Franco Flandoli and Massimiliano Gubinelli- Statistics of a Vortex Filament Model

8/3/2019 Franco Flandoli and Massimiliano Gubinelli- Statistics of a Vortex Filament Model

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E l e c t ro n

ic

Jo u

r na l

o f

Pr

o b a b i l i t y

Vol. 10 (2005), Paper no. 25, pages 865-900.

Journal URLhttp://www.math.washington.edu/ejpecp/

Statistics of a Vortex Filament Model

Franco Flandoli and Massimiliano Gubinelli

Dipartimento di Matematica Applicata

Universita di Pisa, via Bonanno 25 bis

56126 Pisa, Italy

Emails: [email protected] and [email protected]

Abstract

A random incompressible velocity field in three dimensions composed by Poisson

distributed Brownian vortex filaments is constructed. The filaments have a random

thickness, length and intensity, governed by a measure . Under appropriate as-

sumptions on we compute the scaling law of the structure function of the field and

show that, in particular, it allows for either K41-like scaling or multifractal scaling.

Keywords: Homogeneous turbulence, K41, vortex filaments, multifractal randomfields.

MSC (2000): 76F55; 60G60

Submitted to EJP on April 15, 2005. Final version accepted on June 1, 2005.

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1 Introduction

Isotropic homogeneous turbulence is phenomenologically described by several theories,which usually give us the scaling properties of moments of velocity increments. If u(x)denotes the velocity field of the fluid and Sp (), the so called structure function, denotesthe p-moment of the velocity increment over a distance (often only its longitudinalprojection is considered), then one expect a behavior of the form

Sp() = |u(x + ) u(x)|p p. (1)

Here, in our notations, is not the dissipation energy, but just the spatial scale parameter(see remark 5). Let us recall two major theories: the Kolmogorov-Obukov scaling law

(K41) (see [14]) says that2 =

2

3

probably the best result compared with experiments; however the heuristic basis of thetheory also implies p =

p3 which is not in accordance with experiments. Intermittency

corrections seem to be important for larger ps. A general theory which takes them intoaccount is the multifractal scaling theory of Parisi and Frisch [11], that gives us p in theform of a Fenchel-Legendre transform:

p = infhI

[hp + 3 D(h)].

This theory is a sort of container, which includes for instance the striking particular caseof She and Leveque [18]. We do not pretend to go further in the explanation of this topicand address the reader to the monograph [9].

The foundations of these theories, in particular of the multifractal one, are usuallymathematically poor, based mainly on very good intuition and a suitable mental image(see the beginning of Chapter 7 of [9]). Essentially, the scaling properties of Sp() aregiven a priori, after an intuitive description of the mental image. The velocity field of thefluid is not mathematically described or constructed, but some crucial aspects of it aredescribed only in plain words, and then Sp() is given (or heuristically deduced).

We do not pretend to remedy here to this extremely difficult problem, which ultimately

should start from a Navier-Stokes type model and the analysis of its invariant measures.The contribution of this paper is only to construct rigorously a random velocity fieldwhich has two interesting properties: i) its realizations have a geometry inspired by thepictures obtained by numerical simulations of turbulent fluids; ii) the asymptotic as 0ofSp() can be explicitly computed and the multifractal model is recovered with a suitablechoice of the measures defining the random field. Its relation with Navier-Stokes modelsand their invariant measures is obscure as well (a part from some vague conjectures, see[7]), so it is just one small step beyond pure phenomenology of turbulence.

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Concerning (i), the geometry of the field is that of a collection of vortex filaments,

as observed for instance by [19] and many others. The main proposal to model vortexfilaments by paths of stochastic processes came from A. Chorin, who made several con-siderations about their statistical mechanics, see [5]. The processes considered in [5] areself-avoiding walks, hence discrete. Continuous processes like Brownian motion, geomet-rically more natural, have been considered by [10], [16], [6], [8], [17] and others. We donot report here numerical results, but we have observed in simple simulations that thevortex filaments of the present paper, with the tubular smoothening due to the parameter (see below), have a shape that reminds very strongly the simulations of [3].

Concerning (ii), we use stochastic analysis, properties of stochastic integrals and ideasrelated to the theory of the Brownian sausage and occupation measure. We do not knowwhether it is possible to reach so strict estimates on Sp() as those proved here in thecase when stochastic processes are replaced by smooth curves. The power of stochasticcalculus seem to be important.

Ensembles of vortex structures with more stiff or artificial geometry have been consid-ered recently by [1] and [12]. They do not stress the relation with multifractal models andpart of their results are numerical, but nevertheless they indicate that scaling laws can beobtained by models based on many vortex structures. Probably a closer investigation ofsimpler geometrical models like that ones, in spite of the less appealing geometry of theobjects, will be important to understand more about this approach to turbulence theory.Let us also say that these authors introduced that models also for numerical purposes, sothe simplicity of the structures has other important motivations.

Finally, let us remark about a difference with respect to the idea presented in [5] andalso in [4]. There one put the attention on a single vortex and try to relate statisticalproperties of the paths of the process, like Flory exponents of 3D self-avoiding walk, withscaling law of the velocity field. Such an attempt is more intrinsic, in that it hopes toassociate turbulent scalings with relevant exponents known for processes. On the contrary,here (and in [1] and [12]) we consider a fluid made of a multitude of vortex structures andextract statistics from the collective behavior. In fact, it this first work on the subject,we consider independent vortex structures only, having in mind the Gibbs couplings of[8] as a second future step. Due to the independence, at the end again it is just the singlefilament that determines, through the statistics of its parameters, the properties of Sp().However, the interpretation of the results and the conditions on the parameters are more

in the spirit of the classical ideas of K41 and its variants, where one thinks to the 3Dspace more or less filled in by eddies or other structures. It is less natural to interpretmultifractality, for instance, on a single filament (although certain numerical simulationon the evolution of a single filament suggest that multifractality could arise on a singlefilament by a non-uniform procedure of stretching and folding [2]).

