Thibaut Possompès
email: thibaut.possompes@fr.ibm.com

Christophe Dony
email: dony@lirmm.fr

Marianne Huchard
email: huchard@lirmm.fr

Chouki Tibermacine
email: tibermacin@lirmm.fr

Design of a UML profile for feature diagrams and its tooling implementation

This paper proposes an instrumented solution to integrate feature diagrams with UML models to be used as part of a general approach for designing software product lines and for product generation. The contribution is implemented in IBM R Rational Software Architect (RSA). It is intended to be used in the context of large, complex and multidomain projects, and at allowing model transformations to derive products. Our RSA implementation makes it possible to link feature diagrams with UML model artifacts. It allows traceability between feature models and other different kinds of models (requirements, class diagrams, sequence or activity diagrams, etc.). It is used in a project dedicated to create smart building optimization systems.

I. INTRODUCTION

IT projects are closely related to business domains. However, there exist few techniques to gather, link and manage the knowledge related to each domain. Nowadays projects are particularly confronted to problems where instrumentation of specific business domains is required to enhance their efficiency. Examples of these projects are smart buildings, grids, water management, health care or food systems. The research presented in this paper is done in the context of the RIDER project 1 , which brings together several actors who works on improving energy efficiency of buildings. This context requires to be able to link several domains, such as building system components, IT infrastructure, air flow modeling, building thermal management, database models, etc. Each domain can be modeled by a different stakeholder, or sometimes being based upon a standard. Software product line approach is perfectly appropriate to manage the variations that can be found in building instrumentation systems. The solutions presented in this paper are applied to the RIDER project and benefit of its feedback. Our approach is used as a part of a general approach for software product lines and for product generation.

There exist several commercial tools such as [START_REF] Gears | [END_REF], [START_REF] Eriksson | The PLUSS toolkit: extending telelogic DOORS and IBM-rational rose to support product line use case modeling[END_REF], [21] that allow linking features to model artifacts extracted from various modeling tools; eclipse plug-ins [START_REF]ProS Labs -Moskitt Feature Modeler[END_REF], [START_REF] Antkiewicz | FeaturePlugin: feature modeling plug-in for eclipse[END_REF], [START_REF]Generative Software Development Lab[END_REF], [START_REF]Feature-Oriented Software Development Research[END_REF], [27], [START_REF] Hydra | [END_REF], [START_REF]fmp2rsm Plug-in[END_REF]; and standalone feature modeling tools [START_REF] Requiline | [END_REF], [START_REF]S2T2 -An SPL of SPL Techniques and Tools[END_REF], [START_REF]SPLOT -Software Product Line Online Tools[END_REF]. These tools do not implement the latest feature modeling concepts and are not well integrated in UML modeling tools. There also exist several UML profiles [START_REF] Clauss | Untersuchung der Modellierung von Variabilität in UML[END_REF], [START_REF] Ziadi | Towards a UML profile for software product lines[END_REF] and meta-models [START_REF] Zhang | A propositional logic-based method for verification of feature models[END_REF] implementations but they address neither the latest UML specification nor all feature model concepts that are useful in our context. This paper is organized as follows. Section II presents a new synthetic and expressive feature diagram model based on the state of the art. Section III presents our feature meta-model which synthesizes this state of the art. Section IV explains how the corresponding UML profile has been created and implemented using RSA, and shows an excerpt of the project feature model. Section V will sum up what has been presented and suggests some perspectives for further study.

II. SYNTHESIS OF EXISTING FEATURE DIAGRAM MODELS

Various feature diagram semantics and implementations have been proposed since the initial one given in FODA [START_REF] Kang | Feature-Oriented domain analysis (FODA) feasibility study[END_REF]. We use the following criteria to analyze them and to define our model: 1 -Feature definition. 2 -Feature relationships. We identify four criteria to classify existing proposals regarding relations among features. Consists of FORM [START_REF] Kang | Feature-Oriented product line engineering[END_REF] Composed of Generalization Implemented by Specialization Fey et al. [START_REF] Fey | Feature modeling: A Meta-Model to enhance usability and usefulness[END_REF] Refine Provided by Zhang et al. [START_REF] Zhang | A propositional logic-based method for verification of feature models[END_REF] Decompose Specialize Detail Czarnecki et al. [START_REF] Czarnecki | Formalizing cardinality-based feature models and their specialization[END_REF] Relation

A. Feature Definitions

The initial definition of features was introduced by Kang et al. in FODA [START_REF] Kang | Feature-Oriented domain analysis (FODA) feasibility study[END_REF] which is extended by the FORM [START_REF] Kang | FORM: a feature-oriented reuse method with domain-specific reference architectures[END_REF] method. Features are defined as being essential characteristics of applications, described with domain vocabulary.

FORM has introduced four different perspectives to enrich the semantics of feature diagrams: capability, operating environment, domain technology and implementation technique. Indeed, features are also able to model knowledge of the various domain experts involved in the project. Fey et al. [START_REF] Fey | Feature modeling: A Meta-Model to enhance usability and usefulness[END_REF] add the pseudo feature concept in order to allow for specialization with non-exclusive alternatives and to avoid feature redundancy thanks to implicit inheritance. Zhang et al. define features [START_REF] Zhang | Feature-driven requirement dependency analysis and high-level software design[END_REF], [START_REF] Zhang | A propositional logic-based method for verification of feature models[END_REF] as being essentially a cohesive set of individual requirements representing the uservisible capability of a software system. Czarnecki et al. [START_REF] Czarnecki | Formalizing cardinality-based feature models and their specialization[END_REF], [START_REF] Czarnecki | Formalizing cardinality-based feature models and their staged configuration[END_REF] consider that features are system properties relevant for some stakeholder, and that any kind of functional or non-functional characteristic of a described system can be represented by a feature. This extends the intention definition presented by Zhang et al. [START_REF] Zhang | Feature-driven requirement dependency analysis and high-level software design[END_REF].

The definition proposed by Czarnecki et al. allows more freedom of expressiveness in feature diagrams. Its combination with FODA perspectives allows us to apply feature diagrams to the numerous domains involved in a project. The property concept introduced by Fey et al. and the attribute concept presented by Czarnecki et al. are semantically very close. It could be assessed that a property expresses the same information that an attribute, i.e., a measurable characteristic. We will use the property concept to describe specific feature typed values (e.g., a room volume, etc.) and how one property can influence another one.

We extend this concept to allow the customer to choose the property value. Therefore, we will add the necessary semantics to restrict the possible choices to consistent ones.

B. Feature Relationships 1) Hierarchical Relationships: Features and sub-features can be bound by either decomposition or specialization. Table I illustrates the different variations on these concepts. The FODA consists-of relationship [START_REF] Bontemps | Semantics of FODA feature diagrams[END_REF], [START_REF] Kang | Feature-Oriented domain analysis (FODA) feasibility study[END_REF], [START_REF] Schobbens | Generic semantics of feature diagrams[END_REF] is designed to represent decomposition. One parent feature can have several sub-features grouped either by an AND or XOR decomposition semantics. FORM [START_REF] Kang | FORM: a feature-oriented reuse method with domain-specific reference architectures[END_REF], [START_REF] Kang | Feature-Oriented product line engineering[END_REF] introduces the relations Composed-of, to describe the constituents of a feature; Generalization / Specialization, to specialize or generalize a feature; and Implemented-by, to define how a high-level feature can be implemented by a lower-level one. Fey et al. [START_REF] Fey | Feature modeling: A Meta-Model to enhance usability and usefulness[END_REF] uses two kinds of hierarchical links between features: refine, to detail a given feature at a lower abstraction level; and provided-by, to link a pseudofeature with the features which it realizes. Pseudo-features express an abstract functionality, quality or characteristic. Fey et al. make it possible to build directed acyclic graphs, which is impossible with FODA, by allowing one feature to refine several ones. Zhang et al. [START_REF] Zhang | A propositional logic-based method for verification of feature models[END_REF] identify three different kinds of hierarchy relationships: decomposition, to refine a feature into its constituents; detailization, to identify feature attributes; and specialization, to add further details into a feature. Czarnecki et al. [START_REF] Czarnecki | Formalizing cardinality-based feature models and their specialization[END_REF], [START_REF] Czarnecki | Formalizing cardinality-based feature models and their staged configuration[END_REF], [START_REF] Czarnecki | Staged configuration using feature models[END_REF] decided not to consider relationships between features in favor of entity-relationship or class diagrams.

We have chosen to keep the hierarchy relationships categorized in Table I. with the following concepts: decomposition, which consists in detailing the sub-features that compose the parent feature; specialization, which encompasses the concepts of enrichment for sub-features that add functions to the parent feature, and Realization (or implementation) that describes how a feature can be implemented.

2) Feature Choice Constraint Relations: FODA [START_REF] Kang | Feature-Oriented domain analysis (FODA) feasibility study[END_REF] uses the concept of composition rules to describe how features relate to one another: one feature can require another one, or two features can be mutually exclusive. Riebisch et al. [START_REF] Riebisch | Modelling Variability for Object-Oriented Product Lines[END_REF] introduce the hint relationship which recommend features to the user.

We have kept the constraints require and conflict to ensure a coherent product generation. Furthermore the hint relation is also convenient to make recommendation to the user during the feature selection process.

