Alexandre Boritchev March
email: alexandre.boritchev@gmail.com

Decaying Turbulence in Generalised Burgers Equation

We consider the generalised Burgers equation

.

Note that we are dealing with decaying, as opposed to stationary turbulence.

Thus, our estimates are not uniform in time. However, they hold on a time in-

where T 1 and T 2 depend only on f and the initial condition, and do not depend on the viscosity.

These results allow us to obtain a rigorous theory of the one-dimensional Burgers turbulence in the spirit of Kolmogorov's 1941 theory. In particular, we obtain two results which hold in the inertial range. On one hand, we explain the bifractal behaviour of the moments of increments, or structure functions. On the other hand, we obtain an energy spectrum of the form k -2 . These results remain valid in the inviscid limit.

1 Introduction

Setting

The Burgers equation

∂u ∂t + u ∂u ∂x -ν ∂ 2 u ∂x 2 = 0, (1)
where ν > 0 is a constant, appears in many elds of physics and other branches of science: see the reviews [START_REF] Bec | Les Houches 2000: New Trends in Turbulence[END_REF][START_REF] Bec | Burgers turbulence[END_REF] and references therein.

The Burgers equation has been mentioned for the rst time by Forsyth [START_REF] Forsyth | Theory of dierential equations. Part 4. Partial dierential equations[END_REF] and Bateman [START_REF] Bateman | Some recent researches on the motion of uids[END_REF], in 1906 and 1915 respectively. However, it only became wellknown in the physical community around 1950, due to the work of the physicist whose name was given to it (see the monograph [START_REF] Burgers | The nonlinear diusion equation: asymptotic solutions and statistical problems[END_REF] and references therein).

Burgers considered this equation as a toy model for hydrodynamics: indeed, the incompressible Navier-Stokes equations and (1) have similar nonlinearities and dissipative terms, so this equation can be seen as the most natural onedimensional model for Navier-Stokes.

The equation ( 1) can be transformed into the heat equation by the Cole-

Hopf transformation [START_REF] Cole | On a quasilinear parabolic equation occurring in aerodynamics[END_REF][START_REF] Hopf | The partial dierential equation u t +uu x = µu xx[END_REF]. However, this transformation will not be used in this paper for two dierent reasons. On one hand, the resulting representation of the solution is very singular as ν → 0 + , and interpreting this singularity rigorously is highly non-trivial. On the other hand, we want to be able to study the Burgers equation with u ∂u/∂x replaced by a more general nonlinearity; see

(2)-(3).

For ν 1, solutions of the Burgers equation display non-trivial small-scale behaviour, often referred to as decaying Burgers turbulence or Burgulence [START_REF] Burgers | The nonlinear diusion equation: asymptotic solutions and statistical problems[END_REF][START_REF] Chorin | Lectures on turbulence theory[END_REF][START_REF] Kida | Asymptotic properties of Burgers turbulence[END_REF]. The language of the Kolmogorov 1941 theory [START_REF] Kolmogorov | Dissipation of energy in locally isotropic turbulence[END_REF][START_REF] Kolmogorov | On degeneration (decay) of isotropic turbulence in an incompressible viscous liquid[END_REF][START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous uid for very large Reynolds number[END_REF] is traditionally used to describe this behaviour.

For simplicity, from now on we consider the space-periodic setting, i.e. x ∈ S 1 = R/Z. In this setting, the solutions of (1) remain of order 1 during a time of order 1. On the other hand, for t → +∞ the solutions decay at least as Ct -1 in any Lebesgue space L p , 1 ≤ p ≤ +∞, uniformly in ν (cf. for instance [START_REF] Kruzhkov | The Cauchy Problem in the large for nonlinear equations and for certain quasilinear systems of the rst-order with several variables[END_REF]). Note that in the limit ν → 0, the diusive eect due to the second derivative vanishes and this upper bound becomes sharp [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF]Theorem 11.7.3]. Thus, the solutions display smooth ramps and sharp clis [START_REF] Bec | Les Houches 2000: New Trends in Turbulence[END_REF]. In the limit ν → 0, they have the N -wave behaviour, i.e. solutions are composed of waves similar to the Cyrillic capital letter I (the mirror image of N). In other words, at a xed (large enough) time t the solution u(t, •) alternates between negative jump discontinuities and smooth regions where the derivative is positive and of the order 1 (see for instance [START_REF] Evans | Partial dierential equations[END_REF]). This is a clear manifestation of the small-scale intermittency in space [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]. For 0 < ν 1 the solutions are still highly intermittent: there are zones where the derivative is small and positive, called ramps, and zones where the derivative is large in absolute value and negative, called clis.

For a typical initial data u 0 (e.g. for max |u 0 | ∼ 1 and max |(u 0 ) x | ∼ 1) and for t > 1/(min(u 0 ) x ), t ∼ 1, it is numerically observed [START_REF] Aurell | On the multifractal properties of the energy dissipation derived from turbulence data[END_REF] that a solution u(t, •) has the following features (cf. Figure 1):

• Amplitude of the solution: ∼ 1.

• Number of clis per period: ∼ 1.

• Vertical drop at a cli: ∼ -1.

• Width of a cli: ∼ ν.

It is easy to verify that for the prototypical N -wave, i.e. for the 1-periodic function equal to x on (-1/2, 1/2], the Fourier coecients satisfy |û(k)| ∼ k -1 . Thus, it is natural to conjecture that for ν small and for a certain range of wave numbers k, the energy-type quantities 1 2 |û(k)| 2 behave, in average, as k -2

[10, [START_REF] Fournier | L'équation de Burgers déterministe et stastistique[END_REF][START_REF] Kida | Asymptotic properties of Burgers turbulence[END_REF][START_REF] Kraichnan | Lagrangian-history statistical theory for Burgers' equation[END_REF].

In the physical space, the natural analogues of the small-scale quantities

1 2 |û(k)| 2 are the structure functions S p ( ) = S 1 |u(x + ) -u(x)| p dx.
For p ≥ 0, the description above implies that for ν 1, these quantities behave as max (1,p) : in other words, we have a bifractal behaviour [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]Chapter 8]. Indeed, as observed in [START_REF] Aurell | On the multifractal properties of the energy dissipation derived from turbulence data[END_REF], there are three possibilities for the interval [x, x+ ] (below, C denotes constants of order 1, and we have to keep in mind that 1): Since the number of clis per period is of order 1 and is larger than a cli, the probability of this event is of order . In this case:

u(x + ) -u(x) ∼ -C "clif f " + C "ramps" ∼ -C; |u(x + ) -u(x)| p ∼ C(p).
• [x, x + ] covers a small part of a "cli".

The contribution due to this possibility is negligible.

• [x, x + ] does not intersect a "cli".

Since is smaller than the length of a ramp, the probability of this event is of order 1 -C ∼ 1. In this case:

u(x + ) -u(x) ∼ C ramp ; |u(x + ) -u(x)| p ∼ C(p) p .
Thus, for ν 1,

S p ( ) ∼ C(p) + p ∼ C(p) p , 0 ≤ p ≤ 1.
C(p) , p ≥ 1.

Burgers equation and turbulence

From now on, we consider the generalised one-dimensional space-periodic Burgers equation

∂u ∂t + df (u) dx -ν ∂ 2 u ∂x 2 = 0, x ∈ S 1 = R/Z, (2)
where f is C ∞ -smooth and strongly convex, i.e. f satises the property f (y) ≥ σ > 0, y ∈ R.

(

) 3
The classical Burgers equation (1) corresponds to f (u) = u 2 /2. The physical arguments justifying the small-scale estimates which are given above still hold in that setting.

For the sake of simplicity, we only consider solutions to ( 2)-( 3) with zero space average for xed t:

S 1 u(t, x)dx = 0, ∀t ≥ 0. (4)
For the generalised Burgers equation, some upper estimates for small-scale quantities have been obtained previously. Lemma 5.1 of our paper is an analogue in the periodic setting of the one-sided Lipschitz estimate due to Oleinik, and the upper estimate for S 1 ( ) follows from an estimate for the solution in the class of bounded variation functions BV . For references on these classical aspects of the theory of scalar conservation laws, see [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF][START_REF] Lax | Hyperbolic Partial Dierential Equations[END_REF][START_REF] Serre | Systems of Conservation Laws I[END_REF]. For some upper estimates for small-scale quantities, see [START_REF] Bardos | Equations de Navier-Stokes et modèle de la turbulence[END_REF][START_REF] Kreiss | Fourier expansions of the solutions of the NavierStokes equations and their exponential decay rate[END_REF][START_REF] Tadmor | Total variation and error estimates for spectral viscosity approximations[END_REF].

