Frédéric Mallet
email: fmallet@sophia.inria.fr

Clock Constraint Specification Language Specifying Clock Constraints with UML/MARTE

Keywords: Unified Modeling Language, Time model, Constraints

The OMG UML Profile for Modeling and Analysis of Real-Time and Embedded systems (MARTE) aims at using the general-purpose modeling language UML in the domain of Real-Time and Embedded (RTE) systems. To achieve this goal, it is absolutely required to introduce inside the mainly untimed UML an unambiguous time structure which MARTE model elements can rely on to build precise models amenable to formal analysis. MARTE Time model has defined such a structure. We have also defined a non-normative concrete syntax called CCSL to demonstrate what can be done based on this structure. This paper gives a brief overview of this syntax and its formal semantics and shows how existing UML model elements can be used to apply this syntax in a graphical way and benefit from the semantics.

Introduction

The Unified Modeling Language (UML) [START_REF]Object Management Group: Unified Modeling Language, Superstructure[END_REF] aims at being a unified and general-purpose modeling language. Its semantics is purposely loose to cover a large domain and introduces so-called semantic variation points that provide for extensions to refine (or even define) a semantics when required for a specific domain. These extensions are to be defined in the context of a UML Profile. In the domain of real-time and embedded (RTE) systems, the Object Management Group (OMG) has recently adopted the UML Profile for Modeling and Analysis of Real-Time and Embedded systems (MARTE) [START_REF]Object Management Group: UML Profile for MARTE, beta 2[END_REF], which is currently in the finalization phase. In its foundations, MARTE defines a broadly expressive Time Model to provide for a generic timed interpretation of UML models. The idea is to precisely define a semantics within the Profile rather than allowing tools for giving their own, possibly incompatible with other tools of the same domain.

MARTE Time Structure is heavily inspired by the Tagged Signal Model [START_REF] Lee | A framework for comparing models of computation[END_REF], which intends to define a common framework for comparing several Models of Computation and Communication in the RTE domain, and from various works around synchronous languages [START_REF] Benveniste | The synchronous languages twelve years later[END_REF] and more generally polychronous/multiclock languages well-suited to specify Globally Asynchronous and Locally Synchronous (GALS) systems. The concrete syntax of our language, called Clock Constraint Specification Language (CCSL), is part of MARTE Profile but is not normative and not based on any existing language to let tool vendors choose their own technology. Our goal has been to use explicit keywords that denote usual concepts of the domain (periodic, sporadic, sampling. . . ).

A comprehensive informal description of CCSL has previously been presented in [START_REF] André | Modeling time(s)[END_REF] and a partial formal declarative description is available in [START_REF] André | Clock constraints in UML/MARTE[END_REF]. Using a declarative mathematical description allows for being language independent. When constraints are not incompatible they should enforce a causal relationship between UML model elements and thus provide a support to build a real-time UML simulator. To implement CCSL and produce acceptable executions, i.e., compatible with all constraints, it may be interesting to transform it into equivalent formalisms that already benefit from analysis tools. After a brief overview of the semantics of a core CCSL constraint subset, this paper proposes several graphical UML-compatible possible representations of these constraints. These graphical representations are not part of MARTE yet and may be proposed to the second OMG Finalization Task Force for being integrated in next MARTE revision.

Section 2 starts with a brief overview of MARTE Time structure, the official OMG profile. Then, Section 3 describes main aspects of the non-normative constraint language CCSL. Section 4 compares three different visual representations of the constraints.

MARTE Time subprofile

Time Structure

A Clock is a 5-tuple I, ≺, D, λ, u where I is a set of instants (possibly infinite), ≺ is a quasi-order relation on I, named strict precedence, D is a set of labels, λ : I → D is a labeling function, u is a symbol, standing for a unit. In this paper, we only consider the clock temporal structure (or pure clock ), i.e., the ordered set I, ≺ and the values are never mentionned. ≺ is a total, irreflexive, and transitive binary relation on I.

A discrete-time clock is a clock with a discrete set of instants I. Since I is discrete, it can be indexed by natural numbers in a way that respects the ordering on I: let N = N \ {0}, idx : I → N , ∀i ∈ I, idx(i) = k if and only if i is the k th instant in I. We restrict the discussion to discrete-time clocks and do not consider dense time at all. For all operators, we always assume that clocks are discrete, whereas these operators may have a more general semantics when applied to dense clocks.

For any discrete-time time structure c = Ic, ≺c , c[k] denotes the k th instant in Ic (i.e., k = idxc (c[k])). For any instant i ∈ Ic, °i is the unique immediate predecessor of i in Ic and i°is the unique immediate successor of i in Ic, if any. To simplify computations, we assume a virtual instant c[0], so that c[0] ≡ °c [START_REF] André | Clock constraints in UML/MARTE[END_REF]. Note, these definitions of predecessor and successor are only possible with a discrete structure.

A It is also useful in electronic design with manycore architectures or even monocore where several time domains are defined (main clock, bus clock. . . ). Locally, it is often better to consider these time domains/bases (clocks) as independent so each part can be designed separately. However, when integrating it is required to understand the relations between these clocks to deal with inter-domain communications.

Clock constraints have been defined with that objective. They make explicit relations amongst clocks. The stereotype ClockConstraint extends the metaclass Constraint and the Clock Constraint Specification Language (CCSL) gives a possible non-normative concrete syntax to specify constraints. CCSL is briefly introduced in the next section.

Clock Constraints

The Time structure defines relations between instants. However, a clock being an infinite set of instants it is neither realistic nor practical to specify constraints one by one.

Based on instant constraints, it is easy to build more powerful relations that define infinitely many instant relations. We call these relations clock constraints. We can classify clock constraints into three families: coincident-based, precedence-based and mixed constraints.

Coincidence-based constraints

The coincidence-based clock constraints defines infinitely many coincidence instant relations. Most of the time, this kind of constraint defines a subclock from a given superclock, i.e., a clock less frequent. For instance we can select every third instant to create a periodic subclock of period 3. But as most clock constraints are relations, as opposed to functions, they can also do the contrary and define a superclock from one (or a set of) subclock(s). For example, to model a phase-lock loop (PLL) system, one may wish to oversample four times a given clock. The most frequently used constraint of this family is isPeriodicOn whose semantics is given in a mathematical declarative way by Eq. 1.

isPeriodicOn B period =P offset =δ ⇐⇒

(∀k ∈ N )(A[k] ≡ B[(k -1) * P + δ + 1]) (1)
Figure 2 illustrates this relation with a period of 3 and an offset of 5. This family describes synchronous relations inspired from operators defined in synchronous languages [START_REF] Benveniste | The synchronous languages twelve years later[END_REF].

Precedence-based constraints

The precedence-based clock constraints define infinitely many precedence instant relations. They characterized asynchronous relations. The most frequently used constraint of this family is alternatesWith. The relation alternatesWith represents alternation between two clocks. A ∼ B means that each occurrence of A is followed by an occurrence of B before any other occurrences of A. The weak form of this relation allows the i th occurrence of B to be simultaneous (coincident) with the i th occurrence of A, whereas the strict form requires A and B to be disjoint.

Typically, an asynchronous communication implies an alternation between sending and receiving. Let A be the sender and B the receiver. The data is received after having been sent. No other communication can start before the previous one completes. The weak form allows the sender to receive data simultaneously with the emission, but do not force the synchronization. The strict form is used to forbid instantaneous communications. The semantics of the strict form is given by Eq. 2 whereas the semantics of the weak form is given by Eq. 3.

A strictly alternatesWith B

⇐⇒

(2)

(∀k ∈ N )(A[k] ≺ B[k] ≺ A[k + 1]) A alternatesWith B ⇐⇒ (3)
(∀k ∈ N )(A[k] B[k] ≺ A[k + 1])

Mixed constraints

Mixed constraints combine both precedence and coincidence relations. There are useful when modeling communications from an asynchronous part of a design to a synchronous part. The most frequently used constraint of this family is sampledOn. The relation sam-pledOn represents sampling, it can be used to model time-triggered communications or for synchronizing asynchronous inputs. A = B C defines a subclock of C that occurs only after an occurrence of B. The strict form of sampledOn does not instantaneously sample an occurrence of B when it is synchronous with an occurrence of C. In that case, the sampling is postponed. Figure 3 shows one possible scenario involving the clock relation sampledOn with both forms weak and strict. Signal B counts its occurrences and signal A contains the value actually sampled from B.

With both forms the first sample has the value 1. However, with the weak form the first sample occurs on the first occurrence of C whereas it occurs on the second occurrence of C with the strict form. The second sample has the value 3 and the input 2 has been lost in both cases. The third sample occurs at the same time, whatever the form, but does not carry the same value in both cases. The semantics of the strict form is given by Eq. 4 whereas the semantics of the weak form is given by Eq. 5.

A = B strictly sampledOn C ⇐⇒ (4) (∀a ∈ N )(∃b, c ∈ N ) ((A[a] ≡ C[c]) ∧ (C[c -1] B[b] ≺ C[c])) Fig. 3 A=B sampledOn C A = B sampledOn C (A = B C) ⇐⇒ (5) (∀a ∈ N )(∃b, c ∈ N ) ((A[a] ≡ C[c]) ∧ (C[c -1] ≺ B[b] C[c]))
4 Applying Constraints to UML models

Using UML constraints

To apply these constraints, one must first create clocks and therefore clock types. Some very useful clock types are provided in MARTE library, like IdealClock that represents a perfect dense chronometric clock. Other clocks, with flaws like jitter, drift, can be derived from IdealClock or rather from one of its instances (a clock). The clock idealClk is also provided in MARTE library and is an instance of IdealClock. idealClk can be discretized with a given resolution to build, for instance, a chronometric discrete clock whose frequency is 100Hz (see Eq. 6).

c 100 = idealClk discretizedBy 0.01 [START_REF]Object Management Group: Unified Modeling Language, Superstructure[END_REF] More generally, MARTE also provides a support to use logical clocks, i.e., clocks not directly related to the physical time. Anything that has to be compared (before, after or simultaneous) to something else can be considered as a logical clock. For instance, to give timing information about thread dispatch, a class Thread can be defined and the stereotype ClockType can be applied to it. Doing so, instances of Thread or properties/parts/ports of type Thread may become clocks. Note that, being a ClockType does not prevent from being something else, like a SchedulableResource, it only provides for a support to build clock constraints and express causality relations with other clocks.

Figure 4 illustrates such a case where two periodic threads (t1, t3) are mixed with an aperiodic thread (t2). The two clock constraints make the two threads (clocks) periodic relative to clock c 100 . The two threads are harmonic since they refer to the same clock with an offset 0 and the ratio of their period is an integer. Thread t2 is However, a more obvious visual notation may be more appropriate. In particular, constraints are visually explicit when all the constrained elements appear on the same diagram. One purpose of the Time model is to bring consistency between different diagrams through clocks. Different model elements from different diagrams are connected together by clock constraints, which constrain them to behave in a consistent way. The two following subsections propose alternative notations where all clocks are shown on the same diagram.