3) Mandatory and Optional Features Identification: In the FODA [START_REF] Kang | Feature-Oriented domain analysis (FODA) feasibility study[END_REF] and FORM [START_REF] Kang | Feature-Oriented product line engineering[END_REF] specifications and tools, all features are mandatory by default; optionality is represented as a feature property as in [START_REF] Riebisch | Modelling Variability for Object-Oriented Product Lines[END_REF], [START_REF] Griss | Integrating feature modeling with the RSEB[END_REF], [START_REF] Zhang | A propositional logic-based method for verification of feature models[END_REF]. Czarnecki et al. [START_REF] Czarnecki | Formalizing cardinality-based feature models and their staged configuration[END_REF] use cardinalities to express optionality. For instance, a (1,1) cardinality describes a mandatory feature and (0,1) describes an optional one. Setting a cardinality greater than 1 specify how often the sub-features of a parent can be duplicated.

We keep for our model the cardinality based relationship introduced by Czarnecki et al. [START_REF] Czarnecki | Formalizing cardinality-based feature models and their staged configuration[END_REF] that brings an interesting enhancement to the initial definition, useful to describe more complex products from a product line.

4) Sub-feature Selection Semantics: FODA [START_REF] Kang | Feature-Oriented domain analysis (FODA) feasibility study[END_REF] and FORM [START_REF] Kang | Feature-Oriented product line engineering[END_REF] methods propose two ways to manage subfeature selection. A simple link between the parent and its sub-features means that there is no choice constraint; equivalent to an or binary choice. A semi circle drawn across the links means that there is an alternative choice; equivalent to the xor binary choice. Likewise, Fey et al. [START_REF] Fey | Feature modeling: A Meta-Model to enhance usability and usefulness[END_REF] express with a simple link that the choice is an or, but the alternative choice is expressed with a complete graph of mutually exclusive constraints among all sub-features. Riebisch et al. [START_REF] Riebisch | Modelling Variability for Object-Oriented Product Lines[END_REF] and Czarnecki et al. [START_REF] Czarnecki | Formalizing cardinality-based feature models and their specialization[END_REF] use cardinalities to define how many sub-features can be chosen.

The most convenient semantics relies on cardinality based selection semantics. It allows a maximum flexibility to describe how many sub-features can be selected. Furthermore, we chose to keep the "or", "and", and "xor" groups to ease the feature models semantics understanding for non-IT specialists end-users.

C. Feature Logical Groups

The FORM [START_REF] Kang | Feature-Oriented product line engineering[END_REF] method classifies features into four categories called layers, from a functional point of view. Riebisch et al. [START_REF] Riebisch | Modelling Variability for Object-Oriented Product Lines[END_REF] use logical groups to represents aspects valuable to the customer and explain that abstract features could be used to encapsulate features related to a given concept. Fey et al. [START_REF] Fey | Feature modeling: A Meta-Model to enhance usability and usefulness[END_REF] present feature-sets to group features from an arbitrary point-of-view. Zhang et al. [START_REF] Zhang | A propositional logic-based method for verification of feature models[END_REF] present the binding-time concept to represent the phases of the software life-cycle in which each feature must be chosen. Czarnecki et al. [START_REF] Czarnecki | Formalizing cardinality-based feature models and their staged configuration[END_REF], [START_REF] Czarnecki | Formalizing cardinality-based feature models and their specialization[END_REF] use abstract features to reference other feature diagrams to reuse a set of features. They also introduce different types of features that can be considered as feature groups.

The binding-time concept can be extended according to the software development process chosen by the final user to allow him to assign features or groups of features to a development phase. It can be modeled by the feature set concept presented by Fey et al. The different kinds of features presented by Riebisch et al. [START_REF] Riebisch | Modelling Variability for Object-Oriented Product Lines[END_REF] could be easily modeled by sub-layers. Hence, we choose to keep the layer concept to organize features in logical groups and sub-groups accordingly to the type of information they represent. We propose to enhance Fey et al. feature-set concept by adapting Zhang et al. [START_REF] Zhang | A propositional logic-based method for verification of feature models[END_REF] constraints metamodel to describe constraints inside a group of features, and constraints between two groups of features. The featuresets could also be used along with the Kano method [START_REF] Kano | Attractive quality and must-be quality[END_REF] to help the user choosing a set of product features that yield high customer satisfaction. The customer preference categories can be modeled as feature-sets encompassing the corresponding features. We also keep the feature-sets idea to reference sub-parts of a feature diagram. Hence, a featureset could be a leaf of the diagram that encompasses another feature hierarchy. Staged configuration can be modelled by creating one feature-set for each configuration stage, and associating it with a group of stakeholders that have the same business concern. Czarnecki et al. [START_REF] Czarnecki | Formalizing cardinality-based feature models and their staged configuration[END_REF], [START_REF] Czarnecki | Formalizing cardinality-based feature models and their specialization[END_REF] have presented four types of features that have been integrated in our meta-model proposal: concrete features can be stored in the implementation layer; aspectual features can be stored in a sub layer of implementation layer; abstract features, e.g. performance requirements, can be represented by feature properties; and grouping features can be modeled as a feature set.

D. Product and Implementation Information

Riebisch et al. [START_REF] Riebisch | Modelling Variability for Object-Oriented Product Lines[END_REF] argue that the feature hierarchy must be organized to make easier the choice of features by using composition relations and require associations. Zhang et al. [START_REF] Zhang | A propositional logic-based method for verification of feature models[END_REF] present a feature attribute for representing the feature binding-state. It must be used in a binding-time context, i.e. when features are implemented in the software product at a given software life-cycle phase. Mathematical relationships have been presented to describe the relative impact of one feature to another.

III. A SYNTHESIS META MODEL FOR FEATURE DIAGRAMS

This section describes a meta model synthesizing the choices we presented and motivated in the previous section. This work extends the work of Asikainen et al. [START_REF] Asikainen | A unified conceptual foundation for feature modelling[END_REF] in order to apply our work in the context of a rich industrial project.

As depicted in Figure 1, a product line contains features, and a feature belongs to one product line. A product belongs to one product line and can be composed of as many features as needed. Features associated to a product must be analyzed in order to check whether all constraints, like mutual exclusion between features or require relations, are satisfied. Feature properties (Figure 2) are used to describe either a feature parameter related to its inner requirements (e.g. the bandwidth capacity of a network) or a characteristic chosen by the user during the product definition (e.g. the frequency of automatic backups of a word processing software).

Features can have a variability type (VariabilityKind) which is further described below:

• fixed, A property value is fixed throughout all products of the product line. • variable, A property value can change, within a product, depending on other features properties, e.g. the text buffer size of a text field in the user interface can vary accordingly to the type of information we want to store (i.e., name or address). • family-variable, A property can vary from product to product accordingly to the selected features, e.g. the phone book capacity depends on the presence of internal memory property and the sim card capacity. • user-defined, A property value can be freely chosen in a given product, e.g., the frequency of automated backups in a word processing software. The hierarchy relationships and sub-feature groups are depicted in Figure 3. The relations between a feature and its sub-features are grouped by the RelationshipGroup class that contains the cardinalities necessary to restrict the number of sub-features to choose. Cardinalities can be chosen freely, but some groups have fixed cardinalities: OrGroup, (0,*); AndGroup, (*,*); and XorGroup, (0,1). The DirectedBina-ryRelationship class represents the kind of association that links a parent feature and a sub-feature. It is specialized either by Enrich, Implement, or Detail classes.

Figure 4 shows how layers and feature sets can be associated with the project stakeholders. A stakeholder represents any kind of people (e.g. domain experts, IT architects, etc.) allowed to choose features. Feature sets and layers can be attached to a concern specific to the project, e.g. network architecture, or business requirements. A Layer represents a specific view onto the software application, a feature can be in only one layer at a time. A FeatureSet inherits from the Feature class, and allows grouping features from an arbitrary point of view, e.g., a business domain, or representing the features that must be implemented to fulfill a norm.

Figure 5 depicts how constraints can be applied to feature sets: mutex, when only one feature can be selected in the feature set; None, when there is no constraint among features; All, when all features or none of them can be selected. The ConstraintRelation class describes the relation between two feature sets: one feature set can require another one, two feature sets can mutually require each other, or be mutually exclusive. A BindingPredicate is used to represent how a constraint must be applied on each constrained feature-set. The choice of the specialized class must be made according to the kind of feature-set.

IV. UML PROFILE IMPLEMENTATION

A. Description

Our profile integrates the previously described feature meta-model into the UML meta-model hierarchy with an appropriate semantics. We describe here which UML metaclasses we have chosen to extend, which information has been added with the stereotypes and how we have restricted the semantics of extended meta-classes thanks to OCL constraints. However, the profile could be implemented in different ways by choosing to extend different meta-classes. We chose the meta-classes that had the closest semantics to our concepts, added the required information with the stereotypes and restricted the initial semantics of metaclasses to what is necessary for our profile with the Object Constraint Language (OCL) constraints.

Contrary to [START_REF] Clauss | Untersuchung der Modellierung von Variabilität in UML[END_REF] we have based our feature diagram profile on the Components UML meta-class. A component, as a The port meta-class allows us to represent the interaction of a feature with other elements. It can be linked to other ports or components. The modification of a property value can be modeled by textual or OCL constraints placed upon the relationship between two properties.

