Aymen Kamoun

Saïd Tazi

Khalil Drira

FADYRCOS, a semantic interoperability

Keywords: collaboration, interoperability, semantic adaptation, dynamic reconfiguration

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Introduction

Collaboration between enterprises requires more interoperability of services in dynamic ways that support the growing complexity of business processes. The collaborative development of complex products provides a wide range of new challenges in distributed systems. Teams of engineers have different roles and experiences; they must deal with numerous types of information both on the product being developed, on the technology used such as the development tool, but also on the organization adopted for work. Due to the size and the complexity of groups and products, the process may require several iterations of design, development and test and hence requires sophisticated modes of connections. Moreover it is important to guarantee transparent collaboration of actors respecting the different roles they are playing and despite the distance and the disparate tools they are using for collaboration and communication.

Teams need to be connected in a manner they feel themselves that the collaboration is continuous and fluid. It is important to ensure the continuity of communication services even if changes happen such as the absence, the presence or the role change of a collaborator. A global model of collaboration is necessary to guarantee the quality of communication and to ensure adaptability and interoperability between tools whatever may happen of such events. A global model of collaboration is necessary to guarantee the quality of communication and to ensure dynamic interoperability between tools. The design of an appropriate system to deliver intelligently the communication flows between the different parts in all situations ensuring the continuum of collaboration is one of CSCW (Computer-Supported Collaborative Work) research challenge [START_REF] Carstensen | Computer supported cooperative work: New challenges to systems design[END_REF]. For example if a collaborator joins/leaves or changes his role within a project, how the system may enable/disable automatically the communication flow adapted to its context of use such as its role, optimized Quality Of Service parameters? How his collaborators may continue their work despite this change? And how the system may ensure the continuum of collaboration by providing new flow of communication regarding the role of each collaborator and their tasks? In this perspective, we propose to explore promising research methodology grounding on analysis of semantic collaboration to ensure interoperability and thus the communication. Semantic interoperability is the combination of the meaning of data and process actions shared in a collaborative work to ensure interchange between human being implied in the collaboration and between the systems that may be the subject or the means of collaboration.

In this paper, we present FADYRCOS (A Framework for DYnamic Reconfiguration of networked COllaborative Systems) that supports semantic adaptation enabling the awareness of the presence/absence, roles and tasks of collaborators. This framework is based on a generic multi-level modeling approach that ensures multi-level adaptation. A generic collaboration model, based on Semantic Web technologies is proposed in order to support real-time collaboration between groups of engineers working together in different enterprises. The framework defines common interfaces for collaborative systems to enable the management of cooperative actions. In the present work we focus on collaboration between individuals belonging to several teams working on different projects. Usually, individuals participating to the collaborative activities are organized within sessions. A session is composed of a set of users that share common interests. These users may get connected to the session from different places. They manipulate data shared through the groupware tools that support collaborative activities. Sessions can be synchronous or asynchronous. In a synchronous session, user co-presence is required to realize the collaborative work. Users communicate interactively and handle the collaborative tools and data of the session in real-time. In an asynchronous session, user co-presence is not required. As a consequence, users can not perform interactive exchanges, and communication is made through asynchronous media. Swapping between asynchronous and synchronous mode is now possible in recent collaboration support tools. For instance, collaborative production of an artifact can occur in an asynchronous mode, but synchronous sessions can be created to synchronize and coordinate artifact production. Sessions are called explicit when all session configurations are defined offline, before session enactment. Relations between members are clearly defined. Such sessions, with their evolution in time, are firstly defined by a privileged user called session designer. Then, systems handle the defined session instances. Other users can join the defined sessions or they can be invited by the session designer. The three main elements of sessions (i.e. users, hardware devices, and software tools) are explicitly modeled.

In this paper, we focus on architectural adaptation in collaborative environments in which participants have different roles and belong to one or many groups. Therefore, they are organized within sessions dynamically depending on their roles or tasks. In order to achieve this adaptation, a set of interconnected components must be deployed in the system. Since sessions are dynamic and participants may change during the collaborative activities, an adaptive components deployment must be considered in such systems to preserve this collaboration. This deployment is needed for establishing multimedia sessions between participants to support collaboration. Adaptation actions in these systems involve components (re)deployments and exchanged flows reconfigurations. For instance, if a new participant wants to join a group in a specific session and needs to collaborate in audio, the system should be able to deploy an audio component in his machine. An audio flow must be setup between this component and the other components deployed in the devices of participants with whom he can communicate. Therefore, a new deployment schema is generated by the system to ensure the communication.

The rest of this paper is organized as follows. Section 2 presents related works, we focus especially on collaborative activities and session management. Section 3 presents the proposed multi-level modeling approach and the framework FARDYCOS for adaptive group communication. Section 4 discusses the maintaining of the interoperability of different applications domain. Section 5 presents a case study and section 6 concludes the paper.

Related work

Significant research has been carried out on collaborative activities and session management. A majority of these solutions deal with different aspects of collaboration and communication. However, as far as we know, very few works treat specifically the problem of providing tools for building context-aware collaborative applications with dynamic reconfiguration of components in runtime.

In CSCW, a groupware is defined as "Computer-based systems that support groups of people engaged in a common task to reach a common goal". They provide an interface to a shared environment [START_REF] Ellis | GROUPWARE -SOME ISSUES AND EXPERIENCES[END_REF]. There are many technical and social challenges that have to be faced in designing groupware products. Examples of technical challenges are conflict of operations, notifications, etc. Many issues has been resolved such as coherence and consistency of actions [START_REF] Ellis | GROUPWARE -SOME ISSUES AND EXPERIENCES[END_REF], face to face interaction [START_REF] Nagasundaram | Ambiguity in Human Communication and the Design of Computer Mediated Communication Systems[END_REF], unpredictable actions that can be carried out in several places [START_REF] Grudin | [END_REF]. Several taxonomies of CSCW have been proposed according to many criteria like functionality [START_REF] Karsenty | Le collecticiel: de l'interaction homme-machine à la communication hommemachine-homme[END_REF], purpose [START_REF] Lewkowicz | Collecticiels pour la construction collective du sensdéfinition, principes de conception[END_REF] and domain of application [START_REF] Laurillau | Clover architecture for groupware[END_REF]. Most of published collaborative tools focus on communication (messaging) and coordination features (approval forms, video-conference), but few of them deal with collaboration amongst actors [START_REF] Yesilbas | Towards a knowledge repository for collaborative design process: focus on conflict management[END_REF]. Some research deal with the content of communication, studies such as [START_REF] Robillard | Measuring cognitive activities in software engineering[END_REF] show that the co-designers spend much time clarifying what it was written in shared documents According to Dourish et al. [START_REF] Dourish | Awareness and coordination in shared workspaces[END_REF], "Awareness of individual and group activities is critical to successful collaboration and is commonly supported in CSCW systems". In CSCW, the notion of group awareness presents a significant feature. Dourish et al. [START_REF] Dourish | Awareness and coordination in shared workspaces[END_REF] defined it as "an understanding of the activities of others, which provides a context of your own activity". This concept depends on the group topology such as members who have strong or weak relationships or members who have different or same languages and experiences [START_REF] Palmer | Multigroup Decision-Support Systems in CSCW[END_REF]. Moreover, as members are not co-present in virtual workspace, it is prominent to create a group awareness model based on different methods such as providing a clear synthesis of all information from different parts of system.

Communication presents a fundamental feature in collaborative workspace. Teeni [START_REF] Te''eni | Review: a cognitive-affective model of organizational communication for designing IT[END_REF] defines it as an act of building a mutual understanding between the sender and the receiver. Communication issues can increase when the sender and the receiver are distant. During team meetings, the members share documents and they can explain directly what they mean. But, communication problem becomes more complex when participants are distant collaborating in an asynchronous mode. As a result, communication matters can have a direct drawback on the shared knowledge. This can have thereafter an impact on project progress. Many studies have so addressed mediated communication issues between designers [START_REF] Lewkowicz | Misunderstandings in Global Virtual Engineering Teams: Definitions, Causes, and Guidelines for Knowledge Sharing and Interaction[END_REF][START_REF] Edwards | Putting computing in context: An infrastructure to support extensible contextenhanced collaborative applications[END_REF].

Baudin et al. [START_REF] Baudin | A model-driven approach for synchronous dynamic collaborative e-learning[END_REF] has thoroughly studied graph-based explicit session management models and services in order to support collaboration inside groups of human users. The goal was to explicitly model relationships of information exchange between users in order to keep a tight coupling between communication and network layers. The proposed graph-based collaboration model is based on data producer/consumer relationships. Three elements have to be managed: users, tools and data flows. Vertices represent users and the labeled edges model the dataflow relationships between users. Therefore, data exchanges for synchronous and interactive work sessions are represented and modeled. Such sessions handle interactive data flows (e.g. video, real time audio). The dynamics of the session is expressed in terms of users entries/exists and user role changes. The session designer explicitly selects collaboration graph structures where collaborative work can occur. Advantages of this collaboration model are twofold. Firstly, this model is simple enough to be easily handled by session designers for various collaborative configurations. Secondly, instances of this model can be automatically taken into account by services or platforms that can be configured by the model. The sessions explicitly designed are therefore managed by model-based platforms. Sessions management services are used in order to manage all session's elements over collaborative systems. These services provide also data flow management and tool management. Deployment of collaboration tools is important in order to ensure session's enactment and collaboration between individuals.

Implicit sessions emerge from the observation of users' actions and their context. When the system detects a potential situation of collaboration, such as human presences, it creates an implicit session and invites involved users to join it. As far as we know, few works were tackling implicit sessions. However, we can cite [START_REF] Edwards | Session management for collaborative applications[END_REF][START_REF] Texier | Automatic Management of Sessions in Shared Spaces[END_REF] where models are based on set formalisms or [START_REF] Rusinkiewicz | Towards a Cooperative Transaction Model -The Cooperative Activity Model[END_REF] where models are based on first-order logic formalism, in order to describe unstructured sessions.

