@let@token singularity formation in the 3d euler equations?

Singularity formation in the 3D Euler equations?

J. D. Gibbon : Imperial College London

Levico Terme 2012

J. D. Gibbon : Imperial College London () Singularity formation in the 3D Euler equations? Levico Terme 2012 1 / 40

Summary of this talk

Themes1 Some elementary introductory remarks on the 3D incompressible

Euler equations & vortex stretching ;2 The Beale-Kato-Majda theorem & variations with an elementary

sketch-proof.3 The search for singularities : a list of numerical experiments.4 Vortex stretching in 3D incompressible, stratified, rotating Euler

equns ; including Ertel’s Theorem and the role of potentialvorticity q = ω · ∇θ – going up to the next derivative.

5 Euler with a passive scalar on a finite domain : a road to blow-up?6 Yet another look at singularity formation ; the very recent

numerical work of Kerr.7 Other models? 2D s-QG ; Restricted Euler ; Tetrad model.


Vortex “stretching & folding” in Euler

For an incompressible fluid (div u = 0), the Euler equations are [1, 2]

DuDt

= −∇p withDDt

= ∂t + u · ∇

Alternative formulationWith the vorticity as ω = curl u, an alternative is (Sij = 1

2

(ui,j + uj,i

))

∂tu − u × ω = −∇(p + 1

2 u2)∂t ω = curl

(u × ω

), or

DωDt

= ω · ∇u = Sω .

Alignment of ω with e-vectors of S leads to “stretching & folding” of thevorticity field depending on the signs of λS(x , t). This processproduces the fine-scale “crinkles” in the vorticity field & may endin a finite time singularity.


3D Euler: Why the interest in singularities?

Physically their formation may signify the onset of invicidturbulence & may be a mechanism for energy transfer to smallscales.Numerically they require special methods – a challenge to CFD.Mathematically their onset would rule out global existence altho’there are results on very weak solutions [3, 4, 5, 6, 7, 8, 9, 10].

Question : is there a single quantity that controls 3D Euler solns?

1 Like the 3D NS equations, it was thought originally that controlover the H1-norm (‖ω‖2) was sufficient – see Chorin [11].

2 Problem : solutions can remain bounded in ‖ω‖2 but not in ‖ω‖∞.3 This has been addressed through a combination of rigorous proof

(Beale-Kato-Majda Theorem [12]) and numerical experiment.


Beale-Kato-Majda-Theorem (1984)

Theorem (BKM theorem [12])

Let u ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−1) be a global solution of the 3D Eulerequns in R3for s > 5/2. Suppose ∃ a time T ∗ such that the solution cannotbe continued to T = T ∗, then if T ∗ is the first such time∫ T∗

0‖ω(· , τ)‖∞ dτ =∞ .

Moreover, if∫ T

0 ‖ω(· , τ)‖∞ dτ <∞ for every T > 0 then ∃ a global solution ofthe Euler equations in [0, T ].

VariationsPonce (1985) has proved

∫ t0 ‖S(· , τ)‖∞ dτ <∞ controls solutions [14].

Ferrari (1993) used BCs u · n = 0 [15]. K-T (2000) [16] & Constantin [2, 17].

CorollaryIf a singularity ‖ω(· , t)‖∞ ∼ (T ∗− t)−β is observed in a numerical experiment,for the singularity to be genuine & not an artefact of the numerics then β ≥ 1.


Sketch of BKM theorem proof I

Existence & uniqueness local in time goes back to Kato [13]. The BKM Thmdetermines conditions on whether regularity is lost. Proof by contradiction :assume

∫ T∗

0 ‖ω‖∞ dτ <∞. From this we prove solutions are regular at T ∗

thus contradicting the hypothesis that T ∗ is the first time regularity is lost.

Step 1 : Define Hn =∫V |∇

nu|2dV , then on R3 (or periodic) BCs

12 Hn = −

∫V∇nu · ∇n (u · ∇u +∇p) dV

= −∫V

u · ∇φdV −n−1∑k=0

∫V

Cn−1k (∇nu) (∇k+1u) (∇n−k u) dV

where φ = 12 |∇nu|2. With a Holder inequality 1

p + 1q = 1

2

12 Hn ≤ H1/2

n

n−1∑k=0

Cnk ‖∇k+1 u‖p‖∇n−k u‖q

Using the Gagliardo-Nirenberg inequalities : 1P = j

d + A( 1

r −md

)+ 1−A

Q

‖∇ju‖P ≤ c ‖∇mu‖Ar ‖u‖1−A

QJ. D. Gibbon : Imperial College London () Singularity formation in the 3D Euler equations? Levico Terme 2012 6 / 40

Sketch of BKM theorem proof II

Let θ = ∇u so with 1p = k

d + A(

12 − n−1

d

)we have

‖∇kθ‖p ≤ c ‖∇n−1θ‖A2 ‖θ‖1−A

∞

Likewise with 1q = n−k−1

d + B(

12 − n−1

d

)we also have

‖∇n−k−1θ‖q ≤ c ‖∇n−1θ‖B2 ‖θ‖1−B

∞

Note, however, that because 1p + 1

q = 12 , then A + B = 1, giving us

12 Hn ≤ H1/2

n

n−1∑k=0

Cnk H

12 (A+B)

n ‖∇u‖2−A−B∞

LemmaFor n ≥ 1, the Hn satisfy

12 Hn ≤ cn‖∇u‖∞Hn


BKM proof III : A bound for ‖∇u‖∞ in terms of ‖ω‖∞

Step 2 : It is always true that ‖ω‖∞ ≤ ‖∇u‖∞ and

‖∇u‖p ≤ cp‖ω‖p 1 < p <∞ .

but for p =∞ BKM proved a difficult technical result [12] in R3

‖∇u‖∞ ≤ c ‖ω‖∞ (1 + ln H3) + ‖ω‖2 .