We choose to create a Rational Software Architect plug-in to leverage its UML modeler, modeling editors, views and tools. All managed models are instances of EMF models. Hence using Rational Software Architect allows simplifying tasks like creating a specific plug-in for integrating feature modeling capabilities into standard EMF-based UML models and diagrams. The plug-in is currently used in the RIDER project.

Table II gives an abstract of our specialization choices, but due to space constraints we did not explicit all our design choices.

B. Example

Figure 6 shows an excerpt of a RIDER feature diagram expressed, using our profile, by two stakeholders having different concerns on a RIDER smart building project. One focusing on HVAC (Heating Ventilation Air Conditioning) domain and the second on data centers and office building optimization problems. Each feature set encompasses several features that are not shown in this example because of space limitation. The example shows in particular require high level constraints applying on groups of features (feature sets). The example also shows how properties can be used to add further information on the product context (such as the building volume).

The feature selection process is achieved by showing feature diagrams to the user accordingly to his concerns thanks to model transformations.

V. CONCLUSION AND PERSPECTIVES

In this paper we have presented a new feature diagram model and its profile implementation in UML2. This feature diagram model is first based on a synthesis of existing ones, and it has then been improved to fit the requirements of the RIDER smart buidings project in which it is applied. Thanks to Rational Software Architect, we have generated a tool making it possible to produce feature diagrams conforms to our model. This synthesis is achieved by classifying the existing concepts into categories. This approach allows a full integration of feature diagrams into UML models and facilitates model transformations. In comparison with [START_REF] Asikainen | A unified conceptual foundation for feature modelling[END_REF] we add several concepts such as layers, stakeholder concerns, feature-sets, and group constraints.

For the time being, we still need to guide users to organize features into layers and sub-layers in order to best integrate them into the software development life-cycle. Hence, the next steps will be to create a framework able to use the full potential of our feature meta-model, and to develop automated model transformation functionalities to automatically generate UML models.

Figure 1 .

 1 Figure 1. Product lines relation to features and products

Figure 2 .

 2 Figure 2. Feature properties

Figure 3 .

 3 Figure 3. Groups and hierarchy relationships

Figure 4 .

 4 Figure 4. Stakeholders concerns

Figure 5 .

 5 Figure 5. Constraints on feature sets

Figure 6 .

 6 Figure 6. Stakeholders and feature sets

The RIDER project ("Research for IT as a Driver of EneRgy efficiency") is led by a consortium of several companies and research laboratories, including IBM and the LIRMM laboratory, interested in improving building energy efficiency by instrumenting it.


Euler C F Teixeira Ppgee -Ufmg

Mauricio A Loureiro
email: mauricioloureiro@ufmg.br

Hani C Yehia

Methodological Aspects of the Research in Musical Expressiveness Based on Corporal Movement Information

This project is focused on empirical analysis of musical expressiveness, based on information of musical content extracted from audio and video information of the musical performance, seeking to understand and to quantify the expressive intentions of the player.

I. INTRODUCTION

Musical expressiveness is a concept witch is difficult to describe quantitatively, using objective analysis methods. In the last few decades, this problem has been studied using a method that aims at describing the expressive intentions of the musician, based on the audio analysis of his performances and more recently, also on the video analysis of these musical performances. The goal is to establish relations between the musician's expressive intentions and the acoustical and visual parameters extracted from the analysis.

The objective of this work is to propose a methodology that can be used to analyze corporal movement, finding relations between the physical gestures of the musician and his expressive intentions in a given musical performance. Clarinetists corporal movements were tracked, as they played classical pieces in their instrument, and then coherent and recurrent physical gestures were defined, based on a morphological and kinesthetic segmentation of these movements. The gestures found were then used to compare different interpretations of a musical piece, and also different parts of a musical piece.

The final objective is to incorporate a corporal analysis method into an acoustical musical expressiveness analysis tool, developed in earlier studies [START_REF] Loureiro | Expan: a tool for musical expressiveness analysis[END_REF].

II. DATA ACQUISITION

In this study, different performances of pieces of the classical clarinet repertoire were used. The pieces chosen were Mozart's Quintet for Clarinet and Strings, Kv 581 and Clarinet Concerto in A Major, Kv 622. Both pieces were performed solo by 3 professional clarinetists. The musicians were instructed to play the pieces is 3 different ways. First freely, as in a real concert situation. Secondly, following the beat of a metronome marking a quarter note and then following the beat of a metronome marking a half note.

The musician's body movements were tracked using a 3D high-end motion capture device, the Optotrak Certus. The audio was also captured synchronously using the same device.

For now, the analysis was restricted to the head movements of the clarinetists. Four markers were used to define a rigid body representing the musician's head. They were mechanically coupled, using a light wooden frame, and attached to the musician's head. This system provides a set of coordinates representing the position of the rigid body in 6 degrees of freedom, 3 describing the position of it's centroid in cartesian space, and 3 describing it's rotation around each of it's axis. The sample rate used for the motion markers was set to 60 Hz and the audio sample rate was set to 10kHz.

III. DATA PROCESSING

A. Musical Note Segmentation

The precise detection of note onsets and offsets is not a trivial problem, even in monophonic musical signals, since the subjectivity in note discrimination cannot be ignored. The note onsets and offsets were detected using the RMS energy envelope curves, with a 23 ms analysis window and an adaptive energy threshold [START_REF] Depoli | From audio to content[END_REF]. The extraction of the note fundamental frequencies can also aid the detection of the onsets and offsets, especially in the case of legato notes. The presence of transients during note transitions also made possible the use of spectral flow to aid the detection of note onsets and offsets. A combination and cross validation of these 3 methods provided a robust tool for note segmentation on the audio signals.

B. Movement Segmentation

The motion capture data was processed and used to extract several parameters of the clarinetists head movements, such as velocity, acceleration, displacement, distance, duration and trajectory centroids. The goal is to define coherent and recurrent physical gestures, based on patterns of the temporal and spatial evolution of these parameters, and then relate them to the musical score and to the musician's expressive intentions. This way it's possible to find out where the expressiveness information lies in the musician's corporal motion and how it relates to the structure of the musical piece been performed [START_REF] Dahl | Visual perception of expressiveness in musicians' body movements[END_REF], [START_REF] Wanderley | The musical significance of clarinetists' ancillary gestures: An exploration of the field[END_REF].

The musicians performed the musical pieces sitting on a chair. In this condition, it's expected that the musician's head movements produce round trip trajectories around an equilibrium position [START_REF] Mcneill | Gesture and Thought[END_REF]. Considering that the ability to sustain movement in a given direction decreases as the musician's head moves away from the equilibrium position, there should be a drop in resultant velocity, related to a kind of direction shift on the motion. Based on this assumption, local minima on modulus of resultant velocity curve should be able to detect inflection points on the movement trajectories.

This simple method provided an efficient way to establish an initial segmentation of the musician's head movements. The modulus of resultant velocity of the musician's head was estimated using the euclidian distance between the positions of it's centroid along 2 subsequent samples.

Figure 1 shows the segmentation obtained by the velocity minima criteria, in a performance of a segment of Mozart's Quintet for Clarinet and Strings, in the case where the metronome was marking a half note. The upper graph shows the modulus of the velocity. In the lower graph, the blue curve represents the extracted pitch, the red curve represents the audio waveform and the vertical green lines indicate the segmentation points for the head movements of the clarinetist. In this analysis, 7 head movement segments were defined by the velocity minima along the first 2 musical phrases, that correspond to the first 4 bars of the musical piece.

Examining these segments in successive groups of 2 and 3, 3 physical gestures were identified, based on the well defined geometric structure of their trajectories, with clear beginning and ending points, delimited by direction shifts in the motion. The first of these gestures was composed only of the motion segment number 2. The second gesture was composed of the pair of motion segments 3-4 and the third gesture was composed of the pair of motion segments 6-7. The trajectories of each of these 3 gestures is showed in Figure 2, with the spatial coordinates given in millimeters. A red solid circle marks the beginning of each initial motion segment.

In a free performance of this same musical segment, without the presence of the metronome, a similar analysis identified only 2 physical gestures. The first gesture was composed of the first 3 motion segments and the second gesture was composed of the last 2 motion segments. The trajectories of each of these 2 gestures is also showed in Figure 3.

The gestures defined in this analysis presented an important characteristic, that might also be used as another segmentation criteria after further investigation. Their trajectories tend to evolve along a given plane, in all of these cases.

IV. RESULTS

A. Correlation Between Motion and Musical Structure

The segmentation points presented an approximate correspondency to some inflection points in the musical structure, like transitions between notes or phrases. In the first 2 phrases, the motion segments were better defined, with smoother velocity curves and clearer peaks and valleys. In these musical segments, specially in the metronome guided performances, it was also noticed a certain periodicity in the velocity curves, that were also more related to the musical metric. Larger values for mean velocity and total duration time of the motion segments were also observed in this musical segment. The last phrases of this musical piece, on the other hand, presented noisy velocity curves, with small amplitudes an without clear peaks, valleys or evolution patterns. The motion segments in this part of the musical piece also presented smaller values for mean velocity and total duration time.

