Potentially Singular Solutions of the 3D Axisymmetric Euler Equations

Transcription

1 Potentially Singular Solutions of the 3D Axisymmetric Euler Equations Thomas Y. Hou Applied and Computational Mathematics California Institute of Technology Joint work with Guo Luo IPAM Workshop on Mathematical Analysis of Turbulence Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

2 Introduction The Basic Problem The 3D incompressible Euler equations are given by with initial condition u(x, 0) = u 0. u t + u u = p, u = 0, Define vorticity ω = u, then ω is governed by ω t + (u )ω = u ω. 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. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

3 Introduction Previous Work On the theoretical side: Kato (1972): local well-posedness Beale-Kato-Majda (1984): necessary and sufficient blowup criterion Constantin-Fefferman-Majda (1996): geometric constraints for blowup Deng-Hou-Yu (2005): Lagrangian localized geometric constraints Other related work: Constantin-Majda-Tabak (1994): 2D surface quasi-geostrophic (SQG) equations as a model for 3D Euler Cordoba (1998): no blowup of 2D SQG near a hyperbolic saddle Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

4 Previous Work (Cont d) Introduction On the numerical search for singularity: Grauer and Sideris (1991): first numerical study of axisymmetric flows with swirl; blowup reported away from the axis Pumir and Siggia (1992): axisymmetric flows with swirl; blowup reported away from the axis Kerr (1993): antiparallel vortex tubes; blowup reported E and Shu (1994): 2D Boussinesq; no blowup observed Boratav and Pelz (1994): viscous simulations using Kida s high-symmetry initial condition; blowup reported Grauer et al. (1998): perturbed vortex tube; blowup reported Hou and Li (2006): use Kerr s two anti-parallel vortex tube initial data with higher resolution; no blowup observed Orlandi and Carnevale (2007): Lamb dipoles; blowup reported Evidence for blowup is inconclusive and problem remains open Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

5 Introduction The role of convection in 3D Euler and Navier-Stokes In [CPAM 08], Hou and Li studied the role of convection for 3D axisymmetric flow and introduced the following new variables: u 1 = u θ /r, ω 1 = ω θ /r, ψ 1 = ψ θ /r, (1) 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 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 r r + 2 ) z ω1 + ( u1 2 ( r 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. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72 (2)

6 Introduction An exact 1D model for 3D Euler/Navier-Stokes In [Hou-Li, CPAM, 61 (2008), no. 5, ], we derived an excact 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, ( ) (3) (ω 1 ) t + 2ψ 1 (ω 1 ) z = ν(ω 1 ) zz + u1 2, z (4) (ψ 1 ) zz = ω 1. (5) 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, (6) (ṽ) t + 2 ψ(ṽ) z = ν(ṽ) zz + (ũ) 2 (ṽ) 2 + c(t), (7) where ṽ = ( ψ) z, ṽ z = ω, and c(t) is an integration constant to enforce the mean of ṽ equal to zero. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

7 The 1D model is exact! Introduction 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 (3)-(5) 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 (3)-(5) preserves some essential nonlinear structure of the 3D axisymmetric Navier-Stokes equations. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

8 Introduction 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. 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. The convection term cancels one of the nonlinear terms. (ũ 2 z ) t + 2 ψ(ũ 2 z ) z = 4ũũ z ṽ z + 2νũ z (ũ z ) zz, (8) (ṽ 2 z ) t + 2 ψ(ṽ 2 z ) z = 4ũũ z ṽ z + 2νṽ z (ṽ z ) zz. (9) Another cancelltion occurs, which gives rise to ( ) ( ( [ ũz 2 + ṽz ψ ũz 2 + ṽz 2 t )z = ν ũz 2 + ṽz 2 )zz 2ν (ũ zz ) 2 + (ṽ zz ) 2]. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

9 Introduction 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. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

10 Introduction Potential singularity for 3D Euler equation Inspired by the previous study, we discover a class of initial data that lead to potentially singular solutions of 3D Euler equations. Main features of our study: the singularity occurs at a stagnation point where the effect of convection is minimized. strong symmetry (axisymmetry plus odd/even symmetry in z) devises highly effective algorithms for adequate resolution employs rigorous criteria for confirmation of singularity Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