We have analyzed existing synchronous collaborative systems such as TANGO [START_REF] Beca | Tango -a Collaborative Environment for the World-Wide Web[END_REF], HABANERO [START_REF] Chanbert | NCSA Habanero-Synchronous collaborative framework and environment[END_REF], DOE2000 [START_REF]Real-time Collaboration Management Project Summary[END_REF] and DISCIPLE [START_REF] Marsic | Real-Time Collaboration in Heterogeneous Computing Environments[END_REF]. The main disadvantage of these systems is that the group member roles are predefined and cannot be changed dynamically during the collaborative activities. Furthermore, either the "master-Slaves" roles or the same role and identical rights for all participants are supported. Consequently, these systems cannot support complex structured sessions. Thus, model-based approaches are required in order to describe session's organization and to ensure the flexibility in the described systems. Moreover, several approaches propose models controlling only one element of the session such as media [START_REF] Holvoet | Reflective programmable coordination media[END_REF], communication [START_REF] Fussell | Coordination of communication: effects of shared visual context on collaborative work[END_REF][START_REF] Gutzmann | Access Control and Session Management in the HTTP Environment[END_REF], floor control [START_REF] Peralta | An XML on-line session model based on graphs for synchronous cooperative groups[END_REF], quality of service control [START_REF] Peralta Laura | [END_REF] and collaborative services [START_REF] Vissers | Reference Models for Advanced e-services[END_REF]. Other sessions models describe only three components of a session: users, tools and data flows [START_REF] Edwards | Session management for collaborative applications[END_REF][START_REF] Dommel | Group Coordination Support for Synchronous Internet Collaboration[END_REF]. These proposed models create collaborative sessions by monitoring members' activities. Some of them support dynamic change and provide a representation of the sessions, but this representation is too specific to the model, what restricts its use in other systems. Some platforms provide tool-based implementation for the session management, whereas, they don't support the representation of complex structured sessions, and even those which allow complex sessions modeling are still theoretical and they don't provide real implementations. Other awareness tools have been proposed: The TeamScope system [START_REF] Jang | Supporting awareness among virtual teams in a web-based collaborative system: the teamSCOPE system[END_REF] allows participants exchanging files and working on it without real time interaction between them. The Classroom BRIDGE [START_REF] Ganoe | Classroom BRIDGE: using collaborative public and desktop timelines to support activity awareness[END_REF], the Collablogger tool [START_REF] Morse | CollabLogger: A Tool for Visualizing Groups at Work[END_REF] and the CAliF system [START_REF] Guyennet | Proceedings of International Conference on information Systems Analysis and Synthesis[END_REF] support distributed group projects. However, they are not able to support dynamic changes of the session's structure.

Other ontologies-based works are proposed and applied to different problems of CSCW. Gu et al. [START_REF] Gu | Ontology based semantic conflicts resolution in collaborative editing of design documents[END_REF] present an onotology-based system for collaborative editing and design documents. The system aims to avoid semantic conflicts. Yao et al. [START_REF] Yao | An ontology based workflow centric collaboration system[END_REF] introduce an ontology-based system for workflow management. This system can interact with other ontology-based applications. Andonoff et al. [START_REF] Andonoff | A protocol ontology for inter-organizational workflow coordination[END_REF] proposes an ontology of high level protocols for agents conversations. Ontologies are used in order to provide semantic to these protocols and to ensure automation of coordination in distributed systems. Garrido et al. [START_REF] Garrido | Definition and use of Computation Independent Models in an MDA-based groupware development process[END_REF] propose an MDA-based approach for modeling enterprise organization and developing groupware applications. The domain model is formalized through a domain ontology in order to describe concepts and relations between actors sharing knowledge. Tomingas and Luts propose a semantic interoperability framework for data management like web services descriptions and ontologies [START_REF] Tomingas | Semantic Interoperability Framework for EstonianPublic Sector's E-Services Integration[END_REF].

The main disadvantage of classical CSCW systems is the lack of flexibility in sessions which are pre-defined and don't support dynamic changes, what cause a problem of interoperability. Indeed, a given CSCW system which supports collaborative activities in a specific domain and in particular in a specific group organization this system cannot coordinate collaborative activities in another domain; and even in a particular working group, sessions and roles organization cannot be changed. Few works consider implicit sessions which are spontaneous and depends on implicit collaboration activities. In systems represented above, sessions are initiated explicitly and the dynamic of changes is limited to roles changes. Moreover, if a new situation of collaboration is required at runtime, the system is not able to provide the adequate service which satisfies users' needs. Thus, session designer should define new session structure. Furthermore, even if a system defines all possible sessions, it should be implemented and hard-coded. This method limits the code reutilization and its maintenance. Therefore, a generic framework that defines common vocabulary for session modeling is needed. Session designers should easily define and implement new session structures for different groups working together and even for the same group at runtime. Another disadvantage of these systems is the lack of rigid deployment services of components or application that they offer. Components and applications are often deployed manually on participants' machines and fixed at design time by a static way. This method cannot be applied to situations that need a high degree of adaptation, and in which even not known components should be deployed in advance. In our work, we intend to support collaboration in distributed environments where sessions can be implicit, new mechanisms are needed for managing session evolution and role changes. In our view, Semantic Web techniques are well suited to achieve this task. As far as we know, there is no existing collaborative system that use semantic for session management and dynamic deployment service. As thus, in this paper, a semantic driven framework has been developed to enable session management and dynamic components deployment for collaborative systems. The next section will describe our approach towards the Framework we have developed.

FADYRCOS

This section presents our proposed approach for the modeling and the implementation of collaborative activities. We propose a generic framework enabling session management, semantic multi-level adaptation and automatic components deployment for collaborative activities called FADYRCOS (Framework for dynamic reconfiguration of networked collaborative systems).

Multi-Level modeling approach

In order to clearly separate different concerns in our approach, a multi-level modeling architecture is proposed (Figure 1). Each level encapsulates specific problems into a specific model. Models represent sets of linked entities like software components, called configurations. High level configurations represent high level requirements, such as participants' organization and their activities, while low levels models represent the real implementation ensuring such activities. Separation between levels may introduce the concept of multi-level adaptation. Indeed, distinguishing different abstraction levels allows mastering the specification and the implementation of adaptation rules.

A configuration is denoted Cn,i, where n is the abstract level and i is an index to distinguish the configuration. For a given configuration Cn,i at level n, multiple configurations (Cn-1,1,…,Cn-1,p) may be implemented at level n-1.These configurations represent the refinement or a detailed view of Cn,i. When a new adaptation is required at level n, only one configuration is chosen among all configurations of this level implementing the new n-level configuration.

Figure 1. Multi-level architecture modeling

This adaptation requires two actions:



Refinement which determines the set of associated (n-1)-level configurations that implement a given n-level configuration Cn,i. This refinement procedure is called refine (). Figure 2 represents the generic version of this function. This function is generic because the action "implements" depends on the considered level.

Figure 2. Refine function

 Selection, which consists in retaining the optimal configuration among the all possible configurations at a given level. The selection procedure uses the resources context such as the variation of communication networks to eliminate the configuration that cannot be deployed depending on the current network architecture. Among the selected configurations, the choice of the best configuration will depend on a set of predefined policies.

Multi-Level modeling for collaborative systems

In this part, we present how we apply the presented multi-level approach to collaborative systems in order to clearly identify specific problems of each level for the considered system. Therefore, we provide an overview of the retained level and we detail the proposed models for each level. Adaptation at the highest levels should be guided by the evolution of activity requirements. Adaptation at the lowest levels should be driven by execution context constraint changes. The retained levels are represented in Figure 3. These levels are detailed as follows:

Application level

The application level models systems needs collaboration between participants that belong to several groups. The proposed model for this level represents a business view of the collaborating entities and the relations between them. Hence, the system takes into account the extern context of entities when instantiating this model. As this model is highly application-dependent, it must be built by the designers of each collaborative system and instantiated at runtime by the system itself. System designers also have to implement the refinement procedure allowing to obtain the set of collaboration level models that implement a given application model. Only collaboration-related elements of the application level model will be taken into account in the refinement process. Although this business view is often specific to one application, it is possible that several applications belonging to a given domain share the same application model.

Collaboration level

The collaboration level provides a session level abstraction. It describes how the members of a group are organized within sessions where they can send and receive data flows. The associated model provides a generic high-level collaboration schema that responds to the current collaboration needs of the system. Indeed, it manages collaboration sessions and all needed elements that implement such sessions like flows, components, etc. The abstract representation of these elements is independent of their implementation and they can be implemented with several technologies which will be detailed in the messaging level. Thus, this level ensures the interoperability between the collaboration' needs expressed in the application level and the real implementation in the messaging level. The architectural model produced by this level is a graph containing the following elements: nodes, components and data flows (which are organized within one or more sessions). This model is inspired by classic graph-based session description formalisms such as dynamic coordination diagrams [START_REF] Baudin | A model-driven approach for synchronous dynamic collaborative e-learning[END_REF].

Messaging level

The messaging level provides a communication model that masks low-level details (like TCP sockets, UDP datagrams, IP addresses, multicast, etc.) in order to simplify the representation of communication channels. In fact, this level abstracts distributed systems, so they are transparent for upper levels. For instance, this model may be based on abstractions like Peer to Peer [39], Remote Procedure Calls [START_REF] Birrell | Implementing remote procedure calls[END_REF], CORBA [START_REF] Vinoski | Introduction to CORBA (tutorial session)[END_REF], Event Based Communications (EBC) [START_REF] Meier | Taxonomy of Distributed Event-Based Programming Systems[END_REF] etc. The produced model represents a deployment descriptor that contains all needed elements to implement sessions provided by the collaboration level. For instance, if this level is implemented with an Event Based Communication technology, the model contains the event producers, event consumers and channel managers to be deployed on each node, as well as the pull/push links between these elements. If a Peer-to-Peer implementation is considered, model contains peers, super-peers and pipes to link peers.

Multi-level collaborative framework

In order to apply the multi-level modeling approach in our work, we propose a model driven framework enabling implicit session management and adaptive component deployment for collaborative systems.

Framework architecture

The proposed architecture (Figure 4) is composed of four levels.

The lower level (infrastructure level) contains all needed software components and hardware infrastructure that enable to run the collaborative system using the framework. Hence, it ensures communication between participants' devices. In the following, we detail the three main levels of the architecture and the retained models.

Collaboration level

The goal of the central level (collaboration Level) is to provide collaboration schema that enables members belonging to several groups to communicate inside sessions where they can send and receive data flows. Thus, the main issue in this level is to determine which data flows have to be created in order to enable the needed communication. The proposed elements in this level are represented by a generic collaboration meta-model (GCM) which details the structure of one or more collaborative sessions and deployment configuration. Hence, new instances of this meta-model are generated at runtime after every context change of the application level configuration as well as arrivals, changing roles, changing groups of collaborating participants. GCM is a graph expressed in the GraphML language (an XML dialect for representing graphs [START_REF] Brandes | GraphML Progress Report[END_REF]).

Figure 4. Framework architecture

In order to represent GCM, We have chosen ontologies because it constitutes a standard knowledge representation system, allowing reasoning and inference. Moreover, ontologies facilitate knowledge reuse and sharing through formal and real world semantics. Therefore, ontologies are high-level representations of business concepts and relations. We have chosen to describe levels' models in OWL [START_REF] Smith | OWL Web ontology Langage Guide[END_REF], the Semantic Web standard for metadata and ontologies.