Thus for n = 3 we have

12 H3 ≤ c3‖ω‖∞H3 (1 + ln H3)

so integrating

H3(t) ≤ exp exp 2c3

∫ t

0‖ω‖∞ dτ .

Thus for n ≥ 1, each Hn is controlled by∫ t

0 ‖ω‖∞ dτ .

The proof in R3 & periodic BCs uses the Biot-Savart integral : doesn’t work forrigid boundaries u · n = 0 – see later the work of Ferrari [15].


Numerical search for singularities

An up-dated version of a list originally compiled by JDG/Rainer Grauer

1) Morf, Orszag and Frisch (1980): [18, 19, 20] : complex timesingularities of the 3D Euler equations were studied usingPade-approximants. Singularity; yes.

2) Chorin (1982): [11] : Vortex–method. Singularity; yes.

3) Brachet, Meiron, Nickel, Orszag & Frisch (1983): [21, 22] : TaylorGreen i.c.s : Saw vortex sheets. Singularity; no.

4) Siggia 1984 [23] : Vortex–filament method. Singularity; yes.

5) Ashurst & Meiron (’87) [24] : Finite-diff methods. Singularity; yes.

6) Kerr & Pumir (’87) [25]: Pseudo-spectral method. Singularity; no.

7) Pumir & Siggia (1990): [26]: results from their adaptive grid methodshowed a tendency to develop quasi-two-dimensional structures withexponential growth of vorticity. Singularity; no.


Singularity list contd ...

8) Bell & Marcus (1992): [27] : projection method with 1283 meshpoints ; amplification of vorticity by 6. Singularity; yes.

9) Brachet, Meneguzzi, Vincent, Politano & Sulem (1992) : [28]:pseudospectral code, Taylor–Green vortex, with a resolution of 8643.Achieved an amplification of vorticity by 5. Singularity; no.

10) Kerr (1993): [29, 30, 31] : Chebyshev polynomials with anti–parallelinitial conditions ; resolution 5122 × 256. Amplification of vorticity by18. Observed vorticity growth ‖ω‖L∞(Ω) ∼ (T ∗ − t)−1. Singularity; yes.

11) Grauer and Sideris (1991) [32] : 3D axisymmetric swirling flow.Singularity; yes. However, both E and Henderson independentlyconcluded : Singularity; no.

12) Boratav & Pelz (1994,95), [33, 34], Pelz & Gulak (1997) [36] andPelz (1997,2002) [35, 37] performed a series of 10243 grid-pointsimulations under Kida’s high-symmetry condition. Singularity; yes.



13) The FDR-memorial issue for Pelz (2005):

1 Cichowlas and Brachet [39] : Singularity; no.2 Gulak and Pelz [40] : Singularity; yes.3 Pelz and Ohkitani [41] : Singularity; no.4 Kerr [42] : Singularity; yes.

14) Grauer, Marliani & Germaschewski (1998) [43] : using an adaptive meshrefinement of the Bell-Marcus i.c. [27] with 20483 mesh points, achieved anamplification of vorticity of 21. Singularity; yes.

15) Hou & Li (2006): [44] : a 1536× 1024× 3072 pseudo-spectral calculationagreed with Kerr (1993) until the final stage but growth slowed; vorticity grewdouble exponentially. Singularity; no.

16) Orlandi & Carnevale (2007) [45] : using Lamb dipoles as initialconditions, they performed a 10243 finite difference calculation with 2symmetry planes. Found a period of rapid growth of vorticity consistent with‖ω‖∞ ∼ (t∗ − t)−1: Singularity; yes.



17) Kerr & Bustamante (2008) : Found growth of vorticity consistentwith ‖ω‖L∞(Ω) ∼ (T − t)−β for β > 1 : Singularity; yes.

18) Bustamante & Brachet 2012 [47] & [48] : Complex singularities andthe analyticity-strip method to investigate numerically theincompressible Euler singularity problem : Singularity; yes.

19) Kerr (2012) : see later [49] Singularity; yes.

20) Pauls, Matsumoto, Frisch & Bec (2006): [50]: a study of complexsingularities of the 2D Euler equations and contains a good list ofreferences for the student.

21) Further refs :1 See also the refs in JDG [51] and Bardos & Titi [52]2 Other refs [53, 54, 55, 56, 57, 58].


Direction of vorticity : the work of CFM & DHY

a) Constantin, Fefferman & Majda (1996) [17, 59] discussed the ideaof vortex lines being “smoothly directed” in a region of greatestcurvature : no singularity at time T if the velocity is finite in (B4ρ) &

limt→T

supW0

∫ t

0‖∇ξ(· , τ)‖2L∞(B4ρ) dτ <∞

where ξ(x , t) = ω(x , t)/|ω(x , t)|.

Theorem (Deng-Hou-Yu [60])

Let x(t) be a family of points such that for some c0 > 0 the vorticity satisfies|ω(x(t), t)| > c0Ω(t). Assume that for all t ∈ [0,T ) there is another point y(t)on the same vortex line as x(t), s.t. the unit vorticity ξ(x , t) along the vortexline c(s) between x(t) and y(t) is well-defined. If we further assume that∫ y(t)

x(t)(∇ · ξ)(c(s), t) ds ≤ C

for some const C &∫ T

0 |ω(y(t), t)| dt <∞ then no blowup occurs up to timeT .J. D. Gibbon : Imperial College London () Singularity formation in the 3D Euler equations? Levico Terme 2012 13 / 40

Direction of vorticity : DHY2

Theorem (Deng-Hou-Yu [61])

Take the arc length L(t) of a vortex line with unit normal n & curvature κ. Noblow-up can occur at time T if

1 Uξ(t) + Un(t) . (T − t)−A A + B = 1,

2 M(t)L(t) ≤ const > 0

3 L(t) & (T − t)B .

Uξ(t) = maxx,y [(u · ξ)(x , t)− (u · ξ)(y , t)] & Un(t) = maxLt [(u · n] &

M(t) ≡ max(‖∇ · ξ‖L∞(Lt ), ‖κ‖L∞(Lt )

)Grafke & Grauer (2012) [62] have investigated these two thms numerically :for Kida’s vortex dodecapole i.c., they injected Lagrangian tracer particles tomonitor geometric properties of vortex line segments. From high resolutionadaptive numerical simulations they concluded : Singularity ; no.


