Alexandre Boritchev
email: alexandre.boritchev@gmail.com

SHARP ESTIMATES FOR TURBULENCE IN WHITE-FORCED GENERALISED BURGERS EQUATION

come

Sharp Estimates for Turbulence in White-Forced

Generalised Burgers Equation

Introduction

The generalised one-dimensional space-periodic Burgers equation

∂u ∂t + f (u) ∂u ∂x -ν ∂ 2 u ∂x 2 = 0, ν > 0, x ∈ S 1 = R/Z (1)
is a popular model for the Navier-Stokes equation, since both of them have similar nonlinearities and dissipative terms (the classical Burgers equation [START_REF] Burgers | The nonlinear diffusion equation: asymptotic solutions and statistical problems[END_REF] corresponds to f (u) = u 2 /2). For ν 1 and f strongly convex, i.e. satisfying:

f (x) ≥ σ > 0, x ∈ R, (2)
solutions of (1) display turbulent-like behaviour, called "Burgulence" [START_REF] Bec | Burgulence[END_REF][START_REF] Bec | Burgers turbulence[END_REF]. In this paper, we are interested in qualitative and quantitative properties of the Burgulence. The mean value in space is a conserved quantity for solutions to [START_REF] Adams | Sobolev spaces[END_REF]. Indeed, since u is 1-periodic in space, we have:

d dt S 1 u(t, x)dx = - S 1 f (u(t, x))u x (t, x)dx + ν S 1
u xx (t, x)dx = 0.

To simplify presentation, we restrict ourselves to solutions with zero mean value in space:

S 1 u(t, x)dx = 0, ∀t ≥ 0. (3)
In [START_REF] Biryuk | Spectral properties of solutions of the Burgers equation with small dissipation[END_REF], Biryuk considered (1) with f satisfying [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]. He studied solutions u for small values of ν and obtained the following estimates for norms in L 2 of their m-th spatial derivatives:

u(t) 2 m ≤ Cν -(2m-1) , 1 T T 0 u(t) 2 m ≥ cν -(2m-1) , m ≥ 1, ν ≤ ν 0 . (4)
Note that the exponents for ν in lower and upper bounds are the same.

For fixed m, the constants ν 0 , C, c and T depend on the deterministic initial condition u 0 . This dependence cannot be removed. Indeed, [START_REF] Adams | Sobolev spaces[END_REF] is dissipative for the L 2 norm of u, so no non-trivial lower estimate can hold if we take 0 as the initial condition. Moreover, as t → +∞, the solution of the deterministic Burgers equation tends to 0 uniformly in u 0 , so we have no hope of getting a non-trivial lower estimate which would hold uniformly in time. In a recent preprint [START_REF] Boritchev | Note on decaying turbulence in a generalised Burgers equation[END_REF], we formulate the dependence of the estimates (4) on u 0 in a simpler way.

To get results which are independent of the initial data and hold uniformly for large enough t, a natural idea is to introduce a random force and to estimate ensemble-averaged characteristics of solutions. In the article [START_REF] Boritchev | Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation[END_REF], we have considered the case when 0 in the right-hand side of ( 1) is replaced by a random spatially smooth force, "kicked" in time. In this article we consider the equation

∂u ∂t + f (u) ∂u ∂x -ν ∂ 2 u ∂x 2 = η ω , (5)
where η ω is a random force, white in time and smooth in space. This force corresponds to a scaled limit of "kicked" forces with more and more frequent kicks. All forces that we consider have zero mean value in space. Study of Sobolev norms of solutions for nonlinear PDEs with small viscosity (with or without random forcing) in order to get estimates for small-scale quantities such as the spectrum is motivated by the problem of turbulence. This research was initiated by Kuksin, who obtained lower and upper estimates of these norms by negative powers of the viscosity for a large class of equations (see [START_REF] Kuksin | On turbulence in nonlinear Schrödinger equations[END_REF][START_REF] Kuksin | Spectral properties of solutions for nonlinear PDEs in the turbulent regime[END_REF] and references in [START_REF] Kuksin | Spectral properties of solutions for nonlinear PDEs in the turbulent regime[END_REF]), and continued by Biryuk [START_REF] Biryuk | Spectral properties of solutions of the Burgers equation with small dissipation[END_REF] for the Burgers equation. We use some methods and ideas from those works. Note that for the Burgers equation considered in [START_REF] Biryuk | Spectral properties of solutions of the Burgers equation with small dissipation[END_REF][START_REF] Boritchev | Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation[END_REF][START_REF] Boritchev | Note on decaying turbulence in a generalised Burgers equation[END_REF] and in the current paper, estimates on Sobolev norms are asymptotically sharp in the sense that viscosity enters lower and upper bounds at the same negative power. Such estimates are not available for the more complicated equations considered in [START_REF] Kuksin | On turbulence in nonlinear Schrödinger equations[END_REF][START_REF] Kuksin | Spectral properties of solutions for nonlinear PDEs in the turbulent regime[END_REF].

In this work, after introducing the notation and setup in Section 1, we formulate the main results in Section 2. In Section 3, we begin by estimating from above the moments of max ∂u/∂x for solutions u(t, x) of (5) for t ≥ 1. Using these bounds, we obtain estimates of the same type as in [START_REF] Biryuk | Spectral properties of solutions of the Burgers equation with small dissipation[END_REF][START_REF] Boritchev | Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation[END_REF], valid for time t ≥ T 1 = T 0 + 2. Here, T 0 is a constant, independent of the initial condition and of ν. Actually, for t ≥ T 1 , we are in a quasi-stationary regime: all estimates hold uniformly in t, ν and in the initial condition u 0 .

In Section 4 we study implications of our results in terms of the theory of Burgulence. Namely, we give sharp upper and lower bounds for the dissipation length scale, increments, flatness and spectral asymptotics for the flow u(t, x) for t ≥ T 1 . These bounds hold uniformly in ν ≤ ν 0 , where ν 0 is a positive constant which is independent of u 0 .

The results of Section 4 rigorously justify the physical predictions for space increments of solutions u(t, x) and for their spectral asymptotics [START_REF] Aurell | On the multifractal properties of the energy dissipation derived from turbulence data[END_REF][START_REF] Chorin | Lectures on turbulence theory[END_REF][START_REF] Weinan | Probability distribution functions for the random forced Burgers equation[END_REF][START_REF] Kida | Asymptotic properties of Burgers turbulence[END_REF][START_REF] Kraichnan | Lagrangian-history statistical theory for Burgers' equation[END_REF]. Our proof of Theorem 4.9 in this section uses an argument from [START_REF] Aurell | On the multifractal properties of the energy dissipation derived from turbulence data[END_REF]. Note that predictions for spectral asymptotics have been known since the 1950s: in [START_REF] Kraichnan | Lagrangian-history statistical theory for Burgers' equation[END_REF], the author refers to some earlier results by Burgers and Tatsumi.

The rigorous proof of the asymptotics predicted by a physical argument, even for such a relatively simple model as the stochastic Burgers equation, is important since for the 3D or 2D incompressible Navier-Stokes equation there is no exact theory of this type, corresponding to the heuristic theories due to Kolmogorov and Kraichnan. Note that since we study the generalised equation [START_REF] Bec | Burgers turbulence[END_REF] and not only the equation with the classical nonlinearity uu x , we cannot use the Cole-Hopf transformation [START_REF] Cole | On a quasilinear parabolic equation occurring in aerodynamics[END_REF][START_REF] Hopf | The partial differential equation u t + uu x = µu xx[END_REF].

In Section 5, we prove that the stochastic Burgers equation admits a unique stationary measure µ, and we estimate the speed of convergence to µ as t → +∞. It follows that the estimates in Sections 3-4 still hold if we replace averaging in time and probability with averaging with respect to µ.

We are concerned with solutions for (5) with small but positive ν. For a detailed study of the limiting dynamics with ν = 0, see [START_REF] Weinan | Invariant measures for Burgers equation with stochastic forcing[END_REF]. Additional properties for the limit corresponding to t → +∞ in both cases ν = 0 and ν > 0 have been established in [START_REF] Gomes | Viscosity limit of stationary distributions for the random forced Burgers equation[END_REF][START_REF] Iturriaga | Burgers turbulence and random Lagrangian systems[END_REF].

The results of Sections 4-5 also hold in the case of a "kicked" force, for which we have estimates analogous to those in Section 3 [START_REF] Boritchev | Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation[END_REF]. We would also like to note that similar estimates hold in the case of the multidimensional potential randomly forced Burgers equation (see [START_REF] Bec | Burgers turbulence[END_REF] for the physical predictions). Those estimates will be the subject of a future publication.

Notation and setup

Agreement: In the whole paper, all functions that we consider are real-valued.

1.1. Sobolev spaces. Consider a zero mean value integrable function v on S 1 . For p ∈ [1, ∞], we denote its L p norm by |v| p . The L 2 norm is denoted by |v|, and •, • stands for the L 2 scalar product. From now on L p , p ∈ [1, ∞], denotes the space of zero mean value functions in L p (S 1 ). Similarly, C ∞ is the space of C ∞ -smooth zero mean value functions on S 1 .

For a nonnegative integer m and p ∈ [1, ∞], W m,p stands for the Sobolev space of zero mean value functions v on S 1 with finite homogeneous norm

|v| m,p = d m v dx m p .
In particular, W 0,p = L p for p ∈ [1, ∞]. For p = 2, we denote W m,2 by H m and abbreviate the corresponding norm as v m . Note that since the length of S 1 is 1, we have

|v| 1 ≤ |v| ∞ ≤ |v| 1,1 ≤ |v| 1,∞ ≤ • • • ≤ |v| m,1 ≤ |v| m,∞ ≤ . . .
We recall a version of the classical Gagliardo-Nirenberg inequality (see [START_REF] Doering | Applied analysis of the Navier-Stokes equations[END_REF]Appendix]):

Lemma 1.1. For a smooth zero mean value function v on S 1 ,

|v| β,r ≤ C |v| θ m,p |v| 1-θ q ,
where m > β ≥ 0, and r is defined by

1 r = β -θ m - 1 p + (1 -θ) 1 q ,
under the assumption θ = β/m if p = 1 or p = ∞, and β/m ≤ θ < 1 otherwise. The constant C depends on m, p, q, β, θ.