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1.1 Preliminary remarks on a single filament

The rigorous definitions will be given in section 2. Here we introduce less formally a fewobjects related to a single vortex filament.

We consider a 3d-Brownian motion {Xt}t[0,T] starting from a point X0. This is thebackbone of the vortex filament whose vorticity field is given by

single(x) =U

2

T0

(x Xt) dXt. (2)

The letter t, that sometimes we shall also call time, is not physical time but just theparameter of the curve. All our random fields are time independent, in the spirit ofequilibrium statistical mechanics. We assume that (x) = (x/) for a radially symmetricmeasurable bounded function with compact support in the ball B(0, 1) (the unit ball in3d Euclidean space). To have an idea, consider the case = 1B(0,1). Then single(x) = 0outside an -neighboor U of the support of the curve {Xt}t[0,T]. Inside U, single(x) is atime-average of the directions dXt, with the pre-factor U/

2. More precisely, if x U,one has to consider the time-set where Xt B(x, ) and average dXt on such time set.The resulting field single(x) looks much less irregular than {Xt}t[0,T], with increasingirregularity for smaller values of .

When is only measurable, it is not a priori clear that the Stratonovich integralsingle(x) is well defined, since the quadratic variation corrector involves distributionalderivatives of (the Ito integral is more easily defined, but it is not the natural object to

be considered, see the remarks in [6]). Since we shall never use explicitly single(x), thisquestion is secondary and we may consider single(x) just as a formal expression that weintroduce to motivate the subsequent definition of usingle(x). However, at least in someparticular case ( = 1B(0,1)) or under a little additional assumptions, the Stratonovichintegral single(x) is well defined since the corrector has a meaning (in the case of = 1B(0,1)the corrector involves the local time of 3d Brownian motion on sferical surfaces).

The factor U/2 in the definition ofsingle(x) is obscure at this level. Formally, it couldbe more natural just to introduce a parameter , in place ofU/2, to describe the intensityof the vortex. However, we do not have a clear interpretation of , a posteriori, from outtheorems, while on the contrary it will arise that the parameter U has the meaning ofa typical velocity intensity in the most active region of the filament. Thus the choice of

the expression U/2 has been devised a posteriori. The final interpretation of the threeparameters is that T is the length of the filament, the thickness, U the typical velocityaround the core.

The velocity field u generated by is given by the Biot-Savart relation

usingle(x) =U

2

T0

K(x Xt) dXt (3)

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where stands for the vector product in R3, denote as usual Stratonovich integrationof semimartingales and the vector kernel K(x) is defined as

K(x) = V(x) = 14

B(0,)

(y)x y

|x y|3 dy, V (x) = 1

4

B(0,)

(y)dy

|x y| . (4)

The scalar field V satisfy the Poisson equation V = in all R3. Since is radially

symmetric and with compact support we can have two different situations according tothe fact that the integral of : Q =

B(0,1)

(x)dx is zero or not. If Q = 0 then the field

V is identically zero outside the ball B(0, ). Otherwise the fields V, K outside the ballB(0, ) have the form

V(x) = Q3

|x| , K(x) = 2Q3 x

|x|3 , |x| . (5)Accordingly we will call the case Q = 0 short range and Q = 0 long range. The proofof the previous formula for K is given, for completeness, in the remark at the end of thesection.

A basic result is that the Stratonovich integral in the expression of usingle can bereplaced by an Ito integral:

Lemma 1 Ito and Stratonovich integrals in the definition of usingle(x) coincide:

usingle(x) =U

2 T

0

K(x

Xt)

dXt (6)

where the integral is understood in Ito sense. Then usingle is a (local)-martingale withrespect to the standard filtration of X.

About the proof, by an approximation procedure that we omit we may assume Holder continuous, so the derivatives of K exist and are Holder continuous by classicalregularity theorems for elliptic equations. Under this regularity one may compute thecorrector and prove that it is equal to zero, so, a posteriori, equation (6) holds true in thelimit also for less regular . About the proof that the corrector is zero, it can be donecomponent-wise, but it is more illuminating to write the following heuristic computation:the corrector is formally given by

12

T0

(K(x Xt)dXt) dXt.

Now, from the property dXitdXjt = ijdt one can verify that

(K(x Xt)dXt) dXt = (curl K) (x Xt)dt.Since K is a gradient, we have curl K = 0, so the corrector is equal to zero.

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Remark 1 One may verify that div usingle = 0, so usingle may be the velocity fluid of an

incompressible fluid. On the contrary, div single is different from zero and curl usingle isnot single but its projection on divergence free fields. Therefore, one should think of singleas an auxiliary field we start from in the construction of the model.

Remark 2 Let us prove (5), limited to K to avoid repetitions. We want to solve V = with spherically symmetric. The gradient K = V of V satisfies K = and, byspherical symmetry, it must be such that

K(x) =x

|x|f(|x|)

for some scalar functionf(r). By Gauss theorem we haveB(0,r)

K(x)dx =B(0,r)

K(x) d(x)

where d(x) is the outward surface element of the sphere. So

f(r)4r2 =

B(0,r)

(x)dx =: Q(r)

namely f(r) = Q(r)4r2 . Therefore

K(x) = Q(r) x4|x|3

with Q(r) = Q(1) if r 1 (since has support in B(0, 1)).

2 Poisson field of vortices

Intuitively, we want to describe a collection of infinitely many independent Brownian vor-tex filaments, uniformly distributed in space, with intensity-thickness-length parameters(U,,T) distributed according to a measure . The total vorticity of the fluid is the sumof the vorticity of the single filaments, so, by linearity of the relation vorticity-velocity,the total velocity field will be the sum of the velocity fields of the single filaments.

The rigorous description requires some care, so we split it into a number of steps.

