est destinée au dépôt

The current research focuses on multi-scale assessment of the pharmacological and functional properties of spider toxins previously selected by screening various venoms on Na V 1.7 subtype voltage-gated sodium channel, genetically validated painkiller target. After the synthesis of toxin hits, the studies involves electrophysiological approaches such as manual and automated patch-clamp for in vitro studies of mouse and human cells, intracellular micro-electrode recordings of ex vivo mouse neuromuscular junctions, and threshold-tracking techniques with the QTRAC software for in vivo studies of the mouse neuromuscular excitability, as well as structures / functions evaluation. These different approaches will allow:

(1) To characterize the selectivity profile of toxins identified as active in the desired therapeutic targets (i) on cell lines expressing the functional subunit of these targets, but also of various other subtypes of channels and receptors, and (ii) on a more integrated scale in native systems such as cardiomyocytes, brain slices and the neuromuscular system of mammals (mice and /or rats).

(2) To identify their mechanism of action and effectiveness (compared to some reference ligands reported in the literature) on validated targets in the area of pain, using native systems such as (i) cultured neurons of the dorsal root ganglion (DRG) isolated from adult mice, (ii) and human induced pluripotent stem (IPS) cells.

(3) To determine the pharmacophore, i.e. the crucial amino acids in the toxin activity and effectiveness, in order to pave the way for engineering to optimize their functional properties.

(4) To evaluate the potential antinociceptive effect of toxins by studying the perception of pain in vigils animals (mice and/or rats).

A

  multiscale functional approach to appraise the pharmacological interactions between spider venom peptides and Nav1.7, a genetically validated analgesic target Tânia C. Gonçalves, Michel Partiseti, Denis Servent, Evelyne Benoit To cite this version: Tânia C. Gonçalves, Michel Partiseti, Denis Servent, Evelyne Benoit. A multiscale functional approach to appraise the pharmacological interactions between spider venom peptides and Nav1.7, a genetically validated analgesic target. YRLS 2018 -9th Young Researchers in Life Science, May 2018, Paris, France. . hal-04466900 A multiscale functional approach to appraise the pharmacological interactions between spider venom peptides and Na V 1.7, a genetically validated analgesic target Tânia C. Gonçalves a,b , Michel Partiseti a , Denis Servent b , Evelyne Benoit b,c a Sanofi Aventis R & D, Integrated Drug Discovery -High Content Biology, F-94440 Vitry-sur-Seine, France b Service d'Ingénierie Moléculaire des Protéines (SIMOPRO), CEA, Université Paris-Saclay, F-91191 Gif sur Yvette, France c Institut des Neurosciences Paris-Saclay (Neuro-PSI), UMR CNRS/Université Paris-Sud 9197, Université Paris-Saclay, F-91198 Gif sur Yvette, FranceAbstract:


C-A Roulet

M Germano

F Allard

C Ghiaus

POTENTIAL FOR NATURAL VENTILATION IN URBAN CONTEXT: AN ASSESSMENT METHOD

Keywords: Natural ventilation, Potential, GIS, Multicriteria analysis, Morphological Indicators, ELECTRE

The natural ventilation potential (NVP) is the possibility to ensure an acceptable indoor air quality by natural ventilation only. The passive cooling potential (PCP) is the possibility to ensure an acceptable summer comfort using natural ventilation. NVP and PCP hang on many conditions depending on the site (outdoor air quality, outdoor air temperature and moisture, outdoor noise, local winds or global wind and urban structure, etc.) or on the building (indoor pollutant sources, indoor heat sources and stored heat, indoor air quality requirements, position and size of ventilation openings, indoor temperature, orientation of building, internal air path distribution, etc.) These potentials are expressed as a list of quantitative and qualitative characteristics of the site or the building, based on available input data. No detailed modelling is required for this assessment.

The contribution presents the method developed to assess the NVP, and to represent it using geographically referenced information.

INTRODUCTION

A large majority of the publications on natural ventilation deal with its behaviour within a single building, like single and multi-zone models, possibly coupled with thermal models [START_REF] Orme | Applicable Models for Air Infiltration and Ventilation Calculations[END_REF]. The aim of the current article is to assist the designer in the early stages of design, if possible before the construction of a building. More precisely, it aims at providing information on the issue of 'where to build' more than on 'how to build'. This is why an entire part of the urban fabric, rather than a single edifice like in the mentioned publications, has to be considered. In fact, the current investigation studies natural ventilation at mesoscale, i.e. at the scale of the building-environment interaction in urban context : local heterogeneity is hence set aside. Moreover, it avoids too complex modelling such as CFD, because the boundary conditions are never known precisely, because of the chaotic nature of the urban flow, and because it resorts to pressure coefficients, which are seldom known and, if the case arises, are hardly usable to calculate infiltration or ventilation rate.

DEFINITION OF THE NATURAL VENTILATION POTENTIAL

The natural ventilation potential (NVP) is the possibility to ensure an acceptable indoor air quality by natural ventilation only. This definition assumes that "acceptable indoor air quality" is defined. There are many definitions available, issued either by institutions such as WHO or ISIAQ, or research programmes such as COST613 (COST 613, 1991).

The passive cooling potential (PCP) is the possibility to ensure an acceptable indoor thermal comfort using natural ventilation. The acceptable indoor thermal comfort is defined by international standards such as ISO 7730 (EN-ISO-7730, 1993).

DEFINITION OF THE SITE NATURAL VENTILATION POTENTIAL

We define the potential of the site as the potential of having good ventilation, provided that an appropriate building is or will be built on. In other words, it tells whether the site is suitable for the construction of a well naturally ventilated building.

SITE POTENTIAL AND MULTICRITERIA EVALUATION

The NVP of the site depends on several criteria. These are: Meteorological criteria:

• macroscale wind speed distribution and orientation The mentioned morphological indicators (ninth row) are a set of criteria introduced hereafter.

As the enumerated criteria are of very different types, it would not be realistic to gather them in a single equation. This is why a multicriteria evaluation seems more relevant for the assessment of the NVP.

These criteria may be grouped and each of them associated with veto, indifference and preference thresholds and weights. This important point is where physics and physical values come into play.

METHOD

In order to have an assessment of the NVP of the site, one useful tool is geographic information systems (GIS) [START_REF] Maguire | Geographical information systems : principles and applications[END_REF], because the intention is to compare various geographical locations.

GIS will be used at two stages. First, the relevant data for the evaluation of NVP are chosen.

Particularly, they comprehend weather data, the layout and the height of the buildings, the length and the width of the streets, and so on. At a second stage, once the multicriteria evaluation is processed, the NVP at each location can be represented as a new GIS layer.

Between both stages, each GIS information layer is seen as a criterion and takes part in a multicriteria evaluation, whose outcome is the NVP.