These observations suggest that the musician's head movements in the last part of the musical piece reveals less information related to his expressive intentions, if compared to what was observed in the first phrases of the piece. This probably occurs due to the simpler rhythmic and melodic structure of the first 2 phrases, and suggests that technically difficult musical parts might inhibit corporal movements and expressive intentions by the musician. Large amplitude ancillary movements could make the precise execution of the primary movements more difficult, compromising the sound production itself. On the other hand, musical phrases composed of longer notes and smoother transitions, like the first 2 in this piece, provide ideal conditions for expressive intentions to occur, trough the manipulation of note durations, intensities and timbres. This would be reflected in the ancillary movements, producing wider, longer and better defined physical gestures.

B. Correlation Between Motion and Expressive Intentions

Based on comparisons between the different performances of the first 2 phrases of Mozart's Quintet for Clarinet and Strings, it was observed that the motion segments were twice as long in time, without the presence of the metronome. The number of head gestures identified was also smaller in the free performance, and these gestures were also wider in space. The 3 gestures found in the metronome guided performance exhibited total displacements of 86, 98 and 96 millimeters, while the 2 gestures found in the free performance exhibited total displacements of 236 and 237 millimeters. In the metronome guided performance, the first 2 gestures exhibited durations of 1.2 and 2.6 seconds, staying inside the first musical phrase of the piece. In the free performance, on the other hand, the first gesture only, lasted for 6 seconds, going trough almost the whole first 2 musical phrases.

These differences in duration and extension of the gestures between the 2 styles of performance suggest a close relation between the gestures and the musician's expressive intentions. In these first 2 musical phrases, there are 2 inflection points in the musical structure, marked by harmonic resolutions, according to Figure 4. The first one is weaker, culminating at the relative tonic, while the second one is stronger, culminating at the tonic, after a faster melodic movement towards it.

In the metronome guided performance, the first 2 head gestures seem to be related to the first musical inflection point, while the third gesture is disconnected to the first 2, and seems to be related to the second musical inflection point. In the free performance, on the other hand, the long and wide first head gesture seems to be related to a preparation to the second and stronger musical inflection point, ignoring the first inflection point. Close to the second inflection point, occurs the second head gesture, that seems to be prepared by, and is directly connected to the first gesture. This suggests that the musician performed the first musical phrase with an expressive intention that seems to anticipate the second musical inflection point, far away, at the end of the second musical phrase. That would require a broader image of the musical structure, allowing for a previous and more consistent planning of the interpretation he is trying to accomplish. With the presence of the metronome, on the other hand, the musician tends to develop a more locally driven image while performing the piece, also emphasizing the first musical inflection point, at the end of the first phrase. The strong influence exerted by the metronome makes the head movements shorter in time and space, and also more correlated to the musical metric, probably inhibiting any expressive intention that transcends this metric.

V. CONCLUSION Short duration physical gestures were identified, based on a segmentation of the musician's head movements during musical performances. Correlations were also identified between these gestures and the musical structure, as well as between these gestures and the musician's different expressive intentions, induced by the chosen interpretation styles. Morphological and kinesthetic criteria were the basis for the motion segmentation used, but an approach more focused on the geometry of the trajectories, as well as the knowledge on human movement could provide broader results.

Fig. 1 .

 1 Fig. 1. Head movement segmentation by the velocity minima criteria for the half note metronome guided performance of Mozart's Quintet for Clarinet and Strings.

Fig. 2 .

 2 Fig. 2. Spatial trajectories of the 3 head gestures identified in the first 2 phrases of the half note metronome guided performance of Mozart's Quintet for Clarinet and Strings.

Fig. 3 .

 3 Fig. 3. Spatial trajectories of the 2 head gestures identified in the first 2 phrases of the free performance of Mozart's Quintet for Clarinet and Strings.

Fig. 4 .

 4 Fig. 4. Harmonic resolution progressions at the end of the first 2 phrases of Mozart's Quintet for Clarinet and Strings.


Léo Agélas
email: agelasleo@yahoo.fr

No blow-up of the 3D incompressible Euler equations

come

No blow-up of the 3D incompressible Euler equations

Introduction

The Euler equations describe the motion of a fluid in R 3 . These equations have enriched many branches of mathematics, were involved in many area outside mathematical activity from weather prediction to exploding supernova (see [CONST]) and present important open physical and mathematical problems (see [CONST]). For Euler equations, non uniqueness of weak solutions has been demonstrated in [SCH], [SHN]. However, these solutions have not been obtained from a Leray solution of Navier-Stokes equations in the limit as the viscosity tends to zero (zero-viscosity limit of Navier-Stokes equations). Zero-viscosity limits of Navier-Stokes equations have only been shown to exist and to give Euler solutions in some more generalized sense known as measurevalued Euler solutions (see [DM]). Another notion of weak solutions to the Euler equations was introduced in [PLLIONS] called dissipative solutions and it was proven that they coincide with classical Euler solutions, when those exist. Recent studies indicate that the local geometric regularity of vortex lines can lead to dynamic depletion of vortex stretching [ [START_REF] Constantin | Geometric statistic in turbulence[END_REF], [CFM], [DHY], [START_REF] Deng | Improved geometric conditions for non-blowup of 3D incompressible Euler equation[END_REF]]. In particular, the recent results obtained in [DHY], [START_REF] Deng | Improved geometric conditions for non-blowup of 3D incompressible Euler equation[END_REF] show that geometric regularity of vortex lines, even in an extremely localized region containing the maximum vorticity can lead to depletion of nonlinear stretching, thus avoiding finite time singularity formation of the 3D Euler equations. The blow-up problem for the Euler equations is a major open problem of PDE theory, of far greater physical importance than the blow-up problem for the Navier-Stokes equation, which of course is known to non-specialists because it is a Clay Millennium Problem. It is known that if there are no singularities in the solution of the Euler equations with initial data u 0 on the time interval [0, T ], then there can be no singularities in the Navier-Stokes solution with the same initial data and small enough viscosity (see [START_REF] Constantin | Note on loss of regularity for solutions of the 3D incompressible Euler and related equations[END_REF]). The regularity for large enough viscosities is also known. Unfortunately, there is a gap between the two ranges of viscosities, and it is not clear how to close it. The blow-up of smooth solutions to the Euler equations which is one of the most challenging mathematical problems in nonlinear PDE is controlled by the time integral of the maximum magnitude of the vorticity (see [BKM], [KP]). And this criterion have been refined in [CHAE] by replacing the L ∞ norm of the vorticity by weaker norms close to the L ∞ norm, more precisely, the blow-up of smooth solutions is only controlled by the time integral of the Besov space Ḃ0 ∞,1 norm of two components of the vorticity.

Let us now introduce the Euler equations in R 3 given by

∂u ∂t + (u • ∇)u + ∇p = 0, ∇ • u = 0, (1)
in which u = u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)) ∈ R 3 , p = p(x, t) ∈ R denote respectively the unknown velocity field, the scalar pressure function of the fluid at the point (x, t) ∈ R 3 × [0, ∞[,

with initial conditions, u(x, 0) = u 0 (x) for a.e x ∈ R 3 . (2)
In this paper, we prove that the maximum magnitude of the vorticity grows at most one exponential in time. For this, we consider the particle-trajectory mapping X(, t) :

α ∈ R 3 -→ X(α, t) ∈ R 3
, where X(α, t) is the location at time t of a fluid particle initially placed at the point α at time t = 0 and such that the fluid velocity u is tangent to the particle trajectory. After, we fix the time at s > 0 and we study the evolution in time from the integral of the initial fluid vorticity over an initial unit ball of fluid particles, B, to the integral of the vorticity at times s over the domain X(B, s), with the ball B such that X(B, s) contains a point where the maximum magnitude of the vorticity is reached at time s. The equation (E) expressing this evolution is obtained from the vorticity equation and by the fact that vorticity is stretched at time t ∈ [0, s] by ∇ α X(α, t) along particle trajectories (ω(X(α, t), t) = ∇ α X(α, t)ω 0 (α)). After some manipulations, The equation (E) makes appear at the right hand side of the equality, the main obstruction in establishing the regularity of Euler solution, the stretching term which presents itself under the form

I = Ω u(y, t)f t (y)dy such that f t L ∞ ≤ ω 0 L ∞ , Ω f t (y)dy = 0 being
the consequence that ω 0 is a divergence-free vector field and Ω = X(B, t). By interpreting the term I as the integral of the product between an element in BM O r (Ω), the class of functions of bounded mean oscillation only on cubes inside Ω (see Section 3), u(•, t) and an element in H 1 z (Ω), a hardy space on Ω (see Section 3), f t , we are then able to use all the theory about Hardy spaces on domains that has been developed in recent years (see [JSW], [MIYA], [CKS] and [CDS]). After, using a duality result between BM O r (Ω) and H 1 z (Ω), we obtain a fine estimate of I given by the bound J = u(t) BM Or (Ω) f t H 1 z (Ω) . Thanks to Lemmata 3.1 and 3.2, the term J is bounded by C ω(t) L ∞ ω 0 L ∞ , where C > 0 is an absolute constant. Furthermore, on the left hand side of equality of (E), we have the derivative in time of the term X(B,t) ω(y, t) dy. After using an integration in time over [0, s] of equation (E) and using the scaling property of Euler equation, we infer the following inequality,

ω(s) L ∞ ≤ ω 0 L ∞ + C s 0 ω(t) L ∞ ω 0 L ∞ ds.
Thanks to Gronwall Lemma, we conclude that,

ω(s) L ∞ ≤ ω 0 L ∞ exp(C ω 0 L ∞ s),
which proves that there does not exist finite time T > 0 for which

T 0 ω(s) L ∞ ds = +∞.
From [BKM], we know that solution of Euler equation blow up at time T if and only if the time integral of the maximum magnitude of the vorticity diverges at time T ,

T 0 ω(t) L ∞ dt = +∞,
then we deduce that the incompressible Euler equations cannot develop a finite-time singularity from smooth initial data.