In general, ontologies are separated in two levels: a generic ontology and a specific ontology. The former is a domain-wide ontology, but independent of applications. The latter ontology extends the generic one with terms specific to an application category. We have followed the same pattern in our implementation: there is a generic ontology, GCM, (that describes sessions, participants, roles, data flows, nodes, etc.) and an application ontology that extends the collaboration ontology with business-specific concepts and relations. The generic ontology is common to all applications, therefore it is provided within the framework. This ontology which represents our Generic Collaboration Meta-model is detailed in Figure 5. The proposed concepts are represented by round-cornered rectangles, while relations are represented as arrows going from one concept (the domain) to another (the range). The main concept is Session. A session is a set of flows, represented by the concept Flow, which represents communication links between entities. A property, called HasdataType, relates the concept Flow to the DataType concept. Possible values of data type are captured by the DataType instances that can be text, audio, video or an exchanged artifact between participants. During the collaborative activities, flows are exchanged between communicating entities represented by the Node concept. Therefore, Flow is related to Node by the two relations: source and destination representing respectively the source node and destination node. Nodes may represent participants. A given node plays one or more roles determining the type of each activity for all involved participants. For example, a role may be a de developer in a collaborative software engineering. Depending on their roles, entities will have multiple tasks and they need to communicate with different members organized within different groups in order to achieve collaboration goal.

Each role belongs to one or several groups. Therefore, the concept Role is related to the concept Group by the relation belongsToGroup. In order to manage collaboration sessions, a set of session must be defined for each group. Therefore the relation hasSession relates the Group concept to the Session concept. The session definition for each group enables a valid deployment of appropriate sessions depending on participants' roles and groups. Whether a node represents a collaborative entity that communicates with others entities, it has to be hosted by a physical machine represented by the Device concept (relation HostingDevice).

Figure 5. Generic Collaboration Meta-model

Nodes manage exchanged data flow using external components (Tool concept) deployed in the same devices in order to enable the separation between collaboration code (implemented in the external components) and the business code (implemented in entities' components and specific to the application).

Application Level

The retained model in this level is a domain specific ontology that represents concepts and relations modeling the context of the application. Such ontology depends on the application domain, hence it will be provided by the designer of the collaborative system using the framework. In general, this ontology is a specialization of the Generic Collaboration Meta-model presented above in order to enable the addition of specific semantics to the general collaboration concepts. Moreover, this ontology can be more complex depending on the application' needs.

The main generic collaboration elements that will be specialized are: Node, Group, Role and hasSession. The concept Node is specialized into sub-concepts specifying the type of the communicating entity (e.g. Actor). The specialization of the Group concept allows to define all communication group (e.g. Commercial-Team, development-Team, etc.).The system designer will also define all possible roles that can be played by the defined entities. The last element that will be specialized is the relation hasSession in order to define the needed collaboration sessions for each group. Therefore the domain ontology contains elements that inherit from these elements and some rules indicating how roles can communicate, within the specified sessions. At runtime, the domain ontology is instantiated depending in the application context, thus providing knowledge base containing explicit and implicit collaboration aspects about participants, their roles, their group and about the needed sessions for each group.

Thanks to the OWL importation mechanism, both instances of GCM concepts and those of the domain ontology are regrouped into the same instance of this model. This instance represents a business view of the collaborative activities. A reasoning engine is used in order to make explicit the implicit knowledge contained in this instance. Rules trigger the instantiation of the generic meta-model enabling a semantic-driven adaptation that enables managing spontaneous sessions and detecting implicit potential collaboration situations.

Messaging level

The architectural model produced in this level is a detailed deployment schema that contains the needed elements in order to implement collaboration sessions determined by the collaboration level. For this level, we have retained two communication paradigms: the Event Based Communication and the Peer-To-Peer communication. The EBC is based on the Publish/Subscribe pattern and it represents a well-established paradigm for interconnecting loosely coupled components and it provides a one-to-many or a many-to-many communication pattern. The main advantage of the EBC is the opportunity for better scalability than traditional client-server pattern through message caching and network-based routing. Moreover, publishers are loosely coupled to subscribers and they operate independently of each other. The Peer-To-Peer model allows to make some resources available in the network participants without a central coordination. Its structure is privileged for file sharing that represents an important activity in collaborative systems.

The produced models in this level are EBC-based graphs and Peer-To-Peer-based graphs that contain the specific needed elements for each paradigm. The EBC model contains a set of event producers, event consumers and channel managers connected with push and pull links. For every determined session, a CM is created in order to manage, store and deliver exchanged data flow between multiple producers and consumers. On the other hand, the Peer-To-Peer model contains peers, super-peer and pipes that enable a the communication and the file sharing between participants These produced models are represented by graphs expressed in the GraphML language. Since the produced model represents a detailed deployment descriptor, we provide a deployment service that effectively deploys elements on participants' devices. This deployment service takes a deployment descriptor as input in order to install, start and update the required components.

Refinement and selection between levels

Application-Collaboration Refinement

Reasoning based on SWRL rules [START_REF] Horrocks | SWRL: A Semantic Web Rule Language Combining OWL and RuleML[END_REF] is used in order to implement the application-collaboration refinement. SWRL rules are applied to an instance of the domain ontology that extends the GCM. They have the form: a1,…,anb1,…,bm where a1,…,an is the body and b1,…,bm is the head of the rule. The terms a1,…,an,b1,…,bm are SWRL atoms which can be a concept assertion, a relation assertion or built-ins. The semantic of the rule is the following: the interpretation of the head holds only if the interpretation of the body holds. The proposed Generic Collaboration Meta-model includes 6 generic rules1 that express some relations and especially those which allow to infer deployment schema from the sessions presents in the domain ontology instance. However, these rules are not sufficient for complete implementation of the refinement from the domain ontology to the collaboration ontology, because it depends on the collaborative system. Therefore the application designer has to specify additional rules in the domain application model which contains a part of the refinement process. An example of these rules is detailed in the section 5. The processing of the SWRL, which is done by a rule engine such Jess, produces an instance of the Generic Collaboration ontology represented by a collaboration graph expressed in the OWL language. This graph is translated into GraphML by XSLT transformation in order to be provided to the messaging level. The application-collaboration refinement produces a single collaboration model from a given application level model.

Collaboration-Messaging Refinement and selection

As the collaboration level and the messaging level models are represented by graphs, graph grammar theories represent an appropriate formalism to handle the refinement process. Moreover, graph grammar is an intuitive and powerful way of handling complex transformations. In order to refine a given collaboration architecture, a generative graph grammar [START_REF] Rodriguez | A rule-driven approach for architectural self adaptation in collaborative activities using graph grammars[END_REF] is used. In this graph grammar, non-terminal nodes are collaboration entities while terminals nodes are deployment entities. This refinement infers two sets of deployment graph: a set containing all possible EBC-based graphs where terminal nodes are EBC entities and another set containing all possible Peer-To-Peer-based graphs where terminal nodes are Peer-To-Peer entities.

In order to apply transformation's rules of the proposed grammar, Graph Matching and Transformation Engine (GMTE) [START_REF] Rodriguez | A rule-driven approach for architectural self adaptation in collaborative activities using graph grammars[END_REF] is used. In order to select the optimal architecture among those built by the refinement process, the generic selection procedure presented in the Figure 6 is used depending on several parameters. This function allows: (1) to discard configuration that cannot be deployed, (2) to select the best adapted configuration to the current context. At first, this function computes the value assigned by the ContextAdaptation() function for each configuration. These values reflect the degree of adaptation of each configuration to the current network context. Well adapted architectures are those that respond to the global communications' needs. Our criterion of selection for this function is as follows: EBC-based architecture is deployed for open collaboration such as synchronous or asynchronous, point-to-point or multicast communications that may require a central server which manages exchanged data flows. Otherwise, a Peer-To-Peer architecture can be deployed for private communication such as sharing and exchanging produced artifact during the collaborative activities. If the configuration cannot support the current context, its assigned value will be -1. Otherwise, a positive value will be assigned. The candidates having the highest positive value will be added to the set S1. If S1 does not contain any configuration, then there is no adapted configuration that supports the current context. In that case, the select() function return an empty set. If S1 contains one candidate, it will be returned. Otherwise, the indicated policies are used by the select() function in order to retain the optimal configuration.

Figure 6. Selection

In order to ensure the framework robustness, we have retained two policies; however we can add new policies depending on the system requirements. The first policy, called dispersion, uses a function Dispersion() (Figure 7) which computes the dispersion degree of components deployed on participants' nodes. The goal of this policy is to balance resources consumption in order to ensure the collaborative systems robustness. The selection procedure aims to maximize the dispersion of components (line 8, Figure 6). The second policy, called distance, uses a function Distances() (Figure 8) which computes the number of needed redeployments between two n-1-level configurations implementing a given n-level configuration. The selection procedure minimizes (line 13, Figure 6) the distance in order to discard useless redeployments. Once an adaptive communication model is selected, the two proposed policies Dispersion (Figure 7) and Distance (Figure 8), indicated by the parameter Policy in the selection function, are used in order to select the best adaptive deployment graph.

Deployment service

In order to manage dynamic change in collaborative systems, deployment needs to be adaptive at run-time, so a dynamic deployment service is needed. Therefore the result graph produced by the messaging level is a low-level detailed deployment descriptor that is used by the deployment service in order to carry out the effective deployment of components needed for supporting collaborative activities. The system keeps the same configuration until the arrival of new context events. The deployment descriptor implements the low-level elements of the required architecture. For instance, it contains producers, consumers, channel managers and link if the required architecture is EBC. Once receiving this descriptor, the deployment service downloads, installs and starts the required components on each participant' device. The deployment service architecture is detailed in the Figure 9. At the same server where the framework is running, we propose a deploymentService entity and a repository that contains the available components. Once receiving the deployment graph, the deployment service finds the required components for each participant's device. Afterwards, it connects to the evolved devices through the deployment agents (broken lines in the figure) informing them about all needed components and the set of remote components to which the deployment agent must be connected. If the required components don't exist, they will be downloaded from the repository through the HTTP protocol (dashed lines).

Figure 9. Deployment service architecture

The implementation of this deployment service is based on the Open Services Gateway Initiative (OSGi) technology [START_REF]OSGi Service Platform Release 4[END_REF]. Indeed, OSGi offers very promising functionalities such as dynamic code loading and execution. Besides, this specification offers a services management platform (with a dynamic register of services enabling the detection, the invocation, the addition, the cancellation of services at runtime). OSGI offers as well a complete and dynamic component model. In OSGI specification, components are called bundles. Within this approach, deployable components are packaged as OSGi bundles that are handled by the proposed deployment service.

The Framework interoperability

In this section, we explain how to take advantage of the proposed framework in order to maintain the interoperability of enterprise systems in networked environment in the advent of their dynamic. We provide methods that enable the dynamic adaptation of such systems and the optimization of the product development process by managing the collaborative sessions during all development phases of a given product. We also address the discovery capabilities in order to enable the detection of new changes on the situations of collaboration within the systems' network. These changes trigger a notification to our communication framework which launches the adaptation process using the proposed knowledge representation technologies applied to a given domain model. The result of this adaptation is a messaging model that decides in what way should the network nodes react in order to enable the system to evolve for the new interoperable state. In order to carry out the adaptation process during the collaborative activities, the collaborative process modeling requires three phases (Figure 10):

Phase 1

In the first phase, the process designers define a process model which contains specific concepts such as groups, roles, activities, relationship, constraints, automatic collaborative sessions for each group, etc. Application-specific SWRL rules are also defined and will be used by the adaptation process in order to determine to what session should belong a participant having a given role or task. This process model is a specialization of the Generic Collaboration Meta-model detailed above (in the Figure 5). However, it can be provided as an UML diagram, so it must be transformed to a domain ontology (OWL file) via XSLT transformation.