For any s ≥ 0, H s stands for the Sobolev space of zero mean value functions v on S 1 with finite norm

v s = (2π) s k∈Z |k| 2s |v k | 2 1/2 , (6)
where vk are the complex Fourier coefficients of v(x). For an integer s = m, this norm coincides with the previously defined H m norm. For s ∈ (0, 1), v s is equivalent to the norm

v s = S 1 1 0 |v(x + ) -v(x)| 2 2s+1 d dx 1/2 (7)
(see [START_REF] Adams | Sobolev spaces[END_REF][START_REF] Taylor | Partial differential equations I: basic theory[END_REF]). Subindices t and x, which can be repeated, denote partial differentiation with respect to the corresponding variables. We denote by v (m) the m-th derivative of v in the variable x. For shortness, the function v(t, •) is denoted by v(t).

1.2. Random setting. We provide each space W m,p with the Borel σ-algebra. Then we consider an L 2 -valued Wiener process

w(t) = w ω (t), ω ∈ Ω, t ≥ 0,
defined on a complete probability space (Ω, F, P), and the corresponding filtration {F t , t ≥ 0}. We assume that for each m and each

t ≥ 0, w(t) ∈ H m , almost surely. That is, for ζ, χ ∈ L 2 , E( w(s), ζ w(t), χ ) = min(s, t) Qζ, χ ,
where Q is a symmetric operator which defines a continuous mapping Q : L 2 → H m for every m. Thus, w(t) ∈ C ∞ for every t, almost surely. From now on, we redefine the Wiener process so that this property holds for all ω ∈ Ω. We will denote w(t)(x) by w(t, x). For m ≥ 0, we denote by I m the quantity

I m = T r H m (Q) = E w(1) 2
m . For more details on Wiener processes in Hilbert spaces, see [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]Chapter 4] and [START_REF] Kuo | Gaussian measures in Banach spaces[END_REF].

For instance, we can consider the "diagonal" Wiener process:

w(t) = √ 2 k≤-1 b k w k (t) cos(2πkx) + √ 2 k≥1 b k w k (t) sin(2πkx),
where w k (t), k = 0, are standard independent Wiener processes and for every m ≥ 0,

I m = k≥1 b 2 k (2πk) 2m < ∞.
From now on, the term dw(s) denotes the stochastic differential corresponding to the Wiener process w(s) in the space L 2 . Now fix m ≥ 0. By Fernique's Theorem [START_REF] Kuo | Gaussian measures in Banach spaces[END_REF]Theorem 3.3.1], there exist λ m , C m > 0 such that

E exp λ m w(T ) 2 m /T ≤ C m , T ≥ 0. ( 8
)
Therefore by Doob's maximal inequality for infinite-dimensional submartingales [12, Theorem 3.8. (ii)] we have

E sup t∈[0,T ] w(t) p m ≤ p p -1 p E w(T ) p m < +∞, (9)
for any T > 0 and p ∈ (1, ∞). Moreover, applying Doob's maximal inequality to exp(α w(T ) m ) and maximising in α, we prove the existence of C m > 0 such that

P( sup t∈[0,T ] w(t) m ≥ λ) ≤ exp(-λ 2 /2C m T ), T, λ > 0. ( 10
)
Note that analogues of (9-10) still hold, uniformly in τ , if we replace sup t∈[0,T ] w(t) m by sup t∈[τ,T +τ ] w(t) -w(τ ) m .

1.3. Preliminaries. We begin by considering the free Burgers-type parabolic equation [START_REF] Adams | Sobolev spaces[END_REF]. Here, t ≥ 0, x ∈ S 1 = R/Z and the viscosity coefficient satisfies ν ∈ (0, 1]. The function f is C ∞ -smooth and strongly convex, i.e. it satisfies [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]. We also assume that its derivatives satisfy:

∀m ≥ 0, ∃h ≥ 0, C m > 0 : |f (m) (x)| ≤ C m (1 + |x|) h , x ∈ R, (11)
where h = h(m) is a function such that 1 ≤ h(1) < 2 (the lower bound on h(1) follows from (2)). The usual Burgers equation corresponds to

f (x) = x 2 /2.
The white-forced generalised Burgers equation is [START_REF] Bec | Burgers turbulence[END_REF] with η ω = ∂w ω /∂t, where w ω (t), t ≥ 0, is the Wiener process defined above. Definition 1.2. We say that an H 1 -valued process u(t, x) = u ω (t, x) is a solution of the equation

∂u ω ∂t + f (u ω ) ∂u ω ∂x -ν ∂ 2 u ω ∂x 2 = η ω ( 12
)
for t ≥ T if: (i) For every t ≥ T , ω → u ω (t, •) is F t -measurable.
(ii) For every ω and for t ≥ T , t → u ω (t, •) is continuous in H 1 and satisfies

u ω (t) =u ω (T ) - t T νLu ω (s) + 1 2 B(u ω )(s) ds + w ω (t) -w ω (T ), (13)
where B(u) = 2f (u)u x ; L = -∂ xx . For shortness, solutions for t ≥ 0 will be referred to as solutions.

When studying solutions of (12), we always assume that the initial condition u T = u(T, •) is F T -measurable and (except in Section 5) that T = 0 and the initial condition is C ∞ -smooth. For a given u T , (12) has a unique solution, i.e. any two solutions coincide for every ω. For shortness, this solution will be denoted by u. This is proved using a straightforward modification of the arguments in [START_REF] Da Prato | Ergodicity for infinite dimensional systems[END_REF]Chapter 14].

Since the forcing and the initial condition are smooth in space, the mapping t → u(t) is time-continuous in H m for every m, and t → u(t) -w(t) has a space derivative in C ∞ for all t. In this paper, we always assume that u T satisfies (3). Consequently, since the mean value of w(t) vanishes identically, u(t) also satisfies (3) for all times.

Solutions of ( 12) make a time-continuous Markov process in H 1 . For details, we refer to [START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF], where a white force is introduced in a similar setting.

Now consider, for a solution u(t, x) of ( 12), the functional G m (u(t)) = u(t) 2 m and apply Itô's formula [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]Theorem 4.17] to [START_REF] Da Prato | Ergodicity for infinite dimensional systems[END_REF]:

u(t) 2 m = u T 2 m - t T 2ν u(s) 2 m+1 + L m u(s), B(u)(s) ds + (t -T )I m + 2 T t L m u(s), dw(s) (14)
(we recall that

I m = T r(Q m ).) Consequently, d dt E u(t) 2 m = -2νE u(t) 2 m+1 -E L m u(t), B(u)(t) + I m . (15) As u, B(u) = 0, for m = 0 this relation becomes d dt E |u(t)| 2 = I 0 -2νE u(t) 2 1 . (16

Y.

In particular, X Y and X ∼ Y mean that X ≤ CY and C -1 Y ≤ X ≤ CY , respectively. All constants are independent of the viscosity ν and of the initial value u 0 .

We denote by u = u(t, x) a solution of ( 12) with an initial condition u 0 . For simplicity, in Sections 3-4, we assume that u 0 is deterministic. However, we can easily generalise all results to the case of an F 0 -measurable random initial condition independent of w(t), t ≥ 0. Indeed, for any measurable functional Φ(u(•)) we have

EΦ(u(•)) = E Φ(u(•))|u(0) = u 0 µ(du 0 ),
where µ(u 0 ) is the law of u 0 , and all estimates in Sections 3-4 hold uniformly in u 0 .

Moreover, for τ ≥ 0 and u 0 independent of w(t) -w(τ ), t ≥ τ , the Markov property yields:

EΦ(u(•)) = E Φ(u(τ + •))|u(τ ) = u 0 µ(du 0 ).
Consequently, all estimates which hold for time t or a time interval [t, t + T ] for solutions u(t) to [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] actually hold for time t + τ or a time interval [t + τ, t + τ + T ] for u(t) which solves [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] for t ≥ τ , uniformly in τ ≥ 0.

We use the notation g -= max(-g, 0) and g + = max(g, 0). For T 2 > T 1 ≥ 0 and a Sobolev space W m,p , we denote by C(T 1 , T 2 ; W m,p ) the space of continuous functions v from [T 1 , T 2 ] to W m,p equipped with the norm sup s∈[T 1 ,T 2 ] |v(s)| m,p . 1.5. Setting and notation in Section 4. For an observable A, i.e. a real-valued functional on a Sobolev space H m , which we evaluate on the solutions u ω (s), we denote by {A} the average of A(u ω (s)) in ensemble and in time over [t, t + T 0 ]:

{A} = 1 T 0 t+T 0 t EA(u ω (s))ds, t ≥ T 1 = T 0 + 2.
The constant T 0 is the same as in Theorem 3.16.

In this section, we assume that ν ≤ ν 0 , where ν 0 is a positive constant. Next, we define the intervals

J 1 = (0, C 1 ν]; J 2 = (C 1 ν, C 2 ]; J 3 = (C 2 , 1]. ( 17
)
In other words,

J 1 = { : 0 < ν}, J 2 = { : ν 1}, J 3 = { : ∼ 1}. For the values of ν 0 , C 1 and C 2 , see (52).
In terms of the Kolmogorov 1941 theory [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF], the interval J 1 corresponds to the dissipation range, i.e. for the Fourier modes k such that |k| -1 C 1 ν, {|û k | 2 } decreases super-algebraically in k. The interval J 2 corresponds to the inertial range, where layer-averaged quantities such as the energy spectrum E(k) defined by:

E(k) = |n|∈[M -1 k,M k] |û n | 2 |n|∈[M -1 k,M k] 1 (18)
behave as a negative degree of k. Here M ≥ 1 is a large enough constant (cf. the proof of Theorem 4.14). The boundary C 1 ν between these two ranges is the dissipation length scale. Finally, the interval J 3 corresponds to the energy range, i.e. the sum Σ{|û k | 2 } is mostly supported by the Fourier modes corresponding to |k| -1 ∈ J 3 . Actually the positive constants C 1 and C 2 can take any value, provided

C 1 ≤ 1 4 K -2 ; 5K 2 ≤ C 1 C 2 < 1 ν 0 . ( 19
)
Here, K is a positive constant, chosen in (51). Note that the intervals defined by ( 17) are non-empty and do not intersect each other for all values of ν ∈ (0, ν 0 ], under the assumption [START_REF] Gomes | Viscosity limit of stationary distributions for the random forced Burgers equation[END_REF]. By Theorem 3.16 we have {|u| 2 } ∼ 1 and (after integration by parts)

{|û n | 2 } ≤ {|u| 2 1,1 }/(2πn) 2 ∼ 1/n 2 .
We recall that we denote by ûn the complex Fourier coefficients of u. Thus, the ratio

Σ |n| -1 ∈J 3 |û n | 2 Σ n∈Z |û n | 2
tends to 1 as C 2 tends to 0, uniformly in ν. Since there exist couples (C 1 , C 2 ) satisfying [START_REF] Gomes | Viscosity limit of stationary distributions for the random forced Burgers equation[END_REF] such that C 2 is as small as desired, we may for instance assume that

|n|<C -1 2 {|û n | 2 } ≥ 99 100 n∈Z {|û n | 2 }.
For p, α ≥ 0, we consider the quantity

S p,α ( ) = S 1 |u(x + ) -u(x)| p dx α .
The quantity S p,1 ( ) is denoted by S p ( ): it corresponds to the structure function of p-th order, while the flatness F ( ), given by

F ( ) = S 4 ( )/S 2 2 ( ), (20)
measures spatial intermittency (see [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]).

