Multidimensional potential Burgers turbulence

Multidimensional potential Burgers turbulence

Alexandre Boritchev University of Lyon
CNRS UMR 5208
University Claude Bernard Lyon 1
Institut Camille Jordan
43 Blvd. du 11 novembre 1918
69622 VILLEURBANNE CEDEX
FRANCE
E-mail: alexandre.boritchev@gmail.com
July 12, 2019

Abstract. We consider the multidimensional generalised stochastic Burgers equation in the space-periodic setting:

under the assumption that is a gradient. Here is strongly convex and satisfies a growth condition, is small and positive, while is a random forcing term, smooth in space and white in time.
For solutions of this equation, we study Sobolev norms of averaged in time and in ensemble: each of these norms behaves as a given negative power of . These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. These quantities behave as a power of the norm of the relevant parameter, which is respectively the separation in the physical space and the wavenumber in the Fourier space. Our bounds do not depend on the initial condition and hold uniformly in .
We generalise the results obtained for the one-dimensional case in BorW, confirming the physical predictions in BK07; GMN10. Note that the form of the estimates does not depend on the dimension: the powers of are the same in the one- and the multi-dimensional setting.

Abbreviations

• 1d, 2d, multi-d: 1, 2, multi-dimensional

• a.e.: almost every

• a.s.: almost surely

• (GN): the Gagliardo–Nirenberg inequality (Lemma 2.1)

• i.i.d.: independent identically distributed

• r.v.: random variable

1 Introduction

1.1 Burgers turbulence

The multi-dimensional generalised Burgers equation

 (1)

where is a constant (the classical Burgers equation Bur74 corresponds to ) is historically a popular model for the Navier-Stokes equations, since both of them have similar nonlinearities and dissipative terms.
Taking the curl of (1), we see that for a gradient initial condition , the solution remains a gradient for all times. Namely, this solution is the gradient of the solution to the viscous generalised Hamilton–Jacobi equation

 ∂ψ∂t+f(∇ψ)−νΔψ=0 (2)

with the initial condition . For shortness, in this case we will write the Burgers equation as

 (3)

where it is implicitly assumed that the potential satisfies (2). We will do likewise for the equation (3) with a gradient right part instead of , and we will say that we are in the potential case. From now on, unless otherwise stated, we will only consider this case. Moreover, we will only consider the space-periodic setting, i.e.

 x∈Td=(R/Z)d.

The mathematical advantage of the potential case is that the equation (2) can be treated by variational methods (see for instance GIKP05). Moreover, for the equation (3) has become popular as a model in astrophysics: in the limit , it corresponds to the adhesion approximation introduced by Gurbatov and Saichev and developed later by Shandarin and Zeldovich GurSai84; GMN10; ShaZel89. The equation (1) is also relevant for fields as different as statistical physics, geology and traffic modelling (see the surveys BF01; BK07 and references therein; see also Flo48).
For , the equation (3) can be transformed into the heat equation by the Cole-Hopf method Col51; Hop50. In some settings (for instance when considering the Burgers equation with very singular additive noise) this method is extremely helpful (see BK07 and references therein). However, it is harder to make use of this transformation in the setting of our paper, where we are concerned with the quantitative behaviour of solutions in the singular limit . Moreover, the Cole-Hopf method does not allow us to study the Burgers equation for a nonlinearity other than .
When studying the local (in space) fine structure of a function, natural objects of interest are the small-scale quantities, which play an important role in the study of turbulence Fri95. In the physical space, this denomination includes the structure functions (i.e. the moments of increments in space) for small separations. In the Fourier space, an important quantity of interest is the energy spectrum on small scales (i.e. the amount of energy carried by high Fourier modes). It is important to understand the critical thresholds for the relevant parameters (respectively, in the physical space the separation distance and in the Fourier space the wavenumber) between regions where the small-scale quantities exhibit different types of behaviour. These values are referred to as length scales.
The systematic study of small-scale quantities for the solutions of nonlinear PDEs with a small parameter with or without random forcing was initiated by Kuksin. He obtained lower and upper estimates of these quantities by negative powers of the parameter for a large class of equations (see Kuk97GAFA; Kuk99GAFA and the references in Kuk99GAFA). A natural way to study these quantities is through upper and lower bounds for Sobolev norms: for a discussion of the relationship between Sobolev norms and spatial scales, see Kuk99GAFA. For more recent results obtained for the 2D Navier-Stokes equations, see the monograph KuSh12 and the references therein.
Before treating the multi-d case, we recall some facts about the behaviour of the solutions to (1) in the 1d setting. We only consider the case where is strongly convex, i.e. there exists such that

 f′′(x)≥σ, x∈R. (4)