3D incompressible, stratified, rotating Euler equations

The 3D Euler equations for an incompressible, stratified, rotating flow(Ω = k Ω) in terms of the velocity field u(x , t) and temperature θ are

DuDt

+ 2 (Ω× u)︸︷︷︸rotation

+ k θ︸︷︷︸buoyancy

= −∇p ,

DθDt

= 0 .

Information about ∇θ is needed to determine how θ(x , t) mightaccumulate into large local concentrations.

Evolution of ωrot

Now consider the vorticity ω = curl u for which ωrot = ω + 2Ω satisfies

Dωrot

Dt= ωrot · ∇u +∇⊥θ ∇⊥ =

(−∂y , ∂x , 0

)J. D. Gibbon : Imperial College London () Singularity formation in the 3D Euler equations? Levico Terme 2012 15 / 40

The 3D Euler equations and Ertel’s Theorem

Theorem (Ertel’s Theorem (1942) [63])

If ωrot (x , t) satisfies the 3D rotating Euler equations then any arbitrarydifferentiable µ(x , t) satisfies

DDt

(ωrot · ∇µ) = ωrot · ∇(

DµDt

).

Remark : The operations[D

Dt , ωrot · ∇]

= 0 commute.

Proof : Ertel (1942) [63]; Truesdell & Toupin (1960) [64] ; Hoskins et al (1985)[65] ; Ohkitani (1993) [66] ; Kuznetsov & Zakharov (1997) [67] ; Viudez (2001)[68].

DDt

(ωrot · ∇µ) =

(Dωrot

Dt− ωrot · ∇u

)· ∇µ+ ωrot · ∇

(DµDt

)


Potential Vorticity for rotating stratified Euler

Potential Vorticity is defined as

q = ωrot · ∇θ withDθDt

= 0 .

PV is important in GFD : – Hoskins, McIntyre, & Robertson (1985) [65].

TheoremBoth q and θ are materially conserved quantities.

Proof.Take µ(x , t) = θ, and thus

DqDt

=

(Dωrot

Dt− ωrot · ∇u

)· ∇θ + ωrot · ∇

(DθDt

)= ∇⊥θ · ∇θ = 0 .

Thus, because θ is a materially conserved quantity so is q.


Evolution of the B-field

TheoremThe vector B = ∇q ×∇θ satisfies

∂t B = curl (u ×B) ⇒ DBDt

= B · ∇u .

Kurgansky (’87) [69, 70, 71], Yahalom (’96) [72] & JDG-Holm [73, 74]

Proof : see JDG & Holm [73, 74].

∂tB = ∂t (∇q)× (∇θ) + (∇q)× ∂t (∇θ)

= −u · ∇(∇q) + (∇q) · ∇u + (∇q)× ω × (∇θ)

−(∇q)× u · ∇(∇θ) + (∇θ) · ∇u + (∇θ)× ω= −u · ∇B + (∇q)(ω · ∇θ)− (∇θ)(ω · ∇q)

+(∇θ)× (∇q · ∇u)− (∇q)× (∇θ · ∇u)

= curl (u ×B)


Why are we not surprised? – JDG & D2H : J. Phys. A, 43, (2010) 172001

Consider B = ∇q ×∇θ where DqDt = 0 and Dθ

Dt = 0

AAAK

B

θ = const

q = const

∇θ

∇q

B is tangent to the intersection-curve of q = const and θ = const

DBDt

= B · ∇u

which is the equn for the stretching of a line-element B ≡ δ`.


Stretching & folding in the B-field

Because div u = 0 and divB = 0 we have

curl (u ×B) = B · ∇u − u · ∇B

∂t B = curl (u ×B) orDBDt

= B · ∇u ,

The same as for ω & the magnetic B-field in MHD (Moffatt 1978 [75]).

Stretching & folding properties(i) Thus all the “stretching & folding” properties associated with ωor magnetic field-lines lift over to B even though B contains ω,∇ω, ∇θ and ∇2θ in various forms of projection.

(ii) Moreover, for any surface S(u) moving with the flow u, one finds

ddt

∫S(U)

B · dS = 0 .


Helicity in the B-field

Now define the vector potential A such that B = curlA where

A = 12

(q∇θ − θ∇q

)+∇ψ .

The helicity H that results from this definition,

H =

∫VA ·B dV =

∫V

div(ψB)

dV =

∮∂VψB · n dS ,

measures the knottedness of the B field-lines. H = 0 for homog-BCsbut H 6= 0 for realistic topographies. The boundaries may therefore bean important generating source for helicity, thus allowing the formationof knots & linkages in the B-field.

See Ohkitani (2007) [76] for a discussion of helicity-free vorticity fields.

ω = ∇f ×∇g

with Df/Dt = 0 and Dg/Dt = 0.J. D. Gibbon : Imperial College London () Singularity formation in the 3D Euler equations? Levico Terme 2012 21 / 40

Blow-up with a passive scalar? I

Let us consider Euler with a passive scalar θ : this could represent atracer-dye or some dust added to the flow. What do we know?