Main results

In Section 3, we prove sharp upper and lower estimates for a large class of Sobolev norms of u. A key result is proved in Theorem 3.1. Namely, there we obtain that for k ≥ 1,

E max s∈[t,t+1] max x∈S 1 u x (s, x) k k 1, t ≥ 1. ( 21
)
The main estimates are those in the first part of Theorem 3.16. There we prove that for m ∈ {0, 1} and

p ∈ [1, ∞] or for m ≥ 2 and p ∈ (1, ∞], 1
T t+T t E |u(s)| α m,p 1/α m,p,α ∼ ν -γ , α > 0, t ≥ T 0 + 2, T ≥ T 0 , (22)
where γ = max(0, m -1/p), and T 0 is a constant.

In Section 4 we assume that ν ∈ (0, ν 0 ], where ν 0 ∈ (0, 1] is a constant. Then, we obtain sharp estimates for analogues of quantities characterising hydrodynamical turbulence. Although we only prove results for quantities averaged over a time period of length T 0 , those results can be immediately extended to quantities averaged over time periods of length T ≥ T 0 .

As the first application of estimates [START_REF] Iturriaga | Burgers turbulence and random Lagrangian systems[END_REF][START_REF] Kida | Asymptotic properties of Burgers turbulence[END_REF], in Section 4 we obtain sharp estimates for the quantities S p,α , α ≥ 0. Namely, by Theorem 4.10, for ∈ J 1 :

S p,α ( ) p,α ∼ αp , 0 ≤ p ≤ 1. αp ν -α(p-1) , p ≥ 1,
and on the other hand for ∈ J 2 :

S p,α ( ) p,α ∼ αp , 0 ≤ p ≤ 1. α , p ≥ 1.
Consequently, for ∈ J 2 the flatness function F ( ) = S 4 ( )/S 2 2 ( ) satisfies F ( ) ∼ -1 . Thus, solutions u are highly intermittent in the inertial range (see [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]).

On the other hand, we obtain estimates for the spectral asymptotics of Burgulence. Namely, for all m ≥ 1 and k ∈ Z, k = 0 we have:

{|û k | 2 } m (kν) -2m ν,
and by Theorem 4.14 and Remark 4.15 for k such that k -1 ∈ J 2 we have:

|n|∈[M -1 k,M k] |û n | 2 |n|∈[M -1 k,M k] 1 α α ∼ k -2α , α > 0,
for large enough values of M > 1. In particular, in the inertial range the energy spectrum satisfies

E(k) ∼ k -2 .
Finally, in Section 5, we prove that (12) admits a unique stationary measure µ. Consequently, all upper and lower estimates listed above still hold if we redefine the brackets as averaging with respect to µ, i.e.

{f (u)} = f (u)µ(du).
Moreover, as t → +∞, the rate of convergence to µ in the Lipschitzdual distance for Borel probability measures on L 1 is at least of the form Ct -1/13 , where C does not depend on the initial condition or on the viscosity ν.

Estimates for Sobolev norms

3.1. Upper estimates. The following theorem is proved using a stochastic version of the Kruzhkov maximum principle (cf. [START_REF] Kruzhkov | The Cauchy Problem in the large for nonlinear equations and for certain quasilinear systems of the first-order with several variables[END_REF]).

Theorem 3.1. Denote by X t the random variable

X t = max s∈[t,t+1] max x∈S 1 u x (s, x). For every k ≥ 1, we have E X k t k 1, t ≥ 1.
Proof. We take t = 1, denoting X t by X: the general case follows by the argument exposed in Subsection 1.4.

Consider the equation ( 12) on the time interval [0, 2]. Putting v = u -w and differentiating once in space, we get

∂v x ∂t + f (u)(v x + w x ) 2 + f (u)(v x + w x ) x = ν(v x + w x ) xx . (23)
Consider ṽ(t, x) = tv x (t, x) and multiply (23) by t 2 . For t > 0, ṽ satisfies

tṽ t -ṽ + f (u)(ṽ + tw x ) 2 + tf (u)ṽ x + t 2 f (u)w xx = νtṽ xx + νt 2 w xxx . ( 24
)
Now observe that if the zero mean function ṽ does not vanish identically on the domain S = [0, 2] × S 1 , then it attains its positive maximum N on S at a point (t 1 , x 1 ) such that t 1 > 0. At (t 1 , x 1 ) we have ṽt ≥ 0, ṽx = 0, and ṽxx ≤ 0. By [START_REF] Kraichnan | Lagrangian-history statistical theory for Burgers' equation[END_REF], at (t 1 , x 1 ) we have the inequality

f (u)(ṽ + tw x ) 2 ≤ ṽ -t 2 f (u)w xx + νt 2 w xxx . ( 25
)
Denote by A the random variable

A = max t∈[0,2] |w(t)| 3,∞ .
Since for every t, tv(t) is the zero space average primitive of ṽ(t) on S 1 , we get

max t∈[0,2], x∈S 1 |tu| ≤ max t∈[0,2], x∈S 1 (|tv| + |tw|) ≤ N + 2 max t∈[0,2] |w(t)| ∞ ≤ N + 2A.

Now denote by δ the quantity

δ = 2 -h(1). (26)
(cf. ( 11)). Since δ > 0, we obtain that

max t∈[0,2], x∈S 1 |t 2 f (u)w xx | ≤ A max t∈[0,2], x∈S 1 t δ |t 2-δ f (u)| ≤ A max t∈[0,2], x∈S 1 t δ (|tu| + t) 2-δ ≤ CA(N + 2A + 2) 2-δ .
From now on, we assume that N ≥ 2A. Since ν ∈ (0, 1] and f ≥ σ, the relation [START_REF] Kruzhkov | The Cauchy Problem in the large for nonlinear equations and for certain quasilinear systems of the first-order with several variables[END_REF] yields

σ(N -2A) 2 ≤ N + CA(N + 2A + 2) 2-δ + 4A.
Thus we have proved that if N ≥ 2A, then N ≤ C(A + 1) 1/δ . Since by [START_REF] Burgers | The nonlinear diffusion equation: asymptotic solutions and statistical problems[END_REF], all moments of A are finite, all moments of N are also finite. By definition of ṽ and S, the same is true for X. This proves the theorem's assertion.

Remark 3.2. Actually, using [START_REF] Chorin | Lectures on turbulence theory[END_REF], we can prove that there exist β, β > 0 such that

E exp(βX 2δ t ) ≤ E exp β ( max t∈[0,2] |w(t)| 3,∞ + 1) 2 1, t ≥ 1. Corollary 3.3. For k ≥ 1, E max s∈[t,t+1] |u(s)| k 1,1 k 1, t ≥ 1.
Proof. The space average of u x (s) vanishes identically. Therefore

S 1 |u x (s)| = 2 S 1 (u x (s)) + ≤ 2 max x∈S 1 u x (s, x). Corollary 3.4. For k ≥ 1, E max s∈[t,t+1] |u(s)| k p k 1, p ∈ [1, ∞], t ≥ 1.
Now we recall a standard estimate of the nonlinearity L m u, B(u) (see Subsection 1.3 for the definitions of L and B). Lemma 3.5. For every m ≥ 1 there exist C m > 0 and a natural number

n = n (m) such that for w ∈ C ∞ , N m (w) = | L m w, B(w) | ≤ C m (1 + |w| ∞ ) n w m w m+1 . ( 27
)
Proof. Fix m ≥ 1. Denote |w| ∞ by N . Let C denote various expressions of the form C m (1 + N ) n(m) . We have N m (w) = 2 w (2m) , (f (w)) (1) = 2 w (m+1) , (f (w)) (m) ≤C(m) m k=1 1≤a 1 ≤•••≤a k ≤m a 1 +•••+a k =m S 1 w (m+1) w (a 1 ) . . . w (a k ) f (k) (w) ≤C(m) max x∈[-N,N ] max(f (x), . . . f (m) (x)) × m k=1 1≤a 1 ≤•••≤a k ≤m a 1 +•••+a k =m S 1 |w (a 1 ) . . . w (a k ) w (m+1) |.
Using first [START_REF] Cole | On a quasilinear parabolic equation occurring in aerodynamics[END_REF], then Hölder's inequality, and finally Lemma 1.1, we get

N m (w) ≤C(m)(1 + N ) max(h(1),...,h(m)) × m k=1 1≤a 1 ≤•••≤a k ≤m a 1 +•••+a k =m S 1 |w (a 1 ) . . . w (a k ) w (m+1) | ≤C m k=1 1≤a 1 ≤•••≤a k ≤m a 1 +•••+a k =m w (a 1 ) 2m/a 1 . . . w (a k ) 2m/a k w m+1 ≤C w m+1 m k=1 1≤a 1 ≤•••≤a k ≤m a 1 +•••+a k =m ( w a 1 /m m |w| (m-a 1 )/m ∞ ) × . . . • • • × ( w a k /m m |w| (m-a k )/m ∞ ) ≤C (1 + N ) m-1 w m w m+1 = C w m w m+1 . Lemma 3.6. For m ≥ 1, E u(t) 2 m m ν -(2m-1) , t ≥ 2.
Proof. Fix m ≥ 1. We will use the notation

x(s) = E u(s) 2 m ; y(s) = E u(s) 2 m+1 .
As previously, it suffices to consider the case t = 2. We claim that for s ∈ [START_REF] Adams | Sobolev spaces[END_REF][START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF] we have the implication

x(s) ≥ C ν -(2m-1) =⇒ d ds x(s) ≤ -(2m -1)(x(s)) 2m/(2m-1) , (28)
where C ≥ 1 is a fixed number, chosen later. Below, all constants denoted by C are positive and do not depend on C , and we denote by 1) . Indeed, assume that x(s) ≥ Z. By [START_REF] Doering | Applied analysis of the Navier-Stokes equations[END_REF] and Lemma 3.5, we have

Z the quantity Z = C ν -(2m-
d ds x(s) ≤ -2νy(s) + CE (1 + |u(s)| ∞ ) n u(s) m u(s) m+1 + I m ,
with n = n (m). Since by Lemma 1.1 applied to u x , we get

u(s) m ≤ C u(s) (2m-1)/(2m+1) m+1 |u(s)| 2/(2m+1) 1,1 , (29)
we obtain that

d ds x(s) ≤ -2νy(s) + CE (1 + |u(s)| 1,1 ) n +2/(2m+1) × u(s) 4m/(2m+1) m+1 + I m .
Thus by Hölder's inequality and Corollary 3.3 we get d ds x(s) ≤ -2ν(y(s)) 1/(2m+1) + C (y(s)) 2m/(2m+1) + I m .