Using a CCSL-specific profile

UML profile are a way to introduce new concepts but also to change the visual notation. Introducing a new visual language for clock constraints within MARTE is not practical because MARTE is very big and there are some many other matters to address. A solution can be to define aside, a CCSL-specific profile. Each often used CCSL constraint can have its own stereotype associated with a specific icon. The most appropriate meta-class to represent CCSL constraint is probably the metaclass Dependency.

Figure 5 shows a possible profile that could be defined to ease the use of CCSL constraints. The left-hand side part of the figure shows the actual profile. The righthand side part of the figure shows another composite structure diagram about the same three-thread example. In this example, the diagram represents three new clocks (step1, step2 and step3) that are the actual behaviors (subprograms) executed by the threads. These clocks could also have been shown on the previous diagram but this would have resulted in a very heavy construct. Using the defined stereotypes and their icons brings simplicity even though the semantics is exactly the same.

The relation alternatesWith from the threads to their assigned subprogram denotes that every time the thread is dispatched, the subprogram must execute. Additionnally, the subprogram cannot be executed another time before the completion of the previous execution. From step1 to step2, there is also a relation alternatesWith since there is an asynchronous communication. Whenever step1 completes, step2 takes its output data and executes.

Finally, the ternary relation involving step2, step3 and t3 denotes the synchronization of step2 output according to the clock t3. Every time t3 is dispatched, whatever the current status of step2 output, the data available is sampled and step3 executes using this sample. Obviously, this may lead to data loss, if the sampling rate is too low or to the use of the sample data in multiple executions of step3 if the sampling rate is too high.

This representation requires to define a new profile aside MARTE. In this profile, each CCSL constraint has its own stereotype and its properties are determined according to CCSL grammar. An alternative to the creation of another profile is to use parametric diagrams as described in the following subsection.

Using SysML parametric diagrams

The UML profile for System Engineering (SysML) [START_REF]OMG: Systems Modeling Language (SysML) Specification 1[END_REF] is an adopted OMG Specification to be used at the system level. They have selected a subset of UML constructs called UML4SysML and they provide few new extensions amongst which Refinement and Parametric diagrams. The former helps making explicit system-level requirements and tracing their proposed implementations. The latter should be used to represent noncausal relations among values of the system and possibly making explicit within the model, physical laws required for the design.

So, we can use this SysML construct to represent laws related to time, whether physical or logical. SysML recommends building a new "Constraint Block" for each new law and then use it in so-called parametric diagrams to apply this law to relevant design values. In our case, we have a small number of identified relations among logical clocks. Consequently, we can easily construct a library of CCSL-specific constraint block.

Figure 6 illustrates the same example using SysML constraint blocks and parametric diagrams.

The left-hand side part is an excerpt of the library. Three constraint blocks (Periodic, Alternation, Sampling) have been defined for each of the three CCSL relations introduced previously. Each constraint block has two compartments. The bottom one, called The next step is to provide analysis support for such diagrams. We already have implemented an Eclipse-based simulator for CCSL constraints [START_REF] André | Clock constraints in UML/MARTE[END_REF] and we are trying to integrate it with parametric diagram to get a direct visual feedback of a simulation run. We have also made some progress to transform CCSL constraints into other formalisms, like Signal or Petri nets, that have their own analysis techniques and tools [START_REF] Mallet | UML/MARTE CCSL, signal and petri nets[END_REF].

Fig. 1

 1 Fig. 1 Excerpt of MARTE Time subprofile

Fig. 4

 4 Fig. 4 Periodic threads with CCSL

Fig. 5

 5 Fig. 5 Using a CCSL-specific profile

  ClockType extends the metaclass Class. It models classes of compatible clocks, i.e., clocks of the same nature (discrete or dense) and that use the same set of units. The stereotype Clock extends metaclasses InstanceSpecification and Property. It models a set of instants as defined in the previous subsection. Clocks can appear either in instance diagrams to represent a snapshot of a system at a given time or in composite structure diagrams to represent a family of possible behaviors. Starting from there, several identified model elements can be associated with one or several clocks using one the concrete stereotype that specializes the abstract stereotype TimedElement. This association with a clock gives the ability to the model element to embedd expressions or value specifications identifying precisely instants or durations. Having different reference clocks is very useful in distributed systems where different elements use different time bases.

	instant relations: Coincidence (≡	∩ ), Strict precedence (≺	\ ≡), Independence
	(	∪ ), and Exclusion (# ≺ ∪ ).
	2.2 UML Profile
	Figure 1 presents a simplified view of MARTE Time subprofile. The green elements are
	defined outside the profile itself but are useful to understand the profile. The stereo-
	type
		Time Structure is a pair C,	where C is a set of clocks, is a binary relation
	on c∈C Ic, named precedence. is reflexive and transitive. From we derive four new

Acknowledgements Some UML diagrams have been built using Papyrus, an open-source tool available at http://www.papyrusuml.org.

parameters contains typed formal parameters. The upper compartment, called constraints, contains the constraint itself that applies on the parameters. In our case, the constraint is defined in CCSL. However, this library is built once and for all, so end-users need not being entirely familiar with the concrete syntax and only need to be familiar with underlying concepts.

The right-hand side part presents the three-thread example as a SysML parametric diagram. In such a diagram, boxes are properties extracted from the model. Some of the properties are clocks (t1, step1 . . . ), some others have integer values (of f set, t1 p . . . ). These properties may come from different diagrams and different blocks. The rounded rectangles are usages of constraint blocks. Their ports, which represent parameters, are connected with properties using non-causal connectors. Being non-causal means that there is no input or output, whichever value is known causes the other to update. For instance, considering Alternation, if b is known, one can deduce (partially) a but if a is known, then one can deduce (partially) b.

conclusion

The paper has presented some of the clock constraint relations introduced by MARTE Time model together with their formal semantics. These clock constraints brings consistency in the timing information of UML models. Introducing formal models in UML models is a problem. OMG Specifications are usually big and are not the right place to put formal specification. Leaving the formal semantics outside the specification leaves the interpretation to tools, possibly having different interpretations in different tools.

With CCSL, we rely UML constraints. However, constraints are not the best visual representation. We propose here two visual alternative notations that result in more compact diagrams. These two solutions are not specific to CCSL and could be easily extended to other constraint-based languages like OCL. The limitation comes from finding the right icon or shape that makes immediately the link with the constraint. Our first proposition implies the creation of a CCSL-specific profile. The second one relies on SysML parametric diagrams. It is interesting to see that all the proposed notations are composite structure diagrams or derived notations.

These two notations attempts to replace CCSL concrete syntax by a visual representation and to allow a representation of otherwise scattered timing information in a single diagram.


1

SOLVING THE TRAINING DIVIDE Andrea Bassanini

OECD Employment, Labour and Social Affairs Directorate

Published on OECD Observer No. 240/241, pp.43-45, December 2003 http://www.oecdobserver.org/news/printpage.php/aid/1189/Solving_the_training_divide.html Computers are everywhere, except in the productivity statistics. This famous, perhaps overquoted, line from Robert Solow, the US economist, had everyone from MBA students to Nobel wannabes and policymakers scurrying to their databases in search of evidence that a new economy was indeed being fashioned from the emergence of new technology. Some evidence has been uncovered that information and communication technology can, when combined with human capital, drive up firm-level productivity and performance, for instance. But regardless of whether one believes that all of this adds up to a new economy or not, part of Mr Solow's point remains blatantly true: computers are everywhere. In fact, the development and spread of new technology in the workplace is not a myth, but a reality. Everyone's skill set requirement has been upgraded as a result.

But this raises another possible myth that demands our attention: that the new economy would lift everyone's potential, however educated they might be. This is, of course, a contradiction: remember, the improvement in productivity comes from combining technology and human capital, particularly skilled labour. Yet, the world is full of unskilled labour too (anyone educated to secondary level only, with no higher level or specific skills qualification, is defined as unskilled). So, while the frontier of economic possibilities will thankfully always expand, it should come as no surprise that job opportunities for the low-educated are shrinking almost everywhere.

Take employment growth. This has been positive in the last 10 years in OECD countries, yet there has been a marked contraction of jobs that are typically held by low-educated workers (see graph).

The cases of Germany and Italy stand out: in these countries net destruction of unskilled jobs has occurred at a rate greater than 1% per year since 1993, despite overall positive employment growth. Even in countries such as the United States, Luxembourg, the Netherlands, New Zealand and Spain, that have experienced fast employment growth (of the order of 1-1.5% a year or more), jobs for low-educated workers have grown much less quickly.

The progressive disappearance of low-skilled jobs may seem like progress, and indeed fewer mundane tasks is part of the promise of a knowledge society, but it makes for rather gloomy employment prospects for the less-educated, who stand to fall behind in terms of employment performance and wealth. In the OECD area as a whole, the employment rate of workers with lower secondary education or less fell by 0.3 percentage points each year during the 1990s, in spite of growing employment rates for the other groups. In effect, to add to a digital divide, there is a skills divide opening up in the OECD area.

The downward trend of the employment rate of the low-educated reflects the pace of technological change, and in fact has been sharper in countries where technological change is faster. For instance, in Sweden, business R&D expenditure soared from less than 1.9% of GDP in the late 1980s to about 2.7% by the late 1990s, while the employment rate of the low-educated slumped by 13 percentage points between 1991 and 2001. On other hand, in the Netherlands, a country that recorded a fall in the growth of business R&D intensity in the same period, the employment rate of the low-educated grew by about 10 percentage points, thanks also in part to a boom in part-time jobs.

But overall, low-skilled job opportunities are dwindling. Economists have different views on what to do about this. Some think that nothing should be done, as markets will take care of any problems as people respond to new incentives and upgrade themselves. Others emphasise the need for government to assure better adult-oriented education and training, particularly as fast technological change and rapid turnover of technologies can make skills learned at school become quickly obsolete. Moreover, with the workforce ageing and working life lengthening, education beyond the schooling received in youth will become all the more important. The problem is how to go beyond rhetoric and identify what can be done to improve adult workers' skills.

Countries still do not appear to invest enough in the education of under-skilled adults, although the extent of the problem is difficult to quantify and may be eased somewhat by the presence of informal training. Still, more needs to be done to encourage a more efficient sharing of the costs and benefits of training between employers and employees, thereby increasing the incentives to invest in human capital.