Estimating small-scale quantities for nonlinear PDEs with small viscosity from above and from below is motivated by the problem of turbulence. This research was initiated by Kuksin, who obtained estimates 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 the references in [START_REF] Kuksin | Spectral properties of solutions for nonlinear PDEs in the turbulent regime[END_REF]).

In the paper [START_REF] Biryuk | Spectral properties of solutions of the Burgers equation with small dissipation[END_REF], Biryuk obtained lower and upper estimates for the L 2 -Sobolev norms of solutions to [START_REF] Bardos | Equations de Navier-Stokes et modèle de la turbulence[END_REF]. These estimates are sharp, in the sense that the lower and the upper bounds only dier by a multiplicative constant. Moreover, he obtained upper and lower estimates for the energy spectrum which enable him to give the correct value for the dissipation length scale. In [START_REF] Boritchev | Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation[END_REF][START_REF] Boritchev | Sharp estimates for turbulence in white-forced generalised Burgers equation[END_REF],

based on a better understanding of solutions for small values of ν, we obtain sharp results for L p -Sobolev norms, p ∈ (1, ∞], and small-scale quantities. However, in both articles we add a rough in time and smooth in space random forcing term in the right-hand side of equation (2) (a kicked and a white force, respectively). Thus, we change the nature of the equation: the energy injection due to the random forcing now balances the dissipation due to the second derivative.

In other words, we study stochastic stationary Burgulence, which is dierent from decaying Burgulence.

Note that stochastic Burgulence has been studied on a physical level of rigour for a random forcing which is more or less smooth in space, namely by Gotoh and Kraichnan [START_REF] Gotoh | Steady-state Burgers turbulence with largescale forcing[END_REF], Gurarie and Migdal [START_REF] Gurarie | Instantons in the Burgers equation[END_REF] and Polyakov [START_REF] Polyakov | Turbulence without pressure[END_REF]. For more references, see the review by Bec and Khanin [START_REF] Bec | Burgers turbulence[END_REF]. On the other hand, E, Khanin, Mazel and Sinai have obtained rigorous results on the behaviour of solutions for a random spatially smooth forcing in the limit ν → 0, t → ∞ [START_REF] Weinan | Probability distribution functions for the random forced Burgers equation[END_REF][START_REF] Weinan | Invariant measures for Burgers equation with stochastic forcing[END_REF].

Note that it is also possible to study (2) in a deterministic stationary setting, which amounts to considering a deterministic additive random force. However, this is a delicate issue: indeed, for any initial condition u 0 we can build a bad time-independent random force equal to f (u 0 )(u 0 ) x -ν(u 0 ) xx , corresponding to a stationary solution of (2) which manifests no turbulent behaviour.

Here, we prove sharp lower and upper estimates for the dissipation length scale and for the small-scale quantities, i.e. the structure functions and the energy spectrum, which characterise the decaying Burgulence. Thus, we improve signicantly the results of [START_REF] Biryuk | Spectral properties of solutions of the Burgers equation with small dissipation[END_REF]. To our best knowledge, this is the rst such result for the deterministic generalised Burgers equation. Moreover, we extend the results for the L 2 -Sobolev norms obtained by Biryuk to the L p -Sobolev norms, p ∈ (1, ∞]. The powers of ν, , k involved in our estimates turn out to be the same as in the randomly forced case considered in [START_REF] Boritchev | Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation[END_REF][START_REF] Boritchev | Sharp estimates for turbulence in white-forced generalised Burgers equation[END_REF]. Note that our estimates hold in average on a time interval [T 1 , T 2 ], where both T 1 and T 2 do not depend on ν. In other words, we consider a time range during which we have the transitory behaviour which is referred to as decaying Burgers turbulence [START_REF] Bec | Burgers turbulence[END_REF].

This time interval depends only on f and, through the quantity D (see 5)), on u 0 . In particular, it does not depend on ν.

A detailed overview of the results mentioned above is given in Section 3 (for the state of art) and in Section 4 (for the main results in this paper).

Note that when studying the typical behaviour for solutions of nonrandom PDEs, one usually considers some averaging in the initial condition in order to avoid pathological initial data. Indeed, unlike for the stochastic case, now there is no random mechanism to get solutions out of bad regions of the phase space. Here, no such averaging is necessary. This is due to the particular structure of the deterministic Burgers equation: a non-zero initial condition u 0 is as generic as the ratio between the orders of (u 0 ) x and of u 0 itself. This ratio can be bounded from above using the quantity D:

D = max(|u 0 | -1 1 , |u 0 | 1,∞ ) > 1 (5)
(see Subsection 2.1 for the meaning of the notation | • | m,p ). Note that for 0 ≤ m ≤ 1 and 1 ≤ p ≤ ∞, we have:

D -1 ≤ |u 0 | m,p ≤ D. (6)
The physical meaning of D is that it gives a lower bound for the ratio between the amount of energy 1 2 S 1 u 2 initially contained in the system and its rate of dissipation ν S 1 u 2

x .

Now let us say a few words about similarities and dierences between the

Burgulence and real turbulence. It is clear that the geometric structures which are responsible for non-trivial small-scale behaviour are quite dierent for these two models: N -waves do not have the same properties as complex multi-scale structures such as vortex tubes observed in the real turbulence. However, because of the similarity in the form of the Burgers equation and the Navier-Stokes equations, the physical arguments justifying dierent theories of turbulence can be applied to the Burgulence. Indeed, both models exhibit an inertial nonlinearity of the form u • ∇u, and viscous dissipation which in the limit ν → 0 gives a dissipative anomaly [START_REF] Bec | Burgers turbulence[END_REF]. Hence, the Burgers equation is often used as a benchmark for the turbulence theories. It is also used as a benchmark for dierent numerical methods for the Navier-Stokes equations. For more information on both subjects, see [START_REF] Bec | Burgers turbulence[END_REF]. Now consider the generalised Burgers equation with a random regular in space and white in time forcing term η such as in [START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF]. Then the generalised Burgers equation with the natural scaling for this term (needed to counterbalance the energy dissipation due to the viscous term) is of the form:

u t + f (u)u x = νu xx + η,
i.e. the force does not depend on ν [START_REF] Boritchev | Sharp estimates for turbulence in white-forced generalised Burgers equation[END_REF]. This is similar to the conjectured behaviour for real turbulence, and contrasts with the situation for the 2D Navier-Stokes equations, where the corresponding term is of the form ν 1/2 η [START_REF] Kuksin | Mathematics of two-dimensional turbulence[END_REF]. This justies the study of the small-scale quantities for the randomly forced Burgers equation in the limit ν → 0 such as in [START_REF] Boritchev | Sharp estimates for turbulence in white-forced generalised Burgers equation[END_REF]. As it will be shown in Section 6, on a time scale which only depends on the initial condition and on the form of the nonlinearity f (u)u x , the small-scale quantities for the unforced Burgers equation also have a non-trivial behaviour as ν → 0, similar to the behaviour in the stochastic case. This is the main result of the paper. Up to now this question has only been adressed rigorously by Biryuk [START_REF] Biryuk | Spectral properties of solutions of the Burgers equation with small dissipation[END_REF], who obtained less sharp estimates. For more details on his results, see Section 3.

Plan of the paper

We introduce the notation and the setup in Section 2. In Section 3, we give an overview of the state of art, before presenting the main results of our paper in Section 4.

In Section 5, we begin by recalling an upper estimate for the quantity ∂u/∂x.

This result allows us to obtain upper bounds, as well as time-averaged lower bounds, for the Sobolev norms |u| m,p . These bounds depend only on f and on the quanity D dened by [START_REF] Bec | Burgers turbulence[END_REF].