On the other hand, (29), Hölder's inequality and Corollary 3.3 yield

x(s) ≤ C(y(s)) (2m-1)/(2m+1) (E|u(s)| 2 1,1 ) 2/(2m+1) ≤C(y(s)) (2m-1)/(2m+1) ,
and thus (y(s)) 1/(2m+1) ≥ C(x(s)) 1/(2m-1) .

Consequently, since x(s) ≥ C ν -(2m-1) , for C large enough we have

d ds x(s) ≤ -CC 1/(2m-1) + C (x(s)) 2m/(2m-1) + I m .
Thus we can choose C in such a way that (28) holds. Now we claim that x(2) ≤ Z.

(30) Indeed, if x(s) ≤ Z for some s ∈ [START_REF] Adams | Sobolev spaces[END_REF][START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF], then the assertion [START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF] ensures that x(s) remains below this threshold up to s = 2: thus we have proved [START_REF] Landis | Second Order Equations of Elliptic and Parabolic Type[END_REF]. Now, assume that x(s) > Z for all s ∈ [START_REF] Adams | Sobolev spaces[END_REF][START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]. Denote

x(s) = (x(s)) -1/(2m-1) , s ∈ [1, 2] .
Using the implication [START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF] we get dx(s)/ds ≥ 1. Therefore x(2) ≥ 1.

As ν ≤ 1 and C ≥ 1, we get x(2) ≤ Z. Thus in both cases the inequality [START_REF] Landis | Second Order Equations of Elliptic and Parabolic Type[END_REF] holds. This proves the lemma's assertion.

Corollary 3.7. For m ≥ 1, E u(t) k m m,k ν -k(2m-1)/2 , k ≥ 1, t ≥ 2.
Proof. The cases k = 1, 2 follow immediately from Lemma 3.6.

For k ≥ 3, we consider only the case when k is odd, since the general case follows by Hölder's inequality. Setting N = ((2m -1)k + 1)/2 and applying Lemma 1.1, we get

u(t) k m m,k u(t) N |u(t)| k-1 1,1 .
Therefore, by Hölder's inequality, Lemma 3.6 and Corollary 3.3 we get

E u(t) k m m,k (E u(t) 2 N ) 1/2 (E |u(t)| 2k-2 1,1 ) 1/2 m,k ν -(N -1/2) = ν -k(2m-1)/2 . Lemma 3.8. For m ≥ 1, E max s∈[t,t+1] u(s) 2 m m ν -(2m-1) , t ≥ 2.
Proof. We begin by fixing m ≥ 1. As previously, we can take t = 2. In this proof, the random variables Θ i , i ∈ [START_REF] Adams | Sobolev spaces[END_REF][START_REF] Bec | Burgers turbulence[END_REF] are positive and have finite moments. All constants denoted by C are positive and only depend on m. We denote w(t) -w(2) by w(t), and u(t) -w(t) by ũ(t). By [START_REF] Burgers | The nonlinear diffusion equation: asymptotic solutions and statistical problems[END_REF], it follows that it suffices to prove the result with u replaced by ũ. By [START_REF] Da Prato | Ergodicity for infinite dimensional systems[END_REF], for s ≥ 2 we have

ũ(s) 2 m = ũ(2) 2 m - s 2 L m ũ(s ), 2νLu(s ) + B(u(s )) ds = ũ(2) 2 m - s 2 L m u(s ), 2νLu(s ) + B(u(s )) ds + s 2 L m w(s ), 2νLu(s ) + B(u(s )) ds = ũ(2) 2 m - s 2 2ν u(s ) 2 m+1 + L m u(s ), B(u(s )) ds (31)
+ s 2 2ν L m+1 w(s ), u(s ) -2 L m w (s ), f (u(s )) ds . (32)
Let

Θ 1 = 1 + max s ∈[2,3] |u(s )| 1,1 + max s ∈[2,3], x∈S 1 |f (u(s , x))|; Θ 2 = 1 + max s ∈[2,3] | w(s )| 2m+2,1 ; Θ 3 = ν (2m-1) ũ(2) 2 m . (33)
Using Corollary 3.3, Corollary 3.4 and ( 11), we obtain that the random variable Θ 1 has all moments finite. Finiteness of moments for Θ 2 follows from [START_REF] Burgers | The nonlinear diffusion equation: asymptotic solutions and statistical problems[END_REF]. Finally, finiteness of moments for Θ 3 follows from Lemma 3.6, since we have u(2) = ũ(2). Now denote by A 1 (s) and A 2 (s) the righthand sides of ( 31) and (32), respectively. As in the proof of Lemma 3.6, by Lemma 3.5 and Lemma 1.1 we get that for s ∈ [2, 3], we have respectively

| L m u(s), B(u(s)) | ≤ C(1 + |u(s)| ∞ ) n (m) u(s) m u(s) m+1 ≤ C(1 + |u(s)| 1,1 ) n (m) |u(s)| 2/(2m+1) 1,1 u(s) 4m/(2m+1) m+1 ≤ CΘ n (m)+2/(2m+1) 1 u(s) 4m/(2m+1) m+1 , (34)
and

u(s) 2 m+1 ≥ C|u(s)| -4/(2m-1) 1,1 u(s) (4m+2)/(2m-1) m . ( 35
)
Now we claim that there exists a positive random variable of the form

Θ 4 = CΘ a(m) 1 such that u(s) 2 m ≥ Θ 4 ν -(2m-1) =⇒ dA 1 (s) ds ≤ 0. ( 36
)
Indeed, by (35

), if u(s) 2 m ≥ Θ 4 ν -(2m-1) , then we have u(s) 2 m+1 ≥ CΘ -4/(2m-1) 1 Θ (2m+1)/(2m-1) 4 ν -(2m+1) ,
and therefore by (34) we get:

dA 1 (s) ds = -2ν u(s) 2 m+1 -L m u(s), B(u(s)) ≤ u(s) 4m/(2m+1) m+1 (-2ν u(s) 2/(2m+1) m+1 + CΘ n (m)+2/(2m+1) 1 )ν -2m ≤ u(s) 4m/(2m+1) m+1 (-CΘ -4/(4m 2 -1) 1 Θ 1/(2m-1) 4 + CΘ n (m)+2/(2m+1) 1
)ν -2m .

Moreover, if we define the random variable Θ 5 by

Θ 5 = ( Θ 4 + Θ 2 ) 2 ,
then we have

ũ(s) 2 m ≥ Θ 5 ν -(2m-1) =⇒ u(s) 2 m ≥ Θ 4 ν -(2m-1) . (37)
Indeed:

u(s) 2 m = ũ(s) + w(s) 2 m ≥ ũ(s) 2 m -w(s) m 2 .
Now consider the stopping time τ defined by

τ = {inf s ∈ [2, 3] : ũ(s) 2 m ≥ Θ 5 ν -(2m-
ũ(s) 2 m ≤ ũ(τ ) 2 m + max s∈[τ,3] A 2 (s) ≤ max(Θ 3 , Θ 5 )ν -(2m-1) + s ∈[2,3] dA 2 (s ) ds ds . (38)
To prove the lemma's assertion, it remains to observe that we have:

3 2 dA 2 (s ) ds ds ≤ 3 2 2ν | w(s )| 2m+2,1 |u(s )| ∞ +2 | w(s )| 2m+1,1 max x∈S 1 |f (u(s , x))| ds ≤CΘ 1 Θ 2 .
Repeating the proof of Corollary 3.7 we get that for m ≥ 1,

E max s∈[t,t+1] u(s) k m m,k ν -k(2m-1)/2 , k ≥ 1, t ≥ 2. ( 39
)
Denote γ = max(0, m -1/p).

Theorem 3.9. For m ∈ {0, 1} and p ∈ [1, ∞], or for m ≥ 2 and p ∈ (1, ∞],

E max s∈[t,t+1] |u(s)| α m,p 1/α m,p,α ν -γ , α > 0, t ≥ 2.
Proof. We consider only the case when α is an integer: the general case follows by Hölder's inequality.

For m ≥ 1 and p ∈ [2, ∞], we interpolate |u(s)| m,p between u(s) m and u(s) m+1 . By Lemma 1.1 we have

|u(s)| α m,p p ( u(s) α m ) 1-θ ( u(s) α m+1 ) θ , θ = 1 2 - 1 p .
Then we use (39) and Hölder's inequality to complete the proof. We use the same method to prove the case m = 1, p ∈ [1, 2], combining the inequality (39) and Corollary 3.3. We also proceed similarly for m ≥ 2, p ∈ (1, 2), combining Corollary 3.3 and an estimate for u α M,p for a large value of M and some p ≥ 2. Finally, the case m = 0 follows from Corollary 3.4.