1 The potential vorticity q = ω · ∇θ is also passive : thus we have

DθDt

= 0DqDt

= 0

2 B = ∇q ×∇θ satisfies

DBDt

= B · ∇u

3 Do null points (zeros) occcur in |B|? Firstly, i.d. for |B| must befree of null points : then a null can potentially develop either bymaxima or minima developing in θ or q or if ∇q and ∇θmomentarily align at some point.


BKM Theorem on a finite domain

The proofs for V ≡ R3 require FTs & the Biot-Savart integral which are noteasily adaptable to BCs u · n|∂ V = 0. Ferrari [15] circumvented thisdifficulty to prove these results for BCs u · n|∂ V = 0.

Theorem (Ponce [14], Ferrari [15])

There exists a global solution of the 3D Euler equations for s ≥ 3u ∈ C([0, ∞]; Hs) ∩ C1([0, ∞]; Hs−1) if, for every T > 0∫ T

0‖S(τ)‖L∞(R3) dτ <∞ .

Conversely, if there exists a time T for which∫ T

0‖S(τ)‖L∞(R3) dτ =∞ ,

then limt→T ‖S(t)‖L∞(R3) =∞.


The B-theorem

We consider a finite domain V ⊂ R3 with boundary conditions

u · n|∂ V = 0 .

Theorem (JDG & Titi 2012 ArXiv: 1211.3811v1 [77])

On a smooth domain V ⊂ R3 with boundary conditions u · n|∂ V = 0,with initial data for which |B(x , 0)| > 0 and ‖B (x ,0) ‖L∞(V) <∞, ifthere exists a smooth solution of the 3D Euler equations in the interval[0, t∗), then at the earliest time t∗ at which |B(x , t∗)| = 0∫ t∗

0‖S‖L∞(V) dτ =∞ .

Conversely, if∫ T

0 ‖S‖L∞(V) dτ <∞ in any interval [0, T ] then |B(x , t)|cannot develop a null point for t ∈ [0, T ].


Proof of the B-theorem : I

On [0, t∗) take the scalar product with B, where |B| = B

12

D(B2)

Dt= B · ∇u ·B ⇒ 1

2D| ln B|2

Dt= (ln B)

(B · S · B

).

Now multiply by | ln B|2(m−1) for 1 ≤ m <∞

12m

D| ln B|2m

Dt= | ln B|2(m−1)(ln B)

(B · S · B

).

Integrate, invoke the Divergence Thm/BCs & use Holder’s inequality

12m

ddt

∫V| ln B|2mdV ≤

∫V|ln B|2m−1 |S| dV

≤(∫V| ln B|2mdV

) 2m−12m(∫V|S|2mdV

) 12m

.


Proof of the B-theorem : II

Using the standard notation ‖X‖Lp(V) =(∫V |X |

pdV)1/p

ddt‖ ln B‖L2m(V) ≤ ‖S‖L2m(V)

which integrates to

‖ ln B(t)‖L2m(V) ≤ ‖ ln B(0)‖L2m(V) +

∫ t

0‖S(τ)‖L2m(V) dτ .

Since V is bounded we take the limit m→∞ to obtain

‖ ln B(t)‖L∞(V) ≤ ‖ ln B(0)‖L∞(V) +

∫ t

0‖S(τ)‖L∞(V) dτ .

If |B| has no zero in its i.d. the log-singularity at |B| = 0 causes theLHS to blow up at t∗ thus forcing

∫ t∗0 ‖S‖L∞(V) dτ →∞ as t → t∗.

If∫ T

0 ‖S‖L∞(V) dτ remains finite in an interval t ∈ [0, T ] then no nullcan develop in B.


Example of i.d. with no null points : I

Is it easy to construct i.d. such that |B(x , 0)| > 0? We find a simpleexample of a set of i.d. u on a finite domain V ⊂ R3 from initial data onω and θ such that |B| > 0 and ‖B‖L∞(V) <∞ for the elliptic system

curl u = ω , div u = 0 , u · n|∂ V = 0 .

Consider ω = (1, 1, 1)T : ∃ a vel-field v = (z, x , y)T which satisfies

div v = 0 and curl v = (1,1,1)T

but we cannot be sure that it satisfies v · n = 0 for any V : modify thisto satisfy the BCs by introducing some potential φ such that

u = v +∇φ .Note that curl u = (1, 1, 1)T . φ must satisfy the Neumann boundaryvalue problem

∆φ = 0 ,∂φ

∂n

∣∣∣∣∂ V

= −(z, x , y)T · n ,

which always has a solution on any smooth domain V.J. D. Gibbon : Imperial College London () Singularity formation in the 3D Euler equations? Levico Terme 2012 27 / 40

Example of i.d. with no null points : II

Thus we have been able to construct a velocity field u correspondingto ω = (1, 1, 1)T , that satisfies the boundary conditions. For simplicity,now choose

θ = 12 (x2 + y2 + z2)

and calculate q, ∇q and ∇θ

q = x + y + z , ∇θ = (x , y , z)T , ∇q = (1, 1, 1)T ,

and then B

B = (z − y , x − z, y − x)T .

Note that |B| = 0 only on the straight line x = y = z = t for t ∈ R.Hence |B| > 0 on any smooth, finite domain V that avoids this line :this is enough to achieve our goal.


Another new approach to Euler blow up : JDG [78]

Define the set of frequencies based on L2m-norms of ω = curl u

Ωm(t) =

(L−3

∫V|ω|2mdV

)1/2m

1 ≤ m ≤ ∞ .