2.1 Underlying Poisson random measure

Let be the metric space

= {(U,,T,X) R3+ C([0, 1];R3) : 0 <

T 1}

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with its Borel -field B (). Let (, A, P) be a probability space, with expectation denotedby E, and let , , be a Poisson random measure on B (), with intensity (a -finite measure on B ()) given by

d(U,,T,X) = d(U,,T)dW(X).for a -finite measure on the Borel sets of {(U,,T) R3+ : 0 <

T 1} and

dW(X) the -finite measure defined byC([0,1];R3)

(X)dW(X) =R3

C([0,1];R3)

(X)dWx0(X)

dx0

for any integrable test function : C([0, 1];R3) R where dWx0(X) is the Wiener mea-

sure on C([0, 1],R

3

) starting at x0 and dx0 is the Lebesgue measure onR

3

. Heuristicallythe measure Wdescribe a Brownian path starting from an uniformly distributed point inall space. The assumptions on will be specified at due time.

The random measure is uniquely determined by its characteristic function

Eexp

i

()(d)

= exp

(e() 1)(d)

for any bounded measurable function on with support in a set of finite -measure.In particular, for example, the first two moments of read

E

()(d) =

()(d)

and

E

()(d)

2=

()(d)

2+

2()(d).

We have to deal also with moments of order p; some useful formulae will be now given.

2.2 Moments of the Poisson Random Field

Let : R be a measurable function. We shall say it is -integrable if it is -integrable for P-a.e. . In such a case, by approximation by bounded measurablecompact support functions, one can show that the mapping

() is measurable.

If : R is a measurable function with ( = ) = 0, since P( ( = ) = 0) =1, we have that is -integrable and P( () = ()) = 1. Therefore the concept of-integrability and the random variable () depend only on the equivalence class of .

Let be a measurable function on , possibly defined only -a.s. We shall say that itis -integrable if some of its measurable extensions to the whole is -integrable. Neitherthe condition of-integrability nor the equivalence class of () depend on the extension,by the previous observations. We need all these remarks in the sequel when we deal with given by stochastic integrals.

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Lemma 2 Let be a measurable function on (possibly defined only -a.s.), such that

(p

) < for some even integer number p. Then is -integrable, () Lp

(), andE[ ()p] epp(p).

If in addition we have (k) = 0 for every odd k < p, then

E[ ()p] (p).Proof. Assume for a moment that is bounded measurable and with support in a

set of finite -measure, so that the qualitative parts of the statement are obviously true.Using the moment generating function

Ee() = e(e1)

we obtainp=0

p

p!E[()p] =

n=0

1

n!

(e 1)n

= 1 +n=1

1

n!

k=1

k

k!(k)

n

= 1 +n=1

1

n!

k1,...,kn1

k1++kn

k1! kn! (k1) (kn)

= 1 +

p=1p

p!

p

n=1 k1,...,kn1k1++kn=p

p!

n!k1! kn!(k1)

(kn)

Hence we have an equation for the moments:

E[()p] =

pn=1

k1,...,kn1k1++kn=p

p!

n!k1! kn! (k1) (kn) (7)

Since (||k) [(||p)]k/p for k p we haveE[|()|p] E[(||)p]

p

n=1

k1,...,kn1k1++kn=p

p!

n!k1! kn! (||p

)

(||p)pn=1

k1,...,kn0k1++kn=p

p!

n!k1! kn!

= (||p)pn=1

np

n! epp(||p).

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This proves the first inequality of the lemma. A posteriori, we may use it to prove the

qualitative part of the first statement, by a simple approximation procedure for generalmeasurable .

For the lower bound, from the assumption that (k) = 0 for odd k < p, in the sum(7) we have contributions only when all ki, i = 1, . . . , n are even. Then, neglecting manyterms, we have

E[()p] (p).The proof is complete.

2.3 Velocity field

Let be a radially symmetric measurable bounded function onR

3

with compact supportin the ball B(0, 1), and let K be defined as in section 1.1. Then we have that

K1 is Lipschitz continuous. (8)

Indeed by an explicit computation it is possible to show that

K1(x) = 2Q(|x|) x|x|3

with

Q(r) :=

B(0,r)(x)dx

(since has support in B(0, 1) we have Q(r) = Q if r 1). Then

K1(x) = 2Q(|x|) x x|x|4 2Q(|x|)

1

|x|3 3x x|x|5

with Q(r) = dQ(r)/dr and we can bound

|K1(x)| Csince

Q(r) Cr3, Q(r) Cr2.

For any x, x0 R3, , T > 0, the random variable X T0 K(x Xt) dXt isdefined Wx0-a.s. on C([0, 1],R3). We also have, given x R3, , T > 0, that X T0

K(xXt)dXt is a well defined measurable function, defined W-a.s. on C([0, 1],R3).More globally, writing = (U,,T,X) for shortness, for any x R3 we may consider themeasurable R3-valued function

usingle (x) :=U

2

T0

K(x Xt) dXt

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defined -a.s. on . In plain words, this is the velocity field at point x of a filament

specified by .Again in plain words, given , the point measure specifies the parameters and

locations of infinitely many filaments: formally

=N

(9)

for a sequence of i.i.d. random points {} distributed in according to (this fact isnot rigorous since is only -finite, but it has a rigorous version by localization explainedbelow). Since the total velocity at a given point x R3 should be the sum of thecontributions from each single filament, i.e. in heuristic terms

u(x) =N

u

single (x) (10)

we see that, in the rigorous language of , we should write

u(x) =

usingle (x) (d) =

usingle (x)

. (11)

If we show that usingle (x)p < for some even p 2, then from lemma 2, usingle (x)

is -integrable and the random variable

u(x, ) := usingle (x)is well defined.

In some proof below we will use the occupation measure of the Brownian motion inthe interval [0, T], which is defined, for every Borel set B ofR3 as

B LTB :=T0

1XtBdt.

Lemma 3 Given x R3 and p > 0, there exist Cp > 0 such that for every 2 T 1wehave

WW(x)p/2 CppTwhere

W(x) =

T0

|K(x Xt)|2 dt.

Proof. Let us bound K by a multi-scale argument. This is necessary only in the long-range case (see the introduction). If|y| we can bound |K(y)| C. Indeed if|w| 1,|K(w)| C for some constant C and then if |y| < we have |K(y)| = |K1(y/)| C.