Proceedings: Indoor Air 2002

METEOROLOGICAL DATA

The meteorological data used are provided by Swiss Federal Office of Energy's work METEONORM (1999). It comprehends amongst others a software and a data source on a vast number of meteorological data covering Switzerland and other locations around the world.

The software provides an interpolation facility allowing to describe meteorological conditions between weather stations. Monthly GIS maps have been derived for wind speed, wind direction and air temperature. The choice of the wind is justified by the fact that wind pressure is one of the driving forces of natural ventilation. The other driving force is stack effect, resulting from air density differences, in their turn provoked by temperature differences and humidity differences. This is why air temperature was selected, too. Night cooling is considered as well.

MORPHOLOGICAL INDICATORS

If we intend not to work at a too close scale ant not to look at the urban fabric details, an idea is to assimilate the urban fabric to a porous medium with a rigid skeleton. This is a way to get rid of local heterogeneity. The so-called morphological indicators (Adolphe, 1999) fulfil this objective. They are helpful in the sense that they were designed to be compatible with several requirements which allow them to be used in practice, namely completeness, non-redundancy and operationability. As a matter of fact, the morphological indicators model was developed in order to assess NVP, while remaining compatible with the usual existing urban databases. This model is in the process of being validated on three sites in the South of France (Toulouse, Blagnac and Marseille) by comparisons with on-site measurements and with CFD calculations.

Another advantage of these indicators is their help in deducting urban climate conditions from the conditions outside the city. Usually indeed the weather stations are located in the suburbs. The list of the morphological indicators is the following:

• absolute rugosity: mean height of the building These indicators -except solar admittance -have been implemented in MapBasic ® , the programming language of the GIS software MapInfo Professional ® . Each indicator can be implemented over a city area.

MULTICRITERIA EVALUATION

The expected result of a multicriteria evaluation is a decision or a group of decisions to be taken (Schärlig, 1990ab). In our case, the decisions to be taken are 'where to build', which will be expressed here on a map by squares of different colours telling how good the potential for natural ventilation is. In other words, the decisions -or actions -of our multicriteria evaluation consist of the locations in the city.

Proceedings: Indoor Air 2002

As they are numerous, it makes it difficult to use the multicriteria method referred to as ELECTRE III [START_REF] Maystre | Méthodes multicritères ELECTRE[END_REF] -one of the methods usually used for such an evaluation -because it proceeds to a classification of the entire set of decisions at the same time, implying too much calculation.

Another method, ELECTRE TRI (Yu, 1992ab;[START_REF] Maystre | Méthodes multicritères ELECTRE[END_REF] -also referred to as ELECTRE BETA -, seems more relevant because it proceeds to an assignment of the decisions to a category. Namely, two base decisions -or more -are chosen and the other decisions classified by comparison with them. Therefore, the number of decisions can be very high, because they are evaluated independently. This consists in the so-called βmethodology. The drawback of this method is the choice of the two base decisions. They can be chosen in accordance with the experience, i.e. with the knowledge of well-ventilated sites.

Alternatively, a preliminary study can be done with ELECTRE III and a small number of locations to select two appropriate base decisions.

RESULTS

ELECTRE TRI has been implemented in MapBasic and can be run for the time being over the canton of Geneva area, in Switzerland. The user of the program has the possibility to choose the month of the evaluation, the region he intends to evaluate, the size of the squares delimiting each location-decision, the criteria to include in the evaluation and the two base locations-decisions. The result is displayed on two maps. The first one is the outcome of the so-called pessimistic assignment procedure of ELECTRE TRI and the second one that of its optimistic assignment procedure. The pessimistic assignment procedure tries to assign each decision to the lowest category, in our case the category corresponding to the worst NVP, whereas the optimistic does the same towards the highest category.

In Figure 1, the two bold squares stand for the two base decisions, cross-hatched squares for the areas classified below the worst base decision, lined squares for these classified between the two base decisions and blank squares for these classified above the best base decision.

The map corresponding to the pessimistic assignment procedure (not shown here) is quite similar but exhibits a little less blank areas.

DISCUSSION

It has to be made clear that the current work is a first step in the assessment of the natural ventilation potential and that it concerns only the site NVP as it is defined in the fifth section. It does not tell whether an edifice in the area of concern is well ventilated or not, but rather whether it is possible or not to build a well ventilated building there or whether an existing building of this area can be well ventilated at the expense of possible modifications.

The multicriteria analysis ELECTRE TRI gives several sets of indices (in our software in the message window of MapInfo) -concordance indices, global concordance indices, discordance indices and credibility degrees -that should tell the user which of the criteria tipped the scales in favour or against a good NVP. The designer should then be able to choose the appropriate ventilation strategy -e.g. stack effect use -for the erection of a new building or for the refurbishment of an old one. Note that, up to the end of 2001, no validation has been undergone by comparison with on-site measurements.

A domain of applicability has to be defined for the scales that can be used: the morphological indicators model imply that the squares of assessment (Figure 2) must not be too small because it does not make sense to calculate these indicators -i.e. working out a sort of average on the urban fabric -in such a case. On the other hand, they must not be too large because this would generate a loss of the local information of the site.

CONCLUSION AND IMPLICATIONS

The presently proposed methodology and software should be able to help the designer to select the appropriate location to erect a subsequently well ventilated building or the location of an old building having the possibility, after refurbishment, to be well ventilated. It should be kept in mind that a new construction can itself affect the ventilation properties of an area.

However, one of the main drawbacks of this methodology is the lack of GIS data currently available around the world. The case of Geneva was chosen on the grounds of the existence of the required data for this town. Nevertheless, we know that the same type of data as Geneva will soon be available for the whole Switzerland and there is good hope that it will for the main European cities. Even if the case does not arise, there is still the possibility to generate the relevant data manually.

Figure 1 -

 1 Figure 1 -Map of the site natural ventilation potential, optimistic assignment procedure : canton of Geneva, Switzerland

* Contact author email: claude.roulet@epfl.ch Proceedings: Indoor Air 2002

Proceedings: Indoor Air 2002

ACKNOWLEDGMENTS

We would like to thank the Swiss Federal Office for Education and Science, which made this investigation possible by its financial support. We also wish to thank the Service des systèmes d'information et de géomatique of Geneva, which kindly provided us with the entire set of GIS data of the canton of Geneva that we needed, notably with Geneva's building heights.

Proceedings: Indoor Air 2002


Tânia C Gonçalves

Evelyne Benoit

Michael Kurz

Laetitia Lucarain

Sophie Fouconnier

Stéphanie Combemale

Lucie Jaquillard

Brigitte Schombert

Rachid Boukaiba

Gerhard Hessler

Cyriotoxin-1a, the first toxin with antinociceptive properties purified from Cyriopagopus schioedtei spider venom: from identification

Cyriotoxin-1a, the first toxin with antinociceptive properties purified from Cyriopagopus schioedtei spider venom: from identification to structural and functional characterization Inhibitory toxins : potential painkillers targeting Na V 1.7 Na V 1.7 : an antinociceptive target genetically validated Spider venom provides an amount of highly specific inhibitory cystine knots (ICK)-patterned toxins inhibiting Na V 1.7 with a natural selectivity against Na V 1.5 (> 2 logs). The BDS-1 and the Ssm6a are provided from sea anemone and Chinese red-headed centipede venoms, respectively.

