 0.16 0.14 6000 5000 r 0.12 0.1 4000 0.08 3000 0.06 0.04 0.02 2000 1000 0 0.3 0.35 0.4 0.45 0.5 z

Contours of u 1 at t = 0.021, solution of full NSE. Contour of u 1 at t=0.021, N z =2048, N r =1024, t=2.5 10 7, ν=0.001, full NSE 0.2 3000 0.18 0.16 2500 0.14 0.12 2000 r 0.1 1500 0.08 0.06 1000 0.04 0.02 500 0 0.3 0.35 0.4 0.45 0.5 z

Contours of u 1 at t = 0.0235, solution of full NSE. Contour of u 1 at t=0.0235, N z =2048, N r =1024, t=2.5 10 7, ν=0.001, full NSE 0.2 1000 0.18 0.16 0.14 0.12 900 800 700 600 r 0.1 500 0.08 0.06 400 300 0.04 0.02 0 0.3 0.35 0.4 0.45 0.5 z 200 100

Proof of finite time blow-up of the 3D model Theorem 8. [Hou-Shi-Wang, 09]. Consider the 3D inviscid model { u t = 2uψ z, ω t = (u 2 ) z, ( ) x 2 + y 2 + z 2 (19) 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.

Recent progress on full Euler/NSE, joint with Dr. Guo Luo A recent study of the full 3D NSE and Euler equations reveals that for certain axi-symmetric initial data with swirl, the solution could develop a strong dynamic nonlinear alignment in the vortex stretching term. Our preliminary numerical study shows that the solution of the 3D Euler/NSE could experience a strong transitional growth in time with a quadratic nonlinearity, between u 1 and ψ 1z, u 1t + u r (u 1 ) r + u z (u 1 ) z = 2u 1 ψ 1z + ν u 1. An important question is whether the profile near the potentially singular region remains dynamically stable. 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 = 0.02 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 u 1 at t = 0.0445 Figure : Contour plot of u 1 at t=0.0445. Adaptive mesh is uded with smallest mesh proportional to 10 6.

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 ω 1 The following figure shows the rapid, super-double-exponential growth of the maximum of u 1 and ω 1 in time. 1.9 log(log(max u1 )) as a function of t 2.6 log(log(max w1 )) as a function of t 1.85 2.55 1.8 2.5 log(log(max u1 )) 1.75 1.7 1.65 log(log(max w1 )) 2.45 2.4 2.35 1.6 2.3 1.55 2.25 1.5 0.02 0.025 0.03 0.035 0.04 0.045 t 2.2 0.02 0.025 0.03 0.035 0.04 0.045 t Figure : Log log plot of maximum u 1 and ω 1 as a function of time.

The quadratic nonlinearity, u 1t + u u 1 = 2u 1 ψ 1z We observe that the nonlinear u 1 -stretching term, 2ψ 1z, 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 ω 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, 2ψ 1, to u 1 at the maximum of u 1.

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 3.5 x 10 3 z1 2 as a function of t, where z1 is the z coordinate of maxloc u1 3 3 2.5 2.5 2 2 r1 2 z1 2 1.5 1.5 1 1 0.5 0.5 0 0.02 0.025 0.03 0.035 0.04 0.045 t 0 0.02 0.025 0.03 0.035 0.04 0.045 t Figure : Plot of r1 2 and z1 2 as a function of time, where (r 1, z 1) is the point at which u 1 attains its maximum. This suggests that r 1(t) c 1(T t) 1/2 and z 1(t) c 2(T t) 1/2.

Front thickness as a function of time 0.07 Thickness of the front along r as a function of t 0.06 0.05 0.04 lr 0.03 0.02 0.01 0 0.038 0.039 0.04 0.041 0.042 0.043 0.044 0.045 t Figure : The front thickness as a function of time at the point at which u 1 attains its maximum. This seems to suggest that the thickness decreases like O(T t).

Asymptotic scaling for 1/ max u 1 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. This suggests that u 1 L c/(t t).

Asymptotic scaling for 1/ max ω 1 2.5 x 10 3 1/max w1 2/3 as a function of t 2 1/max w1 2/3 1.5 1 0.5 0 0.02 0.025 0.03 0.035 0.04 0.045 t Figure : Plot of 1/ max ω 1 2/3 as a function of time. This seems to suggest that ω 1 L c/(t t) 3/2.

Asymptotic scaling for 1/ max ψ 1 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 ψ 1 2 as a function of time. This seems to suggest that ψ 1 L c/(t t) 1/2. The estimated singularity time is T 0.0447.

Fluid dynamic instability seems to induce front instability Unfortunately (or fortunately), the fluid dynamics instability (Kelvin-Helmholtz) eventually kicks in when the thickness of the vortex sheet like structure is below some critical value. The vortex sheet develops roll-up and destroys the strong nonlinear alignment in the vortex stretching term. The flow subsequently develops many small scales dyanamically and transitions to turbulence.

Contour plots of u 1 at t = 0.04435, Euler equation Figure : Contour plot of u 1 at t=0.04435, the effective adaptive mesh near the most singular region = 10 6.

Contour plots of u 1 at t = 0.04438, Euler equation Figure : Contour plot of u 1 at t=0.04438.

Contour plots of u 1 at t = 0.04441, Euler equation Figure : Contour plot of u 1 at t=0.04441.

Contour plots of u 1 at t = 0.04459, NSE with ν = 10 5 Figure : Contour plot of u 1 at t=0.04459, the effective adaptive mesh near the most singular region = 6.6 10 7.

Contour plots of u 1 at t = 0.04461, NSE with ν = 10 5 Figure : Contour plot of u 1 at t=0.04461.

Contour plots of u 1 at t = 0.04463, NSE with ν = 10 5 Figure : Contour plot of u 1 at t=0.04463.

Contour plots of u 1 at t = 0.04464, NSE with ν = 10 5 Figure : Contour plot of u 1 at t=0.04464.

Contour plots of u 1 at t = 0.044666, ν = 10 5 Figure : Contour plot of u 1 at t=0.044666.

Concluding Remarks Our study shows that convection could play a dual role in the dynamic stability or finite time singularity formation of 3D incompressible flow. One the one hand, we construct an exact 1D model, in which we show that convection plays an essential role to deplete the nonlinear vortex stretching, and the 1D model has global regularity. We also construct a new 3D model by neglecting the convection term. This 3D model shares almost all properties of the Navier-Stokes equations, but could develop finite time singularities. On the other hand, our recent study suggests that convection may not always stablize the nonlinear vortex stretching. For initial data with certain symmetry, the combined effect of convection and vortex stretching could lead to a strong nonlinear alignment. It remains to understand whether the nearly singular behavior we observed numerically is dynamically stable. If so, it may suggest a possible way to construct a singular solution analytically.

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.