11 Introduction Vorticity Kinematics z = 1 4 L z = 0 z = 1 4 L Figure : Vorticity kinematics of the 3D Euler singularity; solid: vortex lines; straight dashed lines: axial flow; curved dash lines: vortical circulation. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

12 Introduction Local Flow Field ũ = (u r, u z ) near q 0 on mesh, t = z r = z_r z 0 r l = r_l 1 r Figure : The 2D flow field ũ = (u r, u z ) T near the maximum vorticity. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

13 Outline Numerical Method Overview 1 Introduction 2 Numerical Method Overview The Adaptive (Moving) Mesh Algorithm 3 Numerical Results Effectiveness of the Adaptive Mesh First Sign of Singularity Confirming the Singularity I: Maximum Vorticity Confirming the Singularity IV: Local Self-Similarity Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

14 Numerical Method Overview 3D Axisymmetric Euler Equations Equations being solved: the 3D axisymmetric Euler (Hou-Li, 2008) u 1,t + u r u 1,r + u z u 1,z = 2u 1 ψ 1,z, ω 1,t + u r ω 1,r + u z ω 1,z = (u 2 1 ) z, [ 2 r + 3 r r + 2 z ] ψ1 = ω 1, where u 1 = u θ /r, ω 1 = ω θ /r, ψ 1 = ψ θ /r. u r = rψ 1,z, u z = 2ψ 1 + rψ 1,r : the radial/axial velocity components. Initial condition: u 1 (r, z, 0) = 100e 30(1 r 2 ) 4 sin ( ) 2πz L, ω1 (r, z, 0) = ψ 1 (r, z, 0) = 0. No flow boundary condition ψ 1 = 0 at r = 1 and periodic BC in z. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

15 Numerical Method Overview Outline of the Method Discretization in space: a hybrid 6th-order Galerkin and 6th-order finite difference method, on an adaptive (moving) mesh that is dynamically adjusted to the evolving solution Discretization in time: an explicit 4th-order Runge-Kutta method, with an adaptively chosen time step Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

16 Outline Numerical Method The Adaptive (Moving) Mesh Algorithm 1 Introduction 2 Numerical Method Overview The Adaptive (Moving) Mesh Algorithm 3 Numerical Results Effectiveness of the Adaptive Mesh First Sign of Singularity Confirming the Singularity I: Maximum Vorticity Confirming the Singularity IV: Local Self-Similarity Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

17 Numerical Method The Adaptive (Moving) Mesh Algorithm Adaptive Methods for Singularity Detection Existing methods for computing (self-similar) singularities: dynamic rescaling (McLaughlin et al. 1986: nonlinear Schrödinger) adaptive mesh refinement (Berger and Kohn 1988: semilinear heat) moving mesh method (Budd et al. 1996: semilinear heat; Budd et al. 1999: nonlinear Schrödinger) However, these methods require knowledge of the singularity discrete approximation of mesh mapping functions can result in significant loss of accuracy Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

18 Numerical Method The Adaptive (Moving) Mesh Algorithm Our Approach: Defining the Adaptive Mesh We observe that vorticity tends to concentrate at a single point in the rz-plane. This motivates the development of the following special mesh adaptation strategy. The adaptive mesh covering the computational domain is constructed from a pair of analytic mesh mapping functions: r = r(ρ), z = z(η), where each mesh mapping function contains a small number of parameters, which are dynamically adjusted so that along each dimension a certain fraction (e.g. 50%) of the mesh points is placed in a small neighborhood of the singularity Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

19 Numerical Method Advancing the Solution The Adaptive (Moving) Mesh Algorithm The Poisson equation for ψ 1 is solved in the ρη-space using a 6th order B-spline based Galerkin method. The evolutionary equations for u 1 and ω 1 are semi-discretized in the ρη-space, where the space derivatives are expressed in the ρη-coordinates and are approximated using 6th-order centered difference scheme, e.g. v r,ij = v ρ,ij r ρ,j 1 r ρ,j (D 6 ρ,0 v i) j the resulting system of ODEs is integrated in time using an explicit 4th-order Runge-Kutta method Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