In-vitro venom screening workflow

The peptide exhibits nanomolar range affinity for Na V 1.7, Na V 1.1-1.3 and Na V 1.6 subtypes and micromolar range affinity for Na V 1.5 and Na V 1.4 subtypes in HEK293-Na V engineered cell lines.

In-vitro selectivity characterization of CyrTx-1a

Take home messages

• As Na V 1.7 is involved in a multitude of genetic pain disorders, it seems obvious that this sodium channel subtype might be an important antinociceptive target.

• Toxins from spider venoms are valuable tools with characteristics of high affinity against Na V 1.7 and high selectivity against Na V 1.5. The studies of all these toxin sequences will help us better understand which amino acids are crucial for the binding on Na V 1.7 and which could increase selectivity against closely-related ion channels.

• CyrTx-1a is an promising toxin targeting Na V 1.7 with a nanomolar potency (selective vs. TTX-R NaV channels) and exhibiting antinociceptive in-vivo effects. However some structure-activity relationship studies are required to exclude the muscle contraction side-effects probably caused by the high affinity of CyrTx-1a for Na V 1.6 expressed in nodes of Ranvier of alpha motoneurons. From the venom library tested, only 8% hits versus hNa V 1.7 came from scorpion venom, 6% from snake while the larger majority (86%) was spider fractions (F). (G) Hits were sub-fractioned and tested again versus hNa V 1.7 and hNa V 1.5 to identify most promising hits to be further characterized using whole-cell patch-clamp aPC (QPatch, Sophion).

ProTx-II

IC 50 (hNa V 1.7) = 0.3 nM Pn3a IC 50 (hNa V 1.7) = 0.9 nM Df1a IC 50 (hNa V 1.7) = 1.9 nM ProTx-III

IC 50 (hNa V 1.7) = 2.5 nM CcoTx-1Tx-1 IC 50 (hNa V 1.7) = 2.0 nM BDS-1 IC 50 (hNa V 1.7) = 3.0 nM GpTx-1 IC 50 (hNa V 1.7) = 4.4 nM Cd1a IC 50 (hNa V 1.7) = 16.0 nM HwTx-IV IC 50 (hNa V 1.7) = 17.0 nM HnTx-IV IC 50 (hNa V 1.7) = 21.0 nM Ssm6a IC 50 (hNa V 1.7) = 25.0 nM GsAFI IC 50 (hNa V 1.7) = 40.0 nM ProTx-I ICv(hNa V 1.7) = 51.0 nM Hd1a ICv(hNa V 1.7) = 110 nM HnTx-III IC 50 (hNa V 1.7) = 232 nM GrTx-1 IC 50 (hNa V 1.7) = 370 nM Phlo 1a, 1b, 2a IC 50 (hNa V 1.7) ≈ 400 nM JzTx-34 IC 50 (hNa V 1.7) = 610 nM

CyrTx-1a

IC 50 (hNa v 1.7) = 59.0 nM (Namudarai et al., 2015) JzTx-V IC 50 (hNa V 1.7) = 0.6 nM

• Evaluation of painkiller effects

Na V 1

 1 .7 is mainly expressed in dorsal root ganglia (DRG) neurons from peripheral to central nerve terminals (PNS). The secondary expression takes place in olfactory sensory nerves (PNS), in the pituitary gland and the subfornical organ as well as several hypothalamic nuclei (CNS) (cartoon from Lai et al. 2006) Human Na V 1.7 genetic alterations are linked to pain : • Na V 1.7 loss of function is the root of congenital insensivity to pain (CIP) or hereditary sensory and autonomic neuropathy (HSAN) type IID. • Na V 1.7 gain of function can cause hypersensibility to pain , primary erythromelalgia (PE), paroxysmal extreme pain disorder (PEPD), as well as small fiber neuropathy.

•

  Evaluation of muscle side-effects The hot plate (A, thermic pain) and Von Frey (B, mechanic pain) assays are acute pain tests. The hot plate and von Frey pain assays showed that mice injected with 367 µg/kg of CyrTx-1a or with 200 µg/kg of HwTx-IV were more resistant to pain than animals injected with the vehicle (PBS). The compound muscle action potential (C) assessment of both toxin reveals a narrow therapeutic window (D) with lethality occurring after the painkiller dose probably due to muscles (including diaphragm) paralysis.

Sanofi

  Aventis R & D, Integrated Drug Discovery -High Content Biology, Vitry-sur-Seine, France a / Synthetic Molecular Design, Frankfurt, Germany b Service d'Ingénierie Moléculaire des Protéines (SIMOPRO), CEA, Université Paris-Saclay, Gif sur Yvette, France c Institut des Neurosciences Paris-Saclay (Neuro-PSI), UMR CNRS/Université Paris-Sud 9197, Université Paris-Saclay, Gif sur Yvette, France d Sanofi R & D, Neuroscience Therapeutic Area, Neurodegeneration Research, F-91385, Chilly-Mazarin, France e Smartox Biotechnology, 6 rue des Platanes, F-38120 Saint-Egrève, France f T. Gonçalves a,c , E. Benoit c,d , M. Kurtz b , L. Lucarain a , S. Fouconnier a , S Combemale d , L. Jaquillard d , B. Schombert a , R. Boukaiba a , G. Hessler d , A. Bohme a , L. Bialy e , S. Hourcade e , R. Béroud f , M. De Waard f , D. Servent c,d , M. Partiseti a & JM. of 0.05 μg of each of the 64 fractions obtained from C. schioedtei spider venom. (I) Each fraction was then separated again to isolate one peptide per well. Sub-fractions were tested again in our IWQ assay. At this stage, hNa V 1.7 (red squares) and hNa V 1.5 (black circles) inhibition was investigated. (A) Collection of venoms was prepared into fractions individually added to 384-well plates (columns 1 and 2 are TTX full block and columns 3 and 4 are maximal current obtained in extracellular buffer) for testing vs. hNa V 1.7 on the Ionworks Quattro (IWQ, Molecular Devices). The train protocol (described as an insert) was applied before and after 10-min fraction incubation. Traces of pre-and post-scan are shown in full (B). Tonic block obtained on hNa V 1.7 has been enlarged (C). Source of the screened crude venom is shown in (D). Percentage of inhibition for all tested fractions was plotted. As shown in (E), a threshold was set at 50% block (red line).

Fractions

  form a positive zone. (K) Superposition of backbone peptide folding of CyrTx-1a and three other toxins of the NaSpTx family 1 previously described to possess analgesic effects (PDB entries of HnTx-IV: 1NIY, ProTx-III: 2MXM and HwTx-IV: 1MB6).