In this setting, the requirement that we are in the potential case implies the vanishing of the space average of the solution.
We consider the regime . Since all other parameters are fixed, in the hydrodynamical language this corresponds to the case of a large Reynolds number. Under these assumptions, the solutions display turbulent-like behaviour, called Burgers turbulence or “Burgulence” Bur74; Cho75; Kid79, which we describe now.
In the limit and for large enough times, we observe -waves, i.e. the graphs of the solutions are composed of waves similar to the Cyrillic capital letter I (the mirror image of ). In other words, at a time the solution stops being smooth, and for times the solution alternates between negative jump discontinuities and smooth regions where the derivative is positive and of the order (see for instance Eva08). Thus, it exhibits small-scale spatial intermittency Fri95, i.e. for a fixed time the excited behaviour only takes place in a small region of space. For the solutions are still highly intermittent: shocks become zones where the derivative is small and positive, called ramps, which alternate with zones where the derivative is large in absolute value and negative, called cliffs (cf. Figure 1).

For the prototypical -wave, i.e. for the -periodic function equal to on , the Fourier coefficients satisfy . On the other hand, for the dissipation gives exponential decay of the spectrum for large values of . This justifies the conjecture that for small and for "moderately large" values of , the energy-type quantities behave, in average, as Cho75; FouFri83; Kid79; Kra68.
In the physical space, the natural analogues of the energy at the wavelength are the structure functions

 Sp(ℓ)=∫S1|u(x+ℓ)−u(x)|p dx. (5)

Heuristically, the behaviour of the solutions which is described above implies that for , these quantities behave as for : see AFLV92 and the introduction to BorD.

Now we consider the potential multi-d case. In the case , in the inviscid limit it is numerically observed that the behaviour of the solution is analogous to what is happening in 1d BK07. Namely, for large enough times one observes a tesselation where cells inside which solutions are smooth are separated by 1-codimension shock manifolds. In particular, in average, 1d projections of the multi-d solution look like the 1d solution (cf. Figure 2).
Thus, it is reasonable to expect a behaviour of the longitudinal structure functions

 S∥p(r)= ∫x∈Td∣∣∣(u(x+r)−u(x))⋅r|r|∣∣∣pdx (6)

for a certain range of values of which is analogous to the behaviour of the structure functions in 1d (at least after averaging with respect to for a fixed value of ). Similarly, we could expect spectral asymptotics of the type

 ∑|n|∼k|^u(n)|2∑|n|∼k1∼k−2,

for a certain range of values of .
By analogy with the 1d case, one can conjecture that we have the same behaviour for strongly convex, which in multi-d means that

 tvD2f(x)v|v|2≥σ>0,v=(v1,…,vd)≠0,x∈Rd, (7)

where is the Hessian matrix and is the norm

 √v21+⋯+v2d.

Now let us say a few words about the similarities and the differences between the multi-dimensional potential Burgulence and the real incompressible turbulence. It is clear that the geometric pictures on small scales are quite different for these two models: the multi-dimensional analogues of -waves created by infinitely strong compressibility do not have the same nature as the complex multi-scale structures modeled by incompressibility constraints such as the vortex tubes. However, the similarity in the form of the potential Burgers equation and the incompressible Navier-Stokes equations implies that some physical arguments justifying different theories of turbulence can be applied to the Burgulence. Indeed, both models exhibit an inertial nonlinearity of the form , and a viscous term which in the limit gives a dissipative anomaly Fri95. Hence, the Burgers equation is often used as a benchmark for turbulence theories, as well as for numerical methods for the Navier-Stokes equations. For more information on both subjects, see BK07.