Unfortunately, the proof of Theorem 3.9 cannot be adapted to the case m ≥ 2 and p = 1. Indeed, Lemma 1.1 only allows us to estimate a W m,1 norm from above by other W m,1 norms: we can only get that , for any λ > 0.

|w| m,1 m,n,k |w| (m-k)/(n-k) n,1 |w| (n-m)/(n-k) k,1 , 0 ≤ k < m < n,
3.2. Lower estimates. For a solution u(t) of ( 12), the first quantity that we estimate from below is the expected value of 1 T t+T t u(s) 2 1 , where t ≥ 1 and T > 0 is sufficiently large. Lemma 3.10. There exists a constant T 0 > 0 such that we have

1 T t+T t E u(s) 2 1 1/2 ν -1/2 , t ≥ 1, T ≥ T 0 .
Proof. For T > 0, by [START_REF] Weinan | Probability distribution functions for the random forced Burgers equation[END_REF] we get

E |u(t + T )| 2 ≥ E(|u(t + T )| 2 -|u(t)| 2 ) = T I 0 -2ν t+T t E u(s) 2 1 .
On the other hand, by Corollary 3.4 there exists a constant C > 0 such that E |u(t + T )| 2 ≤ C . Consequently, for T ≥ T 0 := (C + 1)/I 0 , 1

T t+T t E u(s) 2 1 ≥ T I 0 -C 2T ν -1 ≥ I 0 2(C + 1) ν -1 ,
which proves the lemma's assertion.

This time-averaged lower bound of the H 1 norm yields similar bounds of H m norms for m ≥ 2.

Lemma 3.11. For m ≥ 1, 1 T t+T t E u(s) 2 m 1/2 m ν -(m-1/2) , t ≥ 1, T ≥ T 0 .
Proof. Since the case m = 1 has been treated in the previous lemma, we may assume that m ≥ 2. By Lemma 1.1, we have 1) . Therefore by Hölder's inequality and Corollary 3.3 we get

u 2 1 u 2/(2m-1) m |u(s)| 2 1,1 ) (4m-4)/(2m-
(E u(s) 2 1 ) 2m-1 m (E u(s) 2 m )(E |u(s)| 2 1,1 ) 2m-2 m E u(s) 2 m . (40)
Integrating (40) in time, we get 1

T t+T t E u(s) 2 m m 1 T t+T t (E u(s) 2 1 ) 2m-1 m 1 T t+T t E u(s) 2 1 2m-1
.

Now the lemma's assertion follows from Lemma 3.10.

The following two results generalise Lemma 3.11. We recall that γ = max(0, m -1/p). Lemma 3.12. For m = 0 and p = ∞, or for m ≥ 1 and p ∈ [1, ∞],

1

T t+T t E |u(s)| 2 m,p 1/2 m,p ν -γ , t ≥ 2, T ≥ T 0 .
Proof. In the case m = 1, p ≥ 2, it suffices to apply Hölder's inequality in place of Lemma 1.1 in the proof of an analogue for Lemma 3.11.

In the case m ≥ 2, the proof is exactly the same as for Lemma 3.11 for p ∈ (1, ∞). In the cases p = 1, ∞, Lemma 1.

T t+T t E |u(s)| 2 1,p ≥ 1 T t+T t E u(s) 2 1 2/p × 1 T t+T t E |u(s)| 2 1,∞ (p-2)/p .
Using Lemma 3.10 and Theorem 3.9, we get the lemma's assertion. We proceed similarly for the case m = 0, p = ∞. Indeed, by Lemma 1.

1 we have |u(s)| 1,∞ ≤ C |u(s)| 1/2 ∞ |u(s)| 1/2
2,∞ . Thus, the lemma's assertion follows from Hölder's inequality, the case m = 1, p = ∞ and Theorem 3.9 (case m = 2, p = ∞). Lemma 3.13. For m = 0 and p = ∞, or for m ≥ 1 and p ∈ [1, ∞],

1

T t+T t E |u(s)| α m,p 1/α m,p,α ν -γ , α > 0, t ≥ 2, T ≥ T 0 .
Proof. As previously, we may assume that p > 1. The case α ≥ 2 follows immediately from Lemma 3.12 and Hölder's inequality. The case α < 2 follows from Hölder's inequality, the case α = 2 and Theorem 3.9 (case α = 3), since we have

1 T t+T t E |u(s)| α m,p ≥ 1 T t+T t E |u(s)| 2 m,p 3-α × 1 T t+T t E |u(s)| 3 m,p α-2
. Now we prove that for every p ∈ [1, ∞), in a certain sense, E|u| p is large if and only if E|u| ∞ is large. Lemma 3.14. For t ≥ 1, denote by A the quantity E|u(t)| 2 ∞ . Then there exists a constant C > 0 such that for p ∈ [1, ∞] we have

g(A) := min 3A 8 , 3A 2 16C ≤ E|u(t)| 2 p ≤ A.
Proof. We may take p = 1. Denote by l the quantity

l = min( A/2C , 1),
where C is the upper bound for E X 2 t in the statement of Theorem 3.1. Consider the random point x = x t where |u(t, •)| reaches its maximum. If this point is not unique, let x be the leftmost such point on S 1 considered as [0, 1). Let I be the interval [x, x + l] if u(t, x) < 0, and the interval [x -l, x] if u(t, x) ≥ 0, respectively. We have

E|u(t)| 2 1 ≥ E I |u(t, y)|dy 2 ≥ E l |u(t)| ∞ - l max x∈S 1 u x (t) 2 2 ≥ l 2 3 4 E|u(t)| 2 ∞ - 3l 2 4 E (max x∈S 1 u x (t)) 2 .
By definition of A, C and l, we get

E|u(t)| 2 1 ≥ l 2 3A 4 - 3l 2 C 4 ≥ 3l 2 A 8 = g(A).
Finally we prove the following uniform lower estimate.

Lemma 3.15. We have

E|u(t)| 2 p 1, t ≥ T 0 + 2, p ∈ [1, ∞].
Proof. We can take p = 2. Indeed, the case p ∈ (2, ∞] follows immediately from the case p = 2. On the other hand, the case p ∈ [1, 2) follows from Hölder's inequality, the case p = 2 and the upper estimate for E|u(t)| 2 ∞ in Theorem 3.9, in the same way as in the proof of Lemma 3.13.

Let C denote various positive constants. From Lemma 3.12 (case m = 0 and p = ∞), it follows that for some t in [

E|u(t)| 2 ≤ κ =⇒ d dt E|u(t)| 2 ≥ 0, t ≥ 2,
where κ is a fixed positive number, chosen later. If E|u(t)| 2 ≤ κ, then by Lemma 3.14, E|u(t)| 2 ∞ ≤ g-1 (κ). On the other hand, by Hölder's inequality and Lemma 1.1, we have

E u(t) 2 1 ≤ (E|u(t)| 2 1,∞ ) 1/2 (E|u(t)| 2 1,1 ) 1/2 ≤ C (E|u(t)| 2 ∞ ) 1/4 (E|u(t)| 2 2,∞ ) 1/4 (E|u(t)| 2 1,1 ) 1/2 .
Therefore, by Theorem 3.9, E u(t) 2 1 ≤ C (g -1 (κ)) 1/4 ν -1 , and thus by ( 16), we get:

d dt E|u(t)| 2 ≥ I 0 -2C (g -1 (κ)) 1/4 .
Since g-1 (κ) -→ κ→0 0, choosing κ small enough so that 2C (g -1 (κ))

Main theorem.

The following theorem sums up the main results of Section 3, with the exception of Theorem 3.1. We recall that γ = max(0, m -1/p). Theorem 3.16. For m ∈ {0, 1} and p ∈ [1, ∞], or for m ≥ 2 and p ∈ (1, ∞], we have

1 T t+T t E |u(s)| α m,p 1/α m,p,α ∼ ν -γ , α > 0, t ≥ T 1 = T 0 + 2, T ≥ T 0 . ( 41
)
Moreover, the upper estimates hold with time-averaging replaced by maximising over [t, t + 1] for t ≥ 2, i.e.

E max

s∈[t,t+1] |u(s)| α m,p 1/α m,p,α ν -γ , α > 0, t ≥ 2. ( 42
)
On the other hand, the lower estimates hold for all m ≥ 0 and p ∈ [1, ∞]. The asymptotics (41) hold without time-averaging if m and p are such that γ(m, p) = 0. Namely, in this case,

E |u(t)| α m,p 1/α m,p,α ∼ 1, α > 0, t ≥ T 1 . ( 43
)
Proof. The upper estimates for all cases, as well as the lower estimates in (41) for all cases and in (43) for the case α = 2, follow from the lemmas and theorems above. For α > 2, the lower estimates in (43) follow immediately from the lower estimates for α = 2. For α < 2, these estimates are obtained from Hölder's inequality, the lower estimates for α = 2 and the upper estimates for α = 3 in the same way as in the proof of Lemma 3.13. This theorem yields, for integers m ≥ 1, the relation

{ u 2 m } m ∼ ν -(2m-1) . ( 44
)
By a standard interpolation argument (see ( 6)) the upper bound in (44) also holds for non-integer indices s > 1. Actually, the same is true for the lower bound, since for any integer n > s we have 1) .

{ u 2 s } ≥ { u 2 n } n-s+1 { u 2 n+1 } -(n-s) s ν -(2s-
In all results in this section as well as in Section 4, the quantities estimated for a fixed trajectory of the noise, such as max s∈[t,t+1], x∈S 1 u ω x or maxima in time of Sobolev norms, can be replaced by their suprema over all smooth initial conditions (taken before considering the expected value). For instance, the quantity

E max s∈[t,t+1] |u ω (s)| α m,p can be replaced by E sup u 0 ∈C ∞ max s∈[t,t+1] |u ω (s)| α m,p .
For the lower estimates, this is obvious. For the upper ones, this follows form the following pathwise version of Theorem 3.9, and analogous pathwise versions of Theorem 3.1 and of the upper estimates in Section 4. To prove these statements, it suffices to recast the original proofs in a pathwise setting (i.e., to work for a fixed ω instead of using the expected values).

Theorem 3.17. For m ∈ {0, 1} and p ∈ [1, ∞], or for m ≥ 2 and p ∈ (1, ∞], there exist constants β(m, p), m (m, p) > 0 such that we have:

max s∈[t,t+1] |u ω (s)| m,p m,p (1 + max s∈[t-1,t+1] w ω (s) m ) β ν -γ , t ≥ 2, ω ∈ Ω. ( 45
)
On the other hand, in the results of this section and of Section 4 the expected values (and not the quantities themselves) can be replaced by their infima over all smooth initial conditions. For instance, the quantity E max

s∈[t,t+1] |u(s)| m,p can be replaced by inf u 0 ∈C ∞ E max s∈[t,t+1] |u(s)| m,p .

Estimates for small-scale quantities

In this section, we estimate small-scale quantities which characterise Burgulence in physical space (increments, flatness) as well as in Fourier space (energy spectrum). We fix t satisfying t ≥ T 1 . Its precise value is not important, since all estimates in Section 3 hold uniformly in t provided that t ≥ T 1 and the same is true for all estimates in this section. For the notation used here, see Subsection 1.5.