Peng Gao

Sergei Kuksin

LAW FOR STOCHASTIC BURGERS EQUATION

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The 4/5 law

The Kolmogorov theory of turbulence, known as the K41 theory (see in [START_REF] Landau | Fluid Mechanics[END_REF][START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]), examines homogeneous turbulence, corresponding to velocity fields u(t, x) which are random fields, stationary in time t, homogeneous and isotropic in the space-variable x. Using dimension analysis and arguing on physical level of rigour, Kolmogorov made a number of remarkable predictions concerning small-scale properties of velocity fields with large Reynolds numbers R, corresponding to increments u(t, x + r) -u(t, x). Here the vectors r are such that their lengths r belong to the inertial range, which is an interval in R + , formed by real numbers which are "small but not too small" in term of R and the rate of dissipation of energy ε = νE ∇u(t, x) 2 , where ν > 0 is the kinematic viscosity of the fluid. One of these predictions is the 4/5-law, stating that for large R and for r from the inertial range (1.1) E u(t, x + r) -u(t, x) ⋅ r r 3 = -4 5 ε r .

Later the law was intensively discussed by physicists and was re-proved, using physical arguments, always related to the original Kolmogorov's arguing (see [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]Sec. 6.2] and [START_REF] Falkovich | Fluid Mechanics[END_REF]Sec. 2.2.2]). Rigorous verification of the 4/5-law and other laws of the K41 theory remains an outstanding open mathematical problem. Recently a progress was achieved in [START_REF] Bedrossian | A sufficient condition for the Kolmogorov 4/5 law for stationary martingale solutions to the 3d Navier-Stokes equations[END_REF]. There -as it is often the case since 1960's -3d turbulent flows are modelled by solutions of the stochastic 3d NSE with small viscosity ν > 0 on the torus T 3 . It is known that the latter equation has stationary solutions. Taking such a solutions u ν (t, x) and assuming that it meets the assumption νE u(t) 2 L 2 = o(1) as ν → 0 they prove that (1.1) holds after averaging in r over a sphere of radius r . The relation is established for all r from an interval in R + whose left end goes to zero with ν, but whose relation with the inertial range is not clear.

In order to understand better turbulence and the laws of the K41 theory, starting 1940's physicists use stochastic 1d models, where the most popular one is given by the stochastic space-periodic Burgers equation, see [START_REF] Fournier | L'équation de Burgers déterministe et statistique[END_REF][START_REF] Bec | Les Houches 2000: New Trends in Turbulence[END_REF][START_REF] Bec | Burgers turbulence[END_REF]. The equation describes fictitious 1d "burgers fluid" and turbulence in it, called by U. Frisch burgulence. Our goal in this work is to rigorously derive for this equation two relations, which may be regarded as weak and strong forms of the 4/5-law (1.1).

In the next Section 2 we discuss the stochastic space-periodic Burgers equation, following the book [START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF]. There we develop the notation and state properties of the equation's solutions, used in the rest of the paper. Then in Section 3 we prove a weak form of the 4/5-law for the Burgers equation. There we show that for any solution u(t, x) of the equation, its cubed increments (u(t, x + l) -u(t, x)) 3 with l from the inertial range for burgulence, averaged in x, in ensemble and locally averaged in time, behave as -Const l, uniformly in small non-negative viscosities. See relations (3.4), (3.5).

In Section 4 we prove a strong form of the 4/5-law for the Burgers equation. Namely, there in Theorem 4.1 we show that if u ν st (t, x) is a stationary in time solution of the equation with a positive viscosity ν 1 and ε B = νE ∫ u x (t, x) 2 dx is its rate of dissipation of energy, then, for any t,

(1.2) E (u(t, x + l) -u(t, x) 3 dx = -12ε B l + o(l) as ν → 0.
The relation is proved to hold for l from a strongly inertial range, which is "just a bit smaller" than the inertial range for burgulence. The latter is the segment [c 1 ν, c], where the constants c 1 , c 2 depend on the force, applied to the "burgers fluid" (see a discussion after Theorem 3.1), and the strongly inertial range is defined in (4.8). In Corollary 4.2 we show that (1.2) as well holds for any solution of Burgers equation, asymptotically as t → ∞. We also prove in Theorem 4.1 that a stationary solution of the inviscid Burgers equation satisfies (1.2) with removed limit "ν → 0" and o(l) replaced by O(l 3 ). Relation (1.2) is the form in which the "4/5-law for burgulence" appears in works of physicists, justified by heuristic arguments. E.g. see [START_REF] Polyakov | Turbulence without pressure[END_REF][START_REF] Diamond | A couse on fluid dynamics[END_REF].

In Section 5 we discuss the criticism of possible universality of the K41 theory, made by Landau. It implies (on the physical level of rigour) that in the frame of K41 the only universal relation for moments of increments u(t, x + r) -u(t, x) is the 4/5-law (1.1) for cubic moments. We state a reformulation for burgulence of the Landau claim and rigorously prove it.

Notation. For a metric space M we denote by P(M ) the set of probability Borel measures on M , for a random variable ξ, valued in M , denote by D(ξ) ∈ P(M ) its distribution, and for a function f and a measure µ on M denote ⟨f, µ⟩ = ∫ M f dµ. The arrow ⇀ stands for weak convergence of measures.

Stochastic Burgers equation

2.1. The setting and well posedness of the equation. The stochastic Burgers equation under periodic boundary conditions has the following form:

u t (t, x) + u(t, x)u x (t, x) -νu xx (t, x) = ∂ t ξ(t, x), t ≥ 0, x ∈ S 1 ∶= R Z, (2.1)
where the viscosity coefficient satisfies ν ∈ (0, 1], and

(2.2) ξ(t, x) ∶= s∈Z * b s β s (t)e s (x).
Here {b s } are real numbers, {β s } are standard independent Wiener processes defined on a probability space (Ω, F, P), and {e s (x), s ∈ Z * = Z∖{0}} is an usual trigonometric basic in the space of 1-periodic functions with zero mean-value. Namely, for a k ≥ 1, e k (x) = √ 2 cos(2πkx) and e -k (x) = √ 2 sin(2πkx). We suppose that the process ξ is non-zero and is sufficiently smooth in x:

(2.3) B 0 > 0, B M < ∞ for some M ≥ 5, where B m ∶= 2πs 2m b 2 s ≤ ∞. Since always ν ≤ 1, then below ν > 0 stands for 0 < ν ≤ 1.
1 All such solutions have the same distribution.

As the space-meanvalue of ξ(t, x) is zero, then ∫ S 1 u(t, x) dx is an integral of motion for the equation, and we always assume that it vanishes: u(t, x) dx ≡ 0.

