Blowup or No Blowup? The Interplay between Theory and Numerics

Thomas Y. Hou1, and Ruo Li2

1Applied and Computational Math, 217-50, Caltech, Pasadena, CA 91125, USA.2LMAM&School of Mathematical Sciences, Peking University, Beijing 100871, China.

The question of whether the 3D incompressible Euler equations can develop a finite time sin-gularity from smooth initial data has been an outstanding open problem in fluid dynamics andmathematics. Recent studies indicate that the local geometric regularity of vortex lines can leadto dynamic depletion of vortex stretching. Guided by the local non-blow-up theory, we have per-formed large scale computations of the 3D Euler equations on some of the most promising blow-upcandidates. Our results show that there is tremendous dynamic depletion of vortex stretching. Thelocal geometric regularity of vortex lines and the anisotropic solution structure play an importantrole in depleting the nonlinearity dynamically and thus prevent a finite time blow-up.

PACS numbers: 47.32.C-,47.11.Kb

I. INTRODUCTION

The question of whether the 3D incompressible Eu-ler equations can develop a finite time singularity fromsmooth initial data is one of the most outstanding openproblems in fluid dynamics and mathematics. This openproblem is closely related to the Clay Millennium OpenProblem on the 3D Navier-Stokes equations. The under-standing of this problem could improve our understand-ing on the onset of turbulence and the intermittencyproperties of turbulent flows. A main difficulty in an-swering this question is the presence of vortex stretching,which gives a formal quadratic nonlinearity in vorticity.There have been many computational efforts in search-ing for finite time singularities of the 3D Euler equations,see e.g. [25, 1113, 17, 18, 2123]. For a more compre-hensive review of this subject, we refer the reader to thebook by Majda and Bertozzi [20] and the excellent reviewarticle by J. Gibbon in this issue [10].

Computing Euler singularities numerically is an ex-tremely challenging task. First of all, it requires hugecomputational resources. Tremendous resolutions are re-quired to capture the nearly singular behavior of the Eu-ler equations. Secondly, one has to perform careful con-vergence study. It is dangerous to interpret the blowupof an under-resolved computation as an evidence of finitetime singularities for the 3D Euler equations. Thirdly, ifwe believe that the numerical solution we compute leadsto a finite time blowup, we need to demonstrate the val-idation of the asymptotic blowup rate, i.e. is the blowuprate L

C(Tt) asymptotically valid as t T ?

One also needs to check if the blowup rate of the numeri-cal solution is consistent with the Beale-Kato-Majda non-blowup criterion [1] and other non-blowup criteria [79].The interplay between theory and numerics is clearly es-sential in our search for Euler singularities.

There has been some interesting development in the

Electronic address: hou@acm.caltech.edu

theoretical understanding of the 3D incompressible Eu-ler equations. It has been shown that the local geometricregularity of vortex lines can play an important role indepleting nonlinear vortex stretching [69]. In particular,the recent results obtained by Deng, Hou, and Yu [8, 9]show that geometric regularity of vortex lines, even in anextremely localized region containing the maximum vor-ticity, can lead to depletion of nonlinear vortex stretch-ing, thus avoiding finite time singularity formation of the3D Euler equations. To obtain these results, Deng-Hou-Yu [8, 9] explore the connection between the stretchingof local vortex lines and the growth of vorticity. In par-ticular, they show that if the vortex lines near the regionof maximum vorticity satisfy some local geometric reg-ularity conditions and the maximum velocity field is in-tegrable in time, then no finite time blow-up is possible.These localized non-blowup criteria provide stronger con-straints on the local geometry of a potential finite timesingularity. They can be used to re-examine some of thewell-known numerical evidences for finite time singulari-ties of the 3D Euler equations.

II. A BRIEF REVIEW

We begin with a brief review on the subject. Due to theformal quadratic nonlinearity in vortex stretching, onlyshort time existence is known for the 3D Euler equations.One of the most well-known results on the 3D Euler equa-tions is due to Beale-Kato-Majda [1] who show that thesolution of the 3D Euler equations blows up at T if and

only if T0

(t) dt = , where is vorticity.