1.2 State of the art and setting

For the unforced Burgers equation, some upper estimates for Sobolev norms of solutions and for small-scale quantities are well-known. For references on classical aspects of the theory of scalar (viscous or inviscid) conservation laws, see Daf10; Lax06; Ser99. For some upper estimates for small-scale quantities, see Kre88; Tad93. To our best knowledge, rigorous lower estimates were not known before Biryuk’s and our work.
In Bir01, Biryuk considered the unforced generalised Burgers equation (1) in the 1d space-periodic case, with satisfying (4). He obtained estimates for Sobolev norms of the -th spatial derivatives of the solutions:

 1T∫T0∥u(t)∥2m∼ν−(2m−1),m≥1, ν≤ν0.

The constants and and the multiplicative constants implicitly contained in the symbol depend on the deterministic initial condition as well as on . Biryuk also obtained almost sharp spectral estimates which allowed him to give the correct value of the dissipation scale, which equals (see Section 2.6 for its definition). We can explain Biryuk’s method by a dimensional analysis argument, considering the quantity

 Am=∥u(t)∥m∥u(t)∥m+1

(see Kuk97GAFA; Kuk99GAFA). Indeed, after averaging in time one gets

 Am∼ν,m≥1,

as .
In BorD, we generalised Biryuk’s estimates to the Lebesgue norms of the -th spatial derivatives for . Moreover, we improved Biryuk’s estimates for small-scale quantities, obtaining sharp -independent estimates. In particular, for , we proved that

with defined by (5), and for such that , we obtained that

 1T2−T1∫T2T1∑n∈[M−1k,Mk]|^un|2∑|n|∈[M−1k,Mk]1dt∼k−2.

The constants , , and , as well as the different strictly positive constants denoted by and the multiplicative constants implicitly contained in the symbol depend on the deterministic initial condition as well as on . Note that here again, the upper and the lower estimates only differ by a multiplicative constant. Moreover, we rigorously prove that is the threshold parameter which corresponds to the transition between algebraic (in ) and super-algebraic behaviour of the energy spectrum.
To get results independent of the initial data, a natural idea is to introduce random forcing and to average with respect to the corresponding probability measure. In the articles BorK; BorW, we have considered the 1d case with in the right-hand side of (1) replaced by a random spatially smooth force, “kicked” and white in time, respectively. In the “kicked” model, we consider the unforced equation and at integer times, we add i.i.d. smooth in space impulsions. The white force corresponds, heuristically, to a scaled limit of “kicked” forces with more and more frequent kicks. On a formal level, this can be explained by Donsker’s theorem, since by definition a white force is the weak derivative in time of a Wiener process.
In the random case, the estimates for the Sobolev norms and for the small-scale quantities seem at first sight to be almost word-to-word the same as in the unforced case. However, there are two major differences. The first one is that along with the averaging in time we also need to take the expected value. The second one is that we have estimates which hold uniformly with respect to the starting time for intervals of fixed length on which we consider the averaged quantities; moreover, the constants in the bounds do not any more depend on the initial condition.
To explain the second difference, we observe that in the unforced case, no energy source is available to counterbalance the viscous dissipation, whereas in the forced case the stochastic term provides such a source. Thus, the existence of a stationary measure which is nontrivial (i.e., not proportional to the Dirac measure ) is possible in the randomly forced case, as opposed to the unforced case where we have a decay to of the solutions at the speed . In the language of statistical physics, this corresponds to the existence of a non-trivial non-equilibrium steady state Gal02. Indeed, in BorW we prove the existence and the uniqueness of the stationary measure for the generalised white-forced Burgers equation; our arguments also apply to the kick-forced case. For more details on Biryuk’s and our work on 1d Burgulence, see the survey BorS.
In this paper, we study the white-forced equation