20 Outline Numerical Results Effectiveness of the Adaptive Mesh 1 Introduction 2 Numerical Method Overview The Adaptive (Moving) Mesh Algorithm 3 Numerical Results Effectiveness of the Adaptive Mesh First Sign of Singularity Confirming the Singularity I: Maximum Vorticity Confirming the Singularity IV: Local Self-Similarity Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

21 Numerical Results Effectiveness of the Adaptive Mesh Effectiveness of the Adaptive Mesh rz-plot of ω on mesh, t = x z r Figure : The vorticity function ω on the mesh at t = ; plot shown in rz-coordinates with 1 : 10 sub-sampling in each dimension. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

22 Numerical Results Effectiveness of the Adaptive Mesh Effectiveness of the Adaptive Mesh (Cont d) ρη-plot of ω on mesh, t = x η 0 0 ρ Figure : The vorticity function ω on the mesh at t = ; plot shown in ρη-coordinates with 1 : 10 sub-sampling in each dimension. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

23 Numerical Results Effective Mesh Resolutions Effectiveness of the Adaptive Mesh Table : Effective mesh resolutions M, N near the maximum vorticity at selected time t. Mesh size M t = N Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

24 Numerical Results Effectiveness of the Adaptive Mesh Backward Error Analysis of the Linear Solve Table : Backward errors of the linear solve Ax = b associated with the elliptic equation for ψ 1 at t = Mesh size t = ω 1 κ ω1 ω 2 κ ω2 δx / x The linear system is solved using a parallel sparse direct solver called PaStiX package. Here ω 1, ω 2 are the componentwise backward errors of first and second kind, and κ ω1, κ ω2 are the componentwise condition numbers of first and second kind. It can be shown that (Arioli 1989) δx x ω 1 κ ω1 + ω 2 κ ω2. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

25 Outline Numerical Results First Sign of Singularity 1 Introduction 2 Numerical Method Overview The Adaptive (Moving) Mesh Algorithm 3 Numerical Results Effectiveness of the Adaptive Mesh First Sign of Singularity Confirming the Singularity I: Maximum Vorticity Confirming the Singularity IV: Local Self-Similarity Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

26 Maximum Vorticity Numerical Results First Sign of Singularity Table : Maximum vorticity ω = u at selected time t. Mesh size ω t = 0 t = t = Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

27 Numerical Results Maximum Vorticity (Cont d) First Sign of Singularity log(log ω ) on and mesh log(log ω ) t x 10 3 Figure : The double logarithm of the maximum vorticity, log(log ω ), computed on the and the mesh. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

28 Numerical Results First Sign of Singularity Resolution Study on Primitive Variables Table : Sup-norm relative error and numerical order of convergence of the transformed primitive variables u 1 at selected time t. Mesh size t = Error Order Sup-norm Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

29 Numerical Results First Sign of Singularity Resolution Study on Primitive Variables (Cont d) Table : Sup-norm relative error and numerical order of convergence of the transformed primitive variables ω 1 at selected time t. Mesh size t = Error Order Sup-norm Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

30 Numerical Results First Sign of Singularity Resolution Study on Primitive Variables (Cont d) Table : Sup-norm relative error and numerical order of convergence of the transformed primitive variables ψ 1 at selected time t. Mesh size t = Error Order Sup-norm Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

31 Numerical Results First Sign of Singularity Resolution Study on Conserved Quantities Table : Kinetic energy E, minimum circulation Γ 1, maximum circulation Γ 2 and their maximum (relative) change over [0, ]. Mesh size t = δe,t δγ 1,t δγ 2,t Init. value Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

32 Outline Numerical Results Confirming the Singularity I: Maximum Vorticity 1 Introduction 2 Numerical Method Overview The Adaptive (Moving) Mesh Algorithm 3 Numerical Results Effectiveness of the Adaptive Mesh First Sign of Singularity Confirming the Singularity I: Maximum Vorticity Confirming the Singularity IV: Local Self-Similarity Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