For short we will write the solutions u ν (t, x) of (2.1) as u ν (t) or just u(t). If u 0 is a random field, independent from ξ (e.g. u 0 is non-random), then by u ν (t, x; u 0 ) = u ν (t; u 0 ) we will denote a solution of (2.1), equal u 0 at t = 0. By H we denote the space of L 2 -functions on S 1 with zero mean, given the L 2 -norm ⋅ (so {e s } is its Hilbert basis). For m ∈ N ∪ {0} we denote by H m the Sobolev space

H m = {v ∈ H ∶ ∂ m v ∈ H},
given the homogeneous norm u m = ∂ m u (so u 0 = u ). By ⋅ p , 1 ≤ p ≤ ∞, we denote the L p -norm on S 1 , and abbreviate L p (S 1 ) to L p .

It is well known that under assumption (2.3) eq. ( 2.1) is well posed in spaces H m , 1 ≤ m ≤ M , and defines in spaces H m with 1 ≤ m ≤ M -1 Markov processes with continuous trajectories. E.g. see [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF][START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF] and references in [START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF]. Moreover, if

u 0 ∈ H 1 is non-random, then (2.4) E exp(σ u ν (t; u 0 ) 2 ) ≤ C
for some σ, C > 0 (depending on u 0 , ν and the random force

). If u 0 ∈ H m , 1 ≤ m ≤ M , then (2.5) E u ν (t; u 0 ) 2 m ≤ C m ∀ t ≥ 0,
where again C m depends on u 0 , ν and ξ. A solution u ν = u ν (t; u 0 ) may be constructed by Galerkin's method. Then u ν is obtained as a limit of Galerkin's approximations u

(N ) (t) ∈ span {e s , s ≤ N } ⊂ H, which also satisfy estimates (2.4), (2.5). If u 0 ∈ H m with 1 ≤ m < M , then a.s. (2.6) u (N ) (t) → u ν (t) in H m-1 as N → ∞, uniformly in t ∈ [0, T ],
for any finite T > 0. See in [START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF] Theorems 1.4.2 and 1.4.4.

Main estimates for solutions.

A remarkable property of solutions u ν of eq. (2.1) is given by Oleinik's estimates: If u 0 ∈ H 1 is a r.v., independent from ξ, then for each 0 < q < ∞ and 0 < θ ≤ 1 there exists C = C(q, B 4 ) > 0 such that for every ν > 0 and t ≥ θ the solution

u ν (t) = u ν (t; u 0 ) satisfies (2.7) E u ν + x (t) q ∞ ≤ Cθ -q , (2.8) E u ν (t) q ∞ + u ν x (t) q 1 ≤ Cθ -q (in (2.7) u ν + x (t) = u ν + x (t, x) is a positive part of the function u ν x (t, x)).
We stress that C does not depend on ν and u 0 . These relations imply crucial lower and upper estimates on Sobolev norms of solutions. To state them we need a definition: for a random process f ω (t) and σ ≥ 0, T > 0 we denote

(2.9) ⟨⟨f ⟩⟩ ∶= ⟨⟨f ⟩⟩ T +σ T = 1 σ T +σ T Ef (s) ds.
The following result is proved in [7, Section 2.3]:

Theorem 2.1. For each θ > 0, any N ∋ m ≤ M and for every random variable 1) for t ≥ θ, where C ′ m depends on θ and B max(4,m) .

u 0 ∈ H 1 , independent from ξ, solution u ν (t) = u ν (t; u 0 ) satisfies 1) E u ν (t) 2 m ≤ C ′ m ν -(2m-
2) There exists T * (θ) > 0 such that

(2.10) C -1 m ν -(2m-1) ≤ ⟨⟨E u ν 2 m ⟩⟩ ≤ C m ν -(2m-1) for some C m ≥ 1.
Here the averaging ⟨⟨⋅⟩⟩ = ⟨⟨⋅⟩⟩ T +σ T corresponds to arbitrary constants σ ≥ θ and T ≥ T * . The constants C m depend on θ, T * and ξ, but not on σ ≥ θ, T ≥ T * and ν ∈ (0, 1].

We will write A ∼ B if the two quantities satisfy C 1 B ≤ A ≤ C 2 B for some C 1 , C 2 > 0 which do not depend on ν and σ, T as in the theorem. Then (2.10) may be written as 1) . For m = 0 relations for the norm u ν (t; u 0 ) 0 are different. Namely from (2.8) it follows that E u ν (t) 2 ≤ C ′ 0 for t ≥ θ, and it is shown in [START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF] that for the brackets as in the theorem's assumptions, ⟨⟨E u ν 2 ⟩⟩ ∼ 1. The quantity 1 2 E u ν (t) 2 is the (averaged) energy of the "1d flow" u ν (t, x). Formally applying Ito's formula to 1 2 u ν (t) 2 and taking the expectation we get the balance of energy relation (2.11) 2.4. Inviscid limit. When ν → 0, a solution u ν (t; u 0 ) converges to a limit, known as an entropy solution of eq. (2.1) ν=0 . This result may be established in a number of different ways. In the form, given in the theorem below, it is proved in [7, Chapter 8], following Kruzkov's approach [START_REF] Kruzhkov | First order quasilinear equations in several independent variables[END_REF], based on a version of Oleinik's estimates (2.7), (2.8).

⟨⟨E u ν 2 m ⟩⟩ ∼ ν -(2m-
1 2 E u ν (t) 2 -1 2 E u ν (0) 2 = -E t 0 ν u ν (s)
Theorem 2.2. Let T > 0, u 0 ∈ H 2 be a non-random function and u ν = u ν (t; u 0 ). Then there exists a random field u 0 (t, x) ∶= u 0 ω (t, x; u 0 ) such that almost surely, for any

1 ≤ p < ∞, 1) u 0 ∈ L ∞ ([0, T ] × S 1 ) ∩ C([0, T ]; L p ); 2) u ν (t) → u 0 (t) in L p , uniformly in t ∈ [0, T ];
3) for any t ≥ θ > 0 and 0 < q < ∞, E u 0 (t) q p ≤ C(q, B 4 )θ -q ; 4) u 0 (0, x) = u 0 and u 0 (t, x) satisfies eq. (2.1) in the sense of generalised functions.

Eq. (2.1) ν=0 with a prescribed initial data has many generalised solution, but its entropy solution, defined by the limiting construction above, exists and is unique. Entropy solutions extend to a mixing process in L 1 : Theorem 2.3. Solutions u 0 (t; u 0 ) extend by continuity in u 0 to a Markov process in L 1 such that for every u 0 ∈ L 1 , u 0 (t; u 0 ) a.s. is continuous in t and meets the estimate in item 3) of Theorem 2.2. This process is mixing: there is a measure µ 0 ∈ P(L 1 ), satisfying µ 0 (∩ q<∞ L q ) = 1, such that for every r.v. u 0 ∈ L 1 , independent from ξ, (2.13) Du 0 (t; u 0 ) ⇀ µ 0 in P(L p ) as t → ∞, for any p < ∞. If Du 0 = µ 0 , then solution u 0 (t; u 0 ) is stationary: Du 0 (t; u 0 ) ≡ µ 0 .