Notations : For any function ϕ defined on R 3 × [0, +∞[, for all t ≥ 0, we denote by ϕ(t) the function defined on R 3 by x -→ ϕ(x, t). For any vector x = (x 1 , x 2 , x 3 ) ∈ R 3 , we denote by i . For all vectors v = (v i ) 1≤i≤3 and w = (w j ) 1≤j≤3 , we denote by v ⊗ w the matrix with ij-elements given by (v i w j ) 1≤i,j≤3 . For any bounded subset Ω ⊂ R 3 , we denote by |Ω| the measure of Ω and by diam(Ω) the diameter of Ω defined by diam(Ω) = sup

x,y∈Ω |x -y|. For all Ω ⊂ R 3 , we denote by χ Ω , the function such that χ Ω (x) = 1, ∀x ∈ Ω and χ Ω (x) = 0, ∀x ∈ Ω. For any x 0 ∈ R 3 and r > 0, we denote by B(x 0 , r), the ball of R 3 of center x 0 and radius r. For the sake of simplicity, for any q ≥ 1, we denote by L q (resp W r,q ) indifferently the Sobolev space L q (R 3 ) 3 (resp W r,q (R 3 ) 3 ) and L q (R 3 ) 3×3 (resp W r,q (R 3 ) 3×3 ). We denote by BC the class of bounded and continuous functions and by BC m the class of bounded and m times continuously derivable functions. We denote by div the differential operator given by, div =

3 i=1 ∂ ∂x i .
2 Preliminary Definition 2.1 If ∆ denotes the Laplace operator, the Bessel potential space L p s , 1 < p < ∞, -∞ < α < ∞ can be defined as the space of functions (or distributions) f such that (I -∆) It is a theorem of A.P. Calderón that for positive integers α and 1 < p < ∞ the space coincides (with equivalence of norms) with the Sobolev space W p α of functions all of whose derivatives (in the distributional, or weak sense) of order at most α are functions in L p .

For α > 0 the elements of L p α are themselves L p -functions, which can be represented as Bessel potentials of L p -functions. In fact, the function (1 + |ξ| 2 ) -α 2 is then the Fourier transform of an integrable function, the Bessel kernel G α (x). In other words, f ∈ L p α , 1 < p < ∞, α > 0 if and only if there is a g ∈ L p such that f (x) = G α (x -y)g(y)dy, where the integral is taken over all of R 3 with respect to the Lebesgue measure. In what follows, we denote W r,q as the space L q r .

Definition 2.2 We denote by P the matrix Leray's projection operator with components,

P i,j = δ i,j - ∂ ∂x i ∂ ∂x j ∆ -1 = δ i,j -R j R k , (3)
where R j are the Riesz transform given by R j = ∂ ∂x j (-∆) -1 2 = 1 4π

x j |x| 4 ⋆ (see [STE] for more details), ∆ -1 is the inverse of Laplace operator given by ∆

-1 = - 1 4π|x|
⋆ , with ⋆ the convolution operator.

The operator P, which acts on vector-valued functions, is a projection : P 2 = P, annihilates gradients and maps into solenoidal (divergence-free) vectors; it is a bounded operator on (vectorvalued) L p , 1 < p < ∞ and commutes with translation. We can notice that the operator P can be written under the form,

P = I -∇∆ -1 div, (4)
which yields to Helmholtz decomposition, indeed for all v ∈ L q (R 3 ) 3 , 1 < q < ∞, v = Pv + ∇q, with div Pv = 0, q = ∆ -1 div v.

(5)

3 Hardy space and BMO space in bounded domain in R 3

We recall the definition and some of the main properties of Hardy spaces H p (R 3 ) introduced in [SW] ( for more details on these spaces, see [FS], see also [START_REF] Stein | Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals[END_REF]).

Definition 3.1 Let 0 < p < ∞ and let Ψ ∈ S(R 3 ) the Schwartz class, satisfying R 3 Ψ(x) dx = 1.
A tempered distribution f belongs to the Hardy space

H p (R 3 ) if, f * (x) = sup t>0 |(Ψ t ⋆ f )(x)| ∈ L p (R 3 ), ( 6
)
where

Ψ t (x) = t -3 Ψ(t -1 x).
The (quasi)-norm of H p (R 3 ) is defined, up to equivalence, by,

f H p (R 3 ) = f * L p (R 3 ) for all f ∈ H p (R 3 ). ( 7
)
In what follows, in Definition 3.1, we choose Ψ ∈ C ∞ 0 (B(0, 1)) such that

R 3 Ψ(x)dx = 1.
Furthermore, an L 1 -function f on R 3 belongs to H 1 (R 3 ) if and only if its Riesz transforms R j f all belong to L 1 (R 3 ) and

f H 1 (R 3 ) ∼ = f L 1 (R 3 ) + 3 j=1 R j f L 1 (R 3 ) (equivalent norms). ( 8
)
Notice that all function f ∈ H 1 (R 3 ) satisfy

R 3 f (x) dx = 0.
In [CKS], two Hardy spaces are defined on domains Ω of R 3 , one which is reasonably speaking the largest, and the other which in a sense is the smallest. The largest, H 1 r (Ω), arises by restricting to Ω arbitrary elements of H 1 (R 3 ). The other, H 1 z (Ω), arises by restricting to Ω elements of H 1 (R 3 ) which are zero outside Ω. Norms on these spaces are defined as follows,

f H 1 r (Ω) = inf F H 1 (R 3 ) the infinimum being taken over all functions F ∈ H 1 (R 3 ) such that F |Ω = f , f H 1 z (Ω) = F H 1 (R 3 )
where F is the zero extension of f to R 3 . From [CHA] (see also [AR], Theorem 1 (b2) combined with Theorem 5 (e)), the dual of H 1 z (Ω) is BM O r (Ω), a space of locally integrable functions with

f BM Or(Ω) = sup Q⊂Ω 1 |Q| Q |f (x) -f Q | dx < ∞, (9)
where

f Q = 1 |Q| Q f (y)dy
, and the supremum is taken over all cubes Q in the domain Ω.

Lemma 3.1 Let Ω ⊂ R 3 a closed bounded domain and f ∈ H 1 (R 3 )∩L ∞ (R 3 ) such that supp f ⊂ Ω, then, we have, f H 1 (R 3 ) ≤ 2|Ω| f L ∞ (R 3 ) + f L 1 (R 3 ) . Proof. The continuous embedding of H 1 (R 3 ) ֒→ L 1 (R 3 ) give us that f ∈ L 1 (R 3 ). Since B(0, ǫ) → {0} as ǫ → 0, then, we can choose 0 < δ < 1 such that, |B(0, δ) + Ω | ≤ 2|Ω|. ( 10
)
From Definition 3.1, we have,

f H 1 (R 3 ) = R 3 f * (x) dx = R 3 sup t>0 |(Ψ t ⋆ f )(x)| dx ≤ I + J, (11)
where,

I = R 3 sup 0<t<δ |(Ψ t ⋆ f )(x)| dx J = R 3 sup t≥δ |(Ψ t ⋆ f )(x)| dx.
Let us estimate the term I. For all t > 0, we have, supp

Ψ t ⋆ f ⊂ supp f + supp Ψ t .
Since for all t > 0, supp Ψ t ⊂ B(0, t), then, for all 0 < t < δ, we have supp Ψ t ⊂ B(0, δ) and therefore supp Ψ t ⋆ f ⊂ Ω + B(0, δ). This implies that supp h ⊂ Ω + B(0, δ), where h is the function defined by,

h(x) = sup 0<t<δ |(Ψ t ⋆ f )(x)|, ∀x ∈ R 3 . ( 12
)
Moreover, for all 0 < t < δ, we have

Ψ t ⋆ f L ∞ ≤ Ψ t L 1 f L ∞ = f L ∞ , which implies h L ∞ ≤ f L ∞ .
Then, we obtain,

I = R 3 h(x) dx = Ω+B(0,δ) h(x) dx ≤ h L ∞ (|Ω + B(0, δ)|) ≤ 2 f L ∞ |Ω|, (13)
where we have used (10).