33 Numerical Results Confirming the Singularity I: Maximum Vorticity The Beale-Kato-Majda (BKM) Criterion The main tool for studying blowup/non-blowup: the Beale-Kato-Majda (BKM) criterion (Beale et al. 1984) Theorem Let u be a solution of the 3D Euler equations, and suppose there is a time t s such that the solution cannot be continued in the class u C([0, t]; H m ) C 1 ([0, t]; H m 1 ), m 3 to t = t s. Assume that t s is the first such time. Then ts 0 ω(, t) dt =, ω = u. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

34 Numerical Results Confirming the Singularity I: Maximum Vorticity Applying the BKM Criterion The standard approach to singularity detection: 1 assume the existence of an inverse power-law ω(, t) c(t s t) γ, c, γ > 0 2 estimate t s and γ using a line fitting: [ ] 1 d dt log ω(, t) 1 γ (t s t) 3 estimate c using another line fitting: log ω(, t) γ log(ˆt s t) + log c, where ˆt s is the singularity time estimated in step 2 Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

35 Our Criteria Numerical Results Confirming the Singularity I: Maximum Vorticity Our criteria for choosing the fitting interval [τ 1, τ 2 ]: τ 2 is the last time at which the solution is still accurate choose the fitting interval [τ 1, τ 2 ] in the asymptotic regime. Our criteria for a successful line fitting: both τ 2 and the line-fitting predicted singularity time ˆt s converge to the same finite value as the mesh is refined; the convergence should be monotone, i.e. τ 2 t s, ˆt s t s τ 1 converges to a finite value that is strictly less than t s as the mesh is refined Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

36 Numerical Results Confirming the Singularity I: Maximum Vorticity Applying the Ideas: Computing the Line Fitting x 10 5 line fitting of y = [ d dt log ω ] 1 on mesh fitting area y(t) ˆγ 1 1 (ˆt s t) y t x 10 3 Figure : Inverse logarithmic time derivative [ d dt log ω 1 ] and its line fitting ˆγ 1 1 (ˆt s t), computed on the mesh. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

37 Numerical Results Confirming the Singularity I: Maximum Vorticity Applying the Ideas: Computing the Line Fitting 3 x 10 7 line fitting of y = [ d dt log ω ] 1 on mesh fitting area y(t) ˆγ 1 1 (ˆt s t) extrapolation area y t x 10 3 Figure : A zoom-in view of the line fitting ˆγ 1 1 (ˆt s t). Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

38 Numerical Results Confirming the Singularity I: Maximum Vorticity Applying the Ideas: Computing the Line Fitting inverse power-law fitting of ω on mesh ω(, t) ĉ(ˆt s t) ˆγ2 fitting area ω t x 10 3 Figure : Maximum vorticity ω and its inverse power-law fitting ĉ(ˆt s t) ˆγ 2, computed on the mesh. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

39 Numerical Results Confirming the Singularity I: Maximum Vorticity Applying the Ideas: the Best Fitting Interval Table : The best fitting interval [τ 1, τ 2 ] and the estimated singularity time ˆt s. Mesh size τ 1 τ 2 ˆt s Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

40 Numerical Results Confirming the Singularity I: Maximum Vorticity Applying the Ideas: Results of the Line Fitting Table : The best line fittings for ω computed on [τ 1, τ 2 ]. Mesh size ˆγ 1 ˆγ 2 ĉ : ˆγ 1 is computed from [ d dt log ω ] 1 γ 1 (t s t). : ˆγ 2 is computed from log ω γ log(ˆt s t) + log c. Conclusion: the maximum vorticity develops a singularity ω c(t s t) γ at t s (recall t e ) Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