Holder’s inequality insists that Ωm ≤ Ωm+1. The Euler equations areinvariant under : x ′ = εx ; t ′ = ε2t ; u = εu′ ; p = ε2p′. If the domainlength L is also re-scaled as L′ = εL then Ωm re-scales as Ωm = ε2Ω′m.If, however, L is not re-scaled but kept fixed then Ωm re-scales as

Ωαmm = εΩ

′ αmm , αm =

2m4m − 3

.

Therefore definine the the set of dimensionless quantities

Dm(t) =($−1

0 Ωm

)αm, 1 ≤ m ≤ ∞ .

Instead of $ 0 = νL−2 for NS, use $ 0 = ΓL−2 where Γ is the circulationaround some i.d.


A remark on the Dm

Consider the boundedness of the integral∫ t

0D2

m dτ (∗)

based on the continuum lying between

D1 = $−20 L−3

∫V|ω|2 dV . . . → . . . D∞ =

($−1

0 ‖ω‖∞)1/2

.

1 When m = 1 (α1 = 2) the criterion (*) is exactly a NS-criterion forregularity ; namely

∫ t0 H2

1 dτ <∞.2 When m =∞ (α∞ = 1

2 ) the criterion (*) is exactly the BKMcriterion for Euler regularity ; namely

∫ t0 ‖ω‖∞ dτ <∞.


A differential inequality for Dm : I

Lemma (JDG [78])

Provided solutions of the three-dimensional Euler equations exist, for1 ≤ m <∞ the Dm formally satisfy the following differential inequality

Dm ≤ cm $ 0

(Dm+1

Dm

)ξm

D3m , ξm = 1

2 (4m + 1) .

Proof : Ωm obeys

L3Ω2m−1m Ωm =

12m

ddt

∫|ω|2mdV

≤∫ t

0ω2m|∇u|dV

≤(∫|ω|2mdV

) 12(∫|ω|2(m+1)dV

) m2(m+1)

(∫|∇u|2(m+1)dV

) 12(m+1)

≤ L3c1,m Ωm+1m+1 Ωm

m


A differential inequality for Dm : II

In the above we have used ‖∇u‖p ≤ cp‖ω‖p, for 1 ≤ p <∞ so thecase p =∞ is excluded because ‖∇u‖∞ ≤ c ‖ω‖∞ (1 + ln H3).

Ωm ≤ c1,m

(Ωm+1

Ωm

)m+1

Ω2m , 1 ≤ m <∞ .

We wish to convert the inequality to one in terms of Dm(Ωm+1

Ωm

)m+1

Ωm = $ 0

(Dm+1

Dm

) m+1αm+1

D2m = $ 0

(Dm+1

Dm

) 12 (4m+1)

D2m

having used the fact that(1

αm+1− 1αm

)βm = 2 , βm = 4

3 m(m + 1) .

Substitution completes the proof.


1st route of integration

Firstly if a finite time singularity is suspected then divide by D3−εm with

0 ≤ ε < 2 to obtain

[Dm(t)]2−ε ≤ 1[Dm(t0)]−(2−ε) − F1,ε(t)

where (ξm = 12 (4m + 1))

F1,ε(t) = cm(2− ε)$ 0

∫ t

t0

(Dm+1

Dm

)ξm

Dεm dτ .

For instance, for ε = 0 we have

D2m(t) ≤ 1

[Dm(t0)]−2 − F1,0(t)


1st route of integration contd ...

where F1,0 ≡ Fm is defined as

Fm(t) = 2cm$ 0

∫ t

t0

(Dm+1

Dm

)ξm

dτ .

A singularity in the upper bound of the inequality is not necessarilysignificant. What is more significant is whether the solution tracks thissingular upper bound. This suggests the following numerical test :

1 Is F1,0 = Fm linear in t?2 If so, then test whether

D2m (Tc,m − t)→ Cm with Tc,m → Tc

uniformly in m? If such behaviour occurs it suggests, but does notprove, that the Dm may be blowing up close to the singular upperbound.


Kerr’s initial condition [49]

Figure: The figure shows the NS evolution of very long anti-parallel vortices with a localizedperturbation in a turbulent flow at three times. The first at t = 16, which is the time of the firstreconnection, is shown on the left ; t = 96 lies in the middle and is the time of the second set ofreconnections. The formation of the first vortex rings at t = 256 is on the right and shows thecumulation of this process after two sets of secondary reconnections have released two sets ofvortex rings. The full domain in the vortical y direction reaches from y = −16π ≈ −50 toy = 16π ≈ 50. The partial domains shown are |y | ≤ 10 for t = 16, 0 ≤ y ≤ 25 for t = 96 and0 ≤ y ≤ 16π ≈ 50 for t = 256. The inset of the upper left vorticity near y = 0 at t = 16demonstrates that on the y = 0 symmetry plane the usual head-tail configuration forms.


The recent results of Kerr I [49]

Figure: Fm =∫$0(Dm+1/Dm)ξm dt for Euler anti-parallel i.d. with resolution

1024× 512× 4096. The observed ordering & linear increase as m→∞would permit at least super-exponential growth of the Dm for both Euler & NSup to t = 14.5.


The recent results of Kerr II [49]

Figure: Estimated the singular time of Dm : Tm(t) = (d log D2m/dt)−1 + t . Only

m odd are shown to reduce clutter & a curve with t = t is added to clarify thetrend. Linear extrapolations to t = 15.75 of the m > 1 curves, based on thelast two values, have dashed lines. These extra polations appear to becrossing the t = t line at about Tc ≈ 15.8.


Other models? Connection with the 2D surface QG equations?

Let q = z = const and θ = const be material surfaces :

B = ∇z ×∇θ = k ×∇θ = −∇⊥θ .