Figure 10. Collaborative process modeling

The process designers formalize the process depending on the needed structure that enable inter and intra-enterprise collaboration. For example, they formalize the collaboration between participants having a common experience, performing common tasks and belonging to one or more enterprises. The main advantage of this architecture is that the process model can be updated at runtime without performing actions on the system's code implementation. This update may affect sessions' organization, groups' definition, possible roles; etc. For example, the process designers can add or remove one or more collaborative session of a given groups. Therefore, during the adaptation process, the system's server has a permanent HTTP access to the defined process model which can be updated.



Quit (userId): A user quits the system. The system core API should call this function to notify the communication framework. The participant and all his roles will be removed from the application descriptor.  AddToGroup (userId, group): A connected user is added to a group on the system server. In order to determine if the requested service has a potential conflict, a consistency check mechanism is used. Indeed, all these services depend on the process model which defines actors, groups, roles, etc. For instance, if a participant request to have two roles r1 and r2 and to join a working group g1, the communication framework refers to the defined process model in order to check if the two roles r1 and r2 and the group g1 are already exist. If his request contains a conflict, the system will be informed by the communication framework.

Once the application descriptor is created using the proposed services, the communication framework carry into effect the adaptation process. Firstly, a collaboration schema that responds to the high level needs is generated by the collaboration level through reasoning using SWRL rules defined both by the Generic Collaboration Meta-model and the process model. After that, the Graph Matching and Transformation Engine (GMTE) infers a set messaging graph representing all possible communication models such as EBC and Per-To-Peer models. Then, the communication framework selects the well adapted messaging graph that ensures the needed communication between participants performing the collaborative activities. Finally, the deployment service takes charge of the effective deployment of components on participants' device.

Sessions on demand API are also defined in order to enable spontaneous sessions between participants. The proposed API provides the following services:

Case study

The software prototype environment has been implemented in Java. In this section, we consider a software engineering project concerned with the development of complex software products. The ultimate goal is to ease the development of such products by assisting engineers, whose are organized within different teams, in their collaborative work. The present case study is implemented in the context of the ANR Galaxy project2 . We detail the three required phases: Phase 1: In this phase, the process designer of the Galaxy project defines the process model which contains the groups, the possible roles and collaboration sessions for each group. Figure 11

Figure 11. Domain ontology

Having one or more roles, engineers can belong to one or more working groups. The defined working groups for this project, workGroupA and workGroupB, are two instances of the Group concept of GCM (all instances are represented with dashed lines). For each working group are defined the associated collaboration sessions. Engineers who belong to the workGroupA are organized within two sessions: DesignerSession which regroups all project designers and DesignerDeveloperSession which regroups designers and developers. On the other hand, engineers who belong to the workGroupB are organized within the two sessions: integratorDeveloperSession which regroups the integration manager and developers when integrating the developed codes and DeploymentMgrDeveloperSession which regroups the deployment manager and the developers when a bug is found. These collaboration sessions are instances of the Session concept of the GCM. In order to assign sessions to the groups, the hasSession relation of GCM is specialized into four relations: hasDesigner_S, hasDesigner_Developer_S, hasIntegrator_Developer_S and hasDeploymentMgr_Developer_S.

In addition to the specific concept and relations, the domain ontology which describes the process model must also contain the application-specific SWRL rules that allow rule engines to create sessions and data flows between engineers according to their roles. As an example, let us consider the following rule (Figure 12). This rule states that, whenever a simple designer and a designers leader are connected to the system and they belong to the same working group and they need to communicate in audio, an audio flow will be created having the node of the simple designer as source and the node of the designer leader as destination and another audio flow will be created having the node of the simple designer as destination and the node of the designer leader as source. Similar rules are defined to handle other data flows enabling the communication between other members. This domain ontology can be eventually updated at runtime by the process designer. Indeed, other sessions can be added for a given working group, other groups or roles can also be defined during a software product development.

Figure 12. An application-specific SWRL rule

Phase 2: After the process model definition, an enactable process model is created instantiating the defined process model. For the present case study, this instance contains the two working groups and their associated collaboration sessions.

Phase 3: The participants-context awareness is taken into account in this phase during which the adaptation process is needed after each action. Let us consider the following scenario (Figure 13): John is connecting to the Galaxy system and he has both the designer role and the integrator role. The galaxy core API will update the instance of the domain ontology as above using the proposed API and especially using the service Connect. Hence, the process adaptation infers a collaboration schema through SWRL rule execution allowing John to collaborate within the DesignerSession and DesignerDeveloperSession of the workGroupA as a designer and the IntegrationDeveloperSession of the workgroupB as an integrator. Then it produces an adaptive deployment graph that models the needed components to ensure his collaboration. After the automatic deployment of components, John will be able to collaborate with all designers of workGroupA, on one hand, and with developers and integrators of the workGroupB, on the other hand.

Whenever the instance that represents the new changes is created, the proposed communication API checks the consistency of the required action. For instance, if the process designer removes the workGroupA, then the Galaxy server will be informed that this group does not exist.

Conclusion

In this paper, a semantic-driven, adaptive architecture has been developed to manage messaging for collaborative engineering activities within networked enterprises. This architecture is based on a Generic Collaboration Meta-model (GCM) which supports the implementation of session management services. GCM is a domain-independent ontology that can be extended for modeling collaboration knowledge in different application domains. This work provides a framework for session management and adaptive component deployment. Knowledge representation technologies, such as ontologies and SWRL rules, are used to ensure awareness of collaboration situations and to decide when sessions have to be adapted. The proposed architecture provides a generic level of collaboration. It enables the integration of different applications and deals with application-specific information as sub models that are plugged into the framework. The developed prototype system establishes an open and scalable architecture to support inter and intra-enterprise collaboration. Indeed, a process designer defines a process model which represents the participants, their roles, their groups and the associated sessions. The process model is instantiated using the proposed services that are able to check the consistency of the requested actions. The main benefit from this approach is ensuring the continuity of the communication services when changes occur in the process model. The framework supports a dynamic component deployment using the deployment service which uses the produced deployment graph. The type of deployed components depends both on the selected communication model and on the needed communication mode such as synchronous and asynchronous communication. We can conclude that semantic representation and Web semantic technologies are useful to enable dynamic reconfiguration to ensure interoperability. The work presented is being tested on real case in the context of Galaxy project. Future work will consists in performing generic security functionalities and access control, at the communication framework level, which rely on the involved collaborative system.

Figure 3 .

 3 Figure 3. Collaborative systems levels

  Tools are composed of one or several components (e.g. a sender component and a receiver component) represented by the concept Component. The Tool concept is related to the Component concept through the relation hasComponent. Components manage several flows. Therefore, the relation manages relates Component to Flow. Components have the same data type of the managed flows (relation hasDataType). Each component is deployed on one device (relation isDeployedOn which links Component and Device). The component' type depends on the handled data type (text, audio, video, files, artifacts …) and on the communication mode (real-time communication, asynchronous communication).

Figure 7

 7 Figure 7. Dispersion function



  createSession(String userId, String name): A user creates a new communication session.  closeSession(String userId, String name): A user closes a communication session. Only the creator should be allowed to close this kind of sessions.  joinSession(String userId, String name): invites a user to join an open communication session.  quitSession(String userId, String name): removes a user from an open communication session.

Figure 13

 13 Figure 13. Scenario

  The core API should call this function to notify the communication framework.

		RemoveFromGroup (userId, group): A connected user is removed from a workgroup on the system
		server. The core API should call this function to notify the communication framework.
		AddRole (userId, role): A new role is added to a connected user on the system. The core API should call
		this function to notify the communication framework. Then the new role is added to all groups belonged by
		the user.
		RemoveRole (userId, role): A role is removed from a user connected on the system. The core API
		should call this function to notify the communication framework.
		ChangeRole (userId, previous Role, new Role): A connected user changes one of his roles.

  describes a proposed domain ontology (expressed by an OWL file) that represents the collaboration aspect of the defined process model. In this figure, concepts and relations of the Generic Collaboration Meta-model are represented with broken lines, while specific concepts and relations are represented in white. The Node concept is specialized into the Actor concept which models a galaxy participant. For each actor, a set of possible roles, which specialize the Role concept, are defined:

		Designer: a designer can be:
		o	Simple designer: designs and writes architecture description models.
		o	Designers' leader: collaborates with the simple designers in order to produce the detailed
			architecture of the system.
		Integration manager: decides when features and fixes are ready for production, and merges them while
		being sure that the result is functional and reasonably bug-free.
		Developer: a developer can be:
		o	Code developer: writes codes.

o Test developer: writes integration tests for the system under development.



Deployment manager: deploys the project to production, monitors execution, and reports errors back to developers.

An example of these rules is available at: http://homepages.laas.fr/akamoun/sessions.owl#same_group

http://www.irit.fr/GALAXY

Acknowledgement

This article was funded partially by the ANR Project Galaxy and the IST IP IMAGINE.

Phase 2

Once the process model is detected, the system's server instantiates the process model and allocates to the project all resources (project plan). This instance represents the initialization of the application descriptor that will be the entry of the adaptation process. This phase produces an enactable process model that contains the defined specific concepts and relations defined in the process model such as all potential roles that may be played by participants, all possible groups that may represents sub-projects and mainly the defined collaboration sessions for all groups.

Phase 3

Once the process model is defined by the process designers and initialized by the system's server, participants will be assisted in performing the collaborative project according to the enactable process model, to their roles, to the teams' organization and to the defined collaboration scenarii. Therefore, the system's server should enable the needed adaptation after each actions (such as change involving connection, disconnection , changing roles, joining or leaving a given working group) performed by each participants. Hence, the process model which has been instantiated and initialized in the phase 2 must be updated in order to represent all the performed actions. In our communication framework, this model represents the application descriptor which is the entry of each adaptation process. In order to create dynamically the required application descriptors that represent the current collaboration state, services are proposed to the system's server enabling its communication with the proposed framework. The proposed services are as follows:



Connect (user Id, device IP, list Roles, list Groups): A user connects to system. The system core API should call this function to notify the communication framework (FADYRCOS). Each role will be added to all groups.

Appendix. List for acronyms in this paper

CSCW


Olivier Glass

Franck Sueur

Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations

It was recently proven by De Lellis, Kappeler, and Topalov in [17] that the periodic Cauchy problem for the Camassa-Holm equations is locally well-posed in the space Lip(T) endowed with the topology of H 1 (T). We prove here that the Lagrangian flow of these solutions are analytic with respect to time and smooth with respect to the initial data.