4.1.

Results in physical space. We begin by proving upper estimates for the functions S p,α ( ). In the proofs of the two following lemmas, constants denoted by C depend only on p, α.

Lemma 4.1. For α ≥ 0 and ∈ [0, 1], S p,α ( ) p,α αp , 0 ≤ p ≤ 1. αp ν -α(p-1) , p ≥ 1.
Proof. We begin by considering the case p ≥ 1. We have

S p,α ( ) = S 1 |u(x + ) -u(x)| p dx α ≤ max x |u(x + ) -u(x)| p-1 S 1 |u(x + ) -u(x)|dx α .
By Hölder's inequality we get

S p,α ( ) ≤ S 1 |u(x + ) -u(x)|dx αp 1/p × max x |u(x + ) -u(x)| αp (p-1)/p .
Since the space average of u(x + ) -u(x) vanishes, we obtain that

S p,α ( ) ≤ 2 S 1 (u(x + ) -u(x)) + dx αp 1/p × max x |u(x + ) -u(x)| αp (p-1)/p ≤C α max x |u(x + ) -u(x)| αp (p-1)/p , (46)
where the second inequality follows from Theorem 3.1. Finally, by Theorem 3.16 we get 1) .

S p,α ( ) ≤ C α ( |u| 1,∞ ) αp (p-1)/p ≤ C αp ν -α(p-
The case p < 1 follows immediately from the case p = 1 since now S p,α ( ) ≤ S 1,αp ( ), by Hölder's inequality.

For ∈ J 2 ∪ J 3 , we have a better upper bound if p ≥ 1.

Lemma 4.2. For α ≥ 0 and ∈ J 2 ∪ J 3 , S p,α ( )

p,α αp , 0 ≤ p ≤ 1. α , p ≥ 1.
Proof. The calculations are almost the same as in the previous lemma. The only difference is that we use another upper bound for the right-hand side of (46). Namely, we have

S p,α ( ) ≤ C α max x |u(x + ) -u(x)| αp (p-1)/p ≤ C α (2|u| ∞ ) αp (p-1)/p ≤ C α ,
where the third inequality follows from Theorem 3.16.

To prove lower estimates for S p,α ( ), we need a lemma. Loosely speaking, this lemma states that with a probability which is not too small, during a period of time which is not too small, several Sobolev norms are of the same order as their expected values. Note that in the following definition, (47-48) contain lower and upper estimates, while (49) only contains an upper estimate. The inequality |u(s)| ∞ ≤ max u x (s) in (47) always holds, since u(s) has zero mean value and the length of S 1 is 1. Definition 4.3. For a given solution u(s) = u ω (s) and K > 1, we denote by L K the set of all (s, ω)

∈ [t, t + T 0 ] × Ω such that K -1 ≤ |u(s)| ∞ ≤ max u x (s) ≤ K (47) K -1 ν -1 ≤ |u(s)| 1,∞ ≤ Kν -1 (48) |u(s)| 2,∞ ≤ Kν -2 . ( 49
) Lemma 4.4. There exist constants C, K 1 > 0 such that for all K ≥ K 1 , ρ(L K ) ≥ C.
Here, ρ denotes the product measure of the Lebesgue measure and P on [t, t + T 0 ] × Ω.

Proof. We denote by A K , B K and D K the set of (s, ω) satisfying "The upper estimates in (47-49) hold for a given value of K", "The lower estimates in (47-48) hold for a given value of K" and "The lower estimate in (48) holds for a given value of K", respectively. Note that for K ≤ K , L K ⊂ L K , and similarly for A K , B K and

D K . By Lemma 1.1 we get |u| ∞ ≥ C |u| -1 2,∞ |u| 2 1
,∞ for some constant C > 0. Thus, for K ≥ max(C , 1)K 3 , we have A K ∩ D K ⊂ B K , and therefore:

A K ∩ D K ⊂ A K ∩ B K = L K . Consequently: ρ(L K ) ≥ ρ(A K ) + ρ(D K ) -T 0
. By Theorem 3.1, Theorem 3.16 and Chebyshev's inequality, the measure of the set A K tends to T 0 as K tends to +∞. So to prove the lemma's assertion, it remains to show that there exists C > 0 such that for K large enough we have ρ(D K ) ≥ C. Using the upper estimate for {|u| 2 1,∞ } in Theorem 3.16, we get

{|u| 1,∞ 1(|u| 1,∞ ≥ Kν -1 )} ≤ CK -1 ν -1 .
Here, 1(A) denotes the indicator function of an event A. On the other hand, we clearly have

{|u| 1,∞ 1(|u| 1,∞ ≤ K -1 ν -1 )} ≤ K -1 ν -1 .
Now, for K 0 > 0, consider the function

g K 0 = |u| 1,∞ 1(K -1 0 ν -1 ≤ |u| 1,∞ ≤ K 0 ν -1
). The lower estimate for {|u| 1,∞ } in Theorem 3.16 and the relations above yield

{g K 0 } ≥ (C -CK -1 0 -K -1 0 )ν -1 ≥ C 0 ν -1 for some constant C 0 , uniformly for large enough values of K 0 . Since g K 0 ≤ K 0 ν -1 , we get ρ(g K 0 ≥ C 0 ν -1 /2) ≥ C 0 K -1 0 T 0 /2. Since g K 0 ≤ |u| 1,∞ , we obtain that ρ(|u| 1,∞ ≥ C 0 ν -1 /2) ≥ C 0 K -1 0 T 0 /2, which implies the existence of C , K > 0 such that ρ(D K ) ≥ C for K ≥ K .
Definition 4.5. For a given solution u(s) = u ω (s) and K > 1, we denote by O K the set of all (s, ω) ∈ [t, t+T 0 ]×Ω such that the conditions (47), (49) and

K -1 ν -1 ≤ -min u x ≤ Kν -1 (50) hold. Corollary 4.6. If K ≥ K 1 and ν < K -2 1 , then ρ(O K ) ≥ C.
Here, C, K 1 are the same as in the statement of Lemma 4.4.

Proof. For K = K 1 and ν < K -2 1 , the estimates (47-48) tell us that for (s, ω)

∈ L K , max u x (s) ≤ K 1 < K -1 1 ν -1 ≤ |u x (s)| ∞ .
Thus, in this case we have O K = L K , and therefore

ρ(O K ) = ρ(L K ) ≥ C0 .
Finally, we observe that since increasing K while keeping ν constant increases the measure of O K , the corollary's statement still holds for K ≥ K 1 and ν < K -2

1 . Now we fix

K = K 1 , (51)
and choose

ν 0 = 1 6 K -2 ; C 1 = 1 4 K -2 ; C 2 = 1 20 K -4 . (52)
In particular, we have 0 < C 1 ν 0 < C 2 < 1: thus the intervals J i are non-empty and non-intersecting for all ν ∈ (0, ν 0 ]. αp ν -α(p-1) , p ≥ 1.

Proof. By Corollary 4.6, it suffices to prove that the inequalities hold uniformly for (s, ω) ∈ O K with S p,α ( ) replaced by

S 1 |u(x + ) -u(x)| p dx α .
For α = 1, this fact follows from the case α = 1. Indeed, if for (s, ω) ∈ O K , we have

S 1 |u(x + ) -u(x)| p dx p p (resp. p ν -(p-1)
), then we also have

S 1 |u(x + ) -u(x)| p dx α p,α αp (resp. αp ν -α(p-1) ).
Till the end of the proof we assume that (s, ω) ∈ O K .

Case p ≥ 1, α = 1. Denote by z the leftmost point on S 1 (considered as [0, 1)) such that u (z) ≤ -K -1 ν -1 . Since |u| 2,∞ ≤ Kν -2 , we have

u (y) ≤ - 1 2 K -1 ν -1 , y ∈ [z - 1 2 K -2 ν, z + 1 2 K -2 ν]. (53)
Since ≤ C 1 ν = 1 4 K -2 ν, by Hölder's inequality we get

S 1 |u(x + ) -u(x)| p dx ≥ z+ 1 4 K -2 ν z-1 4 K -2 ν |u(x + ) -u(x)| p dx ≥ (K -2 ν/2) 1-p z+ 1 4 K -2 ν z-1 4 K -2 ν |u(x + ) -u(x)|dx p = C(p)ν 1-p z+ 1 4 K -2 ν z-1 4 K -2 ν x+ x -u (y)dy dx p ≥ C(p)ν 1-p z+ 1 4 K -2 ν z-1 4 K -2 ν 1 2 K -1 ν -1 dx p = C(p)ν 1-p p .
Case p < 1, α = 1. By Hölder's inequality we get

S 1 |u(x + ) -u(x)| p dx ≥ S 1 (u(x + ) -u(x)) + p dx ≥ S 1 (u(x + ) -u(x)) + 2 dx p-1 S 1 (u(x + ) -u(x)) + dx 2-p .
Using the upper estimate in (47) we get

S 1 |u(x + ) -u(x)| p dx ≥ S 1 2 K 2 dx p-1 S 1 (u(x + ) -u(x)) + dx 2-p .
Finally, since S 1 (u(• + ) -u(•)) = 0, we obtain that

S 1 |u(x + ) -u(x)| p dx ≥ C(p) 2(p-1) 1 2 S 1 |u(x + ) -u(x)|dx 2-p ≥ C(p) p .
The last inequality follows from the case p = 1, α = 1.

Remark 4.8. To prove this lemma, we do not need Corollary 4.6. Indeed, in its proof we could have considered z such that |u (z)| ≥ K -1 ν -1 : Lemma 4.4 guarantees its existence.

The proof of the following lemma uses an argument from [START_REF] Aurell | On the multifractal properties of the energy dissipation derived from turbulence data[END_REF], which can be made rigorous if we restrict ourselves to the set O K . Lemma 4.9. For α ≥ 0 and ∈ J 2 , S p,α ( )

p,α αp , 0 ≤ p ≤ 1. α , p ≥ 1.
Proof. For the same reason as in the previous proof, it suffices to prove that as long as (s, ω) belongs to O K , the inequalities hold uniformly for α = 1 and for S p,α ( ) replaced by

S 1 |u(x + ) -u(x)| p dx.
Once again, till the end of the proof we assume that (s, ω) ∈ O K .

Case p ≥ 1, α = 1. Defining z in the same way as previously, we have: .