Let us now estimate the term J. For this, we take N ≥ 1 and we bound the term J by two terms J 1 and J 2 ,

J ≤ J 1 + J 2 , (14)
where,

J 1 = R 3 sup δ≤t≤N |(Ψ t ⋆ f )(x)| dx J 2 = R 3 sup t>N |(Ψ t ⋆ f )(x)| dx. The term J 1 is estimate as follows, since Ψ ∈ C ∞ 0 (B(0, 1)), for all t > 0, we have Ψ t ∈ C ∞ 0 (B(0, t)) and then for all δ ≤ t ≤ N , Ψ t ∈ C ∞ 0 (B(0, N )). Therefore, for all δ ≤ t ≤ N , we have Ψ t ⋆ f ∈ C ∞ 0 (Ω + B(0, N )). Moreover, by recalling Ψ t = t -3 Ψ(t -1 •), for all δ ≤ t ≤ N , for all x ∈ R 3 , we have, ∂Ψ t (x) ∂t = -3t -4 Ψ(t -1 x) -t -5 (x • ∇Ψ)(t -1 x)
and since suppΨ ⊂ B(0, 1), the support of the function

x -→ (x • ∇Ψ)(t -1 x) is B(0, t), then, we deduce for all δ ≤ t ≤ N , ∂Ψ t ∂t L ∞ ≤ 3t -4 Ψ L ∞ + t -4 ∇Ψ L ∞ ≤ 3δ -4 ( Ψ L ∞ + ∇Ψ L ∞ ) = 3δ -4 Ψ W 1,∞ .
Then, we obtain,

sup δ≤t≤N ∂Ψ t ∂t L ∞ ≤ 3δ -4 Ψ W 1,∞ , which implies for all δ ≤ t 0 < t 1 ≤ N , Ψ t 1 -Ψ t 0 L ∞ ≤ sup δ≤t≤N ∂Ψ t ∂t L ∞ |t 1 -t 0 | ≤ 3δ -4 Ψ W 1,∞ |t 1 -t 0 |.
Then, we deduce that for all δ ≤ t < t ′ ≤ N ,

Ψ t ⋆ f -Ψ t ′ ⋆ f L ∞ = (Ψ t -Ψ t ′ ) ⋆ f L ∞ ≤ Ψ t -Ψ t ′ L ∞ f L 1 ≤ 3δ -4 Ψ W 1,∞ f L 1 |t -t ′ |.
Let ε > 0 and let us introduce a subdivision of [δ, N ] defined by a finite non-decreasing sequence of reals (τ i ) 0≤i≤n such that τ 0 = δ, τ n = N and for all 1

≤ i ≤ n, |τ i -τ i-1 | ≤ ε. Since for all δ ≤ t ≤ N , we have Ψ t ⋆ f ∈ C ∞ 0 (Ω + B(0, N ))
, then, we get,

J 1 = Ω+B(0,N ) sup δ≤t≤N |(Ψ t ⋆ f )(x)| dx.
For all t ∈ [δ, N ], there exists i t ∈ [0, n] ∩ N such that |t -τ it | ≤ ε, then we deduce for all t ∈ [δ, N ] and x ∈ Ω + B(0, N ),

|(Ψ t ⋆ f )(x)| ≤ |(Ψ t ⋆ f )(x) -(Ψ τ i t ⋆ f )(x)| + |(Ψ τ i t ⋆ f )(x)| ≤ Ψ t ⋆ f -Ψ τ i t ⋆ f L ∞ + |(Ψ τ i t ⋆ f )(x)| ≤ 3δ -4 Ψ W 1,∞ f L 1 |t -τ it | + |(Ψ τ i t ⋆ f )(x)| ≤ 3δ -4 Ψ W 1,∞ f L 1 ε + sup 0≤i≤n |(Ψ τ i ⋆ f )(x)|.
Therefore for all x ∈ Ω + B(0, N ),

sup δ≤t≤N |(Ψ t ⋆ f )(x)| ≤ 3δ -4 Ψ W 1,∞ f L 1 ε + sup 0≤i≤n |(Ψ τ i ⋆ f )(x)|.
Then, we obtain,

J 1 ≤ 3|Ω + B(0, N )|δ -4 Ψ W 1,∞ f L 1 ε + sup 0≤i≤n Ω+B(0,N ) |(Ψ τ i ⋆ f )(x)| dx
Since, for all 0 ≤ i ≤ n, we have,

Ω+B(0,N ) |(Ψ τ i ⋆ f )(x)| dx ≤ R 3 |(Ψ τ i ⋆ f )(x)| dx = Ψ τ i ⋆ f L 1 ≤ Ψ τ i L 1 f L 1 = f L 1 .
Then, we deduce,

J 1 ≤ 3|Ω + B(0, N )|δ -4 Ψ W 1,∞ f L 1 ε + f L 1 , (15)
which is valid for all ε > 0, then taking the limit in (15) as ε → 0, we get,

J 1 ≤ f L 1 . ( 16
)
It remains to estimate the term J 2 , in fact, we have J 2 → 0 as N → ∞. Indeed, for all t > N , we have,

Ψ t ⋆ f L ∞ ≤ Ψ t L ∞ f L 1 = 1 t 3 Ψ L ∞ f L 1 ≤ 1 N 3 Ψ L ∞ f L 1 .
Then, we infer,

sup t>N Ψ t ⋆ f L ∞ ≤ 1 N 3 Ψ L ∞ f L 1 . ( 17
)
We introduce the sequence of functions (f N ) N >1 defined for all x ∈ R 3 by,

f N (x) = sup t>N |(Ψ t ⋆ f )(x)|.
Then, thanks to (17), we have for all x ∈ R 3 ,

f N (x) → 0 as N → ∞. (18)
Since, for all x ∈ R 3 , 0 ≤ f N (x) ≤ f * (x) and f * ∈ L 1 (R 3 ) due to the fact that f ∈ H 1 (R 3 ), therefore, thanks to Lebesgue's dominated convergence Theorem, we deduce,

R 3 f N (x) dx → 0 as N → ∞, which means, J 2 → 0 as N → ∞. (19)
Then, thanks to (11), ( 13), ( 14), ( 16) and ( 19), we deduce,

f H 1 (R 3 ) ≤ 2|Ω| f L ∞ + f L 1 ,
which conclude the proof.

Littlewood-Paley decomposition : To prove Lemma 3.2, we need to introduce the usual dyadic unity partition of Littlewood-Paley decomposition. To this end, we take an arbitrary radial function Φ in the Schwartz class S(R 3 ) whose Fourier transform Φ is such that,

supp Φ ⊂ {ξ ∈ R 3 , |ξ| ≤ 1} and Φ(ξ) ≥ 1 2 for |ξ| ≤ 5 6 .
We choose also a radial function ϕ in the Schwartz class S(R 3 ) whose Fourier transform φ is such that,

supp φ ⊂ {ξ ∈ R 3 , 1 2 ≤ |ξ| ≤ 2} and φ(ξ) ≥ 1 2 for 3 5 ≤ |ξ| ≤ 5 3 ,
and define ϕ j (x) = 2 3j ϕ(2 j x) so that φj (ξ) = φ(2 -j ξ) for j ∈ Z. We may assume,

∀ξ ∈ R 3 , Φ(ξ) + j≥0 φj (ξ) = 1 and j∈Z φj (ξ) = 1.
For any f ∈ S ′ , we denote by,

∆ -1 f = F -1 ( Φ f ) = Φ ⋆ f, ∆ j f = F -1 ( φj f ) = ϕ j ⋆ f for all j ≥ 0. Lemma 3.2 Let u ∈ PW r,s (R 3 ) 3 with 1 < s < ∞, r > 1 + 3
s , for all x ∈ R 3 and y ∈ R 3 , there exists an absolute constant C > 0 such that,

|u(x) -u(y)| ≤ C ω L ∞ |x -y|,
where ω = ∇ × u is the vorticity of u.

Proof. Thanks to the Sobolev embedding,

W m,s (R 3 ) ֒→ BC(R 3 ) for all m > 3 s , we have u ∈ BC 1 (R 3 ) 3 . By the Biot-Savart law (see Proposition 1.3.1 in [CHE]), we have for all x ∈ R 3 , u(x) = - 1 4π R 3 y |y| 3 × ω(x + y) dy. ( 20
)
Denoting K the kernel being in the Biot-Savart law (20), we re-write (20) as u = K ⋆ ω.