 (8)

under the additional convexity and growth assumptions (7, 17) on . We obtain estimates for the Sobolev norms and the small-scale quantities which are (up to some changes in definitions due to the multi-dimensional setting) word-to-word the same as those proved in Bir01; BorK; BorW; BorD, with the same exponents for . The only small difference between the results in 1d and in this article is that we do not obtain upper estimates for the norms. Moreover, we obtain results on the existence and the uniqueness of the stationary measure for the equation (8) as well as the rate of convergence to . Thus, we generalise the 1d results in BorW.
The assumption that is a gradient plays a crucial role, since it allows us to generalise the 1d arguments from the papers Bir01; BorW, in particular for the upper estimates; see Theorem 4.2. On the other hand there is a major difficulty specific to the multi-d case. Namely, the energy balance is much more delicate to deal with than in 1d; see Section 5. This is the reason why here, unlike in 1d, we assume that the noise is “diagonal”: in other words, there is no correlation between the different Fourier modes. This allows us to use a more involved version of the "small-noise zones" argument (see for instance IK03). Roughly speaking, this argument tells that if the noise is small during a long time interval, then the solution of the generalised Burgers equation goes to , roughly at the same rate as if there was no noise at all, i.e. at least as . Note that by classical properties of Wiener processes, such an interval will eventually occur with probability : see (BorW, Formula (10)) for a quantitative version of this statement.
In Bir04, Biryuk studied solutions of the space-periodic multi-d Burgers equation without the assumption that is a gradient. He obtained upper and lower estimates which are non-sharp, in the sense that there is a gap between the powers of for the upper and the lower estimates. In a setting very similar to ours, Brzezniak, Goldys and Neklyudov BGN14; GN09 have considered the multi-d Burgers equation both in the deterministic and in the stochastic case, obtaining results on the well-posedness both in the whole-space and in the periodic setting. Moreover, in the potential space-periodic case those authors have obtained estimates which are uniform with respect to the viscosity coefficient ; however, those estimates are not uniform in time, unlike the ones proved in our paper.
We are concerned with solutions for small but positive . For a study of the limiting dynamics with , see EKMS97; EKMS00 for the 1d case, GIKP05; IK03 for the multi-d case, and DV; DS05 for the case of multi-d scalar conservation laws with nonconvex flux.
In Bir01; BorK; BorW; BorD as well as in our paper, estimates on Sobolev norms and on small-scale quantities are asymptotically sharp in the sense that enter lower and upper bounds at the same power. Such estimates are not available for the more complicated equations considered in Kuk97GAFA; Kuk99GAFA; KuSh12. Another remarkable feature of our estimates is that the powers of the quantities are always the same as in 1d. Thus, those estimates are in agreement with the physical predictions for space increments (BK07, Section 7) and for spectral asymptotics GMN10 of the solutions .
The results of our paper extend to the case of a “kicked” force, under some restrictions. Namely, while the upper estimates hold in a very general setting, to prove the lower estimates we seem to need some non-trivial assumptions on the support of the kick, since the dissipation relation for the energy has an additional trilinear term compared to the 1d case. For the same reason, the results in the unforced case are expected to be less general than in 1d.
To prove our results on the existence and the uniqueness of the stationary measure and the rate of convergence to it, we use a quantitative version of the "small-noise zones" argument IK03, a coupling argument due to Kuksin and Shirikyan KuSh12 and -contractivity for the flow of the Hamilton-Jacobi equation satisfied by the potential .

1.3 Plan of the paper

After introducing the notation and the setup in Section 2, we formulate the main results in Section 3. In Section 4, for and for a vector with integer coefficients, we begin by estimating from above the moments of the quantities

 maxs∈[t,t+1], x∈Td(k⋅∇)2ψ(s,x)

for the potential corresponding to the solution of (8).
In Sections 4-6 we get estimates for the Sobolev norms of the same type as those obtained in Bir01; BorK; BorW; BorD with the same exponents for , valid for time ; the only small difference with the 1d case is that here we do not obtain sharp upper bounds for the norms. Here, is a constant, independent of the initial condition and of . Actually, for , we are in a quasi-stationary regime: all the estimates hold uniformly in and in the initial condition .
In Section 7 we study the implications of our results in terms of the theory of Burgulence. Namely, we give sharp upper and lower bounds for the dissipation length scale, the increments and the spectral asymptotics for the flow . These bounds hold uniformly for , where is a constant which is independent of the initial condition. One proof in this section uses (indirectly) a 1d argument from AFLV92.
In Section 8, we prove the existence and the uniqueness of the stationary measure for the equation (8), and we give an estimate for the speed of convergence to this stationary measure.