There have been some interesting recent theoretical de-velopments. In particular, Constantin-Fefferman-Majda[7] show that local geometric regularity of the unit vortic-ity vector can lead to depletion of the vortex stretching.Let = /|| be the unit vorticity vector and u be thevelocity field. Roughly speaking, Constantin-Fefferman-Majda show that if (1) u is bounded in a O(1) re-

gion containing the maximum vorticity. (2) t02

d

is uniformly bounded for t < T , then the solution of the

2

3D Euler equations remains regular up to t = T .There have been some numerical evidences which sug-

gest a finite time blowup of the 3D Euler equations. Oneof the most well known examples is the finite time col-lapse of two anti-parallel vortex tubes by R. Kerr [17, 18].In Kerrs computations, he used a pseudo-spectral dis-cretization in the x and y directions, and a Chebyshevdiscretization in the z direction with resolution of order512256192. His computations showed that the maxi-mum vorticity blows up like O((T t)1) with T = 18.9.In his subsequent paper [18], Kerr showed that the max-imum velocity blows up like O((T t)1/2) with T beingrevised to T = 18.7. It is worth noting that there is still aconsiderable gap between the predicted singularity timeT = 18.7 and the final time t = 17 of Kerrs computa-tions which he used as the primary evidence for the finitetime singularity.

Kerrs blowup scenario is consistent with theBeale-Kato-Majda non-blowup criterion [1] and theConstantin-Fefferman-Majda non-blowup criterion [7].But it falls into the critical case of Deng-Hou-Yus localnon-blowup criteria [8, 9]. Below we describe the localnon-blowup criteria of Deng-Hou-Yu.

III. THE LOCAL NON-BLOWUP CRITERIA OFDENG-HOU-YU [8, 9]

Motivated by the result of [7], Deng, Hou, and Yu [8]have obtained a sharper non-blowup condition which usesonly very localized information of the vortex lines. As-sume that at each time t there exists some vortex linesegment Lt on which the local maximum vorticity is com-parable to the global maximum vorticity. Further, wedenote L(t) as the arclength of Lt, n the unit normalvector of Lt, and the curvature of Lt.

Theorem 1. (Deng-Hou-Yu [8], 2005) Assume that (1)maxLt(|u | + |u n|) CU (T t)

A with A < 1, and(2) CL(T t)

B L(t) C0/ maxLt(||, | |) for0 t < T . Then the solution of the 3D Euler equationsremains regular up to t = T if A + B < 1.

In Kerrs computations, the first condition of Theorem1 is satisfied with A = 1/2 if we use u C(Tt)

1/2

as alleged in [18]. Moreover, since both and arebounded by O((T t)1/2) in the inner region [18], wecan choose a vortex line segment of length (T t)1/2

(i.e. B = 1/2) so that the second condition is satisfied.However, we violate the condition A+B < 1. Thus Kerrscomputations fall into the critical case of Theorem 1. Ina subsequent paper [9], Deng-Hou-Yu improved the non-blowup condition to include the critical case, A + B = 1.

Theorem 2. (Deng-Hou-Yu [9], 2006) Under the sameassumptions as Theorem 1, in the case of A+B = 1, thesolution of the 3D Euler equations remains regular up tot = T if the scaling constants CU , CL and C0 satisfy analgebraic inequality, f(CU , CL, C0) > 0.

We remark that this algebraic inequality can be

checked numerically if we obtain a good estimate of thesescaling constants. For example, if C0 = 0.1, which seemsreasonable since the vortex lines are relatively straight inthe inner region, Theorem 2 would imply no blowup upto T if 2CU < 0.43CL. Unfortunately, there was no esti-mate available for these scaling constants in [17]. One ofour original motivations to repeat Kerrs computationsusing higher resolutions was to obtain a good estimatefor these scaling constants.