We have also,

∂u ∂x j = R j (R × ω) for j = 1, 2, 3, (21)
where R = (R 1 , R 2 , R 3 ) and R j are the Riesz transform. Thanks to Littlewood-Paley decomposition, we have,

u = ∞ j=-1 ∆ j u. (22)
Thanks to Bernstein inequalities (see Chapter 3 [LEM], see also [MEY]), there exists a constant C > 0 such that for all j ∈ N,

∆ j u L ∞ ≤ C2 -j ∆ j ∇u L ∞ . ( 23
)
Thanks to (21) and using the same arguments as the proof of (iii) in [MIY], pages 439-440, we get that there exists a constant C 2 > 0 such that for all j ∈ N,

∆ j ∇u L ∞ = ∆ j (R ⊗ (R × ω)) L ∞ ≤ C 2 ∆ j ω L ∞ . (24)
Then, using ( 23) and ( 24), we obtain that there exists a constant C 1 > 0 such that for all j ∈ N,

∆ j u L ∞ ≤ C 1 2 -j ∆ j ω L ∞ . ( 25
)
We borrow some arguments used in [VIS] to obtain our proof. Let x ∈ R 3 , y ∈ R 3 and 3 < p < ∞, First case : if |x -y| < 1, then, thanks to ( 22), ( 24) and ( 25), we get that for all N ≥ -1,

|u(x) -u(y)| ≤ N j=-1 |∆ j u(x) -∆ j u(y)| + 2 ∞ j=N +1 ∆ j u L ∞ ≤ N j=-1 ∆ j ∇u L ∞ |x -y| + 2C 1 ∞ j=N +1 2 -j ∆ j ω L ∞ ≤ ∆ -1 ∇u L ∞ |x -y| + C 2 N j=0 ∆ j ω L ∞ |x -y| + 2C 1 ∞ j=N +1 2 -j ∆ j ω L ∞ .
Thanks to Young inequality (see Theorem IV.30 in [BRE]), there exists a real c p > 0 depending only on p, Φ such that,

∆ -1 ∇u L ∞ = Φ ⋆ ∇u L ∞ ≤ c p ∇u L p . ( 26
)
Thanks to Theorem 3.1.1 in [CHE], there exists a real d p > 0 depending only p such that,

∇u L p ≤ d p ω L p . (27)
Using ( 26) and ( 27), there exists a real C p > 0 depending only on p, Φ such that,

∆ -1 ∇u L ∞ ≤ C p ω L p . ( 28
)
Let 3 < q < ∞ and q ′ > 1 such that 1 q ′ + 1 q = 1, Thanks to Young inequality (see Theorem IV.30 in [BRE]), we have for j ≥ 0,

∆ j ω L ∞ = ϕ j ⋆ ω L ∞ ≤ ϕ j L q ′ ω L q = 2 3j q ϕ L q ′ ω L q (29)
Thanks to ( 28) and ( 29), we deduce,

|u(x) -u(y)| ≤ C p ω L p |x -y| + C 2 ϕ L q ′ ω L q |x -y| N j=0 2 3j q + 2C 1 ϕ L q ′ ω L q ∞ j=N +1 2 -j(1-3 q ) = C p ω L p |x -y| + ϕ L q ′ ω L q (C 2 |x -y| 2 3 q (N +1) + 2C 1 2 -(1-3 q )(N +1) )
We choose N + 1 = [-log 2 |x -y|] and since q > 3, we deduce that there exists a constant

C 3 > 0 such that, |u(x) -u(y)| ≤ C p ω L p |x -y| + C 3 ϕ L q ′ ω L q |x -y| 1-3 q . ( 30
)
Then, taking the limit as q → ∞ in (30), we obtain,

|u(x) -u(y)| ≤ C p ω L p |x -y| + C 3 ϕ L 1 ω L ∞ |x -y|. (31)
Second case : if |x -y| ≥ 1, thanks to thanks to ( 22) and ( 25), we have,

|u(x) -u(y)| ≤ |∆ -1 u(x) -∆ -1 u(y)| + 2 ∞ j≥0 ∆ j u L ∞ ≤ ∆ -1 ∇u L ∞ |x -y| + 2C 1 ∞ j≥0 2 -j ∆ j ω L ∞ ≤ ∆ -1 ∇u L ∞ |x -y| + 2C 1 ∞ j≥0 ω L ∞ ϕ j L 1 2 -j ≤ C p ω L p |x -y| + 4C 1 ϕ L 1 ω L ∞ ,
by using ( 28) and the fact that

ϕ j L 1 = ϕ L 1 . Since |x -y| ≥ 1, we infer, |u(x) -u(y)| ≤ C p ω L p |x -y| + 4C 1 ϕ L 1 ω L ∞ |x -y|. (32)
Then, using ( 31) and ( 32), deduce that there exists a real C p > 0 depending only on p, Φ and an constant C 4 > 0 such that for all x ∈ R 3 and y ∈ R 3 , we have,

|u(x) -u(y)| ≤ C p ω L p |x -y| + C 4 ϕ L 1 ω L ∞ |x -y|. (33)
Now, we use the scaling of Inequality (33), for this, we consider the function u λ = u(λ•) with λ > 0 a non-negative real. Since ∇ • u = 0, then we have also ∇ • u λ = 0 and ω λ the vorticity of u λ is given by ω λ = λω(λ•). Let x ∈ R 3 and y ∈ R 3 . We set x λ = x λ and y λ = y λ , then using (33) with u λ instead of u, we obtain,

|u λ (x λ ) -u λ (y λ )| ≤ C p ω λ L p |x λ -y λ | + C 4 ϕ L 1 ω λ L ∞ |x λ -y λ |. (34)
Using the expression of u λ , ω λ , x λ and y λ , from (34), we deduce,

|u(x) -u(y)| ≤ C p λ -3 p ω L p |x -y| + C 4 ϕ L 1 ω L ∞ |x -y|. (35)
Then, taking the limit as λ → ∞ in (35), we deduce,

|u(x) -u(y)| ≤ C 4 ϕ L 1 ω L ∞ |x -y|. ( 36
)
Since also ϕ is fixed, we conclude the proof.

Global regularity of solution of Euler equations

Using P the matrix Leray operator, Euler equations ( 1)-( 2) can be re-written as follows,

∂u ∂t + P(u • ∇)u = 0, (37)
with initial condition,

u(0) = u 0 . ( 38
)
For u solution of ( 37)-( 38), ω = ∇×u, the vorticity of u, formally satisfies the vorticity equation,

∂ω ∂t + (u • ∇)ω -(ω • ∇)u = 0, ( 39
)
with initial conditions,

ω(0) = ω 0 , ( 40
)
where ω 0 = ∇ × u 0 is the vorticity of u 0 .

Let us give now the proof of our Theorem.

Theorem 4.1 Let u 0 ∈ PW r,q (R 3 ) 3 with 1 < q < ∞, r > 2 + 3 q , then there exists an unique strong solution u ∈ C([0, +∞[, PW r,q (R 3 )) 3 ∩ C 1 ([0, +∞[, PW r-1,q (R 3 )) 3 to Euler equations ( 37)-( 38) and moreover, there exists a constant C > 0 such that for all t ≥ 0,

ω(t) L ∞ ≤ ω 0 L ∞ exp(C ω 0 L ∞ t).

Proof.

Since u 0 ∈ PW r,q (R 3 ) 3 and thanks to Theorem I in [START_REF] Kato | Well-posedness of the Euler and Navier-Stokes equations in the Lebesgue spaces L p s (R 2 )[END_REF], Theorem 3.5 and 4.7 in [KP], Theorem 1 in [BB] or the results obtained in [BKM], we deduce that there exists a maximal time of existence strictly positive T * > 0 such that there exists an unique strong solution u ∈ C([0, T * [, PW r,q (R 3 )) 3 to the Euler equations ( 37)-( 38) and if T * < +∞, then we get,

T * 0 ω(t) L ∞ dt = +∞. ( 41
)
Since u ∈ C([0, T * [, PW r,q (R 3 )) 3 , the operator P a bounded operator in L q (R 3 ) and thanks to Lemma X4 in [KP], from Euler equation (37), we deduce that u ∈ C 1 ([0, T * [, PW r-1,q (R 3 )) 3 . From Helmholtz decomposition (5), we retrieve the pressure p from the velocity u with the formula,

p = -∆ -1 div((u • ∇)u).
Let us assume that T * < +∞. We introduce T a non-negative real such that 0 < T < T * . Thanks to Sobolev embedding,

W m,q (R 3 ) ֒→ BC(R 3 ) for all m > 3 q , ( 42
)
and since u ∈ C([0, T ], W r,q (R 3 )) 3 ∩ C 1 ([0, T ], PW r-1,q (R 3 )) 3 with r > 1 + 3 q , we deduce that u ∈ BC 1 ([0, T ] × R 3 ) 3 .
Equation (1) represents the Eulerian description of the flow. The Lagrangian formulation of Euler equations (1) describes the flow in term of a volume preserving diffeomorphism, the time dependent map X : R 3 -→ R 3 , α -→ X(α, t), X(α, 0) = α.

These maps represent marked fluid particle trajectories, α is the label of the particle, which can be seen as the location of the particle at time t = 0. The fact that the particle travel with velocity u is expressed in the system of ordinary differential equations,

∂ ∂t X(α, t) = u(X(α, t), t), X(α, 0) = α. ( 43
)
The Lagrangian formulation of Euler equations and incompressibility condition given by ( 1) are respectively the Newton's second law ∂ 2 t X + ∇p(X, t) = 0 and det(∇ α X) = 1 (see Proposition 1.4 in [MB]), where det(∇ α X) denotes the determinant of ∇ α X the jacobian matrix of X.

The map X defined by Equation ( 43) is well a volume preserving C 1 -diffeomorphism from R 3 on itself, indeed, since u ∈ BC 1 ([0, T ] × R 3 ) 3 , then thanks to Cauchy-Lipschitz Theorem, we deduce that the differential equation ( 43) admits a unique solution X ∈ C 1 (R 3 × [0, T ]) 3 and for all t ∈ [0, T ], the map X(t) is a C 1 -diffeomorphism from R 3 on itself (see Lemma 2.3.2 in [LERN] in particular the end of its proof, see also Theorem 2.10 combined with Theorem 2.17 in [TES]). Thanks to Proposition 1.4 in [MB], we have for all t ∈ [0, T ], the mapping X(t) : α -→ X(α, t) is volume preserving , and det(∇ α X(α, t)) = 1.