2 Notation and setup

2.1 Functions, indices, derivatives

All functions that we consider are real-valued or, if written in bold script, vector-valued. When giving formulas which hold for functions which can be scalar or vector-valued, we use the usual script. We denote by the canonical vector basis of . We assume that . Note that all of our estimates still hold for : see BorW.
The subscript denotes partial differentiation with respect to the variable . When we consider a scalar-valued function , the subscripts , which can be repeated, denote differentiation with respect to the variables , respectively. Since the only scalar-valued functions for which the notation will be used are infinitely differentiable, by Schwarz’s lemma we will always have

 vi1,…,ik=vπ(i1),…,π(ik)

for any permutation of the subscripts.
For a -dimensional vector and a (vector or scalar)-valued function , the notation means that we fix all coordinates except one, i.e. we consider

 v(x1,…,xi−1,⋅,xi+1,…,xd).

Accordingly, the notation means that we integrate over the variables

 x1,…,xi−1,xi+1,…,xd,

for a fixed value of . For shortness, a function is denoted by . The norm of an -dimensional vector is defined by

 |v|=√v21+⋯+v2N.

It should not be confused with the norm of a function, which will be introduced in the next subsection and is also denoted by : the meaning of the notation will always be clear from the context. We use the notation and .

2.2 Sobolev spaces

For , consider an integrable -valued function on .We only study spatial Sobolev norms for functions considered at a fixed moment of time. We do not always assume that : for instance, we will study functions of the type which are defined on . The dimensions are always clear from the context, and thus are not specified in the notation for Sobolev norms.
For , we denote the Lebesgue norm of a scalar-valued function by . For a vector-valued function , we define this norm as the norm in of the function , and denote it by . We denote the norm by , and the corresponding scalar product by . From now on denotes the space of functions in . Similarly, is the space of -smooth functions on .
Except in Appendix 1, we only study Sobolev norms for zero mean functions. Thus, in the following, we always assume that . In particular, we never study directly the Sobolev norms of the potential : either we consider the mean value function or the partial derivatives of .
For a nonnegative integer and , stands for the Sobolev space of zero mean functions on with finite homogeneous norm

 |v|m,p=∑|α|=mm!α1!…αk!∣∣∣(dmv1dxα,…,dmvNdxα)∣∣∣p. (9)

Here and from now on, denotes the norm of the multi-index

 α=(α1,…,αd′),

defined by

 α1+⋯+αd′.

In particular, for . For , we denote by , and abbreviate the corresponding norm as .
We recall a version of the classical Gagliardo–Nirenberg inequality (see (DG95, Appendix)). We will refer to this inequality as (GN).

Lemma 2.1.

For a smooth zero mean function on , we have

 |v|β,r≤C|v|θm,p|v|1−θq,

where , and is defined by

 d′r=β−θ(m−d′p)+(1−θ)d′q,

under the assumption if is a nonnegative integer, and otherwise. The constant depends on .

Let us stress that we only use this inequality in cases when it gives the same value of as in 1d. Actually, the only place where we use it in a multi-d setting is when we mention that the proof of Lemma 4.8 is word-to-word the same as in 1d.
We will use a norm denoted by , which is defined for and is equivalent to the norm defined above. For its definition, see Corollary 4.10. By analogy with the notation , we will abbreviate as the norm .
For any , we define as the Sobolev space of zero mean functions on with finite norm

 ∥v∥′s=(⟨v, (−Δ)sv⟩)1/2=(2π)s(∑n∈Zd′|n|2s|^v(n)|2)1/2, (10)

where are the complex Fourier coefficients of . For integer values of , this norm is equivalent to the previously defined norm . For , is equivalent to the norm

 ∥v∥′′s=(∫x∈Td′, |r|≤1|v(x+r)−v(x)|2|r|2s+d′ dx dr)1/2. (11)

Moreover, for all integers we have the embedding

 |v|m,∞≤C(s)∥v∥′m+s, s>d/2 (12)

Finally, it should be noted that the integer , defined by:

 s0=(d+1)/2 if d even; d/2+1 if d odd (13)

plays a crucial role in the study of the well-posedness for (19) (see Section 2.4 and Appendix 1).