41 Numerical Results Confirming the Singularity I: Maximum Vorticity Comparison with Other Numerical Studies Table : Comparison of our results with other numerical studies. K: Kerr (1993); BP: Boratav and Pelz (1994); GMG: Grauer et al. (1998); OC: Orlandi and Carnevale (2007); τ 2 : the last time at which the solution is deemed well resolved. Studies τ 2 t s Effec. res. Vort. amp. K BP GMG OC Ours ( ) : According to Hou and Li (2008). Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

42 Numerical Results Confirming the Singularity I: Maximum Vorticity Nonlinear alignment of vortex stretching The vorticity direction ξ = ω/ ω could also play a role! Recall the vorticity equation ω t + u ω = α ω, where α = ξ u ξ is the vorticity amplification factor α = ξ u ξ = ξ Sξ, S = 1 2( u + u T ), thus the growth of α depends on the eigenstructure of S Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

43 Spectral Dynamics Numerical Results Confirming the Singularity I: Maximum Vorticity Due to symmetry, S has 3 real eigenvalues {λ i } 3 i=1 (assuming λ 1 λ 2 λ 3 ), and a complete set of orthogonal eigenvectors {w i } 3 i=1 We discover, at the location of the maximum vorticity, that: the vorticity direction ξ is perfectly aligned with w 2, i.e. λ 2 = α = d dt log ω c 2 (t s t) 1 the largest positive/negative eigenvalues satisfy λ 1,3 ± 1 2 ω ±c(t s t) Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

44 Numerical Results Confirming the Singularity I: Maximum Vorticity The DHY Non-blowup Criterion Essential ideas of DHY: no blowup if, among other things, the divergence of ξ, ξ, and the curvature κ = ξ ξ, along a vortex line do not grow too fast compared with the diminishing rate of the length of the vortex line Similar in spirit to CFM but more localized Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

45 Numerical Results Checking Against the DHY Criterion Confirming the Singularity I: Maximum Vorticity deriv bound derivative bounds of ξ near q 0 on mesh min D (t) ξ max D (t) ξ max D (t) κ t x 10 3 Figure : The maximum/minimum of ξ and κ in a local neighborhood D (t) of the maximum vorticity. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

46 Numerical Results Confirming the Singularity I: Maximum Vorticity Geometry of the Vorticity Direction (Cont d) ξ z (r, z) near q 0 on mesh, t = e z e 011 r Figure : The z-component ξ z of the vorticity direction ξ near the maximum vorticity. Note the rapid variation of ξ z in z. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

47 Outline Numerical Results Confirming the Singularity IV: Local Self-Similarity 1 Introduction 2 Numerical Method Overview The Adaptive (Moving) Mesh Algorithm 3 Numerical Results Effectiveness of the Adaptive Mesh First Sign of Singularity Confirming the Singularity I: Maximum Vorticity Confirming the Singularity IV: Local Self-Similarity Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

48 Numerical Results Locally Self-Similar Solutions Confirming the Singularity IV: Local Self-Similarity Solutions of the 3D Euler equations in R 3 have special scaling properties: u(x, t) λ α u(λx, λ α+1 t), λ > 0, α R Can this give rise to a (locally) self-similar blowup? u(x, t) 1 ( ) x t s t U x0 [t s t] β, x R 3 Recent results by D. Chae (2007,2010,2011) seem to give a negative answer under some strong (exponential) decay assumption on the self-similar profile U. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

49 Numerical Results Confirming the Singularity IV: Local Self-Similarity Self-Similar Solutions with Axis-Symmetry In axisymmetric flows, self-similar solutions naturally take the form ( ) x x0 u 1 ( x, t) (t s t) γu U, l(t) ( ) x x0 ω 1 ( x, t) (t s t) γω Ω, l(t) ( ) ψ 1 ( x, t) (t s t) γ ψ x x0 Ψ, x x 0, t ts, l(t) where x = (r, z) T and l(t) [δ 1 (t s t)] γ l is a length scale, and the exponents satisfy γ ω = 1, γ ψ = 1 + 2γ l, γ u = γ l. This would give rise to u(, t) c(t s t) γu γ l. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