See [START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF]Chapter 8.5] and see [START_REF] Khanin | Invariant measures for Burgers equation with stochastic forcing[END_REF][START_REF] Bec | Burgers turbulence[END_REF] for another approach to stationary solutions of the inviscid Burgers equation (2.1).

3. Moments of increments u ν (t, x + r) -u ν (t, x) and a weak form of the 4/5-law for Burgers equation.

Let us adopt some more notation. For a function v(x) on S 1 and l < 1 we denote v l = v(x + l) and set δ l v(x) = v l (x) -v(x). The absolute moments of increments in x of a solution u ν (t, x) for (2.1) with respect to the brackets ⟪⋅⟫ and averaging in x are

⟨⟨ δ l u ν (t, x) p dx ⟩⟩ = ⟨⟨ δ l u ν (t) p p ⟩⟩ =∶ S p,l = S p,l (u ν ) < ∞, p > 0. Obviously if u ν (t) is a stationary solution, then S p,l (u ν ) = E ∫ δ l u ν (t, x) p dx . The function (p, l) ↦ S p,l (u ν ) is called the structure function of a solution u ν . Since the function v(x) ↦ δ l v p p is continuous on L max(1,p)
, then in view of item 3) of Theorem 2.2 the structure function S p,l (u 0 ) for entropy solutions u 0 (t, x) also is well defined and finite.

Careful analysis of solutions u ν with ν ≥ 0 and of estimates (2.7), (2.8), (2.10) implies that for any random initial data u 0 ∈ H 1 (independent from ξ) the structure function S p,l (u ν ) obeys the following law: Theorem 3.1. There exist constants c * , c > 0 and c 1 ≥ 1, and for each p > 0 there exists C p ≥ 1, all depending only on the random force in eq. (2.1) and on θ, T * as in Theorem 2.1, such that for each ν ∈ [0, c * ] and p > 0 we have:

1) if l ∈ [c 1 ν, c], then (3.1)
C -1 p l min(1,p) ≤ S p,l (u ν ) ≤ C p l min(1,p) ; 2) if l ∈ [0, c 1 ν), then C -1
p l p ν 1-max(1,p) ≤ S p,l (u ν ) ≤ C p l p ν 1-max(1,p) (for ν = 0 this assertion is empty).

We see from this result that statistically the increments δ l u ν (t) , ν > 0, with l ∈ [0, c 1 ν] behave "linearly in l", while for l ∈ [c 1 ν, c] their behaviour is non-linear. So the interval [0, c 1 ν] is the dissipation range for burgulence, described by the Burgers equation (2.1), while [c 1 ν, c] is the inertial range. The frontier c 1 ν between the two interval is the dissipation or inner scale of the flow. The constants c 1 and c, depending on the random force, may change from one group of results to another.

On the physical level of rigour the first assertion of the theorem was proved in [START_REF] Aurell | On the multifractal properties of the energy dissipation derived from turbulence data[END_REF]. Rigorously for ν > 0 it was established in [START_REF] Boritchev | Sharp estimates for turbulence in white-forced generalised Burgers equation[END_REF], using some ides from [START_REF] Aurell | On the multifractal properties of the energy dissipation derived from turbulence data[END_REF]. For a complete proof of assertions 1) and 2) see [START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF]Chapter 7.2]. Now we start to discuss moments (not absolute ones) of increments δ l u ν of solutions u ν (t; u 0 ) as above:

(3.2)
S s p,l (u ν ) = ⟨⟨s p,l (u ν (t))⟩⟩, s p,l (v) = (δ l v ν (x)) p dx, ν > 0, 0 ≤ l < 1, where p ∈ N (the upper index s stands for "skew"). If p is an even number then S s p,l = S p,l , but for an odd p the two moments are different. As S s 1,l = 0, then the first non-trivial skew moment is the third one. Let us examine it. Since any real number x may be written as

x = 2x + -x , then (3.3) S s 3,l = -S 3,l + 2⟨⟨ δ l u ν ) + 3 dx⟩⟩.
But for any x, δ l u ν (x)

+ ≤ ∫ x+l x (u ν x ) + (t, y)dy ≤ l (u ν x ) + ∞ .
So in view of (2.7) the second term in the r.h.s. of (3.3) is bounded by Cl 3 , uniformly in ν > 0. From here, (3.3) and (3.1) with p = 3 we get that for a suitable c ′ > 0 and all ν ∈ (0, c * ],

(3.4) -C ′ 1 l ≤ S s 3,l (u ν ) ≤ -C ′ 2 l if l ∈ [c 1 ν, c ′ ], for some C ′ 1 ≥ C ′ 2
> 0, independent from ν > 0 and solution u ν . Since in view of (2.8) with p = 4 the family of functions ∫ (δ l u ν (t, x)) 3 dx, ν > 0, is uniformly integrable in variables (t, ω) ∈ [σ, σ + T ] × Ω, then passing to the limit in (3.4) as ν → 0 using item 2) of Theorem 2.2 with p = 3 we find that entropy solutions u 0 = u 0 (t; u 0 ), u 0 ∈ H 2 , satisfy (3.5), valid for all small ν ≥ 0, is a weak form of the 4/5-law (1.1) for Burgers equation. Literally the same argument shows that the two relations hold for all moments S s p,l with odd p ≥ 3 (the constants C ′ 1 , C ′ 2 should be modified). In the next section we will see that some ideas, originated in K41, allow to strengthen the asymptotic behaviour (3.4) for S s 3,l (as ν → 0) with l in the inertial range to a real asymptotic, if l belongs not to the whole inertial range, but to some large part of the latter.

(3.5) -C ′ 1 l ≤ S s 3,l (u 0 ) ≤ -C ′ 2 l if l ∈ [0, c ′ ]. Relation (3.4) +

Strong form of the 4/5-law for Burgers equation

Assuming that in (1.1) the velocity field u is homogeneous and isotropic in x (not necessarily stationary in t), for any p ∈ N define the p-th moment S p (t, r) of a longitudinal increment (u(t, x + r) -u(t, x)) ⋅ (r r ) as E u(t, x + r) -u(t, x) ⋅ (r r ) p (so the l.h.s. of (1.1) is S 3 (t, r)). Following Kolmogorov, proofs of the 4/5-law in physical works, e.g. in [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF][START_REF] Falkovich | Fluid Mechanics[END_REF], as well as in the rigorous paper [START_REF] Bedrossian | A sufficient condition for the Kolmogorov 4/5 law for stationary martingale solutions to the 3d Navier-Stokes equations[END_REF], 2 crucially use the Karman-Howard-Monin formula (which is rather a class of formulas, see for them the references above and relation (5.5.5) in [START_REF] Batchelor | The theory of homogeneous turbulence[END_REF]).