2.3 Random setting

We provide each space

 Wm,p(Td), m≥0, p∈[1,∞]

of scalar-valued functions with the Borel -algebra. Then we consider a random process , valued on the space of zero mean value functions in and defined on a complete probability space . We suppose that defines a smooth in space Wiener process with respect to a filtration , in each space . Moreover, we assume that the process is diagonal in the sense that its projections on the Fourier modes are independent weighted Wiener processes. In other words, we assume the following:
i) The process can be written as

 w(t,x)=∑n∈˜Zd(anwn(t)cos(2πn⋅x)+bn~wn(t)sin(2πn⋅x)), (14)

where

 ˜Zd= {n∈Zd | n1>0}∪{n∈Zd | n1=0,n2>0}… ∪{n∈Zd | n1=0,…,nd−1=0,nd>0},

, are independent Wiener processes and for any we have . Without loss of generality, we can assume that for all , we have .
ii) The process is non-trivial: in other words, at least one of the coefficients is not equal to .
Thus, for

where is the correlation operator defined by

which defines a continuous mapping from into for each .
Note that since we have for every , a.s., we can redefine the Wiener process so that this property holds for all . We will denote by . For more details on the construction of infinite-dimensional Wiener processes, see (DZ92, Chapter 4).
For , we denote by the quantity

 Im=TrHm(Q)=E∥w(1)∥2m.

From now on, the term denotes the stochastic differential corresponding to the Wiener process in the space .
Now fix . By Fernique’s Theorem (Kuo75, Theorem 3.3.1), there exist such that for ,

 Eexp(ιm∥w(T)∥2m/T)≤Cm. (15)

Therefore by Doob’s maximal inequality for infinite-dimensional submartingales (DZ92, Theorem 3.8. (ii)) we have the following inequality, which holds uniformly in :

 Esupt∈[τ,T+τ]∥w(t)−w(τ)∥km ≤(kk−1)kE∥w(T+τ)−w(τ)∥km (16) =C(m,k)Tk/2<+∞,

for any and .
Note that the estimates in this subsection still hold for the successive spatial derivatives of , which are also smooth in space infinite-dimensional Wiener processes.

2.4 Preliminaries

We begin by considering the viscous Hamilton-Jacobi equation (2). Here, , and the viscosity coefficient satisfies . The function is strongly convex, i.e. it satisfies (7), and -smooth. We also assume that for any the -th partial derivatives of satisfy

 ∃h≥0, Cm>0: ∣∣∑|α|=m∂mf(x)∂xα∣∣≤Cm(1+|x|)h,x∈Rd, (17)

where is a function such that (the lower bound on follows from (7)). The usual Burgers equation corresponds to .
The white-forced generalised Hamilton-Jacobi equation is (2) with the random forcing term

 ηω=∂wω/∂t,

added on the right-hand side. Here, is the Wiener process with respect to defined above.

Definition 2.2.

We say that an -valued process (for the definition of see (13)) is a solution of the equation

 {∂u∂t+(∇f(u)⋅∇)u−νΔu=∇ηωu=∇ψ (18)

if for every and for every , , where satisfies the following properties:
i) For , is -measurable.
ii) The function is continuous in (its gradient is therefore continuous in ) and satisfies

 ψω(t)=ψω(0)−∫t0(νLψω(s)+f(∇ψω)(s))ds+wω(t), (19)

where .