50 Numerical Results Identifying a Self-Similar Solution Confirming the Singularity IV: Local Self-Similarity We remark that the recent result of Chae-Tsai on non-existence of self-similar solutions of 3D axisymmetric Euler does not apply to our solution since they assume U(ξ) 0 as ξ. In our case, we found that U(0) = Ψ(0) = Ω(0) = 0, and U(ξ) c 0 ξ β for some 0 < β < 1 as ξ, where β satisfies γ u = γ l β with γ u > 0 and γ l > 0. This gives u(1, z, t s ) c 0 z β at the singularity time. To identify a self-similar neighborhood, consider { } C (t) = (r, z) D : ω(r, z, t) = 1 2 ω(, t) Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

51 Numerical Results Confirming the Singularity IV: Local Self-Similarity Existence of Self-Similar Neighborhood x 10 7 level curves of 1 2 ω on mesh t = z t = r = r 1 1 Figure : The level curves of 1 2 ω in linear-linear scale at various time instants. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

52 Numerical Results Confirming the Singularity IV: Local Self-Similarity Existence of Self-Similar Neighborhood (Cont d) level curves of 1 2 ω (in log-log scale) on mesh 10 6 t = z t = r Figure : The level curves of 1 2 ω in log-log scale (against the variables 1 r and z) at various time instants. Note the similar shapes of all curves. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

53 Numerical Results Confirming the Singularity IV: Local Self-Similarity Existence of Self-Similar Neighborhood (Cont d) rescaled level curves of 1 2 ω on mesh zoom in z r Figure : The rescaled level curves of 1 2 ω. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

54 Numerical Results Indication of Self-Similarity in 2D Confirming the Singularity IV: Local Self-Similarity z x 10 5 contour plot of ω 1 on mesh, t = z = r Figure : The contour plot of ω 1 near the maximum vorticity at t = Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

55 Numerical Results Confirming the Singularity IV: Local Self-Similarity Indication of Self-Similarity in 2D (Cont d) x 10 9 contour plot of ω 1 on mesh, t = z = x z r = r Figure : The contour plot of ω 1 near the maximum vorticity at t = Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

56 Numerical Results Confirming the Singularity IV: Local Self-Similarity Indication of Self-Similarity in 2D (Cont d) x contour plot of ω 1 on mesh, t = z = x z r = r Figure : The contour plot of ω 1 near the maximum vorticity at t = Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

57 Numerical Results The Scaling Exponents Confirming the Singularity IV: Local Self-Similarity Table : Scaling exponents of l, u 1, ω 1, and ψ 1. Mesh size ˆγ l ˆγ u ˆγ ω ˆγ ψ γ l 1: consistent with the a posteriori bound u C Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

58 Consistency Check Numerical Results Confirming the Singularity IV: Local Self-Similarity Table : Consistency check for the scaling exponents. Mesh size ˆγ l 1 + 2ˆγ l ˆγ u ˆγ l Ref. value ˆγ u : ˆγ ψ : ˆγ 1 : ω c(t s t) 2.45 : consistent with Chae s nonexistence results Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

59 Theoretical understanding of the blow-up via a 1D model Recent progress on the blow-up of a 1D model One can gain important understanding of the blowup mechanism by studying a 1D model along the boundary at r = 1. We propose the following 1D model at r = 1 (ρ = u 2 1, u = uz ): ρ t + uρ z = 0, z (0, L), ω t + uω z = ρ z where the velocity u is defined by u z (z) = Hω(z) with u(0) = 0. This 1D model and the 3D Euler equations shares many similar properties, including all the symmetry properties along z direction. Recently, with Drs. K. Choi, A. Kiselev, G. Luo, V. Sverak, and Y. Yao, we have proved the finite time blowup of the 1D model. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

60 Theoretical understanding of the blow-up via a 1D model Comparison between the 1D model and the 3D Euler 12 x comparing angular vorticity ω(z) ω1(1,z) 0 2 comparing axial velocity angular vorticity axial velocity z x v(z) ψ1,r(1,z) z Comparison of numerical solutions of the 1D model with 3D Euler. (a) angular vorticity, and (b) axial velocity. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