In Section 6 we give sharp upper and lower bounds for the dissipation length scale, the structure functions and the energy spectrum for the ow u(t, x), which hold uniformly for ν ≤ ν 0 , and we analyse the meaning of these results in terms of the theory of turbulence. These bounds and the constant ν 0 > 0 only depend on f and on D.

In Section 7 we consider the inviscid limit ν = 0.

Notation and setup

Agreement: In the whole paper, all functions that we consider are real-valued and the space variable x belongs to S 1 = R/Z.

|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 and the mean value of v vanishes, we have: , where m > β, and r is determined by

|v| 1 ≤ |v| ∞ ≤ |v| 1,1 ≤ |v| 1,∞ ≤ • • • ≤ |v| m,1 ≤ |v| m,∞ ≤ . . .
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.

Subindices t and x, which can be repeated, denote partial dierentiation with respect to the corresponding variables. We denote by v (m) the m-th derivative of v in the variable x. The function v(t, •) is abbreviated as v(t).

Notation

In this paper, we study asymptotical properties of solutions to (2) for small values of ν, i.e. we suppose that 0 < ν 1.

We assume that f is innitely dierentiable and satises [START_REF] Bateman | Some recent researches on the motion of uids[END_REF]. We recall that we restrict ourselves to the zero space average case, i.e. the initial condition u 0 := u(0) satises (4). Consequently, u(t) satises ( 4) for all t. Furthermore, we assume that u 0 ∈ C ∞ . We also assume that we are not in the case u 0 ≡ 0, corresponding to the trivial solution u(t, x) ≡ 0. This ensures that the quantity D (see ( 5)) is well-dened.

For the existence, uniqueness and smoothness of solutions to [START_REF] Bardos | Equations de Navier-Stokes et modèle de la turbulence[END_REF], see for instance [START_REF] Kreiss | Initial-boundary value problems and the Navier-Stokes equations[END_REF].

Agreements: From now on, all constants denoted by C with sub-or superindexes are positive. Unless otherwise stated, they depend only on f and on D. By C(a 1 , . . . , a k ) we denote constants which also depend on parameters

a 1 , . . . , a k . By X a1,...,a k Y we mean that X ≤ C(a 1 , . . . , a k )Y . The notation X a1,...,a k ∼ Y stands for Y a1,...,a k X a1,...,a k 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. We denote by u = u(t, x) a solution of (2) for an initial condition u 0 . A relation where the admissible values of t (respectively, x) are not specied is assumed to hold for all t ≥ 0 or t > 0, depending on the context (respectively, all x ∈ S 1 ).

The brackets {•} stand for the averaging in time over an interval [T 1 , T 2 ], where T 1 , T 2 only depend on f and on D (see [START_REF] Kolmogorov | On degeneration (decay) of isotropic turbulence in an incompressible viscous liquid[END_REF] for their denition.)

For m ≥ 0, p ∈ [1, ∞], γ(m, p) is by denition the quantity max(0, m -1/p).

We use the notation g -= max(-g, 0) and g + = max(g, 0).

Notation in Section 6

In that section, we study analogues of quantities which are important for hydrodynamical turbulence. We consider quantities in physical space (structure functions) as well as in Fourier space (energy spectrum). We assume that ν ≤ ν 0 . The value of ν 0 > 0 will be chosen in [START_REF] Oleinik | Discontinuous solutions of non-linear dierential equations[END_REF].

We dene the non-empty and non-intersecting intervals

J 1 = (0, C 1 ν]; J 2 = (C 1 ν, C 2 ]; J 3 = (C 2 , 1]
(see Figure 2) corresponding to the dissipation range, the inertial range and the energy range from the Kolmogorov 1941 theory of turbulence, respectively [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF].

In particular, the upper bound C 1 ν of the dissipation range is the dissipation

{|û(n)| 2 } = (2πn) -2 S 1 e 2πinx u x (x) 2 ≤ (2πn) -2 {|u| 2 1,1 } ≤ Cn -2 , ( 7
)
and C 1 and C 2 can be made as small as we wish (see [START_REF] Polyakov | Turbulence without pressure[END_REF]). Consequently, the proportion of the sum { |û(n)| 2 } contained in the Fourier modes corresponding to J 3 can be made as large as desired. For instance, we may assume that

|n|<C -1 2 |û(n)| 2 ≥ 99 100 n∈Z |û(n)| 2 .
The quantities S p ( ) denote the averaged moments of the increments in space for the ow u(t, x):

S p ( ) = S 1 |u(t, x + ) -u(t, x)| p dx , p ≥ 0, 0 < ≤ 1.
The quantity S p ( ) is the structure function of p-th order. The atness, which measures spatial intermittency [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF], is dened by:

F ( ) = S 4 ( )/S 2 2 ( ). (8)
Finally, for k ≥ 1, we dene the (layer-averaged) energy spectrum by

E(k) = |n|∈[M -1 k,M k] |û(n)| 2 |n|∈[M -1 k,M k] 1 , (9)
where M ≥ 1 is a constant which will be specied later (see the proof of Theorem 6.11).

State of the art

We recall that u = u(t, x) denotes a solution of (2) for an initial condition u 0 . All constants are independent of the viscosity ν (i.e., dependance on ν is always explicitly stated in the estimates). A relation where the admissible values of t (respectively, x) are not specied is assumed to hold for all t ≥ 0 or t > 0, depending on the context (respectively, all x ∈ S 1 ). For more information on the notation, see Section 2.

The estimate

u x (t, x) ≤ (σt) -1 , t > 0, ( 10
)
where σ is the constant in the assumption (3), is a reformulation of Oleinik's E-condition [START_REF] Oleinik | Discontinuous solutions of non-linear dierential equations[END_REF]. This result immediately implies an upper bound for the rst structure function S 1 ( ). Indeed, since the space average of u(t, •) vanishes identically for all t, we have:

S 1 |u(t, x + ) -u(t, x)| = S 1 (u(t, x + ) -u(t, x)) + + S 1 (u(t, x + ) -u(t, x)) - = 2 S 1 (u(t, x + ) -u(t, x)) + ≤ 2 σt .
Moreover, integration by parts gives us the follwing upper estimate for the spectrum:

{|û(n)| 2 } ≤ C(σtn) -2
(see for instance [START_REF] Bardos | Equations de Navier-Stokes et modèle de la turbulence[END_REF]). In a similar setting, exponential upper estimates for the spectrum in the dissipation range have also been obtained; see [START_REF] Kreiss | Fourier expansions of the solutions of the NavierStokes equations and their exponential decay rate[END_REF]. See also [START_REF] Tadmor | Total variation and error estimates for spectral viscosity approximations[END_REF] for upper estimates in a slightly dierent (hyperviscous) setting.

In [START_REF] Biryuk | Spectral properties of solutions of the Burgers equation with small dissipation[END_REF], Biryuk begins by proving upper as well as lower estimates for the H m -Sobolev norms of u:

B -1 ν -(2m-1) ≤ 1 T T 0 u 2 m ≤ Bν -(2m-1) , 0 < ν ≤ ν 0 . (11)
Here, the strictly positive quantities ν 0 and T depend on f and m as well as on the Sobolev norms of the initial condition u 0 . The letter B denotes dierent strictly positive quantities which also depend on these parameters. Since these estimates hold only for a xed value of T , there is no contradiction with the decay in Ct -1 of the solutions as t → +∞.

Let us denote by E s,θ the averaged energy spectrum:

E s,θ = 1 T T 0 |n|∈[ν -s+θ ,ν -s-θ ] |û(n)| 2 |n|∈[ν -s+θ ,ν -s-θ ] 1
, s, θ > 0.

Using the inequalities ( 10)-( 11), Biryuk obtains upper and lower estimates for the spectrum of the solutions, which hold for 0 < ν ≤ ν 0 :

E s,θ ≤ Bν m , m > 0, s > 1 + θ. (12)
E s,θ ≤ Bν 2(s-θ) , s > θ.

(

) E 1,θ ≥ Bν 2+2θ . ( 13
) 14
The quantities ν 0 and T , as well as the dierent strictly positive quantities denoted by B, depend on f and on the Sobolev norms of u 0 , as well as on m, s, θ.