Another, more intractable problem, is how to get training to those who need it most. As it is, vulnerable workers have fewer opportunities to acquire new skills. On average, only 16% of workers with less than upper secondary education participate in formal adult training, which is less than half of that of workers with tertiary degrees (35%). Similar inequalities are found between older workers and prime-age workers, skilled and unskilled, women and men, and immigrants and natives. Very large gaps are also found between small and large firms; the 2003 OECD Employment Outlook showed that the probability of being offered training was 30% lower for a worker in a small firm than for a similar worker in a large one. Similar gaps emerge as regards involuntary part-time workers (with respect to full-time workers) and temporary workers (with respect to workers in permanent jobs).

Are certain workers unwilling to be trained or could it be that firms find it less profitable to train them? This question is fundamental for policy, since in many countries, particularly in Europe, policies are first and foremost designed to increase the incentives of firms to train. Most of these policies seem to be effective in raising the amount of training provided by firms, but are nevertheless skewed to train workers that are already in a good position in the labour market because of their lower training cost. Experience in some countries, such as the Netherlands, shows that it is very difficult to target incentives to train specific groups, since they can generate undesired substitution between different disadvantaged groups, such as older unskilled workers replacing younger unskilled workers, and the like. Moreover, while our studies show that older workers do not seem to be willing to do training beyond what is offered by their own employers, mainly because they would have to pay for it themselves and there is not enough time left before retirement to recoup the cost of that training, most other categories of workers wish they could get more training. For instance, in cases where training is not provided by employers, some 5% more women than men say they would pay for their own training, provided they could afford it. Even more striking, the share of workers that do not receive training from their employers but would like to is 20% greater in the case of involuntary part-time or temporary workers than in the case of full-time workers. So, for certain groups of individuals, the supply of training falls short of employee wishes, and sometimes dramatically. And policies to encourage employers to increase their overall training investment would do little to alter this situation.

For this reason, some countries are experimenting with co-financing policies for individual The key to closing the skills divide seems to be a matter of finance to assure training. By cofinancing learning between employees, employers and, where needed, government, and by creating the conditions where such co-financing becomes efficient, the number of unskilled that risk being left behind would be greatly reduced.

  investments in human capital, to help workers pay for training themselves when they are not supported by their employer. For instance, experiments with publicly-funded training vouchers in the Geneva canton of Switzerland have been effective, particularly in increasing training among immigrants. Similarly, many countries have introduced so-called individual learning accounts, which are bank accounts that benefit from special tax treatment or are publicly funded, and which individuals can co-finance and draw on to pay for future training. Despite these measures, without support from their employer, individuals often find training courses unaffordable, not only because of their direct costs but also because of time constraints. On average about 20% of the workers of OECD countries say they did not do further training because of lack of time. Some training courses may require long breaks from work, sometimes up to several months. Many OECD countries have responded by providing statutory or contractual training leave schemes that guarantee employees the right to return to their jobs after completing the training course, with the study period usually being partially subsidised by the government through loans or grants. Some countries have also facilitated access to training and education on a part-time basis. For instance, in the Australian technical and further education colleges it is possible to study parttime, at distance and on weekends, and access requirements also take into account previous work experience. As a result, in Australia, 12% of the enrolees in the formal education system are aged 35 and over, which is three times the OECD average. There is no shortage of initiatives to help improve skills and so the job prospects of less-educated workers. And the returns from training are estimated to be quite high. Most studies find a positive and persistent effect of training on earnings. For instance, one study, by Loewenstein and Spletzer, estimates that for the United States one week of employer-paid training of newly hired workers leads to 1.4 percentage point higher wage growth in the two years after hiring and that 17% of the average wage growth in the same period can be explained by training. And Booth and Bryan estimate that in the United Kingdom, one week of accredited formal training leads to higher wages at subsequent employers. The effect of continuous training on employment security is less well established, though evidence points to a greater probability of finding a new job quickly in the case of involuntary employment loss if the worker has previously received training.


Billel Guelmame

ON THE BLOW-UP SCENARIO FOR SOME MODIFIED SERRE-GREEN-NAGHDI EQUATIONS

Keywords: AMS Classification: 35Q35, 35B65, 35B44, 76B15 Serre-Green-Naghdi, dispersion, water waves, energy conservation, blowup, shallow water

 to improve the dispersion relation. We present a precise blow-up scenario of the mSGN equations and we prove the existence of a class of solutions that develop singularities in finite time. All the presented results hold, with the same proof, for the Serre-Green-Naghdi system with surface tension.

Water waves are usually described by the Euler equations. Due to their complexity, other models have been proposed in various regimes. For example, in the shallow water regime, it is assumed that the shallowness parameter σ is small, where σ is the ration of the mean water depth h to the wavelength ι (i.e., σ = h/ι 1). The Serre-Green-Naghdi equations are obtained by neglecting all the terms of order O(σ 4 ) from the water waves equations.

We consider in this paper the conservative modified Serre-Green-Naghdi system that was introduced by Clamond et al. [START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF] h t + [ h u ] x = 0, (1a)

[ h u ] t + h u 2 + 1 2 g h 2 + R x = 0, (1b)
R def = 1 3 1 + 3 2 β h 3 -u tx -u u xx + u 2 x -1 2 β g h 2 h h xx + 1 2 h 2 x , (1c)
where h denotes the depth of the fluid, u is the depth-averaged horizontal velocity, g is the gravitational acceleration and β is a free parameter. The classical Serre-Green-Naghdi system is recovered taking β = 0. The aim of this paper is to study the local (in time) well-posedness of [START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF], to obtain a precise blow-up criterion and to build smooth solutions that develop singularities in finite time.

Several modified Serre-Green-Naghdi equations have been derived and studied in the literature to optimise the linear dispersion, some of them fail to conserve the energy and do not admit a variational principle, some other equations do not satisfy the Galilean invariance. In order to improve the dispersion of the classical Serre-Green-Naghdi (cSGN) system conserving its desirable properties, Clamond et al. [START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF] have modified the Lagrangian instead of modifying directly the cSGN equations to obtain the Lagrangian density (see Section 2 for more details)

L β ρ = 1 2 h u 2 + 1 6 1 + 3 2 β h 3 u 2 x -1 2 g h 2 -1 4 β g h 2 h 2 x + φ {h t + [h u] x } , (2)
where φ is a Lagrange multiplier. The Euler-Lagrange equations of (2) lead to [START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF]. Smooth solutions of the modified Serre-Green-Naghdi (mSGN) equations ( 1) satisfy the energy equation

E t + Q x = 0, (3)
with

E def = 1 2 h u 2 + 1 6 1 + 3 2 β h 3 u 2 x + 1 2 g h -h 2 + 1 4 β g h 2 h 2 x , (4)
Q def = u E + R + g h h -h + 1 2 β g h 3 h x u x , (5)
where h is the mean value of the depth h of the fluid. For β > 0 and if h is far from zero (h h min > 0), the H 1 norms of both u and h -h are controlled by the energy. However, the energy of the cSGN equations (β = 0) cannot control the L 2 norm of h x . This crucial property of mSGN (1) for β > 0 is very important to build the small-energy solutions that develop singularities in finite time (Theorem 3 below). Several criteria have been proposed in [START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF] to chose the parameter β. For β = 2/15 ≈ 0.1333, the dispersion relation of ( 1) coincide with the one of the full Euler system up order 4 instead of order 2 for the classical Serre-Green-Naghdi (Section 2 below). To optimise the decay of a particular solitary wave of (1) studied in [START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF], one must take β = 2 3 (12π -2 -1) ≈ 0.1439. The value β ≈ 0.34560 approximates the inner angle of the crest of the solitary wave solution to the exact angle of the limiting solitary wave (120 • ). In the present paper, we consider only the case β > 0, and for the sake of simplicity we introduce

α def = 1 + 3 2 β. (6)
The mSGN equations on the form (1) contains some terms with high-order derivatives and a term with a time derivative in the definition of R. In order to obtain a simpler form of (1), we introduce the linear Sturm-Liouville operator

L h def = h -1 3 α ∂ x h 3 ∂ x (7)
and we apply L -1 h (the invertibility of L h is proved in Lemma 3 below) on the equation (1b) to obtain

u t + u u x + g h x = -L -1 h ∂ x 2 3 α h 3 u 2 x + 1 3 g h 3 h xx -1 4 β g h 2 h 2 x . (8)
In Lemma 3 below, we show that we gain one derivative with the operator L -1 h ∂ x , this is not enough to control the term 1 3 gh 3 h xx in the right-hand side of (8). To get rid of this term, we use [START_REF] Dutykh | Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations[END_REF] to rewrite [START_REF] Glassey | Singularities of a variational wave equation[END_REF] in the equivalent form

h t + [ h u ] x = 0, ( 9a
)
u t + u u x + α-1 α g h x = -L -1 h ∂ x 2 3 α h 3 u 2 x -1 4 β g h 2 h 2 x + g 2 α h 2 . ( 9b
)
The left-hand side of ( 9) is a symmetrisable 2 × 2 hyperbolic system and the right-hand side is a zero-order non-local term. Then, the local well-posedness of (9) in H s with s 2 can be obtained following the proof of symmetrisable hyperbolic systems. To the author's knowledge, the best blow-up criteria that have been obtained for those type of equations is "if a singularity appears in finite time, then h x L ∞ + u x L ∞ blows-up". This criteria does not show which term, or which slope blows-up. In this paper, we improve this criteria and we identify how exactly the blow-up occurs (Theorem 2 and equation (21) below). We also prove, using Riccati-type equations, the existence of a class of arbitrary small-energy initial data, such that the corresponding solutions develop singularities in finite time.

1.2. Other similar equations. In order to improve the dispersion relation of the classical Serre-Green-Naghdi equations, an interesting Whitham-Green-Naghdi (WGN) system have been proposed and studied in [START_REF] Duchêne | A new class of two-layer Green-Naghdi systems with improved frequency dispersion[END_REF][START_REF] Duchêne | Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart[END_REF]. The WGN system can be written as

h t + [ h u ] x = 0, (10a) u -1 3 h ∂ x F h 3 ∂ x F u t + g h x + u u x = u 3 h ∂ x F h 3 ∂ x F u + 1 2 h 2 (∂ x F u) 2
x , (10b) where F is the Fourier multiplier defined by

Fϕ(ξ) def = 3 h ξ tanh( h ξ) -3 ( h ξ) 2 φ(ξ).
The dispersion relation of the WGN system [START_REF] Guelmame | Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom[END_REF] is exactly the same as the one of the full Euler system. Smooth solutions of the WGN system conserve the energy E t + D x = 0 with

E def = 1 2 h u 2 + 1 6 h 3 (∂ x F u) 2 + 1 2 g h -h 2 .
Since the energy E does not control the L 2 norm of h x , the results presented in this paper cannot be generalised directly for [START_REF] Guelmame | Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom[END_REF]. However, the WGN equations [START_REF] Guelmame | Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom[END_REF] deserve to be studied more in the future.