61 Theoretical understanding of the blow-up via a 1D model Blow-up of the 1D model The main blow-up result is stated in the following theorem: Theorem 4 (Choi, Hou, Kiselev, Luo, Sverak and Yao) For any initial data ρ 0 H 2, ω 0 H 1 such that (i) ρ 0 is even and ω 0 is odd at z = 0, 1 2 L, (ii) ρ 0z, ω 0 0 on [0, 1 2L], and (iii) L/2[ 0 ρ0 (z) ρ 0 (0) ] 2 dz > 0, then the solution of the 1D model develops a singularity in finite time. One can show that the solution of the 1D model satisfies: 1 ρ is even and ω, u are odd at z = 0, 1 2L for all t 0; 2 ρ z, ω 0 and u 0 on [0, 1 2L] for all t 0. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

62 Theoretical understanding of the blow-up via a 1D model Sketch of the proof The analysis relies on the two lemmas, which reveal the key properties of the Biot-Savart law due to the strong symmetry of the flow. Lemma 1 Let ω H 1 be odd at z = 0 and let u z = H(ω) the velocity field. Then for any z [0, L/2], u(z) cot(µz) = 1 π L/2 0 K (z, z )ω(z ) cot(µz ) dz, (10) where µ = π/l and K (x, y) = s log s + 1 s 1 with s = s(x, y) = tan(µy) tan(µx). (11) Furthermore, the kernel K (x, y) has the following properties: 1 K (x, y) 0 for all x, y (0, 1 2L) with x y; 2 K (x, y) 2 and K x (x, y) 0 for all 0 < x < y < 1 2 L; 3 K (x, y) 2s 2 and K x (x, y) 0 for all 0 < y < x < 1 2 L. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

63 Theoretical understanding of the blow-up via a 1D model Sketch of the proof continued Lemma 2 Let the assumptions in Lemma 1 be satisfied and assume in addition that ω 0 on [0, 1 2 L]. Then for any a [0, 1 2 L], L/2 a ω(z) [ u(z) cot(µz) ] dz 0. (12) z Now we are ready to prove the finite time blowup of the 1D model. Consider the integral I(t) := L/2 0 ρ(z, t) cot(µz) dz. (13) Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

64 Theoretical understanding of the blow-up via a 1D model Sketch of the proof continued To prove the finite-time blowup of I(t), we consider d dt I(t) = = 1 π L/2 0 L/2 0 u(x)ρ x (x) cot(µx) dx ρ x (x) L/2 0 ω(y) cot(µy)k (x, y) dy dx, where in the second step we have used the representation formula (10) from Lemma 1. By the assumption on the initial data, we have ρ x, ω 0 on [0, 1 2 L]. Moreover, from Lemma 1, we have K 0 for y < x, and K 2 for y > x. Thus, we get d dt I(t) 2 π L/2 0 ρ x (x) L/2 x ω(y) cot(µy) dy dx. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

65 Theoretical understanding of the blow-up via a 1D model Sketch of the proof continued It remains to find a lower bound for the right hand side, which involves some delicate dynamic estimates. With some work, we can show that d dt I(t) 2 t π = 2 π = µ π µ π L/2 0 0 t L/2 0 0 t L/2 0 0 t L/2 0 2µ t πl 0 0 ρ y (y, s) cot(µy) ζ(t) (ρρ y )(y, s) cot(µy) dy ds ρ 2 (y, s) csc 2 (µy) dy ds ρ 2 (y, s) cot 2 (µy) dy ds ( L/2 ρ(y, s) cot(µy) dy 0 0 ρ x (x, t) dx dy ds ) 2 ds = 2 t L 2 I 2 ds. 0 Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