Note that Biryuk's results for the Sobolev norms are sharp, in the sense that in the lower and upper estimates in [START_REF] Cole | On a quasilinear parabolic equation occurring in aerodynamics[END_REF], ν is raised to the same power. Using the same terminology, his results (13-14) can be described as "almost sharp" for s = 1, since they give almost the same lower and upper bounds for E 1,θ with 0 < θ 1 (up to a multiplicative constant and ν raised to a very small power).

Biryuk's spectral estimates may be interpreted in the spirit of Kolmogorov's theory of turbulence. Indeed, relation [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF] implies that the energy spectrum of the k-th Fourier mode averaged around k = K, where K ν -1 , decays faster than any negative degree of K. This suggests that for K ν -1 we are in the dissipation range, where the energy E k decays fast. On the other hand, relations ( 13) and ( 14) yield that the energy E k , averaged around k = ν -1 , behaves as k -2 , which gives a Kolmogorov-type power law [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]. This suggests a dissipation length scale of the order ν.

4

Main results

In our paper, in Section 5, we prove sharp upper and lower bounds for almost all Sobolev norms of u, generalising the estimates [START_REF] Cole | On a quasilinear parabolic equation occurring in aerodynamics[END_REF]. These results for Sobolev norms of solutions are summed up in Theorem 5.10. Namely, for m ∈ {0, 1} and p ∈ [1, ∞] or for m ≥ 2 and p ∈ (1, ∞] we have:

{|u(t)| α m,p } 1/α m,p,α ∼ ν -γ , α > 0. ( 15
)
We recall that by denition, γ(m, p) = max(0, m -1/p), and the brackets {•} stand for the averaging in time over an interval [T 1 , T 2 ] (T 1 , T 2 only depend on f and, through D, on u 0 : see ( 25)). For more information on the notation, see Section 2.

In Section 6 we obtain sharp estimates for analogues of quantities characterising hydrodynamical turbulence. In what follows, we assume that ν ∈ (0, ν 0 ], where ν 0 ∈ (0, 1] depends only on f and on D.

First, as a consequence of ( 10) and ( 15), in Theorem 6.9 we prove that for

∈ J 1 : S p ( ) p ∼ p , 0 ≤ p ≤ 1. p ν -(p-1) , p ≥ 1,
and for

∈ J 2 : S p ( ) p ∼ p , 0 ≤ p ≤ 1.
, p ≥ 1.

Consequently, for ∈ J 2 the atness satises the estimate:

F ( ) = S 4 ( )/S 2 2 ( ) ∼ -1 .
Thus, u is highly intermittent in the inertial range. This intermittency is in good agreement with the physical heuristics presented in Subsection 1.1, due to the particular structure of the solution, where the excited zones correspond to the clis. Cf. [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF] for a discussion of the intermittency for hydrodynamical turbulent ows.

Finally, as a relatively simple consequence of our estimates for the structure function S 2 ( ), we get estimates for the spectral asymptotics of the decaying Burgulence. On one hand, as a consequence of Theorem 5.10, for m ≥ 1 we get:

{|û(k)| 2 } m k -2m u 2 m m (kν) -2m ν.
In particular, {|û(k)| 2 } decreases at a faster-than-algebraic rate for |k| ν -1 .

On the other hand, by Theorem 6.11, for k such that k -1 ∈ J 2 the energy spectrum E(k) satises:

E(k) ∼ k -2 ,
where the quantity M ≥ 1 in the denition of E(k) depends only on f and on D. This result signicantly improves Biryuk's spectral estimates, since it characterises exactly the spectral behaviour in the whole inertial range.

Note that our estimates hold for quantities averaged on a time interval

[T 1 , T 2 ], T 2 > T 1 > 0,
and not on an interval [0, T ] as in Biryuk's paper. This allows us to obtain estimates which depend on the initial condition only through the single parameter D. Moreover, as in Biryuk's paper, this time interval does not depend on the viscosity coecient ν.

As we mentioned in Section 1.2, upper estimates for S p ( ) follow from known results about the Burgers equation. Sharp lower estimates were not known before our work.

Finally, in Section 7 we note that our estimates for the small-scale quantities still hold in the inviscid limit ν → 0, up to some natural modications.

Estimates for Sobolev norms

We recall that u = u(t, x) denotes a solution of (2) for an initial condition u 0 . All constants are independent of the viscosity ν. A relation where the admissible values of t (respectively, x) are not specied is assumed to hold for all t ≥ 0 or t > 0, depending on the context (respectively, all x ∈ S 1 ). For more information on the notation, see Section 2.

We begin by recalling a key upper estimate for u x .

Lemma 5.1 We have:

u x (t, x) ≤ min(D, σ -1 t -1 ).
Proof. Dierentiating the equation ( 2) once in space we get

(u x ) t + f (u)u 2 x + f (u)(u x ) x = ν(u x ) xx .
Now consider a point (t 1 , x 1 ) where u x reaches its maximum on the cylinder S = [0, t] × S 1 . Suppose that t 1 > 0 and that this maximum is nonnegative. At such a point, Taylor's formula implies that we would have (u x ) t ≥ 0, (u x ) x = 0 and (u x ) xx ≤ 0. Consequently, since by (3) f (u) ≥ σ, we get f (u)u 2 x ≤ 0, which is impossible. Thus u x can only reach a nonnegative maximum on S for t 1 = 0. In other words, since (u 0 ) x has zero mean value, we have:

u x (t, x) ≤ max x∈S 1 (u 0 ) x (x) ≤ D.
The inequality

u x (t, x) ≤ σ -1 t -1
is proved in by a similar maximum principle argument applied to the function tu x : cf. [START_REF] Kruzhkov | The Cauchy Problem in the large for nonlinear equations and for certain quasilinear systems of the rst-order with several variables[END_REF].

Since the space averages of u(t) and u x (t) vanish, we get the following upper estimates:

|u(t)| p ≤ |u(t)| ∞ ≤ S 1 u + x (t) ≤ min(D, σ -1 t -1 ), 1 ≤ p ≤ +∞, (16) |u(t)| 1,1 = S 1 u
+ x (t) + S 1 u - x (t) = 2 S 1 u + x (t) ≤ 2 min(D, σ -1 t -1 ). (17)
Now we recall a standard estimate for the nonlinearity v (m) , (

For its proof, we refer to [START_REF] Boritchev | Sharp estimates for turbulence in white-forced generalised Burgers equation[END_REF].

Lemma 5.2 For v ∈ C ∞ such that |v| ∞ ≤ A, we have:

v (m) , (f (v)) (m+1) ≤ C v m v m+1 , m ≥ 1,
where C depends only on m, A and |f | C m ([-A,A]) .

The following result shows that there is a strong nonlinear damping which prevents the successive derivatives of u from becoming too large.

Lemma 5. [START_REF] Bateman | Some recent researches on the motion of uids[END_REF] We have

u(t) 2 1 ν -1 .
On the other hand, for m ≥ 2,

u(t) 2 m m max(ν -(2m-1) , t -(2m-1) ). Proof. Fix m ≥ 1. Denote x(t) = u(t) 2 m .
We claim that the following implication holds:

x(t) ≥ C ν -(2m-1) =⇒ d dt x(t) ≤ -(2m -1)x(t) 2m/(2m-1) , (18)
where C is a xed positive number, chosen later. Below, all constants denoted by C do not depend on C . Indeed, assume that x(t) ≥ C ν -(2m-1) . Integrating by parts in space and using ( 16) (p = ∞) and Lemma 5.2, we get the following energy dissipation relation:

d dt x(t) = -2ν u(t) 2 m+1 -2 u (m) (t), (f (u(t))) (m+1)
≤ -2ν u(t)

2 m+1 + C u(t) m u(t) m+1 . (19)
Applying Lemma 2.1 to u x and then using [START_REF] Forsyth | Theory of dierential equations. Part 4. Partial dierential equations[END_REF], we get:

u(t) m ≤ C u(t) (2m-1)/(2m+1) m+1 |u(t)| 2/(2m+1) 1,1 ≤ C u(t) (2m-1)/(2m+1) m+1 . ( 20
)
Thus, we have the relation

d dt x(t) ≤(-2ν u(t) 2/(2m+1) m+1 + C) u(t) 4m/(2m+1) m+1 . (21)
The inequality [START_REF] Gotoh | Steady-state Burgers turbulence with largescale forcing[END_REF] yields

u(t) 2/(2m+1) m+1 ≥ Cx(t) 1/(2m-1) , (22)
and then since by assumption x(t) ≥ C ν -(2m-1) we get:

u(t)
2/(2m+1) m+1

≥ CC 1/(2m-1) ν -1 .