IV. THE HIGH RESOLUTION 3D EULERCOMPUTATIONS OF HOU AND LI [14, 15]

In [14, 15], we repeat Kerrs computations usingtwo pseudo-spectral methods. The first pseudo-spectralmethod uses the standard 2/3 dealiasing rule to re-move the aliasing error. For the second pseudo-spectralmethod, we use a novel 36th order Fourier smoothingto remove the aliasing error. For the Fourier smoothingmethod, we use a Fourier smoother along the xj direc-tion as follows: (2kj/Nj) exp(36(2kj/Nj)

36), wherekj is the wave number (|kj | Nj/2). The time inte-gration is performed by using the classical fourth orderRunge-Kutta scheme. Adaptive time stepping is usedto satisfy the CFL stability condition with CFL numberequal to /4. In order to perform a careful resolutionstudy, we use a sequence of resolutions: 7685121536,10247682048 and 153610243072 in our computa-tions. We compute the solution up to t = 19, beyond thealleged singularity time T = 18.7 by Kerr [18]. Our com-putations were performed using 256 parallel processorswith maximal memory consumption 120Gb. The largestnumber of grid points is close to 5 billions.

As a first step, we demonstrate that the two pseudo-spectral methods can be used to compute a singular so-lution arbitrarily close to the singularity time. For thispurpose, we perform a careful convergence study of thetwo pseudo-spectral methods in both physical and spec-tral spaces for the 1D inviscid Burgers equation. Theadvantage of using the inviscid 1D Burgers equation isthat it shares some essential difficulties as the 3D Eulerequations, yet we have a semi-analytic formulation forits solution. By using the Newton iterative method, wecan obtain an approximate solution to the exact solutionup to 13 digits of accuracy. Moreover, we know exactlywhen a shock singularity will form in time. This enablesus to perform a careful convergence study in both thephysical space and the spectral space very close to thesingularity time.

We have performed a sequence of resolution study withthe largest resolution being N = 16, 384 [15]. Our ex-tensive numerical results demonstrate that the pseudo-spectral method with the high order Fourier smoothing(the Fourier smoothing method for short) gives a muchmore accurate approximation than the pseudo-spectralmethod with the 2/3 dealiasing rule (the 2/3 dealias-ing method for short). One of the interesting observa-

3

0 200 400 600 800 1000 1200 1400 1600 1800 200010

20

1018

1016

1014

1012

1010

108

106

104

102

100

blue: Fourier smoothinggreen: 2/3rd dealiasingred: exact solutiont=0.9, 0.95, 0.975, 0.9875

FIG. 1: Spectra comparison on different resolutions at a se-quence of moments. The additional modes kept the Fouriersmoothing method higher than the 2/3rd dealiasing methodare in fact correct. The initial condition is u0(x) = sin(x).The singularity time for this initial condition is T = 1.

3 2 1 0 1 2 310

12

1010

108

106

104

102

pointwise error comparison on 2048 grids, t=0.9875: blue(Fourier smoothing), red(2/3rd dealiasing)

FIG. 2: Pointwise errors of the two pseudo-spectral methodsas a function of time using different resolutions. The plot is ina log scale. The error of the 2/3rd dealiasing method is highlyoscillatory and spreads out over the entire domain, while theerror of the Fourier smoothing method is highly localized nearthe location of the shock singularity.

tions is that the unfiltered high frequency coefficients inthe Fourier smoothing method approximate accuratelythe corresponding exact Fourier coefficients. Moreover,we observe that the Fourier smoothing method capturesabout 12 15% more effective Fourier modes than the2/3 dealiasing method in each dimension, see Figure 1.The gain is even higher for the 3D Euler equations sincethe number of effective modes in the Fourier smoothingmethod is higher in three dimensions. Further, we findthat the error produced by the Fourier smoothing methodis highly localized near the region where the solution ismost singular. In fact, the pointwise error decays expo-nentially fast away from the location of the shock singu-larities. On the other hand, the error produced by the2/3 dealiasing method spreads out to the entire domainas we approach the singularity time, see Figure 2.

Next, we present our high resolution computations forthe two-antiparallel vortex tubes [14]. We used the sameinitial condition as the one used by Kerr (see SectionIII of [17], and also [14] for corrections of some typosin the description of the initial condition in [17]). How-ever, there is some difference between our discretizationand Kerrs discretization. We used a pseudo-spectraldiscretization in all three directions, while Kerr used apseudo-spectral discretization only in the x and y di-rections and used a Chebyshev discretization in the z-direction. Some extra filtering was applied after theChebyshev discretization. Unfortunately, this extra filterwas not documented in [17] and could not even be repro-duced by the author himself (private communication).Fortunately, we did not need to apply this extra filtersince we used a uniform grid and the pseudo-spectraldiscretization in all three directions.