Other equations similar to (1) have been studied in the literature. For example, the Serre-Green-Naghdi equations with surface tension [START_REF] Dias | On the fully-nonlinear shallow-water generalized Serre equations[END_REF][START_REF] Lannes | The Water Waves Problem: Mathematical Analysis and Asymptotics[END_REF]

h t + [ h u ] x = 0, ( 11a
) [ h u ] t + h u 2 + 1 2 g h 2 + S x = 0, (11b)
S def = 1 3 h 3 -u tx -u u xx + u 2 x -γ h h xx -1 2 h 2 x , (11c)
where γ > 0 is a constant (the surface tension coefficient divided by the density). Instead of S , the original equations involve

S def = 1 3 h 3 -u tx -u u xx + u 2 x -γ h 1 + h 2 x -3/2 h xx + 1 + h 2 x -1/2 .
Since we consider shallow water waves, any horizontal derivative is of order O(σ), using then Taylor expansion we obtain S = S -γ + O (σ 4 ). Neglecting terms of order O (σ 4 ) we obtain [START_REF] Guelmame | Global Weak Solutions of a Hamiltonian Regularised Burgers Equation[END_REF]. Smooth solutions of ( 11) conserve an H 1 -equivalent energy. Weakly singular peakon travelling wave solutions of [START_REF] Guelmame | Global Weak Solutions of a Hamiltonian Regularised Burgers Equation[END_REF] have been studied in [START_REF] Dutykh | Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations[END_REF][START_REF] Mitsotakis | On weakly singular and fully nonlinear travelling shallow capillary-gravity waves in the critical regime[END_REF] for the critical case γ = g h2 /3. Another similar system is the dispersionless regularised Saint-Venant (rSV) system proposed by Clamond and Dutykh [START_REF] Clamond | Non-dispersive conservative regularisation of nonlinear shallow water (and isentropic Euler) equations[END_REF] that can be obtained replacing R in (1) by

T def = ε h 3 -u tx -u u xx + u 2 x -ε g h 2 h h xx + 1 2 h 2
x , where ε > 0. The classical Saint-Venant equations are recovered by taking ε = 0. The rSV equations have been studied in [START_REF] Clamond | Non-dispersive conservative regularisation of nonlinear shallow water (and isentropic Euler) equations[END_REF][START_REF] Liu | Well-posedness and derivative blow-up for a dispersionless regularized shallow water system[END_REF][START_REF] Pu | Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations[END_REF] and have been generalised recently to regularise the Burgers equation [START_REF] Guelmame | Global Weak Solutions of a Hamiltonian Regularised Burgers Equation[END_REF] and the barotropic Euler equations [START_REF] Guelmame | Hamiltonian regularisation of the unidimensional barotropic Euler equations[END_REF]. In [START_REF] Liu | Well-posedness and derivative blow-up for a dispersionless regularized shallow water system[END_REF], Liu et al. have proved the local well-posedness of the rSV equations and they derived smooth solutions that cannot exist globally in time. The proofs presented in this paper for the mSGN equations can be generalised for the SGN equations with weak surface tension [START_REF] Guelmame | Global Weak Solutions of a Hamiltonian Regularised Burgers Equation[END_REF], for the rSV equations and also for the regularised barotropic Euler system [START_REF] Guelmame | Hamiltonian regularisation of the unidimensional barotropic Euler equations[END_REF]. The blow-up criterion proved here is more precise compared to the blow-up criterion in [START_REF] Liu | Well-posedness and derivative blow-up for a dispersionless regularized shallow water system[END_REF]. The key of the proof of the blow-up results in this paper is Lemma 5 below, a similar lemma have been proved in [START_REF] Liu | Well-posedness and derivative blow-up for a dispersionless regularized shallow water system[END_REF] for a short time (shorter than the existence time in general). In this paper, we show, with a shorter (and different) proof, that the same result holds true as long as the smooth solution exists.

1.3.

Outline. This paper is organised as follows. In Section 2, we present a brief derivation of the mSGN equations [START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF]. The main results are introduced in Section 3. In Section 4, we prove some useful lemmas. Section 5 is devoted to obtain the precise blow-up scenario of strong solutions of the mSGN equations. In Section 6, we prove that some classical solutions cannot exist globally in time.

Derivation

2.1. Derivation. In this section, we recall briefly the derivation of the model (1) presented in [START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF]. The classical Serre-Green-Naghdi equations can be derived from the Lagrangian density

L 0 ρ = 1 2 h u 2 + 1 6 h 3 u 2 x -1 2 g h 2 + φ {h t + [h u] x } . (12)
The Euler-Lagrange equations lead to

h t + [ h u ] x = 0, u t + u u x + g h x = -1 3 h h 3 -u tx -u u xx + u 2 x x .
Those equations describe long waves in the shallow water regime, thus, any horizontal and temporal derivative is of order O(σ). This leads to

[u t + u u x + g h x ] x = -1 3 h h 3 -u tx -u u xx + u 2 x x x = O σ 4 .
In the Serre-Green-Naghdi model, all the terms of order O (σ 4 ) are neglected. Clamond et al. [START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF] modified the Lagrangian ( 12) by adding a neglected term to obtain the modified Lagrangian Lβ

ρ def = L 0 ρ + β h 3 12 [u t + u u x + g h x ] x .
The choice of h 3 in the additional term is not necessary, one can replace h 3 by any increasing function of h (see [START_REF] Guelmame | Hamiltonian regularisation of the unidimensional barotropic Euler equations[END_REF] for more details). Using the conservation of the mass, one can write

h 3 [u t + u u x ] x = [h 3 u x ] t + [h 3 u , u x ] x + 3 h 3 u 2 x , h 3 h xx = [h 3 h x ] x -3 h 2 h 2 x . This leads to Lβ = L β + [• • • ] t + [• • • ] x ,
where L β is the Lagrangian density defined in [START_REF] Clamond | Non-dispersive conservative regularisation of nonlinear shallow water (and isentropic Euler) equations[END_REF]. This means that L β ≡ Lβ = L 0 + O (σ 4 ). Deriving the Lagrangian density L β we obtain [START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF]. Thus, the mSGN system (1) is a suitable modification of the cSGN system that admits good properties and conserves a better energy compared to the cSGN equations for β > 0.

2.2. The dispersion relation. The choice of the free parameter β depends on the desired property. For example, in order to improve the dispersion relation of (1), one can linearise (1) around the constant state h, 0 and consider traveling waves on the form cos {k(x -ct)}. The dispersion relation then becomes

c 2 g h = 2 + β k h 2 2 + 2 3 + β k h 2 = 1 -1 3 k h 2 + 1 9 + β 6 k h 4 + • • • .
This must be compared with the exact relation

c 2 g h = tanh k h k h = 1 -1 3 k h 2 + 2 15 k h 4 + • • • .
Hence, in order to to have the same dispersion relation up to the order k 4 , one must take β = 2/15. Other criteria exits for choosing the parameter β (see [START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF] for more details).

Main results

The first result of this paper is the local well-posedness of the system (9) in the Sobolev space

H s (R) def = f, f 2 H s (R) def = R 1 + |ξ| 2 s | f (ξ)| 2 dξ < +∞ ( 13
)
where s 2 is a real number.

Theorem 1. Let β > 0, h > 0 and s 2, then, for any

(h 0 -h, u 0 ) ∈ H s (R) satisfying inf x ∈ R h 0 (x) > 0 there exists T > 0 and (h-h, u) ∈ C 0 ([0, T ], H s (R))∩C 1 ([0, T ], H s-1 (R)) a unique solution of (9) such that inf (t,x) ∈ [0,T ]×R h(t, x) > 0. ( 14
)
Moreover, the solution satisfies the conservation of the energy

d dt R 1 2 h u 2 + 1 6 α h 3 u 2 x + 1 2 g h -h 2 + 1 4 β g h 2 h 2 x dx = 0. ( 15
)
Remark 1. The solution given in Theorem 1 depends continuously on the initial data, i.e., If (h 0 -h, u 0 ), ( h0 -h, ũ0 ) ∈ H s , such that h 0 , h0 h * > 0, and t min{T, T }, then there exists a constant C(

( h -h, ũ) L ∞ ([0,t],H s ) , (h -h, u) L ∞ ([0,t],H s ) ) > 0, such that (h -h, u -ũ) L ∞ ([0,t],H s-1 ) C (h 0 -h0 , u 0 -ũ0 ) H s . (16)
The proof of Theorem 1 is classic and omitted in this paper. See [START_REF] Guelmame | Hamiltonian regularisation of the unidimensional barotropic Euler equations[END_REF][START_REF] Israwi | Large time existence for 1D Green-Naghdi equations[END_REF][START_REF] Lannes | The Water Waves Problem: Mathematical Analysis and Asymptotics[END_REF][START_REF] Liu | Well-posedness and derivative blow-up for a dispersionless regularized shallow water system[END_REF] and Theorem 3 of [START_REF] Guelmame | Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom[END_REF] for more details.

Remark 2. If (h -h, u) ∈ C 0 ([0, T ], H s (R)
) and h satisfies ( 14) for some T > 0, then Theorem 1 ensures that the solution can be extended over [0, T ]. In other words, if T max is the maximum time existence of the solution, then, we have the blow-up criterion

T max < +∞ =⇒ lim inf t→Tmax inf x ∈ R h(t, x) = 0 or lim sup t→Tmax (h -h, u) H s = +∞. (17)
The blow-up criterion (17) can be improved as in [START_REF] Guelmame | Hamiltonian regularisation of the unidimensional barotropic Euler equations[END_REF][START_REF] Guelmame | Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom[END_REF][START_REF] Liu | Well-posedness and derivative blow-up for a dispersionless regularized shallow water system[END_REF] to

T max < +∞ =⇒ lim inf t→Tmax inf x ∈ R h(t, x) = 0 or lim sup t→Tmax (h x , u x ) L ∞ = +∞. (18)
The last blow-up criterion ensures that if h > 0 is far from zero, then, the blow-up will appear on the L ∞ norm of u x or h x . This result is improved in this paper, and we claim the two more precise criteria for blow-up mechanism.