66 Theoretical understanding of the blow-up via a 1D model Self-similar Singularity of the CKY Model, joint with P. Liu To understand the self-similar singularity of 3D axisymmetric Euler equations observed in our numerical simulation. We consider the 1D CKY model defined on [0, 1], t ω(x, t) + u(x, t) x ω(x, t) = ρ x (x, t), t ρ(x, t) + u(x, t) x ρ(x, t) = 0, (14) u(x, t) = x 1 x ω(y,t) y dy. This model can be viewed as a local approximation of the 3D Euler equations on the solid boundary of the cylinder with ω ω 1, ρ u 2 1. (15) The formation of finite-time singularity of this model under certain initial conditions has been proved by Choi, Kiselev and Yao. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

67 Theoretical understanding of the blow-up via a 1D model We consider the following self-similar ansatz: ( ρ(x, t) = (T t) cρ ρ x ( u(x, t) = (T t) cu U x ω(x, t) = (T t) cw W (T t) c l (T t) c l ( x (T t) c l Plug these ansatz into the equations, we get ), ), ). c w = 1, c u = c l 1, c ρ = c l 2, and a non-linear non-local ODE system (2 c l )ρ(ξ) + c l ξρ (ξ) + U(ξ)ρ (ξ) = 0, W (ξ) + c l ξw (ξ) + U(ξ)W (ξ) ρ (ξ) = 0, U(ξ) = ξ + W (η) ξ η dη. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

68 Theoretical understanding of the blow-up via a 1D model Summary of main findings of the CKY model For this 1D model problem, we get the following results: We prove the existence of a family of self-similar profiles, corresponding to different leading order non-vanishing derivative of ρ (s) (ξ) 0 at ξ = 0. We analyze the far-field behavior of the profiles. They are analytic with respect to a transformed variable θ = ξ 1/c l. This result can explain the Hölder continuity of the velocity field at singularity time observed in numerical simulation. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

69 Theoretical understanding of the blow-up via a 1D model Summary of main findings of the CKY model continued The asymptotic scaling exponents and self-similar profiles we construct agree with those obtained from direct numerical simulation of the CKY model. The self-similar profiles we construct have some stability property based on our numerical simulation. For fixed initial leading order non-vanishing derivative of ρ (s) (x, 0) 0 at x = 0, the solutions converge to the same profile for different initial conditions of ω. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

70 Theoretical understanding of the blow-up via a 1D model Sketch of proof: The Biot-Savart law for this model can be written as a local relation with a global constraint, ( ) U(ξ) = W (ξ) ξ ξ, U (0) + 0 W (η) dη = 0. η We first ignore the global constraint, and construct the local solutions near ξ = 0 using power series. The local solutions can be extended to the whole R +. The global constraint in the Biot-Savart law determines the asymptotic scaling exponent, c l, which depends on s only, where ρ (s) (0) 0. Once c l is fixed, all other scaling exponents for u, ρ and ω can be expressed in terms of c l. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

71 Summary Summary Main contributions of our study: discovery of potentially singular solutions of the 3D Euler equations Similar singularity formation also observed in 2D Boussinesq equations for stratified flows The singularity occurs at a stagnation point where the effect of convection is minimized. Strong symmetry of the solution plus the presence of the physical boundary seem to play a crucial in generating a stable and substainable locally self-similar blowup. Analysis of the corresponding 1D model sheds new light to the blowup mechanism. Analysis of the 2D Boussinesq and 3D Euler is more challenging and is under investigation. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72

72 Summary References G. Luo and T. Y. Hou, Potentially Singular Solutions of the 3D Axisymmetric Euler Equations,, PNAS, Vol. 111, no. 36, pp , Doi: /pnas G. Luo and T. Y. Hou, Toward the Finite-Time Blowup of the 3D Incompressible Euler Equations: a Numerical Investigation. Accepted by SIAM MMS, K. Choi, T. Y. Hou, A. Kiselev, G. Luo, V. Sverak, and Y. Yao On the Finite-time Blowup of a 1D Model fro the 3D Axisymmetric Euler Equations, arxiv: v1 [math.ap]. T. Y. Hou and P. Liu, Self-Similar Singularity for a 1D Model of the 3D Axisymmetric Euler Equations, arxiv: v1 [math.ap]. Thomas Y. Hou (ACM, Caltech) Finite-Time Singularity of 3D Euler IPAM, / 72