In R2 if u is chosen as u = ∇⊥ψ with θ = −(−∆)1/2ψ, then

DBDt

= B · ∇u B = −∇⊥θ

are the 2D surface QG equations discussed by Constantin, Majda & Tabak(1994) [79] – conjectured that fronts might be finite time singularities.

1 Ohkitani & Yamada (1997) [80] found numerically growth that wasdouble exponential in time ;

2 Confirmed by Constantin, Nie & Schorghofer (1998) [81] ;3 Cordoba (1998) [82] then showed the absence of a singularity for a

simple hyperbolic saddle ;4 See Cordoba, Fefferman & Rodrigo (2004) [83] & Rodrigo (2004).


Other models? Restricted Euler equations – modelling the Hessian P

The gradient matrix Mij = ∂uj/∂x i satisfies (tr P = −tr (M2))

DMDt

+ M2 + P = 0 , tr M = 0 .

Attempts have been made to model the Lagrangian averaged pressureHessian by a constitutive closure. Idea goes back to :

Leorat (1975) [84] ; Vieillefosse (1984) [85] ; Cantwell (1992) [86]

who assumed that the Eulerian pressure Hessian P = p,ij is isotropic. Thisresults in the restricted Euler equations with

P = − 13 I tr (M2) , tr I = 3 .

Constantin’s distorted Euler equations (1986) [87] : With the Lagrangianpath map a 7→ X (a, t), N = M X Euler equns are re-written as

∂N∂t

+ N2 + Q(x , t)Tr(N2) = 0 , Qij = RiRj Ri = (−∆)−1/2∂i

‘Distorted Euler equns’ : replace Q(t) by Q(0)⇒ rigorous blow up.J. D. Gibbon : Imperial College London () Singularity formation in the 3D Euler equations? Levico Terme 2012 39 / 40

Other models? The Tetrad model

A : Tetrad model of Chertkov, Pumir & Shraiman (1999) [88], developed byChevillard & Meneveau (2006) [89] : has the assumption that the Lagrangianpressure Hessian is isotropic.

For P to transform as a Riemannian metric & satisfy tr P = −tr (M2)

P = − Gtr G

tr (M2) , G(t) = I ⇒ restricted Euler .

tr P = −tr (M2) is satisfied for any choice of G = GT (chosen flow-law)

P = −[ N∑β=1

cβGβ

tr Gβ

]tr (M2) , with

N∑β=1

cβ = 1 ,

B : Constantin, Lax & Majda (1985) [90] : 1D vorticity model :

ωt = ωπ−1p.v .∫ ∞−∞

ω(y) dyx − y

u(x , t) =

∫ ∞−∞

ω(y , t)dy

where ω blows up but u remains finite.J. D. Gibbon : Imperial College London () Singularity formation in the 3D Euler equations? Levico Terme 2012 40 / 40

[1] Majda A J and Bertozzi A L 2001 Vorticity and Incompressible Flow(Cambridge: Cambridge University Press)[2] Constantin P, On the Euler equations of incompressible fluids, Bull.Amer. Math. Soc., 44, (2007), 603–621.[3] Shnirelman A, On the non-uniqueness of weak solution of the Eulerequation, Commun. Pure Appl. Math., (1997), 50, 1260–1286.[4] Brenier Y, (1999) Minimal geodesics on groups of volume-preservingmaps and generalized solutions of the Euler equations, Comm. PureAppl. Math., 52, 411-452.[5] Bardos C, Titi E S, Euler equations of incompressible ideal fluids,Russ. Math. Surv., (2007), 62, 409–451.[6] De Lellis C, Szekelyhidi L, The Euler equations as a differentialinclusion, Ann. Math., (2009)(2), 170(3), 1417-1436.[7] De Lellis C, Szekelyhidi L, On admissibility criteria for weak solutionsof the Euler equations, Arch. Ration. Mech. Anal., (2010), 195, 225-260.[8] Bardos C, Titi E S, Loss of smoothness and energy conserving roughweak solutions for the 3D Euler equations, Discrete and continuousdynamical systems, (2010), 3 (2), 187–195.


[9] Wiedemann E, Existence of weak solutions for the incompressibleEuler equations, Annales de l’Institut Henri Poincare (C) NonlinearAnalysis, (2011), 28(5), 727–730.[10] Bardos C, Titi E S, Wiedemann E, The vanishing viscosity as aselection principle for the Euler equations : The case of 3D shear flow,Comptes Rendus De L’Academie Des Sciences, Paris, Serie I,Mathematique, (2012), 350(15), 757–760.[11] Chorin A J, The evolution of a turbulent vortex, Commun. Math.Phys. 83 (1982) 517–535.[12] Beale J T, Kato T and Majda A J 1984 Remarks on the breakdown ofsmooth solutions for the 3D Euler equations Commun. Math. Phys. 9461–66[13] Quasi-linear equations of evolution, with application to PDEs,Lecture Notes in Math., 448, Springer, Berlin, 1975.[14] Ponce G 1985 Remarks on a paper by J. T. Beale, T. Kato and A.Majda, Commun. Math. Phys. 98 349–353.[15] Ferrari A 1993 On the blow-up of solutions of the 3D Euler equationsin a bounded domain Comm. Math. Phys. 155 279–294.