Theorem 2. Let β > 0, and let T max be the maximum time existence of the solution given by Theorem 1, then

T max < +∞ =⇒ lim inf t→Tmax inf x ∈ R h(t, x) = 0 or        lim inf t→Tmax inf x ∈ R u x (t, x) = -∞, and lim sup t→Tmax h x (t, x) L ∞ = +∞, ( 19
)
which is equivalent to second criterion

T max < +∞ =⇒ lim sup t→Tmax u x (t, x) L ∞ = +∞ and        lim inf t→Tmax inf x ∈ R h(t, x) = 0, or lim sup t→Tmax h x (t,x) L ∞ = +∞.
(20) Since H 1 → L ∞ and the energy ( 15) is conserved, then, |h -h| is controlled by the energy of the initial data (see Proposition 1 below). This ensures that if the initial energy is small enough then h is uniformly far from zero (min t,x h > 0). In this case, the blow-up criterion (19) becomes

T max < +∞ =⇒ lim inf t→Tmax inf x ∈ R u x (t, x) = -∞ and lim sup t→Tmax h x L ∞ = +∞. (21)
This blow-up criterion ensures that if a blow-up occurs, then u x goes to -∞. Since h is far from zero, then the conservation of the mass implies that the material derivative of the free surface goes to +∞. However, it is not clear if h x blows-up on -∞ or +∞. The following theorem shows that both scenarios are possible. Theorem 3. For any T > 0 and K ∈ 0, g

√ β 3 √ 2
h3 , there exist

• (h 0 -h, u 0 ) ∈ C ∞ c (R) satisfying R E 0 dx K such that the corresponding solution of (9) blows-up in finite time T max T and inf [0,Tmax[×R u x (t, x) = -∞, sup [0,Tmax[×R h x (t, x) = +∞, inf [0,Tmax[×R h x (t, x) > -∞.
• ( h0 -h, ũ0 ) ∈ C ∞ c (R) satisfying R Ẽ0 dx K such that the corresponding solution of (9) blows-up in finite time Tmax T and

inf [0, Tmax[×R ũx (t, x) = -∞, inf [0, Tmax[×R hx (t, x) = -∞, sup [0, Tmax[×R hx (t, x) < +∞.
Remark 3. All the proofs presented in this paper work also for the SGN equations with surface tension [START_REF] Guelmame | Global Weak Solutions of a Hamiltonian Regularised Burgers Equation[END_REF], for the rSV equations [START_REF] Clamond | Non-dispersive conservative regularisation of nonlinear shallow water (and isentropic Euler) equations[END_REF] and also for the regularised barotropic Euler system [START_REF] Guelmame | Hamiltonian regularisation of the unidimensional barotropic Euler equations[END_REF].

Preliminaries

In this section, we recall some classical estimates and we prove some lemmas that are needed to prove Theorem 2 and Theorem 3.

Lemma 1. ([3]) Let F ∈ C m+2 (R)
with F (0) = 0 and 0 s m, then there exists a continuous function F , such that for all f ∈ H s ∩ W 1,∞ we have

F (f ) H s F ( f W 1,∞ ) f H s . ( 22
)
Let Λ be defined such that Λf = (1 + ξ 2 ) 1 2 f , then we have the following estimate.

Lemma 2. ([13]) Let [A, B]

def = AB -BA be the commutator of the operators A and B. If r 0, then ∃C > 0 such that

f g H r C ( f L ∞ g H r + f H r g L ∞ ) , (23)
[Λ r , f ] g L 2 C ( f x L ∞ g H r-1 + f H r g L ∞ ) . (24)
Now, we recall the invertibility of the operator L h defined in [START_REF] Dutykh | Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations[END_REF] and that we gain two derivatives with L -1 h . Lemma 3. Let α > 0 and 0

< h ∈ W 1,∞ with h -1 ∈ L ∞ , then the operator L h is an isomorphism from H 2 to L 2 and ∃C 1 = C 1 α, s, h -1 L ∞ , h -h W 1,∞ > 0, C 2 = C 2 (α, h -1 L ∞ , h L ∞ ) > 0 such that (1) If s 0, then L -1 h ∂ x ψ H s+1 C 1 ψ H s + h -h H s L -1 h ∂ x ψ W 1,∞ , (25a)
L -1 h φ H s+1 C 1 φ H s + h -h H s L -1 h φ W 1,∞ . (25b)
(2) If s 0, then

L -1 h ∂ x ψ H s+1 C 1 ψ H s 1 + h -h H s , (26a)
L -1 h φ H s+1 C 1 φ H s 1 + h -h H s . ( 26b
) (3) If φ ∈ C lim def = {f ∈ C(R), f (+∞), f (-∞) ∈ R}, then L -1 h φ is well defined and L -1 h φ W 2,∞ C 2 φ L ∞ . ( 27
) (4) If ψ ∈ C lim ∩ L 1 , then L -1 h ∂ x ψ W 1,∞ C 2 ( ψ L ∞ + ψ L 1 ) . ( 28
) (5) If h x ∈ L 2 , then L -1 h ∂ x ψ H 1 + L -1 h ψ H 1 C 2 ψ L 2 , (29a)
L -1 h ψ W 1,∞ L -1 h ψ H 2 C 2 1 + h x 2 L 2 ψ L 2 . (29b)
Proof. The proof of the invertibility of L h and the estimates (25), ( 26), ( 27), ( 28) and (29a) can be found in [START_REF] Liu | Well-posedness and derivative blow-up for a dispersionless regularized shallow water system[END_REF]. It remains only to prove (29b). Using the definition of L h we obtain

∂ 2 x L -1 h ψ = ∂ x h -3 L -1 h h 4 ∂ x L -1 h ψ -α 3 ∂ x h 3 ∂ x h 3 ∂ x L -1 h ψ = ∂ x h -3 L -1 h ∂ x h 4 L -1 h ψ -α 3 h 3 ∂ x h 3 ∂ x L -1 h ψ -4 ∂ x h -3 L -1 h h 3 h x L -1 h ψ = ∂ x h -3 L -1 h ∂ x h 3 ψ -4 ∂ x h -3 L -1 h h 3 h x L -1 h ψ = -3 h -2 h x L -1 h ∂ x h 3 ψ -4 L -1 h h 3 h x L -1 h ψ + h -3 ∂ x L -1 h ∂ x h 3 ψ -4 h -3 ∂ x L -1 h h 3 h x L -1 h ψ .
Using (29a) and the embedding H 1 → L ∞ we obtain

∂ 2 x L -1 h ψ L 2 C 2 h x L 2 L -1 h ∂ x h 3 ψ H 1 + L -1 h h 3 h x L -1 h ψ H 1 + C 2 ∂ x L -1 h ∂ x h 3 ψ L 2 + C 2 ∂ x L -1 h h 3 h x L -1 h ψ L 2 C 2 h x L 2 ψ L 2 + h x L 2 L -1 h ψ H 1 + C 2 ψ L 2 + C 2 h x L 2 L -1 h ψ H 1 C 2 1 + h x 2 L 2 ψ L 2 .
This with (29a) imply (29b).

The R defined in (1c) contains some terms with two order derivatives, using (9), we can write R without those high order derivatives involving the operator L -1 h . Then, using the previous lemma we show that the norm R L ∞ is controlled by (h, u, h -1 ) L ∞ . Lemma 4. Let (h -h, u) be a smooth solution of (9), then for any T < T max , there exists

C = C(β, h, (h, u, h -1 ) L ∞ ([0,T ]×R) ) > 0, such that R L ∞ ([0,T ]×R) C. ( 30
)
Proof. From the definition of L h , we obtain that

1 + 1 3 α h 3 ∂ x L -1 h ∂ x Ψ = h 3 ∂ x L -1 h h x -∞ h -3 Ψ (31)
for any smooth function Ψ, such that Ψ(±∞) = 0. Using (1c), ( 9b) and ( 31) we obtain

R = -1 3 α h 3 ∂ x u t + u u x + α-1 α g h x + 2 3 α h 3 u 2 x -1 4 β g h 2 h 2 x = 1 + 1 3 α h 3 ∂ x L -1 h ∂ x 2 3 α h 3 u 2 x -1 4 β g h 2 h 2 x + g h 2 -h2 2 α -g h 2 -h2 2 α (32) = 1 + 1 3 α h 3 ∂ x L -1 h ∂ x 2 3 α h 3 u 2 x -1 4 β g h 2 h 2 x + g 3 h 3 ∂ x L -1 h {h h x } = h 3 ∂ x L -1 h h x -∞ 2 3 α u 2 x -1 4 β g h -1 h 2 x + g 3 h 3 ∂ x L -1 h {h h x } (33)
Using the conservation of the energy (15) we obtain

x -∞ 2 3 α u 2 x -1 4 β g h -1 h 2 x L ∞ 2 3 α u 2 x -1 4 β g h -1 h 2 x L 1 C 3 .
Then, the inequality (30) follows directly from ( 27) and (29b). Since we are considering the Serre-Green-Naghdi equations on the form (9) instead of [START_REF] Glassey | Singularities of a variational wave equation[END_REF], it is more convenient to use the following Riemann invariants 1 R, S and their corresponding speeds of characteristics λ, µ

R def = u + 2 α-1 α g h, λ def = u + α-1 α g h, (34)
S def = u -2 α-1 α g h, µ def = u - α-1 α g h, ( 35
)
1 Those quantities are constants along the characteristics if the right-hand side of ( 9) is zero.

rather that the Riemann invariants of the classical Saint-Venant system. Then, the system (9) can be rewritten as

R t + λ R x = -L -1 h ∂ x 2 3 α h 3 u 2 x -1 4 β g h 2 h 2 x + g 2 α h 2 , ( 36a
)
S t + µ S x = -L -1 h ∂ x 2 3 α h 3 u 2 x -1 4 β g h 2 h 2 x + g 2 α h 2 . ( 36b
)
Defining

P def = R x = u x + α-1 α g h -1 2 h x , Q def = S x = u x - α-1 α g h -1 2 h x ,
we have

u x = P + Q 2 , h x = √ α h 1 2 2 (α -1) g (P -Q) . ( 37
)
Let the characteristics X a , Y a starting from a defined as the solutions of the ordinary differential equations d dt

X a (t) = λ(t, X a (t)), X a (0) = a (38) d dt Y a (t) = µ(t, Y a (t)), Y a (0) = a (39)
Differentiation (36) with respect to x, and using (32) we obtain the Ricatti-type equations

d λ dt P def = P t + λ P x = -3 8 P 2 + 3 8 Q 2 + P Q -3 α -1 h -3 R, (40a)
d µ dt Q def = Q t + µ Q x = -3 8 Q 2 + 3 8 P 2 + P Q -3 α -1 h -3 R, (40b)
where d λ dt , d µ dt denote the derivatives along the characteristics with the speeds λ and µ respectively.