S 1 |u(x + ) -u(x)| p dx ≥
On the other hand, since ≤ C 2 , using the upper estimate in (47) we get

x+ x u + (y)dy ≤ C 2 K ≤ 1 20 K -3 .
Thus,

S 1 |u(x + ) -u(x)| p dx ≥ 1 2 1 16 - 1 20 K -3 p ≥ C(p) .
Case p < 1, α = 1. The result follows from the case p = 1, α = 1 in exactly the same way as in the previous lemma.

Summing up the results above we obtain the following theorem. On the other hand, for s ∈ (1/2, 1), 1) .

{ u 2 s } s ∼ ν -(2s-
The results above and the relation (44) tell us that {|û k | 2 } decreases very fast for |k| ν -1 , and that for s ≥ 0 the sums |k| 2s {|û k | 2 } have exactly the same behaviour as the partial sums |k|≤ν -1 |k| 2s |k| -2 in the limit ν → 0 + . Therefore we can conjecture that for |k| ν -1 , we have

{|û k | 2 } ∼ |k| -2 .
A result of this type actually holds (after layer-averaging), as long as |k| is not too small. To prove it, we use a version of the Wiener-Khinchin theorem, stating that for any function v ∈ L 2 one has

|v(• + y) -v(•)| 2 = 4 n∈Z sin 2 (πny)|v n | 2 . ( 54
)
Theorem 4.14. If M in the definition ( 18) of E(k) is large enough, then for every k such that k -1 ∈ J 2 , we have

E(k) ∼ k -2 .
Proof. We recall that by definition,

E(k) = |n|∈[M -1 k,M k] |û n | 2 |n|∈[M -1 k,M k] 1
.

Therefore proving the assertion of the theorem is the same as proving that

|n|∈[M -1 k,M k] n 2 {|û n | 2 } ∼ k. (55)
The upper estimate is an immediate corollary of the upper estimate for |u| 1,1 in Theorem 3.16 and holds without averaging over n such that |n| ∈ [M -1 k, M k]. Indeed, integrating by parts we get

{|û n | 2 } ≤ (2πn) -2 {|u x | 2 1 } ≤ Cn -2
, which proves the upper bound. Also, this inequality implies that

|n|<M -1 k n 2 {|û n | 2 } ≤ CM -1 k (56) and |n|>M k {|û n | 2 } ≤ CM -1 k -1 . (57)
To prove the lower bound we note that

|n|≤M k n 2 {|û n | 2 } ≥ k 2 π 2 |n|≤M k sin 2 (πnk -1 ){|û n | 2 } ≥ k 2 π 2 n∈Z sin 2 (πnk -1 ){|û n | 2 } - |n|>M k {|û n | 2 } .
Using ( 54) and (57) we get

|n|≤M k n 2 {|û n | 2 } ≥ k 2 4π 2 {|u(• + k -1 ) -u(•)| 2 } -CM -1 k -1 ≥ k 2 4π 2 (S 2 (k -1 ) -CM -1 k -1
). Finally, using Theorem 4.10 we obtain that

|n|≤M k n 2 {|û n | 2 } ≥ (C -CM -1 )k.
Now we use (56) and we choose M ≥ 1 large enough to obtain (55). Remark 4.15. We actually have

|n|∈[M -1 k,M k] |û n | 2 |n|∈[M -1 k,M k] 1 α α ∼ k -2α , α > 0.
The upper bound is proved in the same way as above, and then the lower bound follows from Hölder's inequality and the lower bound in Theorem 4.14.

Stationary measure and related issues

A contraction property.

Contraction properties for solutions of scalar conservation laws have been known to hold since the works of Oleinik and Kruzhkov (cf. [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF] and references therein). In the space-periodic setting, we have the following contraction property in L 1 .

Theorem 5.1. Consider two solutions u, u of (12), corresponding to the same realisation of the random force but different initial conditions u 0 , u 0 in C ∞ . For all t ≥ s ≥ 0, we have

|u(t) -u(t)| 1 ≤ |u(s) -u(s)| 1 .
Proof. We only consider the case s = 0: the general case is proved in exactly the same way. Consider the function v = u -u and define

Φ(t, x) = f (u(t, x)) -f (u(t, x)) u(t, x) -u(t, x) .
Since f is C ∞ -smooth and u, u are continuous in time and C ∞ -smooth in space, by Hadamard's lemma Φ is continuous in time and C ∞smooth in space. The function v is a weak solution of the equation

v t + (Φv) x = νv xx , v(0) = v 0 = u 0 -u 0 , 0 ≤ t ≤ T. (58)
Moreover, since u t -w t and u t -w t are C ∞ -smooth in space, the same is true for v t . Consequently, v is the classical solution of (58). Now we consider the dual parabolic problem

h t + Φh x = -νh xx , h(T, x) = h T (x), 0 ≤ t ≤ T. (59)
For a C ∞ -smooth final condition h T , this problem has a unique classical solution h, C 1 -smooth in time and C ∞ -smooth in space [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]. Integrating by parts in time and in space, we get

v(T ), h T -v 0 , h(0) = T 0 v t (t), h(t) + v(t), h t (t) dt = T 0 -(Φ(t)v(t)) x + νv xx (t), h(t) dt + T 0 v(t), -Φ(t)h x (t) -νh xx (t) dt = 0. ( 60
)
Now we choose a sequence of C ∞ -smooth functions h n T , n ≥ 0, which approximate sgn(v(T )) pointwise and satisfy |h n T | ≤ 1. We consider the solution h n to the problem (59) for h T = h n T . By the maximum principle [START_REF] Landis | Second Order Equations of Elliptic and Parabolic Type[END_REF], we have |h n (t, x)| ≤ 1 for all t ∈ [0, T ], x ∈ S 1 . Now we pass to the limit as n → ∞. By (60), we get:

|v(T )| 1 = lim n→∞ v(T ), h n T = lim n→∞ v 0 , h n (0) ≤ |v 0 | 1 .
5.2. Setting and definitions. Since C ∞ is dense in L 1 , Theorem 5.1 allows us to extend the stochastic flow corresponding to [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] to the space L 1 . Indeed, consider any F 0 -measurable u 0 ∈ L 1 and approximate it in L 1 by a sequence of smooth functions u 0n , n ≥ 1. Let u ω n (t) be the solutions to the equation ( 12) with the corresponding initial data. By Theorem 5.1, for each ω the sequence {u ω n (t)} is fundamental in the space C(0, T ; L 1 ). Its limit u ω (t) does not depend on the sequence u 0n . We will call this limit the L 1 -solution of (12) corresponding to the initial condition u 0 . It is straightforward that Theorem 5.1 remains valid for L 1 -solutions.

• For any T > 0, we can pass to the limit n → ∞ in the relation [START_REF] Da Prato | Ergodicity for infinite dimensional systems[END_REF]. This proves that the L 1 -solutions u(t) are solutions to [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] for t ≥ T . • We can extend the results of Sections 3-4 to L 1 -solutions.

As in the case of smooth solutions, the L 1 -solutions of (12) form a continuous Markov process in the space L 1 . So they define a Markov semigroup S * t , acting on Borel measures on L 1 . Till the end of this section the L 1 -solutions to [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] will be referred to as solutions.

A stationary measure is a Borel probability measure on L 1 invariant by S * t for every t. A stationary solution of ( 12) is a random process v defined for (t, ω) ∈ [0, +∞) × Ω, valued in L 1 , which solves [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF], such that the distribution of v(t, •) does not depend on t. Such a distribution is automatically a stationary measure. Now we consider the question of existence and uniqueness of a stationary measure, which implies existence and uniqueness (in the sense of distributions) of a stationary solution. This fact has been proved in a slightly different setting: see [START_REF] Iturriaga | Burgers turbulence and random Lagrangian systems[END_REF] and references therein; see also [START_REF] Weinan | Invariant measures for Burgers equation with stochastic forcing[END_REF] for the proof in the case ν = 0. Moreover, we obtain a bound for the rate of convergence to the stationary measure in an appropriate distance. This bound does not depend on the viscosity or on the initial condition. Existence of a stationary measure for (12) can be proved using the Bogolyubov-Krylov argument (see [START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF]). Let us give a sketch of the proof.

Let u(s) be a solution of [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]. For s ≥ 1, E|u(s)| 1,1 is uniformly bounded. Since by Helly's selection principle [START_REF] Kolmogorov | Introductory real analysis[END_REF], W 1,1 is compactly On the other hand, for α ≥ 0 and ∈ J 2 , Theorem 5.9. For k such that k -1 ∈ J 2 , we have:

|n|∈[M -1 k,M k] |û n | 2 |n|∈[M -1 k,M k] 1 µ(du) ∼ k -2 .
5.3. Proof of Theorem 5.5. To begin with, we need an auxiliary lemma. The main idea of the proof is similar to that of Theorem 3.1: namely, if the white noise is small during a certain time, then the solution itself becomes small. The technique is also similar: we apply the maximum principle to a well-chosen function. We only give the proof for an initial condition in C ∞ : the general case follows as above by considering smooth approximations. (63)

In this subsection, from now on we denote by C various positive constants, independent of C.

Proof. Assume the converse. We abbreviate w(s) -w(t) as w(s) and we use the notation ṽ(s, x) = (s -t)(u x (s, x) -wx (s, x)); N = max s∈[t,t+τ ], x∈S 1 ṽ(s, x). (64) Since we assumed that (63) does not hold, we have

N > τ (τ -1/2 -K) > τ 1/2 /2. ( 65
)
Now consider a point (t 1 , x 1 ) at which the maximum N is achieved. In the same way as in the proof of Theorem 3.1, we show that at (t 1 , x 1 ) we have f (u)(ṽ + (t 1 -t) wx )

  and thus the upper estimates obtained above cannot be used. However, |u| m,1 ≤ |u| m,1+β for any β > 0. Consequently, the theorem's statement holds for m ≥ 2 and p = 1, with γ replaced by γ + λ, and m,p,α replaced by m,p,α,λ

Lemma 4 . 7 .

 47 For α ≥ 0 and ∈ J 1 , S p,α ( ) p,α αp , 0 ≤ p ≤ 1.

  We have ≥ C 1 ν = 1 4 K -2 ν. Thus, by (53), for x ∈ [z -

Theorem 4 . 10 .

 410 For α ≥ 0 and ∈ J 1 , S p,α ( ) p,α ∼ αp , 0 ≤ p ≤ 1.

  αp ν -α(p-1) , p ≥ 1.