We first illustrate the dynamic evolution of the vortextubes. In Figures 4-5, we plot the isosurface of the 3Dvortex tubes at t = 0 and t = 6 respectively. As we cansee, the two initial vortex tubes are very smooth and rel-atively symmetric. Due to the mutual attraction of thetwo antiparallel vortex tubes, the two vortex tubes ap-proach to each other and become flattened dynamically.By time t = 6, there is already a significant flatteningnear the center of the tubes. In Figure 6, we plot thelocal 3D vortex structure of the upper vortex tube att = 17. By this time, the 3D vortex tube has essentiallyturned into a thin vortex sheet with rapidly decreasingthickness. The vortex lines become relatively straight.The vortex sheet rolls up near the left edge of the sheet.It is interesting to note that the maximum vorticity isactually located near the rolled-up region of the vortexsheet and moves away from the bottom of the vortexsheet. Thus the mechanism of strong compression be-tween the two vortex tubes becomes weaker dynamicallyat later time.

We have performed a convergence study for the twonumerical methods using a sequence of resolutions. Forthe Fourier smoothing method, we use the resolutions7685121536, 10247682048, and 153610243072respectively. Except for the computation on the largestresolution 1536 1024 3072, all computations are car-ried out from t = 0 to t = 19. The computation on thefinal resolution 1536 1024 3072 is started from t = 10with the initial condition given by the computation withthe resolution 1024 768 2048. For the 2/3 dealias-ing method, we use the resolutions 512 384 1024,7685121536 and 10247682048 respectively. Thecomputations using these three resolutions are all carriedout from t = 0 to t = 19. See [14, 15] for more details.

In Figure 3, we compare the Fourier spectra of the en-ergy obtained by using the 2/3 dealiasing method withthose obtained by the Fourier smoothing method. Fora fixed resolution 1024 768 2048, we can see thatthe Fourier spectra obtained by the Fourier smoothingmethod retain more effective Fourier modes than thoseobtained by the 2/3 dealiasing method. This can be seen

4

0 200 400 600 800 1000 120010

30

1025

1020

1015

1010

105

100

dashed:1024x768x2048, 2/3rd dealiasingdashdotted:1024x768x2048, FS solid:1536x1024x3072, FS

FIG. 3: The energy spectra versus wave numbers. The dashedlines and dashed-dotted lines are the energy spectra with theresolution 10247682048 using the 2/3 dealiasing rule andthe Fourier smoothing, respectively. The times for the spectralines are at t = 15, 16, 17, 18, 19 respectively.

FIG. 4: The 3D view of the vortex tube at t = 0.

by comparing the results with the corresponding compu-tations using a higher resolution 1536 1024 3072 (thesolid lines). Moreover, the Fourier smoothing methoddoes not give the spurious oscillations in the Fourier spec-tra. In comparison, the Fourier spectra obtained by the2/3 dealiasing method produce some spurious oscillationsnear the 2/3 cut-off point. More convergence studies in-cluding the convergence of the enstrophy spectra can befound in [14, 15].

It is worth emphasizing that a significant portion ofthose Fourier modes beyond the 2/3 cut-off position arestill accurate for the Fourier smoothing method. Thisportion of the Fourier modes that go beyond the 2/3cut-off point is about 12 15% of total number of modesin each dimension. For 3D problems, the total numberof effective modes in the Fourier smoothing method isabout 20% more than that in the 2/3 dealiasing method.For our largest resolution, we have about 4.8 billions un-knowns. An increase of 20% effective Fourier modes rep-resents a very significant increase in the resolution for alarge scale computation.

FIG. 5: The 3D view of the vortex tube at t = 6.

FIG. 6: The local 3D vortex structures and vortex linesaround the maximum vorticity at t = 17.