A key point to prove Theorem 2 and Theorem 3 is to control the term P 2 in the Ricatti equation (40b) and the term Q 2 in (40a). For that purpose, we prove in the following lemma that the integral of P 2 along the X characteristics and the integral of Q 2 along the Y characteristics are bounded. Lemma 5. Let β > 0, h > 0 and (h 0 -h, u 0 ) ∈ H 2 initial data satisfying inf x ∈ R h 0 (x) > 0 and let (h -h, u) be the corresponding solution of (9) given by Theorem 1, let also t ∈ [0, T max [, then, there exist

A β, h, (h, u, h -1 ) L ∞ ([0,t]×R) , E dx > 0, B β, h, (h, u, h -1 ) L ∞ ([0,t]×R) , E dx > 0,
such that for any x 2 ∈ R, and for

x 1 ∈] -∞, x 2 [ the solution of X x 1 (t) = Y x 2 (t) (see Figure 1) we have t 0 Q(s, X x 1 (s)) 2 ds + t 0 P (s, Y x 2 (s)) 2 ds A t + B. (41)
Remark 4. A similar result have been proved for the so-called variational wave equation with A = 0 and B depends only on the energy of the initial data [START_REF] Glassey | Singularities of a variational wave equation[END_REF]. For the mSGN, additional terms appear, and a uniform (on time) bound cannot be obtained for large data.

x t t x 0 s Y x 2 (s) X x 1 (s) x 2 x 1 Figure 1. Characteristics. Proof. Defining B 1 def = α-1 α g h 1 2 h u 2 + 1 2 g h -h 2 -u R + g h h -h , B 2 def = α-1 α g h 1 2 h u 2 + 1 2 g h -h 2 + u R + g h h -h ,
and using (30), one can prove that the quantity

B 1 L ∞ + B 2 L ∞ is bounded. One can easily check that λ E -D = √ 6 α β g h 7 12 Q 2 + B 1 , -µ E + D = √ 6 α β g h 7 12 P 2 + B 2 . (42)
Since

λ -µ = 2 α-1 α g h 2 α-1 α g h -1 -1 2 L ∞ > 0, then x 1 < x 2 . Integrating (3) on the set {(s, x), s ∈ [0, t], X x 1 (s) x Y x 2 (
s)}, using the divergence theorem, the energy equation ( 15) and (42) one obtains (41).

Blow-up criteria

The aim of this section is to prove Theorem 2, for that purpose, we consider s 2 and (h -h, u) the solution of [START_REF] Guelmame | Hamiltonian regularisation of the unidimensional barotropic Euler equations[END_REF] given by Theorem 1 with the initial data (h 0 -h, u 0 ) and we start by the following lemmas. Lemma 6. For T max < +∞, we consider the following properties

sup (t,x) ∈ [0,Tmax[×R u x (t, x) < +∞, (43a) inf
(t,x) ∈ [0,Tmax[×R h(t, x) > 0, (43b)
(h, u) L ∞ ([0,Tmax[×R) < +∞. (43c)
Then, (43a) ⇐⇒ (43b) and (43b) =⇒ (43c).

Proof. The proof of (43b) =⇒ (43c) follows directly from the conservation of the energy [START_REF] Li | Linear stability of solitary waves of the Green-Naghdi equations[END_REF] and the embedding

H 1 → L ∞ .
Let the characteristic Z a starting from a defined as the solutions of the ordinary differential equation

d dt Z a (t) = u(t, Z a (t)), Z a (0) = a. (44)
Denoting the derivatives along the characteristics with speed u by d u dt and using the conservation of the mass (9a) one obtains

d u dt h def = h t + u h x = -u x h, =⇒ h inf x ∈ R
h 0 e -sup t,x ux(t,x) Tmax . (45)

The proof of (43a) =⇒ (43b) follows directly from the last inequality. It only remains to prove the converse ((43b) =⇒ (43a)). Using the Young inequality

± a b 3 8 a 2 + 2 3 b 2 , (46)
integrating (40a), (40b) along the characteristics, and using (30), (41) one obtains the existence of à > 0, B > 0 which depend on β, h, (h, u, h -1 ) L ∞ ([0,T ]×R) and E dx, such that

P (t, X x 1 (t)) P 0 (x 1 ) + Ã t + B ∀(t, x 1 ) ∈ [0, T ] × R, (47a)
Q(t, Y x 2 (t)) Q 0 (x 2 ) + Ã t + B ∀(t, x 2 ) ∈ [0, T ] × R. (47b)
The last inequalities imply that sup (t,x) ∈ [0,Tmax[×R P (t, x) < +∞, and sup

(t,x) ∈ [0,Tmax[×R Q(t, x) < +∞, (48)
then, (43a) follows directly from (37).

Lemma 7. For T max < +∞, we consider the following properties

inf (t,x) ∈ [0,Tmax[×R u x (t, x) > -∞, (49a)
h x L ∞ ([0,Tmax[×R) < +∞. ( 49b
)
Then, (43b) =⇒ ((49a) ⇐⇒ (49b)).

Proof. We suppose that (43b) and (49a) are satisfied. Using Lemma 6 one obtains that u x L ∞ is bounded. Then (49b) follows directly from (48) and

h x = √ α h 1 2 (α -1) g (u x -Q) = √ α h 1 2 (α -1) g (P -u x ) . (50)
To prove the converse, we suppose that (43b) and (49b) are satisfied, then, using the Young inequality ±ab -1 2 a 2 -1 2 b 2 , (40a) and 30), (41) and Gronwall lemma, we obtain that inf t,x P > -∞. Using again (49b) we obtain (49a). Now, we can prove Theorem 2. Note that Lemma 6 implies the equivalence between ( 19) and (20). Then, it only remains to prove (19). Step 1 is devoted to prove the blow-up criterion [START_REF] Pu | Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations[END_REF]. The proof of ( 19) is given in Step 2.

P 2 = Q 2 + 4 α-1 α g h -1 2 u x h x = Q 2 + 2 α-1 α g h -1 2 h x (P + Q) (51) one obtains d λ dt P -7 8 P 2 -1 8 Q 2 -3 α -1 h -3 R = -Q 2 -7 4 α-1 α g h -1 2 h x P -7 4 α-1 α g h -1 2 h x Q -3 α -1 h -3 R -7 4 α-1 α g h -1 2 h x P -3 2 Q 2 -49 32 α-1 α g h -1 h 2 x -3 α -1 h -3 R. Using (49b), (
Proof of Theorem 2.

Step 1: In order to prove [START_REF] Pu | Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations[END_REF], we suppose that (h x , u x ) L ∞ ([0,Tmax[×R) < +∞, (43b) and we prove that if T max < +∞, then (h -h, u) L ∞ ([0,Tmax[,H s (R) < +∞ which contradicts with the definition of T max . For that purpose, we define

W def = (h -h, u) A(W ) def = α-1 α g 0 0 h , B(W ) def = u h α-1 α g u , F (W ) def = 0 -L -1 h ∂ x 2 3 α h 3 u 2 x -1 4 β g h 2 h 2 x + g 2 α h 2 , the system (9) becomes W t + B(W ) W x = F (W ). (52)
Let (•, •) be the scalar product in L 2 and E(W

) def = (Λ s W, A Λ s W ).
Since AB is a symmetric matrix, straightforward calculations with (52) show that

E(W ) t = -2 ([Λ s , B] W x , A Λ s W ) -2 (B Λ s W x , A Λ s W ) -2 (Λ s F , A Λ s W ) + (Λ s W, A t Λ s W ) = -2 ([Λ s , B] W x , A Λ s W ) + (Λ s W, (A B) x Λ s W ) -2 (Λ s F , A Λ s W ) + (Λ s W, A t Λ s W ) (53)
Defining B def = B( h, 0), and using (24), (22) one obtains

|([Λ s , B] W x , A Λ s W ) | C A L ∞ W H s B x L ∞ W x H s-1 + B -B H s W x L ∞ C1 W 2 H s ,
where C1 is a positive constant that depends on W W 1,∞ and h -1 L ∞ . Using the conservation of the mass (9a) one obtains that

|(Λ s W, (A B) x Λ s W )| + |(Λ s W, A t Λ s W )| C2 W 2 H s . (54)
From the energy conservation [START_REF] Li | Linear stability of solitary waves of the Green-Naghdi equations[END_REF], it is clear that

2 3 α h 3 u 2 x -1 4 β g h 2 h 2 x L 1 + g h 2 -h2 2 α L 2 C3 (55)
Then, using (25), ( 28), (29a), ( 23) and ( 22) we obtain

F H s C4 W H s . ( 56
) Since (h, h -1 ) L ∞ is bounded, then, W H s C5 E(W ).
Combining all the estimates above, one obtains

E(W ) t C E(W ), (57)
where C does not depend on W H s . Then, Gronwall lemma implies that

(h -h, u) L ∞ ([0,Tmax[,H s (R) C5 E(W ) L ∞ ([0,Tmax[) < +∞.
This ends the proof of (18).

Step 2: It remains to prove (19). We suppose that T max < +∞ and (43b) is satisfied. The blow-up criterion [START_REF] Pu | Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations[END_REF]

inf x ∈ R u x (t, x) = -∞ or lim sup t→Tmax h x L ∞ = +∞.
Finally, Lemma 7, insures that if one of the quantities above blows-up, then the other one should also blow-up at the same time. Then

lim inf t→Tmax inf x ∈ R u x (t, x) = -∞ and lim sup t→Tmax h x L ∞ = +∞.

Blow-up results

The goal of this section is to prove Theorem 3, for that purpose, we consider smooth solutions with small energy. In the following proposition we prove that if the energy is small enough, then, the quantity (h, u, h -1 ) L ∞ is uniformly bounded. Proposition 1. For β > 0, h > 0, let E be a positive number such that

0 < E < g √ β 3 √ 2 h3 , Defining h min def = h -3 √ 2 E g √ β h , h max def = h + 3 √ 2 E g √ β h , u max def = -u min def = (3/α) 1 4 √ E/h min .
Then, for any

(h -h, u) ∈ H 1 , if E dx E, we have 0 < h min h h max < 2 h, u min u u max , (58)
Remark 5. The conservation of the energy [START_REF] Li | Linear stability of solitary waves of the Green-Naghdi equations[END_REF] and Proposition 1 insure that if the initial data satisfy E 0 dx E, then, as long the the solution exist, the quantity (h, u, h -1 ) L ∞ is bounded by a constant that depends only on g, γ, h and E. Thence, all the constants given in ( 27), ( 28), (30), ( 41) and ( 47) are universal and do not depend on the initial data and the solution.