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Next, given > and an integer N, consider a sequence {i}i=0,...,N of scales, with = 0 < 1 < ... < N = . Then, for i = 1, . . . , N , if i1 < |y| < i, by the explicitformula for K(y) we have |K(y)| C(/|y|)2. Therefore

|K(y)| C

i1

21i1 we simply bound Ke (y) C(/)2.Summing up,

|K(y)|2 = |K(y)|2

1|y| +N

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Notice now that

N1i=1

i1

2

i

2=

2

2 1

so that, by Cauchy-Schwartz and Jensen inequalities we haveNi=1

i1

4LTB(x,i)\B(x,i1)

p/2

CpN1

i=1

i1

2

i2

ip

(LTB(x,i))p/2 + Cp

2p

LTB(x,)

p/2

.

An upper bound for W(x)p/2 is then obtained as

W(x)p/2 Cpp(LTB(x,))p/2

+ Cpp

2p LTB(x,)

p/2+ Cp

p

2p LTB(x,)c

p/2+ Cp

N1i=1

i1

2

i

2

i

p(LTB(x,i))

p/2

Cpp

(L

T

B(x,))

p/2

+ Cp

p 2p

T

p/2

+ Cpp

N1i=1

i1

2

i

2

i

p(LTB(x,i))

p/2

where we have used again Cauchy-Schwartz inequality.We use now lemma 14 with = 1. For a given (0, 1), we take both and equal

to in (24) and (25), and get

W

LTB(x,)p/2 Cp( T)p( T)2.

Then we obtain

WW(x)p/2 Cp2pT + Cpp

2pTp/2

+ Cpp

N1i=1

i1

2

i

2

i

p(i

T)pi(i

T)2

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and taking the limit as the partition gets finer:

WW(x)p/2 Cp2pT + Cpp

2pTp/2

+ Cpp

u

p(u

T)p(u

T)2ud

u

2The integral can then be computed as

u

p(u

T)p(u

T)2u

2

u3du

= p2TT

du

u2+ Tp/2p+2

T

updu

= p2T[1

T1

] + (p 1)1Tp/2p+2[T(1p)/2 (1p)] pT + (p 1)1p

TT

Using the fact that T and letting we finally obtain the claim.

Remark 3The multiscale argument above can be rewritten in continuum variables fromthe very beggining by means of the following identity: if f : [0, ) R is of class C1 and

has a suitable decay at infinity, thenT0

f(|x Xt|) dt = 0

f (r) LTB(x,r)dr.

This identity can be applied to W(x). The proof along these lines is not essentially shorterand perhaps it is more obscure, thus we have choosen the discrete multiscale argumentwhich has a neat geometrical interpretation.

Corollary 1 Assume(UpT) 0 such that

Wx0(usingle (x)p) Cp Up2pWx0 W(x)p/2 .

Hence

W(

usingle (x)

p

) Cp Up

2pW

W(x)p/2

Cp Up2p 2pT = CpUpT.Therefore

usingle (x)p < by the assumption (UpT) < . The other claims are aconsequence of lemma 2.

Lemma 4 Under the previous assumptions, the law of u(x, ) is independent of x and isinvariant also under rotations:

Ru(x, ) L= u(Rx, ) for every rotation matrixR.

Proof. With the usual notation = (U,,T,X) we have

usingle (x) = u(U,,T,X)single (x) = u

(U,,T,Xx)single (0) = u

xsingle (0)

where x(U,,T,X) = (U,,T,X x). The map x is a measurable transformation of into itself. One can see that is x-invariant; we omit the details, but we just notice that is not a finite measure, so the invariance means

(x) (d) =

() (d)

for every L1 (, ). From this invariance it follows that the law of the random measure is the same as the law of the random measure x. Therefore

usingle (x)

=

uxsingle (0)

= (x)

usingle (0) L

=

usingle (0)

.

This proves the first claim.

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If R is a rotation, from the explicit form of K it is easy to see that

RK(y) = K(Ry)

hence

Rusingle (x) =U

2

T0

RK(x Xt) dRXt

=U

2

T0

K(Rx RXt) dRXt= uRsingle (Rx)

where we have set R(U,,T,X) = (U,,T,RX). Again R = , R L= , so the end ofthe proof is the same as above. We say that a random field u(x, ) is homogeneous if its law is independent of x and

isotropic if its law is invariant under rotations.

Corollary 2 Assume(UpT) <

for every p > 1. Then{u(x, ); x R3} is an isotropic homogeneous random field, withfinite moments of all orders.

This corollary is sufficient to introduce the structure function and state the main results

of this paper. However, it is natural to ask whether the random field {u(x, ); x R3}has a continuous modification. Having in mind Kolmogorov regularity theorem, we needgood estimates ofE[|u(x) u(y)|p]. They are as difficult as the careful estimates we shallperform in the next section to understand the scaling of the structure function. Thereforewe anticipate the result without proof. It is a direct consequence of Theorem 1.

Proposition 1 Assume(UpT) <

for everyp > 1. Then, for every even integer p there is a constant Cp > 0 such that

E[|u(x) u(y)|p

] CpUp |x y| p

T .Consequently, if the measure has the property that for some even integer p and realnumber > 3 there is a constant Cp > 0 such that

Up

pT

Cp for any (0, 1) , (12)

then the random field u(x) has a continuous modification.

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Remark 4 A sufficient condition for (12) is: there are > 3 and > 0 such that for

every sufficiently large even integer p there is a constant Cp > 0 such that[UpT 11 ] Cp for any (0, 1) .

Indeed,

Up

pT

=

Up

pT 11

+

Up

pT 11

[UpT 11 ] +

UppT 11

Cp + p[UpT]so we have (12) with a suitable choice of p. This happens in particular in the multifractalexample of section 3.2, remark 7.

The model presented here has a further symmetry which is not physically correct. Thissymmetry, described in the next lemma, implies that the odd moments of the longitudinalstructure function vanish, in contradiction both with experiments and certain rigorousresults derived from the Navier-Stokes equation (see [9]). The same drawback is presentin other statistical models of vortex structures [1].