The formula relates time-derivative of the second moment S 2 (t, r) with derivatives in r of the third moment S 3 (t, r). Variations of the formula (e.g. see in [START_REF] Falkovich | Fluid Mechanics[END_REF]) instead of the second moments S 2 analyse the correlations Eu(t, x)⋅u(t, x+r), closely related to S 2 . Thus motivated let us examine time-derivatives of correlations of a solution u ν (t, x) = u ν (t, x; u 0 ) of Burgers equation with ν > 0 and a non-random initial data u 0 ∈ H M -1 , i.e. of

u ν (t, x)u νl (t, x) dx =∶ f l (u ν (t)).
Abbreviating u ν (t) to u(t), formally applying the Ito formula to f l (u(t)) and taking the expectation we get:

d dt Ef l (u(t)) = E -df l (u)(uu x ) + ν df l (u)(u xx ) + 1 2 b 2 s d 2 f l (u)(e s , e s ) =∶ E(-I 1 (t) + I 2 (t) + I 3 (t)). (4.1) Since df l (u)v = ∫ (uv l + u l v)dx = ∫ (uv l + uv -l )dx and (4.2) (∂ ∂l)u l (x) = u l x (x),
2 There, to adjust the formula to periodic boundary conditions, it is integrated in dr with suitable densities.

It turns out that thus obtained "weak KHM formula" is sufficient for a "conditional" derivation of relation (1.1).

we get that

I 1 (t) = (uu l u l x + uu -l u -l x )dx = 1 2 ∂ ∂l u(u l ) 2 -u(u -l ) 2 dx = 1 2 ∂ ∂l u(u l ) 2 -u l u 2 dx.
Recalling that the functional s 3,l was defined in (3.2), we have s 3,l (v(x)) = ∫ δ l v(x)

3 dx = 3 ∫ v l v 2 -(v l ) 2 v dx. So I 1 (t) = -1 6 ∂ ∂l s 3,l (u(t)
). Similar, using (4.2) we get

I 2 (t) = ν uu l xx + uu -l xx dx = ν ∂ 2 ∂l 2 uu l + uu -l dx = 2ν ∂ 2 ∂l 2 f l (u(t)). Since d 2 f l (u)(v, v) = 2f l (v)
, then relation (4.1) may be re-written as

(4.3) d dt Ef l (u(t)) = 1 6 E ∂ ∂l s 3,l (u(t)) + 2νE ∂ 2 ∂l 2 f l (u(t)) + B0 (l),
where B0 (l) ∶= ∑ s b 2 s f l (e s ) = ∑ s b 2 s cos(2πsl). We have obtained (4.1) by a formal application of Ito's formula to the infinite-dimensional stochastic process u ν (t; u 0 ) ∈ H M -1 with u 0 ∈ H M -1 . But Galerkin's approximations u (N ) (t) to solutions u ν (t) satisfy finite-dimensional stochastic systems. Estimates (2.4), (2.5) also hold for them and imply the validity of Ito's formula for u (N ) 's. The latter has the form (4.1) with Ito's term

EI 3 (t) modified to E 1 2 ∑ s ≤N b 2 s d 2 f l (u)(e s , e s ).
Then passing to a limit as N → ∞ using (2.6) and the uniform in N estimates we justify the validity of (4.1) for u(t) = u ν (t; u 0 ). (Doing that we write the Ito equation in the integrated in time form.) Cf. [START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF], where the energy balance (2.11) is established in a similar way. Since solutions u ν (t) ∈ H M -1 meet estimates (2.4), (2.5) with m = M -1 ≥ 4, then the given above formal transformation from (4.1) to (4.3) also is rigorous.

Relation (4.3) is a version of the Karman-Howarth-Monin formula for the stochastic Burgers equation. Now let µ ν ∈ P(H M +1 ) be the stationary measure for eq. (2.1) (see Section 2.3), and let u ν st (t) be a corresponding stationary solution. Then u ν st (t) = u ν (t; u 0 ), where Du 0 = µ ν . Using estimate (2.8), where u ν = u ν st , we see that all terms in the integrated in time relation (4.3) with u(t) = u ν (t; u 0 ), are integrable in µ ν (du 0 ). Performing this integration we get that equality (4.3) stays true for u = u ν st (t). Then the l.h.s. of (4.3) vanishes, so the relation takes form

(∂ ∂l)E s 3,l (u ν st (t)) = -12ν(∂ 2 ∂l 2 )E f l (u ν st (t)) -6 B0 (l).
Since s 3,0 (u(x)) ≡ 0 and since by (4.2) (∂ ∂l)f l (u) l=0 = ∫ u(x)u x (x)dx = 0, then integrating this equality in dl we find that (4.4)

E s 3,l (u ν st (t)) = -12ν(∂ ∂l)E f l (u ν st (t)) -6 l 0 B0 (r)dr.
Next, convergence (2.12), estimate (2.8), item 1) of Theorem 2.1 and Fatou's lemma imply that for all ν > 0,

(4.5) E u νst (t) 2 1 ≤ Cν -1 ,
for all ν > 0 and a suitable C. Consider the first term in the r.h.s of (4.4). Dropping the factor -12ν and using (4.2) we write its modulus as

E uu l x dx = E (u(t, x) -u(t, x + l))u x (x + l)dx ≤ E u(t, x) -u(t, x + l) 2 dx 1 2 E u x (t, x) 2 dx 1 2
(we used that ∫ u(x + l)u x (x + l)dx = 0). Since u is a stationary solution, then the first factor in the r.h.s. equals S 1 2

2,l (u ν st ). So in view of Theorem 3.1 and estimate (4.5) the first term in the r.h.s. of (4.4) is O √ l √ ν . In view of (2.3), B0 is an even C 2 -function. Since B0 (0) = B 0 , then ∫ l 0 B0 (r)dr = B 0 l+O(l3 ). Using in (4.4) the estimates for the terms in its r.h.s. which we have just obtained we find that

(4.6) E s 3,l (u ν st (t)) = -6B 0 l + O(l 3 ) + O √ l √ ν .
By relation (8.5.4) in [START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF], µ ν ⇀ µ 0 in P(L 3 ) as ν → 0. Next, by convergence (2.12) and estimate (2.8) (with q = 4), ⟨ u 4 3 , µ ν ⟩ ≤ C uniformly in ν > 0 (e.g. see [START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF]Corollary 11.1.7]). Since functional s 3,l is continuous on L 3 and s 3,l (u) ≤ C u 3 3 , then we derive from here that lim ν→0 ⟨s 3,l , µ ν ⟩ = ⟨s 3,l , µ 0 ⟩. Let u 0 st (t) be a stationary entropy solutions of eq. (2.1) ν=0 , D(u 0 st (t)) ≡ µ 0 (see Theorem 2.3). Then ⟨s 3,l , µ 0 ⟩ = Es 3,l (u 0 st (t)). So passing in (4.6) to the limit as ν → 0, we get that (4.7) E s 3,l (u 0 st (t)) = -6B 0 l + O(l 3 ).