V. DYNAMICS DEPLETION OF VORTEXSTRETCHING

In this section, we present some convincing numeri-cal evidences which show that there is a strong dynamicdepletion of vortex stretching due to local geometric reg-ularity of the vortex lines, which is consistent with thelocal non-blowup criteria of Deng-Hou-Yu [8, 9]. We firstpresent the result on the growth of the maximum vortic-ity in time, see Figure 7. The maximum vorticity in-creases rapidly from the initial value of 0.669 to 23.46at the final time t = 19, a factor of 35 increase fromits initial value. Kerrs computations predicted a finitetime singularity at T = 18.7 [17, 18]. Our computationsshow no sign of finite time blowup of the 3D Euler equa-tions up to T = 19, beyond the singularity time predictedby Kerr. Note that we use three different resolutions inFigure 7 i.e. 768 512 1536, 1024 768 2048, and153610243072 respectively. As we can see, the agree-ment between the resolution 1024 768 2048 and theresolution 1536 1024 3072 is very good with onlymild disagreement toward the end of the computations.This indicates that a very high space resolution is indeedneeded to capture the rapid growth of maximum vorticity

5

0 2 4 6 8 10 12 14 16 180

5

10

15

20

25

t[0,19],768 512 1536t[0,19],1024 768 2048t[10,19],1536 1024 3072

FIG. 7: The maximum vorticity in time using threedifferent resolutions.

0 2 4 6 8 10 12 14 16 180

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6t[0,19],768 512 1536t[0,19],1024 768 2048t[10,19], 1536 1024 3072

FIG. 8: The inverse of maximum vorticity in time usingthree different resolutions.

at the later stage of the computations.

We also study the inverse of the maximum vorticity intime using different resolutions in Figure 8. As we cansee from Figure 8, the inverse of the maximum vorticityapproaches to zero almost linearly in time for 8 t 17. This was one of the strong evidences presented in[17] that suggests a finite time blowup of the 3D Eulerequations. If this trend were to continue to hold up to T ,it would have led to the blowup of the maximum vorticityin the form of O((T t)1). However, as we increaseour resolutions, we find that the curve corresponding tothe inverse of the maximum vorticity starts to turn awayfrom zero around t = 17. This is precisely the time whenKerrs computations began to lose resolution. By t =17.5, the gradients of the solution become very large inall three directions. In order to resolve the nearly singularsolution structure, we use 1536 1024 3072 grid pointsfrom t = 10 to 19. This level of resolution gives about16 grid points across the most singular region in eachdirection at t = 18 and 8 grid points at t = 19.

In order to understand the nature of the dynamicgrowth in vorticity, we examine the degree of nonlinear-ity in the vortex stretching term. In Figure 9, we plot

15 15.5 16 16.5 17 17.5 18 18.5 190

5

10

15

20

25

30

35

|| u||c

1 |||| log(||||)

c2 ||||

2

FIG. 9: Study of the vortex stretching term in time, resolution1536 1024 3072. The fact | u | c1|| log || plusD

Dt|| = u implies || bounded by doubly exponential.

10 11 12 13 14 15 16 17 18 19

1

0.5

0

0.5

1

FIG. 10: The plot of log log vs time, resolution 1536 1024 3072.

the quantity, u , as a function of time, where is the unit vorticity vector. If the maximum vortic-ity indeed blew up like O((T t)1), as alleged in [17],this quantity should have been quadratic as a functionof maximum vorticity. We find that there is tremendouscancellation in this vortex stretching term. It actuallygrows slower than C~ log(~), see Figure 9. Itis easy to show that u C~ log(~)would imply only doubly exponential growth in the max-imum vorticity. Indeed, as demonstrated by Figure 10,the maximum vorticity does not grow faster than doublyexponential in time. We have also generated the similarplot by extracting the data from Kerrs paper [17]. Wefind that log(log()) basically scales linearly with re-spect to t from 14 t 17.5 when his computations arestill reasonably resolved. This implies that the maximumvorticity up to t = 17.5 in Kerrs computations does notgrow faster than doubly exponential in time. This is con-sistent with our conclusion.

Another important evidence which supports the non-blowup of the solution up to t = 19 is that the maximum

6

0 2 4 6 8 10 12 14 16 180.3

0.35

0.4

0.45

0.5

0.55t[0,19],768 512 1536t[0,19],1024 768 2048t[10,19],1536 1024 3072

FIG. 11: Maximum velocity u in time using three differ-ent resolutions.

velocity remains bounded, see Figure 11. This is in con-trast with the claim in [18] that the maximum velocityblows up like O((T t)1/2) with T = 18.7. With the ve-locity field being bounded, the Deng-Hou-Yu non-blowupcriterion (see Theorem 1) can be applied with A = 0 andB = 1/2, which implies that the solution of the 3D Eulerequations remains smooth at least up to T = 19.