Beyond the rigorous formulation, the following property says that the random fieldusingle has the same law as usingle. We cannot use the concept of law since does notlive on a probability space.Lemma 5 Under the hypotesis of Corollary 1, givenx1,...,xn R3, the measurable vector

Un () :=

usingle (x1) ,...,usingle (xn)

has the property

(Un ()) d() =

(Un ()) d()

for every = (u1,...,un) : R3n R with a polynomial bound in its variables..

Proof. With the notation

Xt = XTt, we have

usingle (xk) = U2 T0 K(xk Xt) dXt=

U

2

T0

K(xk Xs) d Xs= uSsingle (xk)

where S(U,,T,X) = (U,,T, X). Since S = , we have the result (using the integra-bility of Corollary 1).

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Lemma 6 If p is an odd positive integer, then

usingle (y) usingle (x) , y xp = 0 for everyx, y R3.

Proof. It is sufficient to apply the lemma to the function

(u1, u2) := u1 u2, y xp

and the points x1 = y, x2 = x, with the observation that

(

u1,

u2) =

(u1, u2) .

2.4 Localization

At the technical level, we do not need to localize the -finite measures of the present work.However, we give a few remarks on localization to help the intuitive interpretation of themodel. Essentially we are going to introduce rigorous analogs of the heuristic expressions(9) and (10) written at the beginning of the previous section. The problem there was thatthe law of should be , which is only a -finite measure. For this reason one has to

localize and .Given A B () with 0 < (A) < , consider the measure A defined as therestriction of to A:

A (B) = (A B)for any B B (). It can be written (the equality is in law, or a.s. over a possiblyenlarged probability space) as the sum of independent random atoms each distributedaccording to the probability measure B B (A) A (B) := (B|A):

A(d) =

NA=1

(d)

where NA is a Poisson random variable with intensity (A) and the family of randomvariables {}N is independent and identically distributed according to A. Moreover if{Ai}iN is a family of mutually disjoint subsets of then the r.v.s {Ai}iN are indepen-dent.

Sets A as above with a physical significance are the following ones. Given 0 < < 1and R > 0, let

A,R = {(U,T,,X) : > , |x0| R}.

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In a fluid model we meet these sets if we consider only vortexes up to some scale (it

could be the Kolmogorov dissipation scale) and roughly confined in a ball of radius R.If we assume that the measure satisfies 0 < ( > ) < for each > 0, then0 < (A,R) < , and for the measure ,R := A,R we have the representation

,R(d) =

N,R=1

(d)

where N,R is P((A,R)) and {}N are i.i.d. with law ,R := A,R .For any x R3 we may consider usingle (x) as a random variable in R3, defined

,R-a.s. on A,R. Moreover we may consider the r.v. u,R(x) on (, A, P) defined as

u,R(x) :=A,R

usingle (x) (d) =

usingle (x) ,R(d). (13)

It is the velocity field at point x, generated by the vortex filaments in A,R B (). Inthis case we have the representation

u,R(x) =

N,R=1

u

single (x) =

N,R=1

U

()2

T0

K(x Xt ) dXt

where the quadruples = (U, , T, X) are distributed according to ,R and areindependent. If () is any one of such quadruples, the random variable u()single (x)

is well defined, since the law of is ,R and the random variable usingle (x) is welldefined ,R-a.s. on A,R. Therefore u,R(x) is a well defined random variable on (, A, P).We have noticed this in contrast to the fact that the definition of u(x) required difficultestimates, because of the contribution of infinitely many filaments.

Given the Poisson random field, by localization we have constructed the velocity fieldsu,R(x) that have a reasonable intuitive interpretation. Connections between u,R(x) andu(x) can be established rigorously at various levels. We limit ourselves to the following

example of statement.

Lemma 7 Assume(UpT) <

for anyp > 0. Then, at any x R3,

lim(,R1)(0+,0+)

E[|u,R(x) u(x)|p] = 0.

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Proof. Since

u,R(x) u(x) =

1A,R 1

usingle (x) (d)

we have

E[|u,R(x) u(x)|p] Cp1A,R 1usingle (x)p

= Cp

1A,R 1 usingle (x)p

Notice that we do not have

Ac,R 0, in general, so the argument to prove the lemma

must take into account the properties of the r.v. usingle (x). We have (by Burkholder-

Davis-Gundy inequality)

1A,R 1 usingle (x)p

=

|x0|R

dx0

l

Wx0(usingle (x)p)d(U,,T)

Cp

|x0|Rdx0

l

Up

2pWx0

W(x)p/2

d(U,,T)

Cp

Up

2pd(U,,T)

|x0|R

Wx0

W(x)p/2

dx0

where W(x) has been defined in lemma 3. Let us show that|x0|R

Wx0

W(x)p/2

dx0 Cp2pT (R) (14)

where (R) 0 as R . The proof will be complete after this result. Recall fromthe proof of lemma 3 that we have

W(x)p/2 Cpp(LTB(x,))p/2 + Cpp

2pTp/2

+ Cp

pN1

i=1 i12

i2

ip

(LTB(x,i)

)p/2.

From lemma 15 we have|x0|R

Wx0

LTB(x,)p/2

dx0

Cp(

T)p(

T)2 exp

R (|x| + )

T

.

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Hence |x0| R

Wx0 W(x)p/2 dx0 Cp2pTexp

R (|x| + )

T

+ Cp

p

2pTp/2

+ Cpp

N1i=1

i1

2

i

2

i

p (i

T)pi(i

T)2 exp

R (|x| + i)

T

.

Repeating the arguments of lemma 3 we arrive at|x0| R

Wx0

W(x)p/2

dx0

Cp2pTexp

R (|x| + )T

+ Cp

p

2pTp/2.