If in (4.6) l belongs to the inertial range [c 1 ν, c], then the norm of third term in the r.h.s. of (4.6) is bounded by Cc -1 2 1 l. Assuming that c 1 is sufficiently big, we obtain from (4.6) another proof of the weak law (3.4) for stationary solutions u ν st (t) (since S s 3,l (u ν st ) = Es 3,l (u ν st (t))). Now let l belongs to a "strongly inertial range", i.e. Theorem 4.1. Let u ν st (t), ν > 0, be a stationary solution of eq. (2.1) and l satisfies (4.8). Then (4.9) S s 3,l (u ν st (t)) = E s 3,l (u ν st (t)) = -6B 0 l + o(l) as ν → 0, where o(l) depends only on the function L(ν) and the random force ξ. While the stationary entropy solution u 0 st (t) satisfies (4.7). 3 Due to the balance relation (2.11), for stationary solution u ν st with ν > 0 the rate of dissipation of energy is given by (4.10)

ε B = 1 2 B 0 .
So (4.9) may be written as (4.11) S s 3,l (u ν st (t)) = -12ε B l + o(l) as ν → 0. Combining the theorem's result with (2.12) and (2.13) we get Corollary 4.2. 1) Let ν > 0, and u 0 ∈ H 1 be a r.v., independent from ξ. Then for any l as in (4.8) we have lim t→∞ E s 3,l (u ν (t; u 0 )) = -6B 0 l + o(l) as ν → 0.

2) If ν = 0 and u 0 ∈ L 1 is a r.v., independent from ξ, then

lim t→∞ E s 3,l (u 0 (t; u 0 )) = -6B 0 l + O(l 3 ) as l → 0.
The proof of Theorem 4.1 crucially uses that the moment which we analyse is cubic. Indeed, for the proof the Ito term I 3 should be a constant, for that the functional f l should be quadratic, and then the term I 1 in (4.11) is of the third order in u. But this does not imply that moments S s p,l with integers p ≠ 3 do not admit asymptotic expansions in l. In the next section we show that an asymptotic expansion of S s p,l with an integer p ≥ 2, p ≠ 3, is not possible if in addition we require that its leading term "is universal". Remark 4.3. While suitable analogies of Theorems 2.1, 2.2, 3.1 and of the weak 4/5-law (3.5) hold for solutions of the free Burgers equation (2.1) ξ=0 with a non-zero smooth initial data (see [START_REF] Boritchev | One-dimensional turbulence and the stochastic Burgers equation[END_REF]Section 10.11]), we see no way to establish for the latter equation a reasonable analogy of Theorem 4.1.

On the Landau objection to universality in K41 and burgulence

The celebrated 2/3-law of K41 states that for turbulent velocity fields u(t, x) as those, treated by the theory, the second moments S 2 (r) 4 of longitudinal increments of u behave as (ε r ) 2 3 . Originally Kolmogorov insisted on the universality of the law and claimed that S 2 (r) = C K (ε r ) 2 3 + o (ε r ) 2 3 , where C K is an absolute constant. But this universality was put in doubt by Landau who suggested a physical argument, implying that a relation for a moment of velocity increment may be universal only if the value of the moment, suggested by the relation, is linear in the rate of energy dissipation ε (like relation (1.1) for the third moment). See in [START_REF] Landau | Fluid Mechanics[END_REF] a footnote at page 126 and see [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]Section 6.4]. The goal of this section is to show that for burgulence, indeed, the only universal relation for the moments S s p,l is relation (4.11) for the cubic one (which is linear in ε B ).

Namely, for a stationary solution u ν st (t, x) of stochastic Burgers equation (2.1) and an integer p ≥ 2 consider the following relation for the p-th moment S s p,l of increments of u ν st :

(5.1) S s p,l (u ν st (t)) = C * (ε B l) q + o(ε B l) q as ν → 0, where l is any number from the inertial range [c 1 ν, c] and q > 0. We address the following question: for which p and q relation (5.1) holds with a universal constant C * , independent from the random force ξ? Theorem 5.1. If relation (5.1) holds for any random force ξ, satisfying (2.3), with a C * independent from ξ, then p = 3, q = 1, C * = -12.

(4. 8 )

 8 l ∈ [L(ν), c],for any fixed function L(ν) such thatL(ν) → 0 and L(ν) ν → ∞ as ν → 0. Then √ l √ ν = o(l)as ν → 0 and we arrive at the main result of this work:

Equivalently, ⟨s 3l , µ0⟩ = -6B0l + O(l 3 ) as l → 0.

The moments do not depend on t by the assumed stationarity of u.

This is in line with the relation u(t, x + r) -u(t, x) ≍ (ε r )1 3 which appears in the theory of turbulence due to a basic dimension argument, without any relation to the equations, describing the fluid. See[15, (32,1)].

Acknowledgement

The authors are thankful to A. Boritchev for discussion and to a referee for his comments and useful suggestions. PG was supported by Natural Science Foundation of Jilin Province (Grant No. YDZJ202201ZYTS306) and the Fundamental Research Funds for the Central Universities. Both authors were supported by the Ministry of Science and Higher Education of the Russian Federation (megagrant No. 075-15-2022-1115).

Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Proof. Let us abbreviate u ν st (t) to u(t). We take some real number µ > 1 and set ξ(τ ) ∶= µ -1 2 ξ(µτ ). This also is a standard Wiener process. Denote w(τ, x) ∶= µu(µτ, x). Then w is a stationary solution of equation

Consider the inertial range J 1 = [c 1 ν, c] for eq. (2.1) and inertial range J µ = [c µ 1 ν, c µ ] for eq. (5.2). For small ν their intersection J = J 1 ∩ J µ is not empty. For l ∈ J relation (5.1) holds for u which solves eq. (2.1) and for w, solving eq. (5.2). Since S p,l (w) = µ p S p,l (u) and as ε B w = µ 3 ε B u in view of (4.10), then from here

) q for l ∈ J and all small ν. As µ > 1, then by this equality q = p 3. 5 On the other hand, it follows from Theorem 3.1 if p is even and from (3.4) and a discussion after (3.5) if p is odd that S p,l (u) ∼ -l for any integer p ≥ 2. Thus in (5.1) q = 1, and so p = 3q = 3. Then by Theorem 4.1 C * = -12 and the theorem is proved.

Remark 5.2. 1) The result of Theorem 5.1 remains true with the same proof if relation (5.1) is claimed to hold not for all l from the inertial range, but only for l from a strongly inertial range as in (4.8). In this form asymptotic (5.1) with p = 3 and q = 1 indeed is valid by Theorem 4.1.

2) We do not know if for some integer p ≥ 2, different from 3, asymptotical expansion for S p,l (u ν st (t)) of the form (5.1), valid for all l from the inertial range (or from a strongly inertial range) may hold with a constant C * which depends on the random force ξ.