It is interesting to ask how the vorticity vector alignswith the eigenvectors of the deformation tensor. Recallthat the vorticity equations can be written as [20]

t + (u ) = S , S =

1

2(u + T u). (1)

Let 1 < 2 < 3 be the three eigenvalues of S. The in-compressibility condition implies that 1 + 2 + 3 = 0.If the vorticity vector aligns with the eigenvector corre-sponding to 3, which gives the maximum rate of stretch-ing, then it is very likely that the 3D Euler equationswould blow up in a finite time.

In Table 1, we document the alignment information ofthe vorticity vector around the point of maximum vor-ticity with resolution 1536 1024 3072. In this table,i is the angle between the i-th eigenvector of S and thevorticity vector. One can see clearly that for 16 t 19the vorticity vector at the point of maximum vorticity isalmost perfectly aligned with the second eigenvector ofS. The angle between the vorticity vector and the secondeigenvector is very small throughout this time interval.Note that the second eigenvalue, 2, is positive and isabout 20 times smaller in magnitude than the largestand the smallest eigenvalues. This dynamic alignmentof the vorticity vector with the second eigenvector of thedeformation tensor is another indication that there is astrong dynamic depletion of vortex stretching.

VI. KIDA-PELZS HIGH-SYMMETRY DATA

Another well-known numerical evidence for finite timeEuler singularities is the Kida-Pelzs high-symmetry ini-

time || 1 1 2 2 3 3

16.012 5.628 -1.508 89.992 0.206 0.007 1.302 89.998

16.515 7.016 -1.864 89.995 0.232 0.010 1.631 89.990

17.013 8.910 -2.322 89.998 0.254 0.006 2.066 89.993

17.515 11.430 -2.630 89.969 0.224 0.085 2.415 89.920

18.011 14.890 -3.625 89.969 0.257 0.036 3.378 89.979

18.516 19.130 -4.501 89.966 0.246 0.036 4.274 89.984

19.014 23.590 -5.477 89.966 0.247 0.034 5.258 89.994

TABLE I: The alignment of the vorticity vector and the eigen-vectors of S around the point of maximum vorticity with res-olution 1536 1024 3072. Here, i is the angle between thei-th eigenvector of S and the vorticity vector.

tial data [3, 19]. Some people have argued that the singu-lar solution of the 3D Euler equations, if it exists, couldbe very unstable. A highly symmetric initial conditionmay have a better chance to produce a finite time sin-gularity. It is also believed that a computer code needsto build in this symmetry property explicitly in order tocapture the potentially unstable singular solution. Thisconsideration motivated Boratav and Pelz to perform nu-merical simulations using a high-symmetry initial condi-tion for the Navier-Stokes equations in [3].

The initial condition that Boratav and Pelz used [3] hasthe rotational symmetry and the permutation symmetry,which was first introduced by Kida [19]. Their simula-tions suggested a possible finite time blowup of the max-imum vorticity in the limit of infinite Reynolds numbers.However, as they realized later, their simulations wereunder-resolved at later times when the solution becamenearly singular. The vortex structure near the region ofmaximum vorticity motivated Pelz to construct a vortexfilament model to understand this singular behavior. In[21], Pelz presented some numerical evidences which sug-gest that his filament model develop a self-similar blowupin a finite time. It is interesting to note that Pelzs selfsingular solution also falls into the critical case of theDeng-Hou-Yus local non-blowup criteria (see Theorem2). To understand if the same initial condition that ledto a finite time blowup in Pelzs filament model wouldlead to a finite time blowup of the full 3D Euler equa-tions, we decide to repeat Pelzs computations.

Pelzs original filament model was designed for the en-tire free space. To perform the numerical simulation ofthe 3D Euler equations in the free space R3 is very expen-sive. As a first step, we derive a corresponding periodicfilament model. The periodic filament model involvesan infinite sum over all the periodic images of the Biot-Savart kernel. This makes the computation of the pe-riodic filament kernel more expensive than the one overthe free space. To reduce the computational cost, weapply the Ewald summation formula, which significantlyreduces the computational cost.