Since T and are smaller than one, and p 2, we also have|x0| R

Wx0

W(x)p/2

dx0

Cp

2pTexp(R (|x| + )) + 2p

.With the choice = R/2 we prove (14). The proof is complete. 3 The structure function

Given the random velocity field u (x, ) constructed above, under the assumption of Corol-lary 2, a quantity of major interest in the theory of turbulence is the longitudinal structure

function defined for every integer p and > 0 as

Sp() = E[e, u(x + e) u(x)]p (15)

where , is the Euclidean scalar product inR

3

, e R

3

is a unit vector and x R

3

, andE, we recall, is the expectation on (, A, P).Remark 5 We warn the reader familiar with the literature on statistical fluid mechanicsthat here is not the dissipation energy, but just the spatial scale parameter. In the physicalliterature, it is commonly denoted by ; however, in our mathematical analysis we needtwo parameters: the scale parameter of the statistical observation, which we denote by ,and a parameter internal to the model that describes the thickness of the different vortex

filaments, that we denote by.

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The moments Sp() depend only on and p, since u (x, ) is homogeneous and isotropic:

if e = R e1 where e1 is a given unit vector and R is a rotation matrix taking e1 on e,using that the adjoint of R is R1, we have

E[u(x + e) u(x), ep] = E[u(e) u(0), ep]= E[Ru(e1) u(0), ep]= E[u(e1) u(0), e1p] .

For this reason we do not write explicitly the dependence on x and e.Let us also recall the (non-directional) structure function

Sp() = E[|u(x + e) u(x)|p]

which as Sp() depends only on and p. We obviously have

Sp() Sp().We shall see that, for even integers p, they have the same scaling properties. At thetechnical level, due to the previous inequality, it will be sufficient to estimate carefullyS

p() from below and Sp() from above.

3.1 The main result

The quantities Sp() and Sp() describe the statistical behavior of the increments of the

velocity field when 0 and have been extensively investigated, see [9]. Both areexpected to have a characteristic power-like behavior of the form (1). Our aim is to provethat, for the model described in the previous section with a suitable choice of , (1) holdstrue in the sense that the limit

p = lim0

log Sp()

log (16)

exists (similarly for Sp()) and is computable. The following theorem gives us the nec-

essary estimates from above and below, for a rather general measure . Then, in thenext subsection, we make a choice of in order to obtain the classical multifractal scalingbehaviour.

Theorem 1 Assume that(UpT) <

for every p > 1. Then, for any even integer p > 1 there exist two constants Cp, cp > 0such that

cp[UpT1


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The proof of this result is long and reported in a separate section.

We would like to give a very rough heuristic that could explain this result. It must besaid that we would not believe in this heuristic without the proof, since some steps aretoo vague (we have devised this heuristic only a posteriori).

What we are going to explain is that

W(|usingle (x + e) usingle (x)|p) Up

pT.

This is the hard part of the estimate.Let us discuss separately the case > from the opposite one. When > the

vortex structure usingle is very thin compared to the length of observation of the dis-

placement, thus the difference usingle (x + e)usingle (x) does not really play a role and thevalue of W(|usingle (x + e) usingle (x)|p) comes roughly from the separate contributionsof usingle (x + e) and usingle (x), which are similar. Let us compute W(|usingle (x)|p).

Consider the expression (6) which defines usingle(x). Strictly speaking, consider theshort-range case, otherwise there is a correction which makes even more difficult theintuition. Very roughly, K(x Xt) behaves like 1XtB(x,), hence, even more roughly,usingle(x) behaves like

usingle(x) U

T0

1XtB(x,)dXt.

When Xt is a smooth curve, say a straight line (at distances compared to ), then the

quantity T0 1XtB(x,)dXt is roughly proportional to if Xt crosses B (x, ), while it iszero otherwise. We assume the same result holds true when Xt is a Brownian motion. Inaddition, Xt crosses B (x, ) with a probability proportional to the volume of the Wienersausage, which is T. Summarizing, we haveT

0

1XtB(x,)dXt

with probability T0 otherwise

.

Therefore usingle(x) takes rougly two values, U with probability T and 0 otherwise. Itfollows that W(|usingle (x)|p) UpT.

Consider now the case < . The difference now is important. Since the gradient of

K is of order one, we have

usingle (x + e) usingle (x) U2

T0

1XtB(x,)dXt.

As above we conclude that usingle (x + e) usingle (x) takes rougly two values, U/ withprobability T and 0 otherwise. It follows that W(|usingle (x)|p) Up

pT. The intu-

itive argument is complete.

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3.2 Example: the multifractal model

The most elementary idea to introduce a measure on the parameters is to take U and Tas suitable powers of , thus prescribing a relation between the thickness and the lengthand intensity. That is, a relation of the form

d(U,,T) = h(dU)a(dT)bd.

Moreover, we have to prescribe the distribution of itself, which could again be given bya power law bd. The K41 scaling described below is such an example.

Having in mind multi-scale phenomena related to intermittency, we consider a super-position of the previous scheme. Take a probability measure on an interval I R+(which measures the relative importance of the scaling exponent h

I). Given two

functions a, b : I R+ with a(h) 2 (to ensure 2 T) consider the measure

d(U,,T) =

I

h(dU)a(h)(dT)

b(h)d

(dh). (18)

Then, according to Theorem 1 , we must evaluate

(UpT1


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the thickness of the structure are comparable (remember that the curves are Brownian),

hence their shape is blob-like, as in the classical discussions of eddies around K41. Thechoice b(1/3) = 4, namely the measure 4d for the parameter , corresponds to theidea that the eddies are space-filling: it is easy to see that in a box of unit volume thenumber of eddies of size larger that is of order 3. Finally, the choice (dh) = 1/3(dh),namely U = 1/3, is the key point that produces p =

p3 ; one may attempt to justify it

by dimensional analysis or other means, but it is essentially one of the issues that shouldrequire a better understanding.

Remark 6 K41 can be obtain from this model also for other choices of the functions a(h)and b(h). For example, we may take (dh) = 1/3(dh), a(1/3) = 0 and b(1/3) = 2. Thevalue choice (dh) = 1/3(dh) is again the essential point. The value a(1/3) = 0 means

T = 1, hence the model is made of thin filaments of length comparable to the integralscale, instead of blob-like objects. The value b(1/3) = 2, namely the measure 2d forthe parameter , is again space filling in view of the major length of the single vortexstructures (recall also that the Hausdorff dimension of Brownian trajectories is 2). Fromthis example we see that in the model described here it is possible to reconcile K41 with ageometry made of thin long vortexes.