These results can be adapted to the higher-order Camassa-Holm equations describing the exponential curves of the manifold of orientation preserving diffeomorphisms of T using the Riemannian structure induced by the Sobolev inner product H l (T), for l ∈ N, l 2 (the classical Camassa-Holm equation corresponds to the case l = 1): the periodic Cauchy problem is locally well-posed in the space W 2l-1,∞ (T) endowed with the topology of H 2l-1 (T) and the Lagrangian flows of these solutions are analytic with respect to time with values in W 2l-1,∞ (T) and smooth with respect to the initial data.

These results extend some earlier results which dealt with more regular solutions. In particular our results cover the case of peakons, up to the first collision.

Introduction

We consider the Cauchy problem for the Camassa-Holm equations: ∂ t u -∂ txx u + 3u∂ x u -2∂ x u∂ xx u -u∂ xxx u = 0, u(0, x) = u 0 (x), (1.1) where x runs over the line R or the one-dimensional torus T. This equation has been derived independently by:

• Fokas and Fuchssteiner [START_REF] Fuchssteiner | Symplectic structures, their Bäcklund transformations and hereditary symmetries[END_REF] where it appears as a member of a whole family of bi-hamiltonian equations generated by the method of recursion operator,

• Camassa and Holm in [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF] as a model for the propagation of water waves in the shallow water regime, when the wavelength is considerably larger than the average water depth (then u represents the velocity, and also, roughly, the height of the water's free surface above a flat bottom see [START_REF] Constantin | The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations[END_REF]),

• Dai [START_REF] Dai | Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod[END_REF] as a model for the propagation of nonlinear waves in cylindrical hyperelastic rods (in this case the function u represents the radial stretch).

This equation attracted a lot of attention for its rich structure:

• it is formally integrable, in the sense that there is an associated Lax pair,

• its solitary waves are solitons, i.e. they retain their shape and speed after the interaction with waves of the same type,

• the Camassa-Holm equation possesses not only solutions that are global in time but models also wave breaking in finite time. Wave breaking is an important physical phenomenon which is not captured by the other standard shallow water equations, as for example the KdV equation, and therefore makes the Camassa-Holm equation particularly interesting in that context.

Applying the operator (∂ 2 x -1) -1 to the equation (1.1) we get the following non local form of the Camassa-Holm equations:

∂ t u + ∂ x ( 1 2 u 2 + P ) = 0, -(∂ 2 x -1)P = u 2 + 1 2 (∂ x u) 2 , (1.2)
which have a sense in S ′ for all u in L 1 loc (R; H 1 ), where H 1 denotes the usual Sobolev space of order 1. This space H 1 may appear as a very natural space to study this equation since, formally, the solutions of (1.2) preserves the H 1 norm. Actually one gets a better insight into this time-invariance of the H 1 norm by introducing the quantity m := u -∂ 2 x u. (1.3) This quantity m plays a special role in the theory. It can be compared to the vorticity for the incompressible Euler equation. Moreover, if the equation is set on R, integrating by parts and using that the relation (1.3) can be inverted into the formula u(x) = 1 2 R m(y)e -|x-y| dy, (1.4) we obtain that the square of the H 1 norm

u 2 H 1 (R) = 1 2 R R m(y)m(x)e -|x-y| dx dy (1.5)
can be reinterpreted as an interaction energy.

Yet the well-posedness of the equation (1.2) for initial data u 0 in H 1 is a difficult issue. The first results of well-posedness were obtained for smoother data. For instance Constantin and Escher [START_REF] Constantin | Global existence and blow-up for a shallow water equation[END_REF] first established the well-posedness of the equation (1.1) for u 0 in the Sobolev space H 3 (R), by applying Kato's theory for a hyperbolic quasi-linear PDE satisfied by m.

It has been subsequently proved by various methods that the problem (1.2) is locally well-posed in the Sobolev space H s (R) with s > 3/2 in [START_REF] Rodríguez-Blanco | On the Cauchy problem for the Camassa-Holm equation[END_REF][START_REF] Li | Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation[END_REF][START_REF] Misio | Classical solutions of the periodic Camassa-Holm equation[END_REF][START_REF] Danchin | A few remarks on the Camassa-Holm equation[END_REF][START_REF] Danchin | A note on well-posedness for Camassa-Holm equation[END_REF]. These results hold only for a finite time interval, as it may appear a point where the profile of u steepens gradually and ultimately the slope becomes vertical. In the context of water waves, this corresponds to the breaking of a wave, see the paper of Constantin and Escher [START_REF] Constantin | Wave breaking for nonlinear nonlocal shallow water equations[END_REF] (see also the recent paper [START_REF] Brandolese | Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces[END_REF]). In [START_REF] Danchin | A note on well-posedness for Camassa-Holm equation[END_REF], Danchin have extended well-posedness to the Besov space B 3 2 2,1 (R). In [START_REF] Misio | Classical solutions of the periodic Camassa-Holm equation[END_REF] G. Misiolek proved that (1.2) is locally well-posed in the space C 1 (T) of continuously differentiable functions of the one-dimensional torus T. Finally De Lellis, Kappeler, and Topalov [START_REF] De Lellis | Low-regularity solutions of the periodic Camassa-Holm equation[END_REF] recently proved that the equation (1.2) set on the torus T is well-posed in W s,p (T) ∩ Lip(T), endowed with the topology of W s,p (T), provided that 1 s < 2 and 1 p < +∞. We rephrase their result here, only in the case p = 2, for sake of simplicity.

Theorem 1 ([17]

). Let u 0 be in Lip(T). Then there exists T > 0 and a unique solution

u ∈ C((-T, T ), (Lip(T), σ H 1 (T) )) ∩ C 1 ((-T, T ); (L ∞ (T), σ L 2 (T) ))
of (1.2). Moreover for any u 0 in Lip(T) there exists T > 0 and a neighborhood

V of u 0 in (Lip(T), σ H 1 (T) ) such that the map V → C((-T, T ); (Lip(T), σ H 1 (T) )) ∩ C 1 ((-T, T ); (L ∞ (T), σ L 2 (T) )), u 0 → the corresponding solution u of (1.2) (1.6)
is well-defined and continuous.

In the statement above the notation (Lip(T), σ H 1 (T) ) and (L ∞ (T), σ L 2 (T) ) means that the space Lip(T) and L ∞ (T) are respectively endowed with the topology of H 1 (T) and L 2 (T). It is shown in [START_REF] De Lellis | Low-regularity solutions of the periodic Camassa-Holm equation[END_REF]Section 1.2] that the solution u given by Theorem 1 may not be continuous with respect to time in the space Lip(T), endowed with its usual strong topology.

The proof of Theorem 1 relies on the Lagrangian description of the Camassa-Holm equation and on the observation that the flow map ξ defined from (-T, T ) × T to T by

ξ(t, x) = x + t 0 u(s, ξ(s, x)) ds (1.7)
and the Lagrangian velocity u(t, ξ(t, x)) are more regular than the Eulerian velocity u(t, x). In particular ξ is C 1 over (-T, T ) with values in Lip(T), endowed with its strong topology. The aim of this paper is to prove that the flow map of these solutions actually benefits from extra smoothness properties. In particular we prove the following result concerning the smoothness in time of the flow map.

Theorem 2. Assume that u 0 is in Lip(T). Then, with the notations above, ξ belongs to the space C ω ((-T, T ); Lip(T)) of the analytic functions from (-T, T ) to Lip(T).

As a corollary of Theorem 2 we infer that the flow is also smooth with respect to the initial data.

Corollary 1. With the previous notations, the map

V → C ω ((-T, T ); Lip(T)), u 0 → ξ (1.8) is of class C ∞ .
Actually our results cover a more general case that we are now going to describe. Let us first recall that the Camassa-Holm equation describes the exponential curves of the manifold of orientation preserving diffeomorphisms of T using the Riemannian structure induced by the Sobolev inner product H 1 (T) (see for instance [START_REF] Lenells | Riemannian geometry on the diffeomorphism group of the circle[END_REF] and references therein).

This can be seen as a counterpart of the celebrated papers [START_REF] Arnold | Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits[END_REF] and [START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF] where classical solutions of the incompressible Euler equations are interpreted as geodesics of a Riemannian manifold of infinite dimension.

In this spirit some higher-order Camassa-Holm equations can be considered using the Sobolev space H l (T), for l ∈ N, l 2, instead of H 1 (T), see for instance [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF][START_REF] Constantin | H k metrics on the diffeomorphism group of the circle[END_REF]. These equations read:

∂ t u + u∂ x u = A -1 l C l (u) (1.9)
where

A l := l j=0 (-1) j ∂ 2j x , C l (u) := -uA l ∂ x u + A l (u∂ x u) -2(∂ x u)A l u.
(1.10)

For these equations we have the following well-posedness result which is a counterpart of Theorem 1.

Theorem 3. Let l ∈ N, l 1. Let u 0 be in W 2l-1,∞ (T). Then there exists T > 0 and only one solution

u ∈ C((-T, T ), (W 2l-1,∞ (T), σ H 2l-1 (T) )) ∩ C 1 ((-T, T ), (W 2l-2,∞ (T), σ H 2l-2 (T) ))
of (1.2). Moreover for any u 0 in W l,∞ (T) there exists T > 0 and a neighborhood V of u 0 in (W 2l-1,∞ (T), σ H 2l-1 (T) ) such that the map

V → C((-T, T ), (W 2l-1,∞ (T), σ H 2l-1 (T) )) ∩ C 1 ((-T, T ), (W 2l-2,∞ (T), σ H 2l-2 (T) )), u 0 → the corresponding solution u of (1.9) (1.11)
is well-defined and continuous.

In the statement above the notation

(W 2l-1,∞ (T), σ H 2l-1 (T) ) and (W 2l-2,∞ (T), σ H 2l-2 (T)
)) stands for the spaces W 2l-1,∞ (T) and W 2l-2,∞ (T) endowed with the topologies of H 2l-1 (T) and H 2l-2 (T) respectively.

We will sketch the proof of this Theorem in Appendix for sake of completeness. Similarly to the case l = 1 this proof also provides that the flow ξ defined from (-T, T ) × T to T by

ξ(t, x) = x + t 0 u(s, ξ(s, x))ds is C 1 over (-T, T ) with values in W 2l-1,∞ (T).
The next result is the main one of the paper; it shows that the smoothness of the flow map is actually much better.

Theorem 4. Assume that u 0 is in W 2l-1,∞ (T). Then, with the notations above, the flow map ξ is in the space

C ω ((-T, T ), W 2l-1,∞ (T)).
Then Corollary 1 extends as follows.

Corollary 2. With the previous notations, the map u

0 ∈ V → ξ ∈ C ω ((-T, T ); W 2l-1,∞ (T)) is C ∞ .
Three remarks are in order.