(

) 23
Combining the inequalities [START_REF] Gurarie | Instantons in the Burgers equation[END_REF][START_REF] Hopf | The partial dierential equation u t +uu x = µu xx[END_REF][START_REF] Kida | Asymptotic properties of Burgers turbulence[END_REF], for C large enough we get: 1) .

d dt x(t) ≤ (-CC 1/(2m-1) + C)x(t) 2m/(2m-
Thus we can choose C in such a way that the implication (18) holds. For m = 1, (6) and [START_REF] Fournier | L'équation de Burgers déterministe et stastistique[END_REF] immediately yield that

x(t) ≤ max(C ν -1 , D 2 ) ≤ max(C , D 2 )ν -1 , t ≥ 0. Now consider the case m ≥ 2. We claim that x(t) ≤ max(C ν -(2m-1) , t -(2m-1) ). (24)
Indeed, if x(s) ≤ C ν -(2m-1) for some s ∈ [0, t], then the assertion [START_REF] Fournier | L'équation de Burgers déterministe et stastistique[END_REF] ensures that x(s) remains below this threshold up to time t. Now, assume that x(s) > C ν -(2m-1) for all s ∈ [0, t]. Denote

x(s) = (x(s)) -1/(2m-1) , s ∈ [0, t] .

By [START_REF] Fournier | L'équation de Burgers déterministe et stastistique[END_REF] we get dx(s)/ds ≥ 1. Therefore x(t) ≥ t and x(t) ≤ t -(2m-1) . Thus in this case, the inequality [START_REF] Kolmogorov | Dissipation of energy in locally isotropic turbulence[END_REF] still holds. This proves the lemma's assertion.

|u(t)| m,p p u(t) 1-θ m u(t) θ m+1 , θ = 1 2 - 1 p .
Then we use Lemma 5.3 and Hölder's inequality to complete the proof.

We use the same method to prove the case m = 1, p ∈ [1, 2], combining (17) and Lemma 5.3. We also proceed similarly for m ≥ 2, p ∈ (1, 2), combining [START_REF] Forsyth | Theory of dierential equations. Part 4. Partial dierential equations[END_REF] and an upper estimate for u(t) α M,p for a large value of M and some p ≥ 2. Finally, the case m = 0 follows from [START_REF] Evans | Partial dierential equations[END_REF].

Unfortunately, the proof of Lemma 5.4 cannot be adapted to the case m ≥ 2 and p = 1. Indeed, Lemma 2.1 only allows us to estimate a W m,1 norm from above by other W m,1 norms: we can only get that

|u(t)| m,1 m,n,k |u(t)| (m-k)/(n-k) n,1 |u(t)| (n-m)/(n-k) k,1 , 0 ≤ k < m < n,
and thus the upper estimates obtained above cannot be used. However, we have:

|u(t)| m,1 ≤ |u(t)| m,1+β
for any β > 0. Consequently, the lemma's statement holds for m ≥ 2 and p = 1, with γ replaced by γ + λ, and m,p replaced by m,p,λ , for any λ > 0.

Now we dene

T 1 = 1 4 D -2 C-1 ; T 2 = max 3 2 T 1 , 2Dσ -1 , (25)
where C is a constant such that for all t, u(t)

2 1 ≤ Cν -1 (cf. Lemma 5.
3). Note that T 1 and T 2 do not depend on the viscosity coecient ν.

From now on, for any function A(t), {A(t)} is by denition the time average

{A(t)} = 1 T 2 -T 1 T2 T1 A(t).
The rst quantity that we estimate from below is {|u(

t)| 2 p }, p ∈ [1, ∞]. Lemma 5.5 For p ∈ [1, ∞],
we have:

{|u(t)| 2 p } 1.
6

Estimates for small-scale quantities

In this section, we study analogues of quantities which are important for the study of hydrodynamical turbulence. We consider quantities in the physical space (structure functions) as well as in the Fourier space (energy spectrum).

For notation for these quantities and the ranges J 1 , J 2 , J 3 , see Subsection 2.3.

Here, provided ν ≤ ν 0 , all estimates hold independently of the viscosity ν. We recall that the brackets {•} stand for the averaging in time over an interval [T 1 , T 2 ]: see [START_REF] Kolmogorov | On degeneration (decay) of isotropic turbulence in an incompressible viscous liquid[END_REF].

We begin by estimating the functions S p ( ) from above. Lemma 6.1 For ∈ [0, 1],

S p ( ) p p , 0 ≤ p ≤ 1. p ν -(p-1) , p ≥ 1.
Proof. We begin by considering the case p ≥ 1. We have:

S p ( ) = S 1 |u(x + ) -u(x)| p dx ≤ S 1 |u(x + ) -u(x)|dx max x |u(x + ) -u(x)| p-1 .
Using the fact that the space average of u(x + ) -u(x) vanishes and Hölder's inequality, we obtain that S p ( ) ≤ 2

where the second inequality follows from Lemma 5.1. Finally, by Theorem 5.10 we get: 1) .

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

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

Lemma 6.2 For ∈ J 2 ∪ J 3 , S p ( ) p p , 0 ≤ p ≤ 1.
, p ≥ 1.

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.

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 becomes quantitative if we restrict ourselves to the set O K . Lemma 6.8 For m ≥ 0 and ∈ J 2 ,

S p ( ) p p , 0 ≤ p ≤ 1.
, p ≥ 1.

Proof. In the same way as above, it suces to prove that the inequalities hold uniformly in t for t ∈ O K , with S p ( ) replaced by

S 1 |u(x + ) -u(x)| p dx,
and we can restrict ourselves to the case p ≥ 1. Again, till the end of this proof, we assume that t ∈ O K . Dene z as in the proof of Lemma 6.7. We have

S 1 |u(x + ) -u(x)| p dx ≥ z z-1 2 x+ x u -(y)dy clif f s - x+ x u + (y)dy ramps p dx. Since ≥ C 1 ν = 1 4 K -2 ν, by (37) for x ∈ [z -1 2 , z] we get: x+ x u -(y)dy ≥ z+ 1 8 K -2 ν z u -(y)dy ≥ 1 16 K -3 .
.

On the other hand, since ≤ C 2 , by ( 30) and ( 35) 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) .
Summing up the results above we obtain the following theorem. Theorem 6.9 For ∈ J 1 , S p ( ) p ∼ p , 0 ≤ p ≤ 1.

p ν -(p-1) , p ≥ 1.

On the other hand, for ∈ J 2 , S p ( )

p ∼ p , 0 ≤ p ≤ 1.
, p ≥ 1.

The following result follows immediately from the denition (8).

Corollary 6.10 For ∈ J 2 , the atness satises F ( ) ∼ -1 .

It remains to prove that, as long as |k| remains in a certain range, after layer-averaging, we have {|û(k)| 2 } ∼ |k| -2 . For this, we use a version of the Wiener-Khinchin theorem, stating that for any function v ∈ L (41)

Now it remains to prove the lower bound. We have:

|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 [START_REF] Tadmor | Total variation and error estimates for spectral viscosity approximations[END_REF] and (41) 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 6.9 we obtain that

|n|≤M k n 2 {|û(n)| 2 } ≥ (C -CM -1 )k.
Now we use (40) and we choose M ≥ 1 large enough to obtain (39).

7

Estimates for small-scale quantities in the inviscid limit

It is a well-known fact (see for instance [START_REF] Kreiss | Initial-boundary value problems and the Navier-Stokes equations[END_REF]) that as ν tends to 0, the solutions of (2) converge to weak entropy solutions of the inviscid equation u t + f (u)u x = 0, for xed t. The convergence takes place for almost every x, and therefore also in L 1 , since solutions are uniformly bounded for all ν. These solutions, denoted u 0 , inherit all previously proved properties which hold uniformly for small enough ν for solutions of (2).