Remark 7 Assume infI > 0 and, for instance, supID < . Then limp p = +.In particular, p > 3 for some even integer p. Since, by (19) and Laplace method,

(UpT1 0, the condition of remark 4 is satisfied. Therefore the velocity field has a

continuous modification.

4 Proof of Theorem 1

Let us introduce some objects related to the structure functions at the level of a singlevortex filament. Let e be a given unit vector, the first element of the canonical basis, tofix the ideas. Since u(x) =

usingle (x)

, then

u(e) u(0), e = [usingle, e]where

usingle := usingle (e) usingle (0)and similarly

|u(e) u(0)| = | [usingle]| .Of major technical interest will be the quantities, of structure function type,

Sep() = W(usingle, ep)Sp() = W(|usingle|p).

They depend also on ,T ,U.

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4.1 Lower bound

As a direct consequence of lemma 6 and lemma 2 we have:

Corollary 3 If k is an odd positive integer, then

Sk() = 0.

Moreover, for any even integer p > 1 there is a constant cp such that

Sp() cp[usingle, ep]= cp

Sep()

.

If we prove that Sep() cpUpTfor every (, 1) and some constant cp > 0, then

Sep() cp[UpT1 0, which implies the lower bound of theorem1. We have

Sep() =

Wx0 [usingle, ep] dx0.Since

usingle, e = U2T0

Ke (e Xt) Ke (0 Xt), dXtwhere

Ke (y) = K (y) eby Burkholder-Davis-Gundy inequality, there is cp > 0 such that

Wx0 [usingle, ep] cpUp

2pWx0

(We )

p/2

where

We = T

0

dt

|Ke (e

Xt)

Ke (0

Xt)

|2.

Here Ke (y) = K(y), e. Therefore

Sep() cpUp

2p

Wx0

(We )

p/2

dx0

= cpUp

2pW

(We )

p/2

.

The proof of the theorem is then complete with the following basic estimate.

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Lemma 8 Given p > 0, there exist cp > 0 such that for every 2 T 1 and

>

we haveW

(We )

p/2

cp2pT.

Proof. Recall that

K1(x) = 2Q x|x|3 for |x| 1.

Consider the function

f(z, ) = Ke1(z) Ke1(z + e)= (K1(z) K1(z + e)) e

defined for z R3 and 1. Let e be any unit vector orthogonal to e. We have

fe, = 2Q 1

e + e e

|e + e|3

= 2Q

1 1

|e + e

|3

fe, 1 = C0Q

for C0 = 2

1 232

. Moreover, K1 is globally Lipschitz, hence there is L > 0 such thatfe + w, fe, K1(e + w) K1(e) + K1(e + e + w) K1(e + e) L |w|

for every w R3 and 1. Therefore, there exists a ball B(e, a) B(0, 2) and aconstant C1 > 0 such that when z B(e, a) we have

|Ke1(z) Ke1(z + e)| C1uniformly in 1. Then reintroducing the scaling factor we obtain that for y B(e, a)

|Ke (y) Ke (y + e)| > C1uniformly in (0, 1) and > . Then we have

|Ke (0 Xt) Ke (e Xt)| C11XtB(e,a).

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Hence, if > we have

We =T0

dt|Ke (0 Xt) Ke (e Xt)|2

C12LTB(e,a).

From the lower bound in (24) proved in the next section,

W

(We )p/2

cppW

LTB(e,a)

p/2 cp2pT.

The claim is proved.

Remark 8 With a bit more of effort is is also possible to prove the bound

Sep() cpUp

pT

valid for every (0, 1) (not only for > ). This would be the same as the upper bound,but we do not need it to prove that the behaviors as 0 of the upper and lower boundis the same.

4.2 Upper bound

Lemma 9 For every even p there exists a constant Cp > 0 such that

Sp() Cp[|usingle|p]= Cp[Sp()] .

Proof. Let [usingle ()]i be the i-th component of usingle (). We have

Sp() Cp3

i=1

E

[usingle ()]ip

and thus, by lemma 2,

Sp() Cp3

i=1

[usingle ()]pi

which implies the claim.

It is then sufficient to prove the bound

Sp() CpUp

pT.

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Again as above, We have

Sp() =

Wx0 [|usingle|p] dx0

where, by Burkholder-Davis-Gundy inequality, there is Cp > 0 such that

Wx0 [|usingle|p] CpUp

2pWx0

Wp/2

where

W =

T0

dt|K(e Xt) K(0 Xt)|2.

Therefore

Sp() Cp Up

2p

Wx0

Wp/2

dx0

= CpUp

2pWWp/2 .

It is then sufficient to prove the bound

W

Wp/2

Cp2p

pT.

For > it is not necessary to keep into account the closeness of e to 0: each termin the difference of W has already the necessary scaling. The hard part of the work hasbeen done in lemma 3 above.

Lemma 10 Given p > 0, there exist Cp > 0 such that for every 2 T 1 and

>

we haveW

Wp/2

Cp2pT.

Proof. SinceW 2W(e) + 2W(0)

where W(x) =T0

|K(x Xt)|2 dt, by lemma 3 we have the result. For need to extract a power of from the estimate ofW

W

p/2

. We essentially

repeat the multi-scale argument in the proof of lemma 3, with suitable modifications.

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Lemma 11 As in the previous lemma, when

we have

WWp/2 CpppT.Proof. Since now is smaller that we bound K(y) K(z) for |y z| as

|K(y) K(z)| Cif |y| 2. If |y| > 2 then |z| and using the explicit form of the kernel K we havethe bound

|K(y) K(z)| C u3 C

u2

where u is the minimum between |y| and |z| and in this case /u < 1. Then, given apartition of [2, ], say 2 = 0 < 1 < ... < N = , as in the proof of lemma 3, we get

|K(y) K(z)|2 = |K(y) K(z)|2

1|y|2 +Ni=1

1i1


30/36


31/36

Let

B(u,) = inf{t 0 : Xt B(u, )}the entrance time in B(u, ) for the canonical process. We continue to denote by Wx0 theWiener measure starting at x0 and also the mean value with respect to it; similarly forW, the -finite measure dWx0dx0.