• This extends some earlier results by [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF][START_REF] Constantin | H k metrics on the diffeomorphism group of the circle[END_REF][START_REF] Kappeler | Analyticity of Riemannian exponential maps on Diff(T)[END_REF] where the initial data u 0 was assumed to be much smoother.

• Theorem 4 can also be seen as a counterpart of the results of analyticity of the trajectories for the incompressible Euler equation, see [START_REF] Serfati | Structures holomorphes à faible régularité spatiale en mécanique des fluides[END_REF], [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF] and the references therein. In particular the proof of Corollary 2 follows the same lines as the proof of [20, Corollary 1] and will be therefore omitted here.

• Let us stress that the results above are only local in time, including the existence of solutions "à la de Lellis-Kappeler-Topalov". We refer here to the papers [START_REF] Mclachlan | Well-posedness of modified Camassa-Holm equations[END_REF][START_REF] Mu | Well-posedness and blow-up phenomena for a higher order shallow water equation[END_REF] about the issue of the finite time blow up of some classical solutions (with a slightly different regularity) of the higher-order Camassa-Holm equations. In the case of the Camassa-Holm equations, when l = 1, the issue of wave breaking in finite time is a longstanding feature, intimately related to the interaction of the peakons. We will discuss this issue more closely in Section 3.

Finally let us mention here the papers [START_REF] Glass | Controllability and asymptotic stabilization of the Camassa-Holm equation[END_REF] and [START_REF] Perrollaz | Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval[END_REF] which deal with the issues of control and of stabilization of the Camassa-Holm equation.

Proof of Theorem 4

This section is devoted to the proof of Theorem 4.

Preliminary material

Let us start with a few remarks.

Rewriting the equation. In the sequel we will simply denote by A the operator A l defined in (1.10). Following [START_REF] Coclite | Well-posedness of higher-order Camassa-Holm equations[END_REF] the equation (1.9) also reads:

∂ t u + u∂ x u + ∂ x P = 0, AP = F [u], (2.1)
where F [u] is a differential polynomial in u of order 2l -1 of the form

F [u] := 0 m1+m2 2l-1 c m1,m2 (∂ m1 x u)(∂ m2 x u), (2.2)
where the c m1,m2 are some real numbers. Denoting the material derivative

D := ∂ t + u∂ x ,
the equation (1.9) takes the form

Du = -∂ x P, AP = F [u].
Elliptic operators. Let us consider several operators associated to A. We define

ξ j := e iπj l+1 , Λ ± := -i∂ x ± ξ l and Λ± := Λ ∓ l-1 j=1 (-i∂ x -ξ j )(-i∂ x + ξ j ) . (2.3)
Then Λ ± is under the form Λ± :=

0 m 2l-1 d m ∂ m x , (2.4)
where the d m are some complex numbers, and

-2i∂ x = Λ + + Λ -and A = Λ + Λ+ = Λ -Λ-. (2.5)
Now the operators A and Λ± have the following property.

Lemma 1. The operator A is bicontinuous from W 2l,∞ (T) to L ∞ (T). The operators Λ± are bicontinuous from

W 2l-1,∞ (T) to L ∞ (T).
In particular this yields that when

u ∈ W 2l-1,∞ (T) then F [u] is in L ∞ (T) and ∂ x P is in W 2l-1,∞ (T).
Lemma 1 is an immediate consequence from the following elementary one.

Lemma 2. Let ξ ∈ C \ 2πZ and j ∈ N. Then the operator -i∂ x -ξ is bicontinuous from W j+1,∞ (T) to W j,∞ (T).

Proof of Lemma 2. It is clear that the operator -i∂ x -ξ is continuous from W j+1,∞ (T) to W j,∞ (T). In order to prove that it is one-to-one, let us solve the equation in f :

(-i∂ x -ξ)f = g, (2.6)
where g ∈ W j,∞ (T) is given. The functions f solution to (2.6) are given by

-if (x) = Ce iξx + e iξx x 0 g(x ′ )e -iξx ′ dx ′ , (2.7)
for some C ∈ R. Now to be one-periodic the function f must satisfy f (0) = f (1) so that

C(1 -e iξ ) = e iξ 1 0 g(x ′ )e -iξx ′ dx ′ ,
which defines C in a unique way since ξ ∈ C \ 2πZ. Now it is clear from the formula (2.7) that the operator (-i∂ x -ξ) has a continuous inverse from W j,∞ (T) to W j+1,∞ (T).

A formal identity. We will use the following identity, which details the lack of commutation between D k and ∂ m x , for m ∈ N * . This approach is inspired by [START_REF] Kato | On the smoothness of trajectories in incompressible perfect fluids[END_REF].

Lemma 3. For k ∈ N * , for m ∈ N * , we have for smooth functions u, ψ ∈ C ∞ ((-T, T ) × T; R),

∂ m x D k ψ = D k ∂ m x ψ + F k,m [u, ψ] where F k,m [u, ψ] := γ∈B k,m c k,m (γ) f (γ)[u, ψ], (2.8)
where

B k,m := γ = (s, α, β) s ∈ N, 2 s k + 1, α = (α 1 , . . . , α s ) ∈ N s , |α| = k + 1 -s, β = (β 1 , . . . , β s ) ∈ (N * ) s , |β| = m + s -1 , f (γ)[u, ψ] := ∂ β1 x D α1 u • . . . • ∂ βs-1 x D αs-1 u • ∂ βs
x D αs ψ, and where the c k,m (γ) are integers satisfying

|c k,m (γ)| (2s) 2(m-1) k!m! α!β! . (2.9)
Above, for a multi-index α, we used the notations |α| := α 1 + . . . + α s and α! := α 1 ! . . . α s !.

Remark 1. The writing of f (γ)[u, ψ] is not unique in the sense that one can have

f (γ)[u, ψ] = f (γ ′ )[u, ψ] with γ = γ ′ .
However, it will be easier in the proof not to regroup identical terms and not to use the commutativity of the multiplication. We will only rely on Leibniz's rule and on

∂ x D = D∂ x + (∂ x u)∂ x .
Proof of Lemma 3. We proceed by iteration on m.

1. The case m = 1, which corresponds to the commutation between D k and ∂ x is given by the following lemma. Since it can be straightforwardly adapted from [START_REF] Glass | Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid[END_REF]Proposition 6], its proof is omitted.

Lemma 4. For k ∈ N * , we have in T, for we have for smooth functions u, ψ ∈ C ∞ ((-T, T ) × T; R),

∂ x D k ψ = D k ∂ x ψ + F k [u, ψ] with F k [u, ψ] := θ∈A k c k (θ) f (θ)[u, ψ],
where

A k := θ := (s, α) s ∈ N with 2 s k + 1, α = (α 1 , . . . , α s ) ∈ N s with |α| = k + 1 -s , f (θ)[u, ψ] := ∂ x D α1 u • . . . • ∂ x D αs-1 u • ∂ x D αs ψ,
and where the c k (θ) are integers satisfying 2. Now let us assume that Lemma 3 holds true up to order m and let us prove that it also holds at order m + 1. We will proceed in two steps. First we are going to prove an identity of the form (2.8), then we will estimate the coefficients involved in this expression.

|c k (θ)| k! α! . ( 2
a. First, we observe that, using Lemma 3 at order m and Lemma 4 with ∂ m

x ψ instead of ψ, we get

∂ m+1 x D k ψ = ∂ x D k ∂ m x ψ + F k,m [u, ψ] , = D k ∂ m+1 x ψ + F k [u, ∂ m x ψ] + ∂ x F k,m [u, ψ]. Moreover F k [u, ∂ m x ψ] = θ∈A k c k ( θ) f ( θ)[u, ∂ m x ψ]
and for θ = (s, α),

f ( θ)[u, ∂ m x ψ] = ∂ x D α1 u • . . . • ∂ x D αs-1 u • ∂ x D αs ∂ m x ψ = ∂ x D α1 u • . . . • ∂ x D αs-1 u • ∂ m+1 x D αs ψ + ∂ x D α1 u • . . . • ∂ x D αs-1 u • ∂ x F αs,m [u, ψ],
using again Lemma 3 at order m with αs instead of k.

It follows that the commutator of ∂ m+1

x and D k can be described as

∂ m+1 x D k ψ -D k ∂ m+1 x ψ = θ∈A k c k ( θ) ∂ x D α1 u • . . . • ∂ x D αs-1 u • ∂ m+1 x D αs ψ + θ=(s, α)∈A k γ∈B αs ,m c k ( θ) c αs,m (γ) ∂ x D α1 u • . . . • ∂ x D αs-1 u • ∂ x f (γ)[u, ψ] + γ∈B k,m c k (γ) ∂ x f (γ)[u, ψ].
(2.11)

We now discuss according to the three terms in the right hand side of (2.11).

The first kind of terms in the right hand side of (2.11) have the desired form: they can written as f (γ)[u, ψ] with γ := (s, α, β) where s = s, α := (α 1 , . . . , αs ) and β = (1, . . . , 1, m + 1) ∈ (N * ) s satisfy |α| = k + 1 -s and |β| = m + s.

Therefore, it remains to prove the same for the second and third kind of terms in (2.11). Concerning the third kind of terms in (2.11), using Leibniz's rule, we see that for γ = (s, α, β) ∈ B k,m , the term ∂ x f (γ)[u, ψ] is a combination of s terms of the form f (γ)[u, ψ] with γ ∈ B k,m+1 , so these terms have also the desired form. The reasoning for the second kind of terms is analogous.

Consequently we obtain that (2.8) holds true at order m + 1 for some coefficients c k,m+1 (γ).

b. Let us now prove the estimate (2.9) on these coefficients c k,m+1 (γ). Let us consider γ ∈ B k,m+1 . According to the analysis above, a term of the form f (γ)[u, ψ] in F k,m+1 [u, ψ] may be originated from one of the three terms in the right hand side of (2.11). We estimate the contribution to the coefficient c k,m+1 (γ) of each of these three types of terms.