[16] Kozono H and Taniuchi Y, Limiting case of the Sobolev inequality inBMO, with applications to the Euler equations, Comm. Math. Phys. 214,(2000), 191–200[17] Constantin P 1994 Geometric statistics in turbulence SIAM Rev. 3673–98.[18] Morf R H, Orszag S, & Frisch U, Spontaneous singularity inthree-dimensional, inviscid incompressible flow. Phys. Rev. Lett., 44, 572,1980.[19] Bardos C, Benachour S & Zerner M, Analyticite des solutionsperiodiques de l’equation d’Euler en deux dimensions, C. R. Acad. Sc.Paris, 282A, 995–998, 1976.[20] Bardos C & Benachour S, Domaine d’analyticite des solutions del’equation d’Euler dans un ouvert de Rn, Ann. Sc. Norm. Super. Pisa, Cl.Sci., IV Ser. 4 (1977) 647–687.[21] Taylor, G I & Green A E, Mechanism of the production of smalleddies from large ones, Proc. R. Soc. Lond. A 158 (1937), 499–521.[22] Brachet M-E, Meiron D I, Orszag S A, Nickel B G, Morf R H & FrischU, Small-scale structure of the Taylor-Green vortex. J. Fluid Mech., 130,411–452, 1983.


[23] Siggia E D, Phys. Fluids, 28, 794, 1984.

[24] Ashurst W & Meiron D, Phys. Rev. Lett. 58, 1632, 1987.

[25] Pumir A & Kerr R M, Phys. Rev. Lett. 58, 1636, 1987.

[26] Pumir A & Siggia E, Collapsing solutions to the 3-D Euler equations.Physics Fluids A, 2, 220–241, 1990.[27] Bell J B and Marcus D L, Vorticity intensification and transition toturbulence in the three-dimensional Euler equations, Comm. Math. Phys.147 (1992) 371–394.[28] Brachet M-E, Meneguzzi V, Vincent A, Politano H & Sulem P-L,Numerical evidence of smooth self-similar dynamics & the possibility ofsubsequent collapse for ideal flows. Phys. Fluids, 4A, 2845–2854, 1992.[29] Kerr R M, Evidence for a singularity of the three-dimensionalincompressible Euler equations. Phys. Fluids A 5, 1725–1746, 1993.[30] Kerr R M, 1995 “The role of singularities in Euler”. In Small-ScaleStructure in Hydro and Magnetohydrodynamic Turbulence, eds. Pouquet,A., Sulem, P. L., Lecture Notes, Springer-Verlag (Berlin).[31] Kerr R M, Vorticity and scaling of collapsing Euler vortices. Phys.Fluids A, 17, 075103–114, 2005.


[32] Grauer R and Sideris T 1991 Numerical computation of 3Dincompressible ideal fluids with swirl Phys. Rev. Lett. 67 3511–3514[33] Boratav O N and Pelz R B, Direct numerical simulation of transitionto turbulence from a high-symmetry initial condition, Phys. Fluids, 6,2757, 1994.[34] Boratav O N and Pelz R B, ”On the local topology evolution of ahigh-symmetry flow, Phys. Fluids 7, 1712 1995.[35] Pelz R B, Locally self-similar, finite-time collapse in a high-symmetryvortex filament model, Phys. Rev. E55, 1617-1626, 1997.[36] Pelz R B & Gulak Y, PRL 79, 4998, 1997.

[37] Pelz R B, Symmetry & the hydrodynamic blow-up problem. J. FluidMech., 444, 299–320, 2001.[38] Special memorial issue for Richard Pelz; Fluid Dyn. Res. 36, 2005.

[39] Cichowlas C and Brachet M-E 2005 Evolution of complexsingularities in Kida–Pelz and Taylor–Green inviscid flows Fluid Dyn.Res. 36 239–248[40] Gulak Y and Pelz R B 2005 High-symmetry Kida flow: Time seriesanalysis and resummation Fluid Dyn. Res. 36 211–220


[41] Pelz R B and Ohkitani K 2005 Linearly strained flows with andwithout boundaries – the regularizing effect of the pressure term, FluidDyn. Res. 36 193–210[42] Kerr R M 2005 Vortex collapse, Fluid Dyn. Res. 36 249-260.

[43] Grauer R & Marliani C & Germaschewski K, Adaptive meshrefinement for singular solutions of the incompressible Euler equations.Phys. Rev. Lett., 80, 4177–4180, 1998.[44] Hou T Y & Li R, Dynamic Depletion of Vortex Stretching andNon-Blowup of the 3-D Incompressible Euler Equations; J. NonlinearSci., (2006), 16, 639–664. published online August 22, 2006. DOI:10.1007/s00332-006-0800-3.[45] Orlandi P & Carnevale G, Nonlinear amplification of vorticity ininviscid interaction of orthogonal Lamb dipoles, Phys. Fluids, 19, 057106,2007.[46] Grafke T, Homann H, Dreher J & Grauer R, Numerical simulations ofpossible finite time singularities in the incompressible Euler equations :comparison of numerical methods. Physica D 237 (2008) 1932.


[47] Bustamante M & Brachet M-E ; On the interplay between the BKMtheorem & the analyticity-strip method to investigate numerically theincompressible Euler singularity problem ; 2012, arXiv1007.2587[48] Bustamante M, 3D Euler equations and ideal MHD mapped toregular systems : probing the finite-time blow-up hypothesis, Physica D,240, pp. 1092-1099 (2011)[49] Kerr R M, Bounds on a singular attractor in Euler using vorticitymoments,, to appear in the Procedia IUTAM volume “Topological FluidDyanmics II”, 2012 http://arxiv.org/abs/1212.1106[50] Pauls W, Matsumoto T, Frisch U & Bec J, Nature of complexsingularities for the 2D Euler equation, Physica D 219, 40–59, 2006.[51] Gibbon J D 2008 The three dimensional Euler equations: how muchdo we know? Proc. of “Euler Equations 250 years on” Aussois June2007, Physica D, 237, 1894–1904.[52] Bardos C and Titi E S 2007 Euler equations of incompressible idealfluids Russ. Math. Surv. 62:3 409–451[53] Chae D, On the Euler Equations in the Critical Triebel–LizorkinSpaces, Arch. Rat. Mech. Anal. 170 (2003) 185–210