Proof of Proposition 1. The Young inequality 1 2 a 2 + 1 2 b 2 ±ab implies that

E R E dy R 1 2 g h -h 2 + 1 4 β g h 2 h 2 x dx g β 2 x -∞ (h -h) h h x dy - +∞ x (h -h) h h x dy g 3 β 2 h -h 2 2 h + h g 3 β 2 h h -h 2 ,
which implies that h min h h max . Doing the same estimates with u one obtains Let T > 0, and let Ã, B be the constants given in (47). From (30) we obtain that |α -1 h -3 R| C2 for some C > 0. If the initial data satisfy E 0 dx E, then, the constants Ã, B and C are universal and depend only on g, β, h and E (Remark 5). We

and we choose the initial data (h 0 -h, u 0 ) ∈ C ∞ c (R) such that there exist x 1 ∈ R satisfying R E 0 dx E, Q 0 ≡ 0, P 0 (x 1 ) < -2 D T + 1 , P 0 (x 1 ) < -8 T . (60)

Let t < min{ T , T max }, then (40b) with the Young inequality P Q

-3 8 P 2 -2 3 Q 2 imply that d µ dt Q -25 24 Q 2 -3 α -1 h -3 R -3 Q 2 + C2 . ( 61
)
The last inequality with (47b) imply that for all x 1 ∈ R, we have

-C -C tan 3 C t Q(t, Y x 2 (t)) Ã T + B, (62)
which implies that Q cannot blow-up before t = min T , T max and Q L ∞ D/4( T + 1). Using (40a) and the Young inequality P Q 2 . The last inequality follows from P (t, X x 1 (t)) 2 4 D( T + 1) 2 , which is true initially and holds because the map t → P (t, X x 1 (t)) is decreasing and negative. Then T max T T and from (47a) we obtain that inf

x dy α 3 h 2 min |u| 2 ,

 2 the last inequality ends the proof of u min u u max .Proof of Theorem 3. Since the proofs of the two parts of Theorem 3 are the same, we only prove the first part.

  Ã2 , B2 , C2 ,

[ 0 ,

 0 Tmax[×R P (t, x) = -∞, sup [0,Tmax[×R P (t, x) < +∞, sup [0,Tmax[×R |Q(t, x)| < +∞.

  1 8 P 2 + 2Q 2 one obtains d λ dt P (t, X x 1 (t)) -1 4 P (t, X x 1 (t)) 2 + 19 8 Q(t, X x 1 (t)) 2 + 3 C2 -1 4 P (t, X x 1 (t)) 2 + D T + 1

2

-

1

8 P (t, X x 1 (t))

References 16

Acknowledgements

The author would like to thank the anonymous referee for his remarks which help improving the paper. Many thanks go to Didier Clamond, Denys Dutykh and Stéphane Junca for the fruitful discussions.


Auriane Boudin
email: auriane.boudin@univ-amu.fr

Roxane Bertrand
email: roxane.bertrand@univ-amu.fr

Magalie Ochs
email: magalie.ochs@univ-amu.fr

Philippe Blache
email: philippe.blache@univ-amu.fr

Stéphane Rauzy
email: stephane.rauzy@univ-amu.fr

Are you Smiling When I am Speaking?

Keywords: Conversational feedback, Smile, Laughter, Corpus study, Generic/Specific feedback 6. Acknowlodgments

The aim of this study is to investigate conversational feedback that contains smiles and laughter. Firstly, we propose a statistical analysis of smiles and laughter used as generic and specific feedback in a corpus of French talk-in-interaction. Our results show that smiles of low intensity are preferentially used to produce generic feedback while high intensity smiles and laughter are preferentially used to produce specific feedback. Secondly, based on a machine learning approach, we propose a two-stage classification of feedback to automatically predict not only the presence/absence of a smile but, also the type of smile according to an intensity-scale (low or high).

Introduction

During conversations, interlocutors switch dynamically between the role of speaker and listener. The speaker produces discourse, giving information to the listener who produces feedback (referred as FB) 1 to show his/her active listening [START_REF] Schegloff | Discourse as an interactional achievement: Some uses of 'uh huh'and other things that come between sentences[END_REF] but also to contribute to the elaboration of the current discourse [START_REF] Bavelas | Listeners as co-narrators[END_REF][START_REF] Horton | Theories and approaches to the study of conversation and interactive discourse[END_REF]. FB promotes alignment between interlocutors, which allows the success of the interaction [START_REF] Pickering | An integrated theory of language production and comprehension[END_REF]. FB production is also studied in part to render human-machine conversations more efficient [START_REF] Glas | Definitions of engagement in human-agent interaction[END_REF]. Following [START_REF] Bavelas | Listeners as co-narrators[END_REF], generic FB (e.g. "mhmh", "okay", nod) is used to show understanding, while specific FB (e.g. "oh really", "that's so nice") is used to show assessment through diverse attitudes [START_REF] Schegloff | Discourse as an interactional achievement: Some uses of 'uh huh'and other things that come between sentences[END_REF][START_REF] Bavelas | Listeners as co-narrators[END_REF][START_REF] Horton | Theories and approaches to the study of conversation and interactive discourse[END_REF]. Both generic and specific FB can be unimodal or bimodal (vocal and/or visual). In this work, we focus on FB that contains smiles and laughter (associated with verbalization and/or nods). In this study, we first propose to explore how smiles and laughter are distributed according to the generic and specific FB dichotomy. Next, we present a twostage classification to automatically predict smile in FB instances. The 1 st stage of classification will predict whether a FB should be realized with a smile or a neutral face. The 2 nd stage of classification will predict for FB with a smile, the intensity of the smile (high or low). We make use of the open-access corpus PACO [START_REF] Amoyal | Paco: A corpus to analyze the impact of common ground in spontaneous face-to-face interaction[END_REF] and Cheese! (Priego-Valverde et al., 2020) to investigate multimodal FB. Through a statistical analysis and a machine learning approach on a conversational corpus, we explore the 2 following hypotheses. (1) High intensity smiles are more salient in the discourse and should be preferentially used to show assessments or specific attitudes rather than un-1 (also called conversational feedback or backchannel) derstanding. Indeed, specific FB is generally more marked than generic FB. Consequently, Neutral Faces (NF) and Low Intensity Smiles (LI Smiles) should be preferentially used to produce generic FB while High Intensity Smiles (HI Smiles) and laughter should be preferentially used to produce specific FB. (2) Listeners are influenced by the main speaker behavior and tend to align during FB production by adopting similar conversational markers (e.g. same smile intensity). Given the mechanism of alignment, the prediction of the smile and the intensity of the smile during a FB realization should be derived from the smile annotation of the speaker [START_REF] Heerey | Predictive and reactive mechanisms in smile reciprocity[END_REF]. In consequence, we expect to observe an important quantity of FB produced by the listener with a smile intensity similar to the one expressed by the main speaker.

Related Works

Feedback shows the collaboration between a speaker and an interlocutor during interactions [START_REF] Schegloff | Discourse as an interactional achievement: Some uses of 'uh huh'and other things that come between sentences[END_REF]. According to [START_REF] Bavelas | Listeners as co-narrators[END_REF], interlocutors can produce two types of FB: generic and specific. Generic FB shows understanding and is mostly realized with a nod and/or short vocalizations (e.g. "yeah", "mhmh"). On their side, specific FB is closely connected to the semantic content. It occurs once the common ground is established, when the listener has enough information to react with particular elements (wince, exclamation, rising tone) that can show surprise, amusement, enthusiasm, etc. [START_REF] Tolins | Addressee backchannels steer narrative development[END_REF]. Specific FB can be realized with variable elements such as lexicalization, laughter, head movements, eyebrow movements, facial expressions, etc.. Following the generic/specific dichotomy, we propose a fine-grained classification for specific FB by adding two sub-levels [START_REF] Boudin | A multimodal model for predicting conversational feedbacks[END_REF]. The 1 st level corresponds to the polarity: positive or negative. This polarity refers to the semantic content produced by the main speaker (e.g. a positive FB can respond to a fun story and a negative FB to a critic). The of the information given to the listener. The expected/unexpected category refers to the transmission of information. The main speaker can refer to the common ground, i.e. the information already shared with the listener (expected) or she/he can also give new information to the listener (unexpected). These two levels of specific FB allow to classify different attitudes expressed by FB (enthusiasm, happiness, humor, compassion, embarrassment, critic). Within each sub-category (positive-expected, positive-unexpected, negative-expected, negative-unexpected), we infer that FB could be realized with some typical patterns (e.g. a rising intonation, with a smile and raised eyebrows for a positive-unexpected FB). In this work we focus on the different types of smiles used within each type and sub-type of FB. To our knowledge, there is few systematic studies on smiling and its role as FB. Smiles have been identified as a part of FB form quite early [START_REF] Brunner | Smiles can be back channels[END_REF]. [START_REF] Duncan | Strategy signals in face-to-face interaction[END_REF] observe that the listener's FB has a greater probability to be produced with a smile if the speaker is actually smiling. [START_REF] Allwood | A study of gestural feedback expressions[END_REF] investigate FB functions and point out that smiles are frequently used to produce acknowledgments and clarification requests. Smiles can also show a reinforcement of a positive attitude. Among few studies, [START_REF] Jensen | Smile as feedback expressions in interpersonal interaction[END_REF] look at smiles and laughter as FB. In their data 33.3% of smiles and 18.6% of laughter is used as FB. Note that as far as we know, there are few research works which attempt to predict automatically smiles in FB production. [START_REF] Kok | When do we smile? analysis and modeling of the nonverbal context of listener smiles in conversation[END_REF] predict 3 types of smiles (amused, polite and embarrassed) during conversation with a Conditional Random Fields (CRF) algorithm. Four models are trained and evaluated to predict smiles and the type of smile. However, the prediction scores remain low, with f-score under 0.20. [START_REF] El Haddad | Towards a listening agent: A system generating audiovisual laughs and smiles to show interest[END_REF] predict smiles and laughter FB with different intensity levels. A CRF model is trained accepting as input features laughter and smiles of different intensities produced both by the speaker and the listener. The predicted FB instances are implemented in a virtual agent and compared with different baselines. A subjective evaluation leads to satisfying and promising results.

Corpus & Method PACO-Cheese!

Corpus We used the French Cheese! and PACO corpora. They contain a total of 7 hours of audio-visual recording of 26 dyadic face-toface interactions, lasting between 15 and 20 minutes. In the current work a subset of 13 dyads (3.6 hours) is used, on which instances of FB have been annotated. The full set of available annotations is described in [START_REF] Priego-Valverde | cheese!": a corpus of face-to-face french interactions. a case study for analyzing smiling and conversational humor[END_REF][START_REF] Amoyal | Paco: A corpus to analyze the impact of common ground in spontaneous face-to-face interaction[END_REF][START_REF] Boudin | A multimodal model for predicting conversational feedbacks[END_REF]. Laughter has been manually annotated during the transcription process. In [START_REF] Amoyal | Smiling for negotiating topic transitions in french conversation[END_REF][START_REF] Rauzy | SMAD: a tool for automatically annotating the smile intensity along a video record[END_REF][START_REF] Amoyal | Paco: A corpus to analyze the impact of common ground in spontaneous face-to-face interaction[END_REF], smiles have been annotated with 5 labels from the smile intensity scale (SIS) proposed by [START_REF] Gironzetti | Smiling, gaze, and humor in conversation: A pilot study[END_REF]: S0 (neutral face), S1 (close mouth smile), S2 (open mouth smile), S3 (wide open mouth smile), S4 (laughing smile). S4 are mainly associated with vocal laughter. Regions between two vocal laughter could also be annotated S4 if the facial posture did not change. On their side, laughter is vocal elements that can be produced with neutral face or smile of lower intensity. Therefore, there is not a one-to-one correspondence between the two entities and we prefer to distinguish both laughter and S4 in the current work. Instances of FB have been annotated following the 5 labels described in section 2: generic, positive-expected, positive-unexpected, negative-expected and negativeunexpected. They are reactions from one speaker to the other speaker's speech and can be composed of verbalization, nods, laughter and smile or a combination of these elements.