Lemma 4 . 13 .

 413 For s ∈ (0, 1/2),

Definition 5 . 3 .Definition 5 . 4 . 1 g 1 g

 535411 Fix p ∈ [1, ∞). For a continuous real-valued function g on L p , we define its Lipschitz norm as |g| L(p) := sup Lp |g| + |g| Lip , where |g| Lip is the Lipschitz constant of g. The set of continous functions with finite Lipschitz norm will be denoted by L(p) = L(L p ). We will abbreviate L(1) as L. For two Borel probability measures µ 1 , µ 2 on L p , we denote by µ 1 -µ 2 * L(p) the Lipschitz-dual distance: µ 1 -µ 2 * L(p) := sup g∈L(p), |g| L(p) ≤1 S (v)µ 1 (dv) -S (v)µ 2 (dv) .

S 1 |u

 1 (x + ) -u(x)| p dx α µ(du) p,α ∼ αp , 0 ≤ p ≤ 1. α , p ≥ 1.

Lemma 5 . 10 .

 510 There exists a constant C ≥ 2 such that if τ ≥ C and if for some t ≥ 0 and ω ∈ Ω, the trajectory of the Wiener process w ω satisfiesK = max s∈[t,t+τ ] |w ω (s) -w ω (t)| 3,∞ ≤ τ -2 ,then the corresponding solution u ω (t, x) to (12) satisfies max x∈S 1 u x (t + τ, x) ≤ τ -1/2 .

  2, T 0 + 2] we have E|u( t)| 2 ∞ ≥ C . Then by Lemma 3.14 we get E|u( t)| 2 ≥ C . Thus it suffices to prove that

  1/4 ≤ I 0 proves the lemma's assertion. Since |u(t)| 1,1 ≥ |u(t)| ∞ , an analogue of Lemma 3.15 also holds for |u(t)| 1,1 .

  Now we denote by u(t), u(t) the solutions of[START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] corresponding respectively to the initial conditions u 0 , u 0 . By the definition of the Lipschitzdual distance, we haveS * t δ u 0 -S * t δ u 0 *To prove the theorem's statement, it suffices to obtain the inequalityP |u(n 13 ) -u(n 13 )| 1 >Here, t denotes the integer part of t, and the first inequality follows from Theorem 5.1. By Theorem 5.1, for every n ≥ 1 we haveP |u(n 13 ) -u(n 13 )| 1 > 2 n = P ∀k ∈ [1, n 11 ] : |u(kn 2 ) -u(kn 2 )| 1 > 2 n .The second inequality holds since the functions u(t, •) and u(t, •) have zero mean value. From Lemma 5.10, it follows that for n ≥ C1/2 we can only have maxx∈S 1 u x (kn 2 ) > 1 n or max x∈S 1 u x (kn 2 ) > 1 P |u(n 13 ) -u(n 13 )| 1 > 2 n ≤ P ∀k ∈ [1, n 11 ] : max t∈[(k-1)n 2 ,kn 2 ] |w(t) -w((k -1)n 2 )| 3,∞ > 1 n 4 .Since the increments of w on the time intervals [(k -1)n 2 , kn 2 ] are independent, we get that for n ≥ C1/2 :P |u(n 13 ) -u(n 13 )| 1 > 2 n

									n if
	L = t∈[(k-1)n 2 ,kn 2 ] max |w(t) -w((k -1)n 2 )| 3,∞ > sup g∈L, g L ≤1 E g(u(t)) -E g(u(t)) 1 n 4 ,
	and therefore we get:	≤ E		sup	g(u(t)) -g(u(t))
						g∈L, g L ≤1
				≤ E min(2, |u(t) -u(t)| 1 ) .	(67)
									2 n	≤	C n	(68)
	≤	P	max
	1≤k≤n 11
			2 n		≤ exp -	n -8 2C n 2	2 t	n 11	≤ e -C n ≤	C n	.
		≤	2 + 2 C t	≤	C t	,
	Thus,
	P |u(n 13 ) -u(n 13 )| 1 >	2 n
	≤ P ∀k ∈ [1, n 11 ] : |u(kn 2 )| 1 >	1 n	or |u(kn 2 )| 1 >	1 n

2

≤ ṽ -(t 1 -t) 2 f (u) wxx + ν(t 1 -t)

2

wxxx . (66) for large enough integers n. Indeed, this inequality yields that for large enough t we have

E min(2, |u(t 13 ) -u(t 13 )| 1 ) ≤ E min(2, |u( t 13 ) -u( t 13 )| 1 ) ≤ 2 t P |u( t 13 ) -u( t 13 )| 1 ≤ 2 t + 2P |u( t 13 ) -u( t 13 )| 1 > ≤ P ∀k ∈ [1, n 11 ] : max x∈S 1 u x (kn 2 ) > 1 n or max x∈S 1 u x (kn 2 ) > 1 n . t∈[(k-1)n 2 ,kn 2 ] |w(t) -w((k -1)n 2 )| 3,∞ > 1 n 4 ,

and then by the inequality (10) we get:

P |u(n 13 ) -u(n 13 )| 1 >

Acknowledgements 41 References

Acknowledgements I would like to thank my Ph.D. advisor S.Kuksin for formulation of the problem, advice and guidance. I am also very grateful to A.Biryuk, U.Frisch and K.Khanin for helpful discussions. Finally, I would like to thank all of the staff and faculty at the CMLS in Ecole Polytechnique for advice and support.

On the other hand, for α ≥ 0 and ∈ J 2 , S p,α ( )

The following result follows immediately from the definition [START_REF] Hopf | The partial differential equation u t + uu x = µu xx[END_REF].

Corollary 4.11. For ∈ J 2 , the flatness satisfies F ( ) ∼ -1 . 4.2. Results in Fourier space. By (44), for m ≥ 1 we have

Thus, for |k| ν -1 , {|û k | 2 } decreases super-algebraically. Now we want to estimate the H s norms of u for s ∈ (0, 1).

Lemma 4.12. We have { u 2 1/2 } ∼ | log ν|. Proof. By [START_REF] Boritchev | Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation[END_REF] we have

By Theorem 4.10 we get

The proof of the following result follows the same lines.

By construction, for every ω, t → u ω (t, •) is continuous in L 1 , and solutions to [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] are L 1 -solutions.

Conversely, for any T > 0, L 1 -solutions are solutions to [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] for t ≥ T . It suffices to prove this in the case of a deterministic initial condition u 0 . We will use the following elementary lemma, inspired by [28, Theorem 1.2.17.].

Lemma 5.2. Let X be a Banach space, and let x n ∈ X be a sequence converging to x. Assume that f : X → R ∪ {+∞} is a Borel functional such that f k : X → R, k ≥ 1, is a sequence of bounded continuous functions converging to f pointwise, and

Proof. It suffices first to let n → ∞, and then to let k → ∞. Now take T 2 > T 1 > 0 and consider ω ∈ Ω, an initial condition u 0 ∈ L 1 , and the corresponding smooth approximations u 0n , n ≥ 1, as above. Let u and u n , n ≥ 1, be the corresponding L 1 -solution (resp., solutions) to [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]. Let X be the space C(T 1 , T 2 ; L 1 ) and consider the functions f k = f • π k with π k the Galerkin projections on the subspace spanned by x → e ilx , |l| ≤ k, and f the Borel functional

We check that f and the f k verify the assumptions of Lemma 5.2. By Lemma 5.1, we have u ω n → u ω in X. On the other hand, by a timerescaled version of Lemma 3.17, we know that there exist constants β(m), m (m) such that we have:

This proves that for every ω, the L 1 -solutions u ω (t) are C ∞ -smooth for t > 0. Moreover, for every m ≥ 0 and T 2 > T 1 > 0, the upper estimates in H m for those solutions are uniform with respect to u 0 and with respect to t ∈ [T 1 , T 2 ]. By interpolation, we can prove that the L 1 -solutions are limits of the corresponding approximations in every Sobolev space H m , m ≥ 0. This has two important implications: embedded in L 1 , the family of measures µ t defined by:

where µ u 0 denotes the measure on L 1 induced by an initial condition u 0 , is tight in L 1 for any initial condition u 0 . Thus, we can extract a subsequence µ tn , converging weakly to a limit µ. It is not hard to check that µ is a stationary measure for ( 12) in L 1 .

The main result of this section is the following theorem, proved in Subsection 5.3.

Theorem 5.5. There exists a positive constant C such that we have

for any probability measures µ 1 , µ 2 on L 1 .

Corollary 5.6. For every p ∈ (1, ∞), there exists a positive constant C (p) such that we have

for any probability measures µ 1 , µ 2 on L p .

Corollary 5.6 is proved similarly to Theorem 5.5, observing that by Hölder's inequality, for any pair of solutions u, u of ( 12) and p ∈ [1, ∞) we have

Note that all estimates in the previous sections still hold for a stationary solution, since they hold uniformly for any initial condition in L 1 for large times, and a stationary solution has time-independent statistical properties. It follows that those estimates still hold when averaging in time and in ensemble (denoted by {•}) is replaced by averaging solely in ensemble, i.e. by integrating with respect to µ. In particular, Theorem 3.16, Theorem 4.10 and Theorem 4.14 imply, respectively, the following results.

Theorem 5.8. For α ≥ 0 and ∈ J 1 ,

αp ν -α(p-1) , p ≥ 1.

On the other hand, by [START_REF] Cole | On a quasilinear parabolic equation occurring in aerodynamics[END_REF] (as in the proof of Theorem 3.1, we use the notation δ = 2 -h(1)) we get

since (t 1 -t)u is the zero space average primitive of ṽ +(t 1 -t) wx . Thus we get

By assumption, we have τ ≥ C and K ≤ τ -2 , and by (65) we have N > τ 1/2 /2. Therefore we have, on the one hand,

and on the other hand,

Thus, N δ ≤ C , and for C large enough we have a contradiction with the fact that N > τ 1/2 .

To prove the following theorem, we use the coupling method [28, Chapter 3]. The situation is actually simpler than for the stochastic 2D Navier Stokes equation, which is the main subject of [START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF]. Indeed, in our setting the "damping time" needed to make the distance between two solutions small does not depend on the initial conditions, and by Theorem 5.1 the flow of ( 12) is L 1 -contracting.

Proof of Theorem 5.5. We can take (µ 1 , µ 2 ) = (δ u 0 , δ u 0 ); the general case follows by Fubini's theorem. Indeed, we have