To begin with, we dene the non-empty and non-intersecting intervals J 2 = (0, C 2 ]; J 3 = (C 2 , 1], which now correspond to the inertial range and the energy range, respectively.

The inviscid Burgers equation does not have a dissipation range, since formally there is no dissipation, despite the presence of an anomaly due to the shocks [START_REF] Bec | Burgers turbulence[END_REF]. The constant C 2 is the same as above.

Then we dene S 0 p , F 0 and E 0 for solutions u 0 (t, x) in the same way as the previously considered quantities S p , F and E for solutions of the viscous equation. By the dominated convergence theorem, we obtain the following results: , p ≥ 1.

Corollary 7.2 For ∈ J 2 , the atness satises F 0 ( ) ∼ -1 . Theorem 7.3 For k such that k -1 ∈ J 2 , we have E 0 (k) ∼ k -2 .

Figure 1 :

 1 Figure 1: Typical solution of the Burgers equation

  We recall a version of the classical Gagliardo-Nirenberg inequality: cf.[13, Appendix]. Lemma 2.1 For a smooth zero mean value function v on S 1 , |v| β,r ≤ C |v| θ m,p |v| 1-θ q

Figure 2 :

 2 Figure 2: Scales for the 1D Burgers solutions

S 1 (

 1 u(x + ) -u(x)) + dx p 1/p max x |u(x + ) -u(x)| p (p-1)/p ≤C max x |u(x + ) -u(x)| p (p-1)/p ,

  2 one has |v(• + y) -v(•)| 2 = 4 n∈Z sin 2 (πny)|v(n)| 2 .

( 38 ) 1 .

 381 Theorem 6.11 For k such that k -1 ∈ J 2 , we haveE(k) ∼ k -2 .Proof. We recall that by denition[START_REF] Burgers | The nonlinear diusion equation: asymptotic solutions and statistical problems[END_REF],E(k) = |n|∈[M -1 k,M k] |û(n)| 2 |n|∈[M -1 k,M k]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.(39)From now on, we will indicate explicitly the dependence on M . The upper estimate holds without averaging over n such that |n| ∈ [M -1 k, M k]. Indeed, by[START_REF] Boritchev | Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation[END_REF] we know that{|û(n)| 2 } ≤ Cn -2 .Also, this inequality implies that|n|<M -1 k n 2 {|û(n)| 2 } ≤ CM -1 k )| 2 } ≤ CM -1 k -1 .

Theorem 7 . 1

 71 For ∈ J 2 , S 0 p ( ) p ∼ p , 0 ≤ p ≤ 1.

  by |v| p . The L ∞ norm is by denition |v| ∞ = ess sup x∈S 1 |v(x)|. 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 nite norm

	2.1	Sobolev spaces
	Consider a zero mean value integrable function v on S 1 . For p ∈ [1, ∞), we
	denote its L p norm
		1/p
		|v| p
		S 1

The

  Lemma 5.4 For m ∈ {0, 1} and p ∈ [1, ∞], or for m ≥ 2 and p ∈ (1, ∞] we have:Proof. For m ≥ 1 and p ∈ [2, ∞], we interpolate |u(t)| m,p between u(t) m and u(t) m+1 . By Lemma 2.1 applied to u (m) (t), we have:

	|u(t)| m,p	m,p	max(ν -γ , t -γ ).

Acknowledgements I would like to thank my Ph.D. advisor S.Kuksin, who originally formulated the problem. I am also very grateful to A.Biryuk, in particular for bringing to my attention some articles of S.Kruzhkov, and to J.Bec, U.Frisch, F.Golse and K.Khanin for many helpful discussions. Parts of the present work were done during my stays at Laboratoire AGM, University of Cergy-Pontoise, INRIA-IRISA-ENS Cachan Bretagne and DPT, University of Geneva. I have been supported respectively by the grants ERC BLOWDISOL, GEOPARDI and BRIDGES: I would like to thank all the faculty and sta, and especially the principal investigators, respectively F.Merle, E.Faou and J.-P.Eckmann, for their hospitality.

Proof. It suces to prove the lemma's statement for p = 1. But this case follows from the case p = 2. Indeed, by Hölder's inequality and [START_REF] Evans | Partial dierential equations[END_REF] we have:

Integrating by parts in space, we get the dissipation identity

(

Thus, integrating in time and using [START_REF] Bec | Burgers turbulence[END_REF] and Lemma 5.3, we obtain that for t ∈ [T 1 , 3T 1 /2] we have the following uniform lower bound:

Thus,

.

Now we prove a key estimate for { u(t)

Proof. Integrating [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous uid for very large Reynolds number[END_REF] in time in the same way as in [START_REF] Kraichnan | Lagrangian-history statistical theory for Burgers' equation[END_REF], we prove that |u(T 1 )| 2 ≥ D -2 /2. Thus, using (16) (p = 2) we get:

{ u(t)

which proves the lemma's assertion.

This time-averaged lower bound yields similar bounds for other Sobolev norms.

Lemma 5.7 For m ≥ 1, 1) .

Proof. Since the case m = 1 has been treated in the previous lemma, we may assume that m ≥ 2. By (17) and Lemma 2.1, we get:

}.

Thus, using Hölder's inequality and Lemma 5.6, we get: 1) .

The following two results generalise Lemma 5.7.

Proof. The case m = 0 is proved in Lemma 5.5. In the case m = 1, p ≥ 2, it suces to apply Hölder's inequality in place of Lemma 2.1 in the proof of an analogue for Lemma 5.7.

In the case m ≥ 2, the proof is exactly the same as for Lemma 5.7 for p ∈ (1, ∞). In the cases p = 1, ∞, Lemma

. By Hölder's inequality we have:

Using Lemma 5.6 and Lemma 5.4, we get the lemma's assertion.

Lemma 5.9 For m ≥ 0 and p

Proof. As previously, we may assume that p > 1. The case α ≥ 2 follows immediately from Lemma 5.8 and Hölder's inequality. The case α < 2 follows from Hölder's inequality, the case α = 2 and Lemma 5.4 (case α = 3), since we have:

The following theorem sums up the main results of this section, with the exception of Lemma 5.1.

Theorem 5.10 For m ∈ {0, 1} and p ∈ [1, ∞], or for m ≥ 2 and p ∈ (1, ∞] we have:

where

The upper estimates in (28) hold without time-averaging, uniformly for t separated from 0. Namely, we have:

On the other hand, the lower estimates hold for all m ≥ 0 and p ∈ [1, ∞].

Proof. Upper estimates follow from Lemma 5.4, and lower estimates from Lemma 5.9.

Proof. The calculations are almost the same as in the previous lemma.

The only dierence is that we use another bound for the right-hand side of [START_REF] Kreiss | Initial-boundary value problems and the Navier-Stokes equations[END_REF].

Namely, by Theorem 5.10 we have:

Remark 6.3 The Lemmas 6.1 and 6.2 actually hold even if we drop the timeaveraging, since in deriving them we only use upper estimates which hold uniformly for t ≥ T 1 .

To prove the lower estimates for S p ( ), we need a lemma. Loosely speaking, this lemma states that there exists a large enough set L K ⊂ [T 1 , T 2 ] such that for t ∈ L K , several Sobolev norms are of the same order as their time averages. Thus, for t ∈ L K , we can prove the existence of a cli of height at least C and width at least Cν, using some of the arguments in [START_REF] Aurell | On the multifractal properties of the energy dissipation derived from turbulence data[END_REF] which we exposed in the introduction.

Note that in the following denition, (30-31) contain lower and upper estimates, while (32) contains only an upper estimate. The inequality |u(t)| ∞ ≤ max u x (t) in ( 30) always holds, since u(t) has zero mean value and the length of S 1 is 1.

Definition 6.4 For K > 1, we denote by L K the set of all t ∈ [T 1 , T 2 ] such that the assumptions

hold.

Lemma 6.5 There exist constants C, K 1 > 0 such that for K ≥ K 1 , the Lebesgue measure of L K satises λ(L K ) ≥ C.