Logistic regression

We used a Logistic Regression algorithm (Logit). The Logit models the probability that a FB occurs with a smile (and its associated intensity level). It allows to evaluate the specific contribution of each feature which facilitates the interpretation of the model. The Logit proves also to be relevant when dealing with small datasets. A binary classifier response is obtained from the Logit probability, by applying a probability threshold filter. While in [START_REF] Boudin | A multimodal model for predicting conversational feedbacks[END_REF] we aimed at predicting the position and the type of FB, in the current research work we consider that the position and the type of the FB is already known and we focus on the prediction of one element of the FB form: smile. For that, we propose a two-stage classification where the 1 st stage predicts the presence or absence of a smile in the FB form (993 FB with a neutral face and 1372 FB with a smile). The 2 nd stage predicts the intensity of the smile (high or low) for the subset of FB containing a smile. In order to obtain balanced classes, the S1 and S2 smiles have been grouped as LI Smiles (361 FB with 'S1' and 284 FB with 'S2') and S3 and S4 as HI Smiles (188 FB with 'S3' and 539 FB with 'S4'). The dataset is composed of all the annotated FB with the associated smile used to produce it. When different smiles are used to produce the FB, we keep only the smile with the highest intensity. A prediction is correct if the item that composed the FB predicted matches with the item that composed the observed FB. A cross-validation has been obtained by running a Monte Carlo cross-validation (on 50 trials with a ratio 80%-20% for the training versus the evaluation sample) for both models. For comparison, two baseline models are computed that randomly predict the class according to the observed corpus frequency for the 2 distributions: smile/no-smile and LI/HI Smiles. Features are extracted from the speaker signal before the listener's FB. The subset of multi-modal features (a total of 16 features) is based on our previous analysis in [START_REF] Boudin | A multimodal model for predicting conversational feedbacks[END_REF] to predict the FB type:

• Pause (presence or absence of silent pauses, before FB). Overlap (FB is produced during the speech of the main speaker) -Binary encoding (0: absence, 1: presence). 20% of FB is produced with more than one intensity of smiles (e.g. "yeah exactly" that begins with a S0, continues with S1 and ends in S4). Among these instances of FB with particular smile's pattern, the majority (66%) shows an increasing smile intensity.

Figure 1 presents the smile intensities and laughter used to produce FB according to their generic/specific type2 Figure 2 details the smiles and laughter for sub-types of specific FB.

All FB: Globally, 42.35% of FB is produced with a Neutral face (NF). S1, S2, S3 and S4 are equally used (27% for S1/S2, 30% for S3/S4). 17.52% of FB is produced with a laughter. Only 9.87% of FB is realized with a smile or a laughter alone. The rest of the time, FB is associated with verbalization, nods or others facial movements. Note that at least 71% of FB annotations in our corpus are multimodal3 .

Generic FB : NF (58.58%) is mostly used to produce generic FB. Regarding FB produced with a smile, the more the intensity of the smile increases, the more its use decreases. Generic FB rarely contains a laughter (1.74%). Specific FB: Only 25.66% of specific FB is produced with a NF. 24.55% are produced with a LI Smile. 48.65% of specific FB is produced with a HI Smile. 33.76% contain a laughter. Laughter and HI Smiles are more present for positive FB, specifically for unexpected ones compared to negative FB. Concerning negative FB, NF is preferentially used, especially for the expected ones. Nonetheless, as we expected, smiles are still present for negative FB since smiles can be used to show embarrassment or compassion. These observations confirm our 1 st hypothesis: NF and LI Smiles are mainly used to produce generic FB whereas LI Smiles and laughter are mainly used to produce specific FB. These observations support our typology of FB, particularly useful to characterize the form of FB.

Speaker & Listener alignment

There are various ways to evaluate alignment between interlocutors [START_REF] Rauzy | A measure of the smiling synchrony in the conversational face-to-face interaction corpus PACO-CHEESE[END_REF]. Herein, we focus on the alignment between the listeners and the speakers by looking at the smiles and laughter produced both as FB (by the listener) and as features (by the speaker in a window of 2s before the FB). For each level of smiles defined above, we compute 3 quantities: the proportion P F B of FB containing the given level among all the FB, the proportion P F eat of features containing the given level and the proportion P F B/F eat of pair of features/FB containing conjointly the given level. The first two quantities allow to compute the proportion of cooccurrences by chance. We define the alignment's ratio as the ratio of the observed proportion P F B/F eat to the proportion expected by chance P F B × P F eat . Figure 3 presents the logarithm of the alignment's ratio and its associated 2σ standard error bars for NF, LI Smiles, HI Smiles and Laughter. We observe a significant alignment for all the group except for the LI Smiles, particularly for laughter and HI Smiles. These results confirm our 2 nd hypothesis about alignment, except for LI Smiles. Nonetheless, LI Smiles are visually more subtle, which can explain that they are less employed in the alignment strategy.

Logit

The 1 st model predicts smiles in FB. The 2 nd model predicts the smile intensity. The performances and the selected features are presented in Table 1 and2. Smile prediction: The 1 st model provides accurate performances, significantly better than the baseline (ttest provided a p-value < 0.001). All smiles intensity levels are selected as features by the Logit and multimodal features appear significant. To estimate the importance of multi-modality, we test the models with only smiles features. For the 1 st model, a t-test (p-value < 0.05) confirms that multimodal features perform better than smile features alone.

Smile intensity prediction:

The 2 nd fine-grained model gives reliable scores, better than the baseline (t-test provided a p-value < 0.001). Only NF and HI Smiles are selected, which are the most extreme smile intensity. This suggests that the most salient markers produced by the main speaker are the most informative for choosing the smile intensity. Removing the other multi-modal features does not significantly alters the performance obtained when using only smile features. These results suggest that not only smiles but also contextual parameters (speaker activity and semantic polarity) are relevant to decide whether a FB should be produced with a smile or not. Once the listener has decided if a smile will compose his/her FB, NF and HI Smile are sufficient enough to choose the smile intensity, through mechanisms of alignment. Finally, these results are in line with our 2 nd hypothesis, indicating that the smiles from the speaker are a good predictor of the smiles produced by the listener.

Conclusion

In this work, we focused on smiles and laughter as conversational FB in French face-to-face conversation. The data reveal that neutral faces (NF), Low Intensity Smiles (LI Smiles) and High Intensity Smiles (HI Smiles) are used to produce both generic and specific FB. Nonetheless, some trends emerge. Our analysis highlights that generic FB is preferentially produced with NF and LI Smiles, while specific FB, especially positive FB, are preferentially produced with laughter and HI smiles. The same behavior is observed for unexpected FB. For negative FB the trend in the different intensity of smiles stays unclear and need deepest investigations. To better understand it, we could analyse the smiles functions (e.g. embarrassment, compassion, showing sympathy) [START_REF] Hoque | Are you friendly or just polite?-analysis of smiles in spontaneous face-to-face interactions[END_REF][START_REF] Mazzocconi | What's your laughter doing there? a taxonomy of the pragmatic functions of laughter[END_REF]. Alignment between the speaker and the listener is measured for NF, HI Smiles and laughter. Laughter is the behavior that is the most reproduced by the listener when it is produced by the speaker. Finally, we presented a hierarchical classifier method to predict smiles and their intensity for FB production, that obtains reliable performances. The model also indicates that the smile intensity features play an important role in the prediction which confirms our results on alignment. The current work come along with a larger project about the prediction of the FB position and the type of FB. Ultimately, it will provide a complete model including the prediction of localization, types and multimodal component of FB allowing the implementation in an effective dialog system, see for example [START_REF] El Haddad | Towards a listening agent: A system generating audiovisual laughs and smiles to show interest[END_REF].

  • Positive, Negative, Concrete tokens (that give potential cues about the FB sub-type) (Bonin et al., 2018) -Categorical encoding: counted since the last FB produced. • Interjection, Discourse markers, Punctuation (Rauzy et al., 2014). Extracted in a previous window of 2 seconds and binary encoding. Number of tokens in the previous 2 seconds -Categorical encoding. • Nod, Smiles (S1, S2, S3, S4, S0), Laughter -Extracted in a previous window of 2 seconds ; binary encoding. 4. Results & Discussion 4.1. Laughter and Smiles for FB Production A total of 2,380 instances of FB was annotated: 1,207 generic and 1,173 specific, including 416 positiveexpected, 550 positive-unexpected, 115 negativeexpected, 92 negative-unexpected. During the 13 interactions, we report a total of 1215 'S0' , 1014 'S1', 944 'S2', 729 'S3', 798 'S4', among smiles 40% are used inside FB. 1051 laughter has been annotated (including 417 as FB).

Figure 1 :

 1 Figure 1: Generic and Specific FB produced with Neutral face (S0), Smiles according to their intensity level (S1, S2, S3, S4) and laughter. When a FB contains plural smiles, only the highest intensity is kept.

Figure 2 :

 2 Figure 2: Specific FB (Positive-Expected, Positive-Unexpected, Negative-Expected, Negative-Unexpected) produced with Neutral face (S0), Smiles according to their intensity level (S1, S2, S3, S4) and laughter. When a FB contains plural smiles, only the highest intensity is kept.

Figure 3 :

 3 Figure 3: Logarithm of alignment's ratio for NF (S0), LI Smile (S1/S2), HI Smile (S3/S4) and Laughter between the speaker and the listener.

Table 1 :

 1 F-score (F), Precision (P) and Recall (R) for the two predictive models and their baseline.

	Pred		F	P	R
	Smile		0.72 0.83 0.64
	Smile Baseline	0.57 0.57 0.57
	Intensity		0.66 0.72 0.61
	Intensity Baseline 0.48 0.48 0.48
	Pred	Features
	Smile	S4, S3, S1, S0, S2, Overlap, Laughter,
		Pause, Positive Token
	Smile intensity S4, Overlap, Discourse marker, S0, S3

Table 2 :

 2 Features selected by the Logit for the two classification tasks: smile/non smile and LI Smile/HI Smile prediction. Features presented are those selected by the Logit and ranked by their order of importance.

When several smiles are used, only the one with the highest intensity is counted.

3 Our annotations did not contain eyebrow movements, nor other head movements than nods, nor facial expressions. With these annotations, the percentage of multimodal FB would be probably be higher

Research supported by grants ANR-16-CONV-0002 (ILCB) and the Excellence Initiative of Aix-Marseille University (A*MIDEX).