As a corollary of this definition, we obtain that satisfies

 (20)

where . When studying solutions of (18), we always assume that the initial potential is -smooth.
For a given initial condition, (19), and therefore (18), has a unique solution, i.e. any two solutions coincide for all . For shortness, this solution (resp., the corresponding potential) will be denoted by (resp., ). To prove this, we can use the same arguments as in 1d (cf. BorPhD). Namely, to prove local well-posedness we use the “mild solution” technique (cf. (DZ96, Chapter 14)) and a bootstrap argument. Finally, global well-posedness follows from uniform bounds of the same type as in Section 4. For more details, see Appendix 1.
Once is fixed, is fixed up to an additive constant. Moreover, if we consider two different initial conditions and , then the difference between the corresponding solutions to (19) will always be equal to . In other words, fixing is equivalent to fixing an equivalence class of initial conditions .
Since the forcing and the initial condition are smooth in space, we can also show that is time-continuous in for every and the spatial derivatives of are in for all . Consequently, is also a strong solution of the equation

 ∂(ψ−w)∂t+f(∇ψ)−νΔψ=0, (21)

and is a strong solution of the equation

 ∂(u−∇w)∂t+(∇f(u)⋅∇)u−νΔu=0. (22)

Solutions of (18) make a time-continuous Markov process in . For details, we refer to KuSh12, where the stochastic 2D Navier-Stokes equations are studied in a similar setting.
Now consider, for a solution of (18), the functional

 Gm(u(t))=∥u(t)∥2m

and apply Itô’s formula (DZ92, Theorem 4.17) to (20). We get

 ∥u(t)∥2m= ∥∥u0∥∥2m−∫t0(2ν∥u(s)∥2m+1+⟨Lmu(s), B(u)(s)⟩)ds +2∫t0⟨Lmu(s), d∇w(s)⟩+Im+1t. (23)

We recall that for , . Consequently,

 ddtE∥u(t)∥2m=−2νE∥u(t)∥2m+1−E ⟨Lmu(t), B(u)(t)⟩+Im+1. (24)

2.5 Agreements

From now on, all constants denoted by with an eventual subscript are positive and nonrandom. Unless otherwise stated, they depend only on and on the distribution of the Wiener process . By we denote constants which also depend on the parameters . By we mean that . The notation stands for

 Ya1,…,ak≲Xa1,…,ak≲Y.

In particular, and mean that and , respectively. All constants are independent of the viscosity and of the initial value .
We denote by the solution to (18) with an initial condition and by the corresponding solution to (19), which is, for a given value of , uniquely defined up to an additive constant.
For simplicity, in Sections 4-7, we assume that is deterministic. However, we can easily generalise all results in these sections to the case of a random initial condition independent of . Indeed, in this case for any measurable functional we have

 EΦ(u(⋅))=∫E(Φ(u(⋅)) | u0=u0(ω))dμ(u0(ω)),

where is the law of the r.v. .
For and independent of , the Markov property yields that

 EΦ(u(τ+⋅))=∫E(Φ(u(⋅)) |u0=uω(τ))dμ(uω(τ)).

Consequently, all -independent estimates which hold for time or a time interval actually hold for time or a time interval , uniformly in . Thus, for , to prove a -independent estimate which holds uniformly for , it suffices to consider the case .

2.6 Setting and notation in Section 7

Consider an observable , i.e. a real-valued functional on a Sobolev space , which we evaluate on the solutions . We denote by the average of in ensemble and in time over :

 {A}=1T0 ∫t+T0tE A(uω(s)) ds, t≥T1=T0+2.

The constant is the same as in Theorem 6.1. In this section, we assume that , where is a positive constant. Next, we define the intervals

 J1=(0, C1ν]; J2=(C1ν, C2]; J3=(C2, 1]. (25)

For the value of , and , see (66). In other words, , , .
In terms of the Kolmogorov 1941 theory Fri95, the interval corresponds to the dissipation range. In other words, for the Fourier modes such that , the expected values of the Fourier coefficients decrease super-algebraically in . The interval corresponds to the inertial range, where quantities such as the (layer-averaged) energy spectrum defined by

 E(k)=k−1∑|n|∈[M−1k,Mk]{|^u(n)|2} (26)

behave as a negative degree of . Here is a large enough constant (cf. the proof of Theorem 7.15). The boundary between these two ranges is the dissipation length scale. Finally, the interval corresponds to the energy range, i.e. the sum is mostly supported by the Fourier modes corresponding to .
The positive constants and can take any value provided that