Lemma 12 For any p > 0, T > 0, > 0, x0, u R3, it holds that

Wx0[B(u,/2) T /2]W0[|LT/2B(0,/2)|p] Wx0[|LTB(u,)|p]

Wx0[B(u,) T]W0[|LTB(0,2)|p]. (20)

Proof. Let us prove the upper bound. Set, for simplicity, = B(u,) T. When t < T we have

Xt B(u, ) Xt X B(0, 2)then

LTB(u,) =

T

1XtB(u,)dt T

1{(XtX)B(0,2)}dt =T0

1(X+tX)B(0,2)dt

Then, taking into account that = T implies LTB(u,) = 0,

LTB(u,) 1{


32/36

then

LTB(u,) T

1XtB(u,)dt T

1XtXB(0,/2)dt

T0

1X+tXB(0,/2)dt

1T/2T0

1X+tXB(0,/2)dt

1T/2T/20

1X+tXB(0,/2)dt.

Then, using again the strong Markov property with respect to the stopping time , weobtain

Wx0[|LTB(u,)|p] Wx0[B(u,/2) T /2]W0[(LT/2B(0,/2))p]and the proof is complete.

Letting p = 1 in the previous lemma and using the scale invariance of BM we obtainan upper bounds for Wx0 [B(u,) T] as

Wx0[B(u,)

T] =

Wx0/

2[B(u/

2,/

2)

T /2]

Wx0/2[LTB(u/2,2)]

W0[LT/2B(0,/2)]and the corresponding lower bound for Wx0[B(u,/2) T /2]:

Wx0[B(u,/2) T /2] = W2x0[B(2u,/2) T] W2x0[LTB(2u,/2)]

W0[LTB(0,2)]

which leads to the following easy corollary:

Corollary 4 For any p > 0, T > 0, > 0, x0, u R3, it holds that

W2x0 [LTB(2u,/2)]W0[|LT/2B(0,/2)|p]

W0[LT/2B(0,2)] Wx0[|LTB(u,)|p]

Wx0/2[LTB(u/2,2)]W0[|LTB(0,2)|p]

W0[LTB(0,/2)](21)

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Lemma 13 Given > 0 andp > 0, there exist constants c, C > 0 such that the following

properties hold true: for every T, satisfying T /2

we havecp W0|LTB(0,)|p/2 Cp

while if T /2 cTp/2 W0|LTB(0,)|p/2 CTp/2.

Proof. Consider first T /2 . In distribution

LTB(0,)L= 2

T/20

1XtB(0,1)dt (22)

and moreover we have that

c W0T/2

0

1XtB(0,1)dt

p C (23)

uniformly in T /2 > 0 (the constants depend on and p). The lower bound isobtained by setting T /2 = while the upper bound is given by the fact that

W0

0

1XtB(0,1)dtp

<

for any p > 0 (see for instance [13], section 3). From (22) and (23) we get the first claimof the lemma.

Next, if T /2 , in distribution:

LTB(0,)L= T

10

1XtB(0,/T)dt

andW0|L1B(0,1/)|p W0|L1B(0,/T)|p 1

so the second claim is also proved.

Lemma 14 Given p > 0, > 0, there are constants c, C > 0 such that, for T 2

cpT W[|LTB(u,)|p/2] CpT (24)

while for T 2cTp/23 W[|LTB(u,)|p/2] CTp/23. (25)

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Proof. Let us prove (24). Using lemma 13, equation (21) becomes

cp2W2x0[LTB(2u,/2)] Wx0[|LTB(u,)|p/2] Cp2Wx0/2[LTB(u/2,2)] (26)

for two constants c,C > 0 (depending on and p). Moreover, as we remarked at thebeginning of the section,

W[LTB(u,)] = 3T.Using this identity in (26), we get (24).

Now consider eq.(25). Assume T 2. Using eq.(21) and lemma 13 we havecT

p21W2x0[LTB(2u,/2)] Wx0 [|LTB(u,)|p/2] CT

p21Wx0/2[LTB(u/2,2)] (27)

and (25) is a consequence of the identityW[LTB(u,)] = 3T.

The proof is complete. The previous lemma solves the main problem posed at the beginning of the section.

We prove also a related result in finite volume that we need for the complementary resultsof section 2.4. We limit ourselves to the upper bounds, for shortness.

Lemma 15 Given p > 0, > 0, u R3, there is a constant C > 0 and a function (R)with limR (R) = 0, such that, for R > |u| + and T 2

|x0|R

Wx0[|LTB(u,)|p/2]dx0 CpTexpR (|u| + )T

while for T 2

|x0|RWx0[|LTB(u,)|p/2]dx0 CTp/23 exp

R (|u| + )

T

.

Proof. Arguing as in the previous lemma, it is sufficient to prove that|x0|R

Wx0 [LTB(u,)]dx0 C3Texp

R (|u| + )T

.

We have|x0|R

Wx0[LTB(u,)]dx0 =B(0,R)c

dx0

T0

dt

B(u,)

dz pt (z x0)

= 3T

T0

dt

T

B(u,)

dz

|B(u, )|B(0,R)c

dx0 pt (z x0)

3T g (R; T ,u ,)

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with (recall that 1)

g (R; T ,u ,) = supt(0,T],|z||u|+

B(0,R)c

dx0 pt (z x0) .

Now (denoting by (Wt) a 3D Brownian motion)

g (R; T ,u ,) = supt(0,T],|z||u|+

P(|z Wt| R)

supt(0,T]

P(|Wt| R (|u| + ))

= supt(0,T]

P|W1| R (|u| + )

t P

|W1| R (|u| + )

T

This completes the proof.

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Franco Flandoli and Massimiliano Gubinelli- Statistics of a Vortex Filament Model

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