[54] Chae D 2003 Remarks on the blow-up of the Euler equations andthe related equations Comm. Math. Phys. 245 539–550[55] Chae D 2004 Local Existence and Blow-up Criterion for the EulerEquations in the Besov Spaces Asymptotic Analysis 38 339–358[56] Chae D 2005 Remarks on the blow-up criterion of the 3D Eulerequations Nonlinearity 18 1021–1029[57] Chae D 2007 On the finite time singularities of the 3Dincompressible Euler equations Comm. Pure App. Math. 60 597–617[58] Cordoba D and Fefferman Ch 2001 On the collapse of tubes carriedby 3D incompressible flows Comm. Math. Phys. 222 293–298[59] Constantin P, Fefferman Ch and Majda A J 1996 Geometricconstraints on potentially singular solutions for the 3D Euler equationComm. Partial Diff. Equns. 21 559–571[60] Deng J, Hou T Y and Yu X 2005 Geometric Properties andNon-blowup of 3D Incompressible Euler Flow Commun. Partial Diff.Equns. 30 225–243[61] Deng J, Hou T Y and Yu X 2006 Improved geometric condition fornon-blowup of the 3D incompressible Euler equation, Commun. PartialDiff. Equns. 31 293–306


[62] Grauer R & Grafke T ; Finite-Time Euler singularities: A Lagrangianperspective, arXiv:1210.2534, 2012.[63] Ertel, H 1942, Ein Neuer Hydrodynamischer Wirbelsatz, Met Z, 59271–281[64] Truesdell C & Toupin R, Classical Field Theories, Encyclopaedia ofPhysics III/1, ed. S. Flugge, Springer (1960): gives the connection withCauchy.[65] Hoskins B J, McIntyre M E, & Robertson A W ; On the use &significance of isentropic potential vorticity maps, Quart. J. Roy. Met.Soc., 111, 877-946, (1985).[66] Ohkitani K, Phys. Fluids, A5, 2576, (1993).

[67] Kuznetsov E & Zakharov V E, Hamiltonian formalism for nonlinearwaves, Physics Uspekhi, 40 (11), 1087– 1116 (1997).[68] Viudez A, On the relation between Beltrami’s material vorticity andRossby-Ertel’s Potential, J. Atmos. Sci., 58, 2509–2517.[69] Kurgansky M V, Tatarskaya M S, The potential vorticity concept inmeteorology : A review. Izvestiya – Atmospheric and Oceanic Physics,(1987), 23, 587–606.


[70] Kurgansky M V, Pisnichenko I, Modified Ertel potential vorticity as aclimate variable, J. Atmos. Sci., (2000), 57, 822–835.[71] Kurgansky M V, Adiabatic Invariants in large-scale atmosphericdynamics, (2002), Taylor & Francis, London.[72] Yahalom A, (1996), Energy principles for barotropic flows withapplications to gaseous disks. Thesis submitted as part of therequirements for the degree of PhD to the Senate of the HebrewUniversity of Jerusalem.[73] Gibbon J D, Holm D D, The dynamics of the gradient of potentialvorticity, J. Phys. A : Math. Theor., (2010), 43, 17200.[74] Gibbon J D, Holm D D, Stretching & folding diagnostics in solutionsof the 3D Euler & Navier-Stokes equations ; Mathematical Aspects ofFluid Mechanics, edited by J. C. Robinson, Rodrigo J. L. and SadowskiW., CUP (2012), pp 201–220.[75] Moffatt H K 1978 Magnetic field generation in electrically conductingfluids CUP Cambridge.[76] Ohkitani K, A geometrical study of 3D incompressible Euler flowswith Clebsch potentials in a long-lived Euler flow and its power-lawenergy spectrum, Physica D 2008, 237 2020–2027


[77] Gibbon J D and Titi E S, The 3D incompressible Euler equations witha passive scalar : a road to blow-up? ArXiv: 1211.3811v1, 2012.[78] Gibbon J D, Dynamics of scaled norms of vorticity for thethree-dimensional Navier-Stokes and Euler equations, to appear in theProcedia IUTAM volume “Topological Fluid Dyanmics II”, 2012,arXiv:1212.0684[79] Constantin P, Majda A J and Tabak ? 1994 Nonlinearity 7,1495–1533.[80] Ohkitani K & Yamada M 1997 Phys Fluids 9.

[81] Constantin, Nie & Schorghofer 1998 Phys Lett A 24.

[82] Cordoba D, 1998 Ann,. Math, 148, 1135–1152.

[83] Cordoba D & Fefferman Ch 2001 J Amer Math Soc 15.

[84] Leorat J, These de Doctorat, Universite Paris-VII, 1975.

[85] Vieillefosse P, Internal motion of a small element of fluid in aninviscid flow, Physica, 125A, 150-162, 1984.[86] Cantwell B J, Exact solution of a restricted Euler equation for thevelocity gradient tensor, Phys. Fluids, 4, 782-793, 1992.


[87] Constantin P, Note on loss of regularity for solutions of the 3Dincompressible Euler and related equations, Comm. Math. Phys.,311-326, 104, 1986.[88] M. Chertkov, Pumir A and Shraiman B I, Lagrangian tetrad dynamicsand phenomenology of turbulence, Phys. Fluids, 11, 2394, 1999.[89] Chevillard L and Meneveau C, Lagrangian Dynamics and StatisticalGeometric Structure of Turbulence, Phys. Rev. Lett., 97, 174501, 2006.[90] Constantin P, Lax P and Majda A J, A simple one-dimensional modelfor the three-dimensional vorticity equation, Comm. Pure Appl. Math., 38,715-724, 1985.


@let@token singularity formation in the 3d euler equations?

Documents