• Terms of the first kind. Let us assume that in the process described above, the term corresponding to γ = (s, α, β) ∈ B k,m+1 was created as a term of the form

∂ x D α1 u • . . . • ∂ x D αs-1 u • ∂ m+1 x D αs ψ coming from F k [u, ∂ m
x ψ], for some θ = (s, α) ∈ A k . Since this term is of the form f (γ)[u, ψ] we have that s = s, α = α and β = (1, . . . , 1, m + 1). Therefore the contribution of this kind of terms to c k,m+1 (γ) can be estimated by

N a := k! α! k!(m + 1)! α!β! (2s) 2(m-1) k! (m + 1)! α! β! .
(2.12)

• Terms of the second kind. Now let us assume that the term corresponding to γ = (s, α, β) is originated as a second term in the right hand side of (2.11), that is, it can be written as a term in

∂ x D α1 u • . . . • ∂ x D αs-1 u • ∂ x f (γ)[u, ψ], (2.13)
for some θ = (s, α) ∈ A k and γ = (j, α, β) ∈ B αs,m . Now we see that since this term is of the form f (γ)[u, ψ], the indices s, α, β and s, α, β, γ are connected as follows:

one has s -1 + j = s, -one has α = (α 1 , . . . , αs-1 , α), -the multi-index β has the form β := (1, . . . , 1, β), for some β ∈ N j , -moreover β equals to β up to a component for which one adds 1 to βi . In other words, if one denotes for r ∈ {1, . . . , j}, R r (β) := (β 1 , . . . , β r-1 , β r -1, β r+1 , . . . , β j ), then for some r ∈ {1, . . . , j}, one has β = R r (β). Now given γ and j, there is a unique (α, α) and at most j different possibilities of β such that this process can generate at f (γ)[u, ψ] term (according to r). Moreover such a term (2.13) comes with a coefficient c k ( θ) c αs,m (γ). It follows that the contribution of these terms to the coefficient c k,m+1 (γ) can be estimated by

N b := s j=2 j r=1 c k (s, α) c αs,m (γ).
Using the previous iterations, Lemma 4 and β = R r (β) we deduce

N b s j=2 j r=1 (2j) 2(m-1) k! α 1 ! . . . α s-1 ! αs ! αs ! m! α! β! s j=2 j r=1 (2j) 2(m-1) k! α 1 ! . . . α s-1 ! m! α! β! β r . With α! := α 1 ! . . . α s-1 ! α! and β! = β! we infer N b s j=2 (2j) 2(m-1) k! m! α! β! |β|.
Now using |β| |β| = m + s, we obtain

N b (m + s) s j=2 (2j) 2(m-1) k! m! α! β! s(m + s)(2s) 2(m-1) k! m! α! β! . (2.14)
• Terms of the third kind. Now assume that the term corresponding to γ comes from

∂ x F k,m [u, ψ].
In that case the contribution to the coefficient c k,m+1 (γ) can be estimated by

N c := s j=1 |c k,m (γ j )|
such terms, where γj := (s, α, βj ) with βj := R j (β) = (β 1 , . . . , β j-1 , β j -1, β j+1 , . . . , β s ) (assuming β j 1). By the induction hypothesis, we get

N c s j=1 (2j) 2(m-1) k! m! α! β! β j (2s) 2(m-1) k! m! α! β! |β| = (2s) 2(m-1) k! m! α! β! (m + s). (2.15)
Finally, gathering (2.12)-(2.15) we obtain

|c k,m+1 (γ)| (2s) 2(m-1) k!m! α!β! (m + 1 + (s + 1)(m + s)) (2s) 2m k!(m + 1)! α!β! ,
which concludes the proof of Lemma 3.

Propagation of inequalities.

Using the above formal identity, one can prove the following result.

∂ m x D j u L ∞ ((-τ,τ )×T) 4V m,j , (2.16
)

with V m,j := m!j! (m + 1) 2 (j + 1) 2 L j V j+1 and V := u L ∞ (-τ,τ ;W 2l-1,∞ (T)) , (2.17)
then, for any m ∈ N such that 0 m 2l -1,

F k,m [u, u] L ∞ ((-τ,τ )×T) γ(L)V m,k . (2.

18)

Proof of Lemma 5. Here we simply denote • for • L ∞ ((-τ,τ )×T) . First, for any γ := (s, α, β) ∈ B k,m , one has, using (2.16),

f (γ)[u, u] 4 s s i=1 V βi,αi = 4 s α!β! s i=1 1 (α i + 1) 2 (β i + 1) 2 L k+1-s V k+1 . Denoting for α ∈ N s , Υ(s, α) := s i=1 1 (1 + α i ) 2 ,
we therefore obtain, using (2.9),

F k,m [u, u] γ∈B k,m |c k,m (γ)| f (γ)[u, u] , k! m! V k+1 k+1 s=2 (2s) 2(m-1) 4 s L k+1-s α / |α|=k+1-s Υ(s, α) β / |β|=m+s-1 Υ(s, β) .
We now use [6, Lemma 7.3.3], which we recall for the reader's convenience.

Lemma 6 ([6]

). For any couple of positive integers (s, m) we have

α∈N s |α|=m Υ(s, α) 20 s (m + 1) 2 .
This yields (2.18) with

γ(L) := sup k k+1 s=2 (2s) 2(m-1) 1600 s L 1-s (k + 1) 2 (m + 1) 2 (k + 2 -s) 2 (m + s) 2 .
It is straightforward to see that γ(L) → 0 as L → +∞.

In the sequel we will use the same notation γ : R + → R + for some other positive decreasing functions such that lim L→+∞ γ(L) = 0, which may change from line to line. Yet these functions will always be, as above, independent of k.

In particular with some slight modifications left to the reader we also get:

Lemma 7.
There exists γ a positive decreasing function, independent of k, with lim L→+∞ γ(L) = 0 such that for any k ∈ N, if for any j, m ∈ N such that 0 m 2l -1 and 0 j k -1,

∂ m x D j u L ∞ ((-τ,τ )×T) 4V m,j and ∂ m x D j Λ ± P L ∞ ((-τ,τ )×T) LVV m,j
then, for any m ∈ N such that 0 m 2l -1,

F k,m [u, Λ ± P ] L ∞ ((-τ,τ )×T) γ(L)LVV m,k .
Let us also provide the following technical lemma relying on Leibniz's rule.

Lemma 8. For any m 1 , m 2 in N, for any k ∈ N, if for any j ∈ N such that 0 j k, for i = 1, 2, one has

D j f i L ∞ ((-τ,τ )×T) 5V mi,j , then k j=0 k j (D j f 1 )(D k-j f 2 ) L ∞ ((-τ,τ )×T) 400VV m1+m2,k .
Proof of Lemma 8. It follows straightforwardly from the following inequality:

k j=0 k j V m1,j V m2,k-j = L k V k+2 k! m 1 ! m 2 ! (m 1 + 1) 2 (m 2 + 1) 2 k j=0 1 (j + 1) 2 (k -j + 1) 2 2L k V k+2 k! (m 1 + m 2 )! (m 1 + m 2 + 1) 2 ⌊k/2⌋+1 j=0 1 (j + 1) 2 (k -j + 1) 2 2L k V k+2 k! (m 1 + m 2 )! (m 1 + m 2 + 1) 2 π 2 6 4 k 2 16VV m1+m2,k .

Core of the proof

Let us consider τ ∈ (0, T ).

We first deal with smooth solutions u of class C ∞ . We will explain at the end of this section how to extend the analysis to the solutions tackled in Theorem 4.

First we obtain the following a priori estimates.

Proposition 1. There exists L > 0 such that the following holds true. Let (u, P ) be some smooth functions satisfying the equation (2.1) on (-τ, τ ). Then for any k ∈ N, for any j ∈ N such that 0 m 2l -1,

∂ m x D k Λ ± P L ∞ ((-τ,τ )×T) LVV m,k , (2.19)
∂ m x D k u L ∞ ((-τ,τ )×T) 4V m,k . (2.20)
Let us recall here that the operators Λ ± are defined in (2.3) and V, V m,k are defined in (2.17).

Proof

-2iD k u = -D k-1 (Λ + + Λ -)P.
Together with the induction hypothesis, we deduce (2.20) holds true at rank k. It remains to prove that (2. [START_REF] Glass | Controllability and asymptotic stabilization of the Camassa-Holm equation[END_REF]) holds true at rank k as well.

b. We begin by estimating D k F [u] in L ∞ ((-τ, τ ) × T). Applying D k to (2.2) and using Leibniz's rule, we get

D k F [u] = 0 m1+m2 2l-1 c m1,m2 k j=0 k j (D j ∂ m1 x u)(D k-j ∂ m2 x u).
Thanks to Lemma 3, we deduce that

D j ∂ m1 x u = ∂ m1 x D j u -F j,m1 [u, u] and D k-j ∂ m2 x u = ∂ m2 x D k-j u -F k-j,m2 [u, u].
Then using Lemma 5 and the fact that (2.20) is true up to rank k, we deduce that for L large enough,

D j ∂ m1 x u L ∞ ((-τ,τ )×T) 5V m1,j and D k-j ∂ m2 x u L ∞ ((-τ,τ )×T) 5V m2,k-j .
Now using Lemma 8 we deduce

k j=0 k j (D j ∂ m1 x u)(D k-j ∂ m2 x u) L ∞ ((-τ,τ )×T)
400VV m1+m2,k .

Now we observe that l being fixed, there exists some constants C 1 , C 2 > 0 such that for any m, k ∈ N with 0 m 2l -1,

C 1 V 2l-1,k V m,k C 2 V 2l-1,k . It follows that D k F [u] L ∞ ((-τ,τ )×T) CVV 2l-1,k . (2.21)
c. Next we estimate Λ± D k Λ ± P in L ∞ ((-τ, τ ) × T). Applying D k to the second equation in (2.1) and using the relations in (2.5), we dget

D k AP = D k Λ± Λ ± P = D k F [u].
Now, with the identity (2.4) and Lemma 3 we deduce

Λ± D k Λ ± P = D k F [u] + 0 m 2l-1 d m F k,m [u, Λ ± P ]. (2.22)
The first term in the right hand side of (2.22) was estimated in (2.21). To estimate the second one we use Lemma 7 and modify again γ to get

F k,m [u, Λ ± P ] L ∞ ((-τ,τ )×T) γ(L)LVV m,k .
Thus, modifying again γ(L), we get

Λ± D k Λ ± P L ∞ ((-τ,τ )×T) γ(L)LVV 2l-1,k .
d. Finally we use Lemma 1, and deduce that

D k Λ ± P L ∞ (-τ,τ ;W 2l-1 (T)) Cγ(L)LVV 2l-1,k .
We choose L large enough to absorb the constant C (independently of k) and deduce that (2.19) holds up to order k. The proof of Proposition 1 is over.

End of the proof of Theorem 2. We still suppose in a first time that u is smooth. Now from the definition of ξ we infer that

∂ k+1 t ξ(t, x) = D k u(t, ξ(t, x)).
As mentioned above, the proof of Theorem 3 provides that the flow ξ is C 1 over (-τ, τ ) with values in W 2l-1,∞ (T). Thus we infer the following estimates for the flow map: there exists L > 0 such that for any k ∈ N,

∂ k+1 t ξ L ∞ (-τ,τ ;W 2l-1,∞ (T)) k!L k (k + 1) 2 u k+1 L ∞ (-τ,τ ;W 2l-1,∞ (T)) . (2.23)
(One can again choose L large enough to absorb the constants depending on m.) In particular this yields the analyticity in time of ξ in the case where u is a C ∞ solution.