Proof. We begin by noting that if K ≤ K , then L K ⊂ L K . By Lemma 5.1 and Theorem 5.10, for K large enough the upper estimates in [START_REF] Kruzhkov | The Cauchy Problem in the large for nonlinear equations and for certain quasilinear systems of the rst-order with several variables[END_REF][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] hold for all t. Therefore, if we denote by B K the set of t such that The lower estimates in [START_REF] Kruzhkov | The Cauchy Problem in the large for nonlinear equations and for certain quasilinear systems of the rst-order with several variables[END_REF][START_REF] Kuksin | On turbulence in nonlinear Schrödinger equations[END_REF] hold for a given value of K, then it suces to prove the lemma's statement with B K in place of L K . Now denote by D K the set of t such that The lower estimate in [START_REF] Kuksin | On turbulence in nonlinear Schrödinger equations[END_REF] holds for a given value of K.

By Lemma 2.1 we have:

Thus if D K holds, then B K holds for K large enough. Now it remains to show that there exists C > 0 such that for K large enough, we have the inequality λ(D K ) ≥ C. We clearly have:

Here, 1(A) denotes the indicator function of an event A. On the other hand, by the estimate for {|u| 2 1,∞ } in Theorem 5.10 we get:

Now denote by f the function

).

The inequalities above and the lower estimate for {|u| 1,∞ } in Theorem 5.10

Thus, since |u| 1,∞ ≥ f , we have the inequality

Let us denote by O K ⊂ [T 1 , T 2 ] the set dened as L K , but with the relation [START_REF] Kuksin | On turbulence in nonlinear Schrödinger equations[END_REF] replaced by

Corollary 6.6 For K ≥ K 1 and ν < K -2 1 , we have λ(O K ) ≥ C.

Proof. For K = K 1 and ν < K -2

1 , the estimates (30-31) tell us that

Thus, in this case we have O K = L K , which proves the corollary's assertion. Since increasing K while keeping ν constant increases the measure of O K , for K ≥ K 1 and ν < K -2

and choose

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 ]. Everywhere below the constants depend on K.

Actually, we can choose any values of C 1 , C 2 and ν 0 , provided: 1) , p ≥ 1.

Proof. By Corollary 6.6, it suces to prove that these upper estimates hold uniformly in t for t ∈ O K , with S p ( ) replaced by

Till the end of this proof, we assume that t ∈ O K . Denote by z the leftmost point on S 1 (considered as [0, 1)

In other words, the interval

corresponds to (a part of ) a cli.

Case p ≥ 1. Since ≤ C 1 ν = 1 4 K -2 ν, by Hölder's inequality we get


domain: sdu math phys phys.astr

text:
Alain Chenciner
email: chenciner@imcce.fr

Bernard Leclerc
email: bernard.leclerc@unicaen.fr

Jean-Marie Souriau

Between two moments

In this short note, we draw attention to a relation between two Horn polytopes which is proved in [CJ] as the result on the one side of a deep combinatorial result in [FFLP], on the other side of a simple computation involving complex structures. This suggested an inequality between Littlewood-Richardson coefficients which we prove using the symmetric characterization of these coefficients given in [CL].

1 Moment maps and Horn's problem Recall (see for example [K]) that the mapping H → [k → T race(iHk)] is a natural identification of the space H of Hermitian matrices with the dual u(n) * of the Lie algebra u(n) of the unitary group U (n). With this identification, the coadjoint action of U (n) on u(n) * becomes the action by conjugation on the space of Hermitian matrices and the orbits O λ are uniquely characterized by the common spectrum λ of their elements. A coadjoint orbit being endowed with the Kostant-Souriau symplectic structure, the moment map becomes the canonical inclusion O λ ⊂ H.

It follows from standard properties of moment maps (see again [K]) that with this identification, the moment map of the diagonal action of U (n) on O λ × Oµ is simply the mapping (A, B) → A + B. This gives the relation with the so-called Horn's problem of describing the inequalities which constrain the spectrum of the sum of two hermitian matrices whose spectra are given (see the beautiful review [F]). Kirwan's theorem asserts that, in the case of the diagonal action above, the intersection of the image of the moment map with the principal Weyl chamber

W + n = ν = {ν1 ≥ ν2 ≥ • • • ≥ νn},
that is the set of ordered spectra of matrices of the form A + B where A ∈ O λ and B ∈ Oµ, is a convex polytope, which we shall call the Horn polytope. Moreover, Horn's conjecture, whose proof follows from the works of Klyachko on the one hand, Knutson and Tao on the other hand (see [F]), gives inequalities defining the faces of the Horn polytope.

Finally, replacing Hermitian matrices by symmetric matrices and the unitary group by the orthogonal group does not change the Horn polytope, namely: the set of ordered spectra ν of the sums A + B of matrices with given spectra λ and µ is the same whether A and B are hermitian or real symmetric (see [F]).

Complex structures and a Horn-type problem

Motivated by a study of the angular momenta of rigid body motions in dimensions higher than 3 (see [C, CJ]), let us consider the following "Horn-like" problem: characterize the set Q (noted ImF in [CJ]) of ordered spectra of sums

J -1 S0J + S0,
where S0 is a given 2p × 2p real symmetric matrix and J ∈ O(2p) is a complex structure, that is an isometry of R 2p (with its standard euclidean structure) such that J 2 = -Id.

Replacing the set of complex structures J by the full group of isometries O(2p), one gets exactly the Horn polytope in the case when n = 2p and the two spectra λ and µ coincide with the one of S0.

The main result of [CJ] is that Q is a convex polytope. The proof is not direct: one encloses Q between two convex polytopes P1 and P2 associated with two Horn's problems, one in dimension p and one in dimension 2p, and one deduces from results due to Fomin, Fulton, Li and Poon [FFLP] and based on deep combinatorial lemmas due to Carré and Leclerc [CL], that P1 and P2 coincide, and hence coincide with Q.

3 The two polytopes P 1 and P 2 Given decreasing sequences

σ = (σ1 ≥ σ2 ≥ • • • ≥ σ2p) and ν = (ν1 ≥ • • • ≥ νp) of real numbers, one defines σ-= (σ1, σ3, • • • , σ2p-1), σ+ = (σ2, σ4, • • • , σ2p), ν (2) = (ν1, ν1, ν2, ν2, • • • , νp, νp).
(1) Now, P1 is the Horn polytope for p×p symmetric matrices a+b with spectrum (a) = σ-and spectrum (b) = σ+, while P2 is the intersection with the set ∆ of "hermitian spectra" (that is spectra of the form ν (2) ) of the Horn polytope for 2p×2p symmetric matrices A + B with spectrum (A) = spectrum (B) = σ.

The inclusion P 1 ⊂ P 2

The inclusion P1 ⊂ P2 is elementary: it follows from the inclusions

P1 ⊂ Q ⊂ P2, ( 2
)
where Q is the set of ordered spectra of matrices of the form S + J -1 SJ, where S = diag σ and J is a complex structure:

J = R -1 J0R, R ∈ SO(2p), J0 = 0 -Id Id 0 .
The second inclusion is obvious (replace J by any element R ∈ SO(2p) and look for the spectra of sums S + R -1 SR which are J-hermitian for some J;

The first one comes from the fact that P1 is exactly the subset of Im F obtained when one takes into account only the complex structures J which send the subspace of R 2p generated by the basis vectors e1, e3, • • • , e2p-1 onto the orthogonal subspace, generated by{e2, e4, • • • , e2p}. More precisely, it comes from the following identity where ρ ∈ SO(p):

           σ-0 0 σ+ + 0 -ρ -1 ρ 0 -1 σ-0 0 σ+ 0 -ρ -1 ρ 0 = σ-+ ρ -1 σ+ρ 0 0 ρσ-ρ -1 + σ+ • (3)

Littlewood-Richardson coefficients

From now on, the decreasing sequences that we consider consist in non negative integers, which allows them to take the name of "partitions". One calls |λ| = λi the "length" of the partition.