 C1≤14K−2;5K2≤C1C2<1ν0. (27)

Here, is a positive constant (see (65)). Note that the intervals defined by (25) are non-empty and do not intersect each other for all values of , under the assumption (27).
The constants and can be made as small as desired. On the other hand, by (75) the ratio

 ∑|n|−1∈J3|^u(n)|2∑n∈Zd|^u(n)|2

tends to as tends to , uniformly in . This allows us to choose so that

 ∑|n|

Now we suppose that , . We consider the averaged directional increments

 Sp,α,i(r)= {(∫x∈Td|ψi(x+r)−ψi(x)|pdx)α}, 1≤i≤d,

the averaged longitudinal increments

 S∥p,α(r)= {(∫x∈Td∣∣∣(u(x+r)−u(x))⋅r|r|∣∣∣pdx)α}

and the averaged increments

 Sp,α(r)= {(∫x∈Td|u(x+r)−u(x)|pdx)α}.

Now, for , we define the averaged moments of the space increments on the scale for the flow :

 Sp,α(ℓ)=c−1dℓ−(d−1)∫r∈ℓSd−1Sp,α(r)dσ(r),

where stands for the surface measure on and is the surface of . The quantity is denoted by ; it corresponds to the structure function of order , while the flatness , given by

 F(ℓ)=S4(ℓ)/S22(ℓ), (28)

measures spatial intermittency at the scale Fri95.

3 Main results

In Sections 4-6, we prove sharp upper and lower estimates for a large class of Sobolev norms of . A key result is proved in Theorem 4.2. Namely, there we obtain that for ,

 E (maxs∈[t,t+1], x∈Tdmax1≤i≤dψii(s,x))kk≲1,t≥1. (29)

This result is generalised to all second derivatives , where is a vector with integer coefficients, in Lemma 4.5.
The main estimates for Sobolev norms are contained in the first part of Theorem 6.1, where we prove that for and , and , or and , we have

 {|u(s)|αm,p}1/αm,p,α∼ν−γ,α>0, t,T≥T0, (30)

where and is a constant. In particular, this result implies that

 {∥u(s)∥m}{∥u(s)∥m+1}∼ν, m≥1.

In the language of the turbulence theory, the characteristic dissipaton scale of the flow is therefore of the order .
For and , we do not have a similar bound, as opposed to the 1d case. However, there exists such that we have the following result, which will play a key role when we estimate the small-scale quantities:

 {∫~xi|ψi(s,~xi)|αm,∞}1/αm,α∼ν−m, α>0, t,T≥T0. (31)

This result tells us that in average, the 1d restrictions of (which is, as we recall, the -th coordinate of ) for fixed values of have the same behaviour as the 1d solutions (see BorW).
In Section 7 we obtain sharp estimates for analogues of the quantities characterising the hydrodynamical turbulence. Although we only prove results for quantities averaged over a time period of length , those results can be immediately generalised to quantities averaged over time periods of length .
We assume that , where is a constant. As the first application of the estimates (29-31), in Section 7 we obtain sharp estimates for the quantities . Namely, by Theorem 7.12, for :

 Sp,α(ℓ)p,α∼{ℓαp, 0≤p≤1.ℓαpν−α(p−1), p≥1,

and for :

 Sp,α(ℓ)p,α∼{ℓαp, 0≤p≤1.ℓα, p≥1.

Consequently, for the flatness function satisfies Thus, solutions are highly intermittent in the inertial range.
On the other hand, we obtain estimates for the spectral asymptotics of Burgulence. Namely, for all and , (71) tells us that

 {|^u(n)|2}m≲(ν|n|)−2mν,

and for such that , by Remark 7.16 we get

 {(k−1∑|n|∈[M−1k,Mk]|^u(n)|2)α}α∼k−2α,α>0.

In particular, in the inertial range the energy spectrum satisfies .
Finally, in Section 8 we prove that (18) has a unique stationary measure