Let us finally briefly explain how to deduce the result for the general flows considered in Theorem 4. Let u 0 be in W 2l-1,∞ (T). By convolution there exists a sequence of smooth functions (u n 0 ) n∈N which satisfies

u n 0 W 2l-1,∞ (T) u 0 W 2l-1,∞ (
T) and converges to u 0 in H 2l-1 (T). To these initial data u n 0 one associates the corresponding smooth flows (ξ n ) n∈N for which the previous analysis can be applied. In particular the estimates (2.23) hold true when one substitutes ξ n instead of ξ in the right hand side. But due to the convergence of u n 0 to u 0 in H 2l-1 (T), we deduce with (1.11) that the corresponding solutions u n converge to the limit solution u in C([-τ, τ ], W 2l-1,∞ (T)). It follows that the flows (ξ n ) n∈N converge uniformly to ξ (locally in time). Hence passing to the limit when n → +∞ we obtain that ξ satisfies the estimates (2.23) as well, and consequently belongs to the space C ω ((-T, T ), W 2l-1,∞ (T)) of the analytic functions from (-T, T ) to W 2l-1,∞ (T). This concludes the proof of Theorem 2.

A concluding remark

The Camassa-Holm equation possesses solutions of a soliton type, which, because of their shape (at the peak, the derivative is discontinuous), have been given the name of peakons. When the equation is set on the whole line, a single peakon is given by u

c (t, x) = ce -|x-ct| , (3.1)
p 2 (t)e -|x-q2(t)| with p 2 (t), q ′ 2 (t) → c 2 when t → -∞, for some c 1 > 0, c 2 < 0, then one has to glue together the trajectories

q 1 (t) = ln c 1 -c 2 c 1 e -c1t -c 2 e -c2t , q 2 (t) = -ln c 1 -c 2 c 1 e c1t -c 2 e c2t ,
for t < 0 with some trajectories of the form q1 (t) = ln c1 -c2 c1 e -c1t -c2 e -c2t , q2 (t) = -ln c1 -c2 c1 e c1t -c2 e c2t , for t > 0. If one looks for the solutions which verify that R u(t, y)dy and u(t, •) H 1 (R) are time independent, then one must have {c 1 , c2 } = {c 1 , c 2 }, that is, with the arbitrary choice that c1 < c2 , c1 = c 1 , c2 = c 2 . It then remains to choose if one prolongs (q 1 (t), q 2 (t)) t<0 by (q 1 (t), q2 (t)) t>0 or (q 2 (t), q1 (t)) t>0 . Yet since lim t→0 -q ′′ 1 (t) = c 1 c 2 and lim t→0 + q′′ 2 (t) = -c 1 c 2 , the only choice which yields analytic trajectories for t running over the full line is the first one.

It should be therefore interesting to determine whether Theorem 2 could be extended to the H 1 conservative theory or not. Let us stress here that Theorem 1 contains the case of the periodic counterpart of the peakon given in (3.1) : the so-called periodic peakons which are of the form

u γ (t, x) = γ n∈Z e -|x+n-ct| , (3.8)
where γ is chosen in such a way that the height of the crest of the peakon is the same as the speed of the crest, that is such that u γ (t, ct) = c (cf. [17, Section 2.1]).

Appendix. Proof of Theorem 3

In this appendix we give a sketch of proof of Theorem 3. We will follow closely the method of [START_REF] De Lellis | Low-regularity solutions of the periodic Camassa-Holm equation[END_REF].

Definition 1. Let us define D l the Banach manifold given by the union of the following two charts:

U 0 := {ξ(x) = x + f (x)/ f ∈ W 2l-1,∞ (T) : |f (0)| < 1/2, essinf f ′ > -1}, U l := {ξ(x) = x + f (x)/ f ∈ W 2l-1,∞ (T) : 0 < f (0) < 1, essinf f ′ > -1}.
Let us introduce here a few notations. We denote by R ξ v the right translation v • ξ of v by ξ and by P ξ the conjugate R ξ P R ξ -1 of a nonlinear operator P . We also define the directional derivative of P ξ in direction w as follows: let t → ξ t be a We introduce

f (ξ, v) := -A -1 ∂ x F l ξ [v] = -(A -1 ∂ x ) ξ h, with h := (F ) ξ [v].
It is sufficient to prove that f is C 1 in a neighborhood of (Id, 0) in D l × W 2l-1,∞ (T) (see [START_REF] De Lellis | Low-regularity solutions of the periodic Camassa-Holm equation[END_REF]Theorem 2]).

Then by scaling properties we can infer that the ODE Cauchy problem (ξ, v) ′ (t) = (v, f (ξ, v)), (ξ, v)(0) = (Id, u 0 ), admits a local-in-time solution in D l × W 2l-1,∞ (T) for any u 0 ∈ W 2l-1,∞ (T). Since A -1 ∂ x = 2i( Λ+ + Λl,-), this follows from the two next propositions.

Proposition 2. Let ξ 0 ∈ C \ 2πZ and j ∈ N. Then the mapping ξ ∈ D l → (-i∂ x -ξ 0 ) -1 ξ ∈ L(W j-1,∞ (T), W j,∞ (T))

is of class C 1 .

. 10 )

 10 Hence to conclude in the case m = 1, it suffices to set F k,1 [u, ψ] := F k [u, ψ] and to observe that γ := (s, α, β) ∈ B k,1 means exactly that θ := (s, α) belongs to A k and β = (1, . . . , 1).

Lemma 5 .

 5 There exists γ : R + → R + a positive decreasing function satisfying lim L→+∞ γ(L) = 0 such that for any k ∈ N, if for any j, m ∈ N such that 0 m 2l -1 and 0 j k -1 one has

C 1 -

 1 path with values in D l such that ξ 0 = ξ and ( d dt ξ t )| t=0 = w. Then we define the directional derivative D w P ξ (f ) := d dt | t=0 (P ξt f ).

  of Proposition 1. We establish (2.[START_REF] Glass | Controllability and asymptotic stabilization of the Camassa-Holm equation[END_REF])-(2.20) by induction on k. Let us now consider k > 0. We assume that (2.[START_REF] Glass | Controllability and asymptotic stabilization of the Camassa-Holm equation[END_REF]) and (2.20) hold true up to the order k -1, and aim at proving them at rank k, for L large enough independent of k.

	1. For k = 0, (2.20) is straightforward. To get (2.19), we observe that
	-Λ + P = Λ-1 + F [u] and -Λ -P = Λ-1 -F [u].
	Now it follows from (2.2) and Lemma 1 that (2.19) at rank 0 holds true for L large enough.
	2. a. First from (2.1) and (2.5) we infer that

Acknowledgements. The authors are partially supported by the Agence Nationale de la Recherche, Project CISIFS, grant ANR-09-BLAN-0213-02. They wish to thank Raphaël Danchin for useful discussions.

with c > 0, whereas it is called an antipeakon when c < 0. The traveling speed is then equal to the height of the peak. By taking a linear combination of peakons one obtains what is called a multipeakon solution. u(t, x) = n i=1 p i (t)e -|x-qi(t)| .

(3.2)

Then the positions q(t) := (q i (t)) 1 i n and the amplitudes p(t) := (p i (t)) 1 i n of the crests are determined in a non-linear fashion by the interaction. Actually plugging the ansatz (3.2) into Eq. (1.2) we obtain the following system of ordinary differential equations:

When u is given by the ansatz (3.2), then m, defined in (1.3), is equal to the following combination of Dirac masses:

and therefore the energy defined in (1.5) is given by

Since the energy is time-independant we obtain that the system (3.

with Hamiltonian

It is straightforward that solutions of (3.3) are analytic with respect to time up to the first collision. Higher peakons move faster than the smaller ones, and when a higher peakon overtakes a smaller, there is an exchange of mass, but no wave breaking takes place, in the sense that the solution remains Lipschitz. Furthermore, the q i (t) remain distinct. However, if some of the p i (0) have some opposite signs, for instance when one peakon and one antipeakon meet, wave breaking may occur.

The problem of continuation beyond wave breaking was recently considered by Bressan and Constantin [START_REF] Bressan | Global dissipative solutions of the Camassa-Holm equation[END_REF] improving earlier results on the H 1 theory (see the introduction of [START_REF] Bressan | Global dissipative solutions of the Camassa-Holm equation[END_REF] for more about the history of this theory). They reformulated the Camassa-Holm equation as a system of ordinary differential equations taking values in a Banach space. This formulation allowed them to continue the solution beyond the collision time, giving a global conservative solution for which the energy is conserved for almost all times. Holden and Reynaud proceed in the same way but with a different set of variables which simply corresponds to the transformation between Eulerian and Lagrangian coordinates see [START_REF] Holden | Periodic conservative solutions of the Camassa-Holm equation[END_REF][START_REF] Holden | Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view[END_REF]. In the context of peakon-antipeakon collisions they considered the solution where the peakons and antipeakons "passed through each other". Local existence of the ODE system is obtained by a contraction argument. Furthermore, their clever reformulation allows for a global solution where all singularities disappear. Going back to the original function u, one obtains a global solution of the Camassa-Holm equation. In addition to the solution u, one includes a family of non-negative Radon measures µ t with density (u + u 2

x )dx with respect to the Lebesgue measure. The pair (u, µ t ) is associated to a continuous semigroup and in particular, one has uniqueness and stability. Very recently, Bressan and Fonte [START_REF] Bressan | An optimal transportation metric for solutions of the Camassa-Holm equation[END_REF] constructed a semi-group of conservative solutions for the Camassa-Holm equation which is continuous with respect to a distance J, defined in term of a optimal transportation problem, which is intermediate between H 1 and L 1 . The semi-group is actually constructed as the uniform limit of multi-peakon solutions.

The investigation of the explicit example of the collision of a peakon and of an antisymmetric antipeakon reveals that the analyticity of the trajectories of the crests still holds even through the collision if one selects the conservative scenario after the collision with an appropriate labelling. Indeed if one look at the collision at (t, x) = (0, 0) of a peakon p 1 (t)e -|x-q1(t)| with p 1 (t), q ′ 1 (t) → c 1 when t → -∞ and of an antipeakon

This can be proved in the same way as [START_REF] De Lellis | Low-regularity solutions of the periodic Camassa-Holm equation[END_REF]Proposition 3].

Sketch of proof. The proposition follows easily from the formula:

and from the fact that F l [u] is a differential polynomial in u of order 2l -1. With these formulas, one deduces that h(ξ, v) is Gâteaux-differentiable, and that the differential is continuous, so that h is indeed of class C 1 .

Once the problem is solved in Lagrangian variables one use the following lemma to deduce the existence part of Theorem 3 by considering u(t, x) := v(t, ξ(t, •) -1 (x)). Lemma 9. If ξ is in D l then ξ is a homeomorphism and ξ -1 is in D l . Moreover the following map

The previous lemma simply follows from Faà di Bruno's formula. Note that the uniqueness in Eulerian variables is inferred from the one in Lagrangian variables, cf. [START_REF] De Lellis | Low-regularity solutions of the periodic Camassa-Holm equation[END_REF]Proposition 5].