The so-called Littlewood-Richardson coefficients have their origin in the decomposition into irreducible holomorphic representations of the tensor product of two irreducible representations of the linear group GL(n, C). For their definition and their many avatars, as well as for the computational rule expressing them in terms of the Ferrer-Young diagrams representing partitions, we refer to [F]. Hence, identifiying ∆ with a subset of R p by the mapping ν (2) → ν:

Let α = {α1 ≥ α2 ≥ • • • ≥ αn}, β = {β1 ≥ β2 ≥ • • • ≥ βn}, γ = {γ1 ≥ γ2 ≥ • • • ≥ γn},
   P1 = ν = (ν1 ≥ ν2 ≥ • • • ≥ νp), c ν σ -σ + = 0 , P2 = ν = (ν1 ≥ ν2 ≥ • • • ≥ νp), c ν (2) σσ = 0 .
(4)

6 Yamanouchi tableaux and the Carré-Leclerc formula

For the convenience of the reader, we reproduce here definitions given in sections 4.1 and 4.2 of [FFLP].

It is classical to define a bijection Λ between strictly increasing sequences

I = (i1 < i2 < • • • < ir) and partitions Λ(I) = (λ1 ≥ • • • ≥ λr) by the formulae λ1 = ir -r, • • • , λr = i1 -1.
Given two strictly increasing sequences, one defines

τ (I, J) = (2i1 -1, • • • , 2ir -1) ∪ (2j1, • • • , 2jr) reordered.
To a partition λ is associated a Young diagram of shape λ: this is a collection of boxes arranged in rows whose lengths from top to bottom (anglo-saxon convention) are λ1, λ2, • • • The induced operation on partitions, (λ, µ) → τ (λ, µ) is described in [FFLP] in terms of Young diagrams in the following way: "If one traces the Young diagram of a partition by a sequence of horizontal and vertical steps moving from Southwest to Northeast in a rectangle containing the diagrams of λ and µ, the diagram of τ (λ, µ) is traced in a rectangle twice as wide in both directions, by alternating steps from λ and µ, starting with the first step of λ". It is also recalled in this paper that partitions τ (λ, µ) are exactly the ones which correspond to dominodecomposable Young tableaux, that is tableaux which can be partitioned into disjoint 1×2 or 2×1 rectangles (the so-called dominoes). A (semi-standard) domino tableau is such a decomposition of a tableau into dominos with a labelling rule of the dominos similar to the one of ordinary semi-standard tableaux: labels weakly increase from left to right along lines and strictly increase along columns from top to bottom. Special domino tableaux, called Yamanouchi, play the leading part: let the reading word of a tableau be obtained by listing the labels column by column from right to left and from top to bottom (a horizontal domino is skipped the first time it is encountered); a tableau is called Yamanouchi if every entry i appears in any initial segment of its reading word at least as many times as any entry j > i. In [CL], the following rule for computing the Littlewood-Richardson coefficients is given in which, in contrast with the original Littlewood-Richardson rule, the lower indices play symmetric roles: let the weight ν of a Yamanouchi tableau be the sequence (ν1, • • • , νn) where νi is the number of labels equal to i. Note that ν1 ≥ • • • ≥ νn.

Proposition 1 ( [CL] Corollary 4.4). The coefficient c ν λµ is equal to the number of Yamanouchi domino tableaux of shape τ (λ, µ) and weight ν.

Pay attention that, while c ν λµ = c ν µλ , the diagrams τ (λ, µ) and τ (µ, λ) have in general different shapes. This will be important in the sequel.

7 Strengthening the inclusion P 1 ⊂ P 2

The following proposition strengthens the inclusion P1 ⊂ P2 while it was suggested by it:

Proposition 2. Given partitions σ = {σ1, σ2, • • • , σ2p} and ν = {ν1, ν2, • • • , νp}, we have c ν σ -σ + ≤ c ν (2)
σσ . Proof. As in [FFLP], we use Proposition 1. We must define an injection of the set of Yamanouchi domino tableaux of shape τ (σ-, σ+) (or of shape τ (σ+, σ-)) and weight ν into the set of Yamanouchi domino tableaux of shape τ (σ, σ) and weight ν (2) . In order to prove the above lemma, the choice of τ (σ+, σ-) is much more convenient because it is simply related to τ (σ, σ) while this is not the case of τ (σ-, σ+), more precisely, if σ = (σ1, σ2,

• • • , σ2p), τ (σ+, σ-) = (2σ1, 2σ2, • • • , 2σ2p), τ (σ, σ) = (2σ1, 2σ1, 2σ2, 2σ2, • • • , 2σ2p, 2σ2p).
In words, the tableau τ (σ, σ) is obtained from the tableau τ (σ+, σ-) by duplicating each line. The injection we are looking for is then simply obtained by dividing vertically each domino of τ (σ+, σ-) and numbering the resulting two pieces respectively 2k -1 and 2k if the original domino was numbered k. One checks immediately that the Yamanouchi property is verified. An example is given in §10 below, see Figure 1 and Figure 2.

8 Concluding remarks 1) As was said at the beginning, the inclusion P1 ⊃ P2, and hence the equality P1 = P2, is proved in [CJ]. It results from the stronger inclusion in P1 of the orthogonal projection on ∆ of the Horn polytope

P = {γ = (γ1 ≥ • • • ≥ γ2p), c γ σσ = 0} .
This last inclusion follows from the inequalities c ν λµ ≤ c τ (ν,ν) τ (λ,µ)τ (λ,µ) , valid as soon as the lengths of the partitions at stake satisfy |ν| = |λ| + |µ|, ([FFLP] Proposition 4.5). The main ingredient of the proof is again the symmetric characterization of the Littlewood-Richardson coefficients in terms of Yamanouchi tableaux given in [CL]. The equality P1 = P2 of the two polytopes has the consequence, not implied by the inequality of lemma 2, that c ν (2) σσ = 0 implies c ν σσ + = 0. Nevertheless, the inequality given by Proposition 2 may be strict. An example is shown in §10, Figure 3.

2) Dividing σ into two partitions λ and µ of length p in an arbitrary way leads to polytopes strictly smaller than P1 = P2. Correspondingly, we have the following inequalities strengthening these inclusions:

c ν λµ ≤ c ν σ -σ + .
These inequalities are a special case of Corollary 14 of [LPP].

1) Find a more conceptual relation between the complex structures and the doubling of Young tableaux.

2) Find a direct proof of the fact that Q is convex, resp. a convex polytope.

3) Find a relation between the inequality which is the object of Proposition 2 and the one in [FFLP] (Proposition 4.5). In the first case, each domino is divided into 2 pieces while in the second one it is divided into 4 pieces.

Examples

We take σ = (5, 3, 2, 0), so that σ+ = (3, 0) and σ-= (5, 2). We then have τ (σ+, σ-) = (10, 6, 4, 0), τ (σ, σ) = (10, 10, 6, 6, 4, 4, 0, 0).

In Figure 1 below, we display the Yamanouchi domino tableaux T1, T2, T3, T4 of shape τ (σ+, σ-), and of respective weights ν1 = (5, 5), ν2 = (6, 4), ν3 = (7, 3), ν4 = (8, 2).

Their respective reading words are w1 = 1112212212, w2 = 1112212112, w3 = 1112112112, w4 = 1111112112.

In Figure 2 we display the corresponding Yamanouchi domino tableaux U1, U2, U3, U4 of shape τ (σ, σ), obtained from T1, T2, T3, T4 via the duplication procedure of Proposition 2. Finally, Figure 3 gives an example which shows that the inequality of Proposition 2 may be strict. Here we take σ = (7, 6, 4, 3) and ν = (10, 8, 2), and we exhibit a Yamanouchi domino tableau T of shape τ (σ, σ) and weight ν (2) which cannot be obtained from a Yamanouchi domino tableau of shape τ (σ+, σ-) and weight ν by means of the duplication procedure described in the proof of Proposition 2. Fig. 3. A Yamanouchi tableau of shape τ (σ, σ) and weight ν (2) which is not obtained by the duplication procedure of Proposition 2.

  be three partitions. It results from the proof of Horn's conjecture that α, β, γ are respectively the spectra of n × n hermitian (or real symmetric) matrices A, B, C with C = A + B if and only if ([F], theorem 11) 1)|γ| = |α| + |β|, 2) the Littlewood-Richardson coefficient c γ αβ does not vanish.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Yamanouchi domino tableaux of shape τ (σ+, σ-).