We solve the periodic filament model using an initialcondition which is qualitatively the same as the one used

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0.5

0

0.5

scaling constant tcrit

using u=c(tcrit

t)1/2

t crit

time step

FIG. 12: The validity check of singularity fitting using theasymptotic expression u =

Ctcritt

. The figure shows

tcrit as a function of the computational steps, with tcrit 0.0257874. Adaptive time stepping is used with the time stepchosen to be proportional to the inverse of u.

10.80.60.40.200.20.40.60.81

FIG. 13: The locations of the filaments at the end of ourcomputation. The figure gives a closeup view of the filamentsaround the origin.

by Pelz [21]. Our numerical computations show that theperiodic filament model indeed develops a finite time self-similar singularity around t = 0.0257874, see Figures 12-13. However, when we use the same initial condition tosolve the full 3D Euler equations, we find that the solu-tion of the 3D Euler equations has a completely differentbehavior from that of the filament model. We observe nofinite time singularity for the 3D Euler equations usingthe same initial condition. We use a sequence of spaceresolutions with the two largest resolutions being 10243

and 20483. More than 100Gb memory is used in ourcomputation on the 20483 computations. As we can seefrom Figures 14-15, the growth of maximum vorticity intime is very mild. The maximum velocity is bounded andbecomes saturated around t = 0.0325. The 3D isosurfaceof the vortex tubes at t = 0.03 also shows that the vor-tex tubes remain quite regular. We remark that Grauerand his coworkers have recently carried out the full Euler

0 0.005 0.01 0.015 0.02 0.025400

420

440

460

480

500

520

540

560

580

600

|| in time on 10243(dashed) and 20483(solid) mesh grids

FIG. 14: Maximum vorticity in time of the full Euler equa-tions with two resolutions: 10243 (dashed line) vs N = 20484

(solid line).

0 0.02 0.04 0.0636.5

37

37.5

38

38.5

39

39.5

|u| in time on 10243 mesh grids. The end part shows a saturation.

FIG. 15: Maximum velocity in time of the full Euler equationswith resolution : 10243. The maximum velocity seems tosaturate at a later time.

simulation using a simplified Pelzs high-symmetry ini-tial condition which consists of 12 straight parallel bars[13]. They find that the vortex tubes become severelyflattened as they approach each other and the growth ofmaximum vorticity is only exponential in time.

Finally, we remark that we have repeated Boratav andPelzs Navier-Stokes computations [3] using the same ini-tial condition, building both the rotational and permuta-tion symmetries of the solution explicitly into our code.Our resolution study shows that their computations areresolved only up to t = 1.6 when the growth of the max-imum vorticity is only exponential in time. The nearlysingular growth of maximum vorticity around t = 2.06seems due to under-resolution.

VII. CONCLUDING REMARKS

Our analysis and computations reveal a subtle dynamicdepletion of vortex stretching. Sufficient numerical res-olution is essential in capturing this dynamic depletion.Our computations for the two antiparallel vortex tubes

8

FIG. 16: The 50% isosurface of |~| at t = 0.03. Full 3D Eulerequations.

initial data and the high-symmetry initial data show thatthe velocity is bounded and that the vortex stretchingterm is bounded by CL log(L). It is natural to

ask if is this dynamic depletion generic? and what is thedriving mechanism for this depletion of vortex stretch-ing? Some exciting progress has been made recently inanalyzing the dynamic depletion of vortex stretching andnonlinear stability for 3D axisymmetric flows with swirl[16]. The local geometric structure of the solution nearthe region of maximum vorticity and the anisotropic scal-ing of the support of maximum vorticity seem to play akey role in the dynamic depletion of vortex stretching.

Acknowledgments

We would like to thank Prof. Lin-Bo Zhang from theInstitute of Computational Mathematics and the Centerof High Performance Computing in Chinese Academy ofSciences for providing us with the computing resourceto perform this large scale computational project. Wealso thank Prof. Robert Kerr for providing us with hisFortran subroutine that generates his initial data. Thiswork was in part supported by NSF under the NSFgrants, FRG DMS-0353838, ITR ACI-0204932 and DMS-0713670. Li was subsidized by the National Basic Re-search Program of China under the grant 2005CB321701.

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