Gathering these results, we obtain the desired result, for any t ∈ [0, T ], the mapping

X(t) is a volume preserving C 1 -diffeomorphism from R 3 on itself. ( 45
)
Let us give an estimate about the length of any segment transformed during the time by the application X.

Let e 0 = (1, 1, 1), from (43), we have for all t ∈ [0, T ] and

α ∈ R 3 , ∇ α X(α, t) = e 0 + t 0 ∇u(X(α, τ ), τ )∇ α X(α, τ ) dτ. (46)
Then, we deduce from ( 46), for all t ∈ [0, T ],

|∇ α X(α, t)| ≤ √ 3 + t 0 ∇u(τ ) L ∞ |∇ α X(α, τ )| dτ.
Then, thanks to Gronwall Lemma, we obtain for all t ∈ [0, T ] and

α ∈ R 3 , |∇ α X(α, t)| ≤ √ 3 exp t 0 ∇u(τ ) L ∞ dτ .
Therefore, we get for all t ∈ [0, T ],

∇ α X(t) L ∞ ≤ √ 3 exp t 0 ∇u(τ ) L ∞ dτ . (47)
Let us show now that for all t ∈ [0, T ], the maximum of the magnitude of the vorticity ω(t) is reached for some points of R 3 .

Since, we have ω ∈ C([0, T ], W r-1,q (R 3 )) 3 with r-1 > 1+ 3 q , then we get ω ∈ C([0, T ]; BC 1 (R 3 )) 3 and also for all t ∈ [0, T ], ω(x, t) → 0 as |x| → +∞. For all t ∈ [0, T ], the application x -→ |ω(x, t)| is a continuous function from R 3 to [0, +∞[ and since |ω(x, t)| → 0 as |x| → +∞, therefore we deduce that the function x -→ |ω(x, t)| reaches its maximum value for some x(t) ∈ R 3 . Then, we have for all t ∈ [0, T ],

|ω(x(t), t)| = ω(t) L ∞ (R 3 ) 3 . (48)
For any s ∈]0, T ], due to (45), there exists an unique α(s) ∈ R 3 such that the flow X starting from α(s) (which means X(α(s), 0) = α(s)) pass through the particle x(s) at time s, that means X(α(s), s) = x(s).

It is well known that for 3D Euler flows (see Proposition 1.8 in [MB] or Proposition page 24 in [CM]), we have for all t ∈ [0, T ] and

α ∈ R 3 , ω(X(α, t), t) = ∇ α X(α, t)ω 0 (α). ( 49
)
Let us fix s ∈]0, T ] and x(s) such that (48) holds. Let R a non-negative real such that 0 < R ≤ 1 and let B(α(s), R) be the ball of center α(s) with radius R. For any t ∈ [0, T ], we introduce the subset of R 3 , V R,s (t) = X(B(α(s), R), t), thanks to (44), we have the conservation of volume,

|V R,s (t)| = |B(α(s), R)|. (50)
For all 0 ≤ t ≤ s, by using the change of variables y = X(α, t) with α ∈ B(α(s), R) and since det(∇ α X(α, t)) = 1, we deduce,

V R,s (t) ω(y, t) dy = B(α(s),R) ω(X(α, t), t) dα. (51)
Therefore, thanks to (51), ( 43) and (39) we deduce that for all 0 ≤ t ≤ s,

d dt V R,s (t) ω(y, t) dy = B(α(s),R) d dt ω(X(α, t), t) dα = B(α(s),R) ((ω • ∇)u)(X(α, t), t) dα. (52)
Since (ω • ∇)u = ∇u ω and thanks to (49), we deduce,

((ω • ∇)u)(X(α, t), t) = ∇u (X(α, t), t) ∇ α X(α, t)ω 0 (α) = ∇(u(X(α, t), t)) ω 0 (α). (53)
Then, using (52), ( 53) and an integration by parts, we obtain for all 0 ≤ t ≤ s,

d dt V R,s (t) ω(y, t) dy = B(α(s),R) ∇(u(X(α, t), t)) ω 0 (α) dα = - B(α(s),R) u(X(α, t), t)) ∇ • ω 0 (α) dα + ∂B(α(s),R) u(X(α, t), t) ω 0 (α) • n(α) dγ(α), (54)
where n(α) is the unit normal outward to B(α(s), R) in α ∈ ∂B(α(s), R).

We can notice that the unit normal outward to any ball of center α(s) in a point α on its boundary, can be expressed as follows,

n(α) = α -α(s) |α -α(s)| . ( 55
)
For α = α(s), we set, n(α) = (1, 0, 0). ( 56)

In what follows, n is considered as a function from R 3 to R 3 defined by ( 55) and ( 56).

Since ∇ • ω 0 = 0, from (54), we obtain for all 0 ≤ t ≤ s,

d dt V R,s (t) ω(y, t) dy = ∂B(α(s),R) u(X(α, t), t) ω 0 (α) • n(α) dγ(α). (57)
We integrate Equation (57) over t ∈ [0, s] and we get,

V R,s (s) ω(y, s) dy = V R,s (0)
ω(y, 0) dy

+ s 0 ∂B(α(s),R) u(X(α, t), t) ω 0 (α) • n(α) dγ(α) dt.
(58) We notice that V R,s (0) = B(α(s), R) and ω(y, 0) = ω 0 (y) for all y ∈ R 3 , then we obtain, u(X(α, t), t) ω 0 (α) • n(α) dγ(α) dt.

(59) We integrate Inequality (59) over R ∈]0, 1] and we obtain,

1 0 V R,s (s)
ω(y, s) dy dR = 1 0 B(α(s),R) ω 0 (y) dy dR

+ s 0 1 0 ∂B(α(s),R) u(X(α, t), t) ω 0 (α) • n(α) dγ(α) dR dt. (60)
For all t ∈ [0, s], we have,

∂B(α(s),R) u(X(α, t), t) ω 0 (α) • n(α) dγ(α) = ∂ ∂R B(α(s),R) u(X(α, t), t) ω 0 (α) • n(α) dα .
(61) Then, we deduce,

1 0 ∂B(α(s),R) u(X(α, t), t) ω 0 (α) • n(α) dγ(α) dR = B(α(s),1)
u(X(α, t), t) ω 0 (α) • n(α) dα.

(62) Owing to (62), from (60), for all t ∈ [0, s], we obtain,

1 0 V R,s (s)
ω(y, s) dy dR =

1 0 B(α(s),R)
ω 0 (y) dy dR

+ s 0 B(α(s),1) u(X(α, t), t) ω 0 (α) • n(α) dγ(α) dt. (63)
Therefore, from (63), we deduce,

1 0 V R,s (s) ω(y, s) dy dR ≤ ω 0 L ∞ 1 0 |B(α(s), R)| dR + s 0 B(α(s),1)
u(X(α, t), t) ω 0 (α) • n(α) dγ(α) dt.

(64)

On one hand, we have,

1 0 V R,s (s)
ω(y, s) dy dR

= 1 0 V R,s (s)
ω(X(α(s), s), s) dy dR

+ 1 0 V R,s (s)
ω(y, s) -ω(X(α(s), s), s) dy dR

≥ 1 0 V R,s (s)
ω(X(α(s), s), s) dy dR -

1 0 V R,s (s)
ω(y, s) -ω(X(α(s), s), s) dy dR

≥ |ω(X(α(s), s), s)| 1 0 |V R,s (s)| dR - 1 0 V R,s (s)
|ω(y, s) -ω(X(α(s), s), s)| dy dR.

(65) On the other hand, we have for all y ∈ V R,s (s),

|ω(X(α(s), s), s) -ω(y, s)| ≤ ∇ω(s) L ∞ diam(V R,s (s))
and also |ω(X(α(s), s), s)| = |ω(x(s), s)| = ω(s) L ∞ , then from (65), we deduce,

1 0 V R,s (s) ω(y, s) dy dR ≥ ω(s) L ∞ 1 0 |V R,s (s)| dR -∇ω(s) L ∞ 1 0 diam(V R,s (s))|V R,s (s)| dR. (66)
From ( 64) and thanks to (66), we deduce,

ω(s) L ∞ 1 0 |V R,s (s)| dR ≤ ω 0 L ∞ 1 0 |B(α(s), R)| dR + ∇ω(s) L ∞ 1 0 diam(V R,s (s))|V R,s (s)| dR + s 0 B(α(s),1)
u(X(α, t), t) ω 0 (α) • n(α) dα dt.

π 3 ω(s) L ∞ ≤ π 3 ω 0 L ∞ + ∇ω(s) L ∞ 1 0 diam(V R,s (s)) 4π 3 R 3 dR + s 0 B(α(s),1)
u(X(α, t), t) ω 0 (α) • n(α) dα dt.

(68)

Let us give now an estimate of diam(V R,s (s)). We recall V R,s (s) = X(B(α(s), R), s), then we can write diam(V R,s (s)) as follows,

diam(V R,s (s)) = sup α,β∈B(α(s),R) |X(α, s) -X(β, s)|. (69)
Thanks to (47), we deduce,

diam(V R,s (s)) ≤ 2 √ 3R exp s 0 ∇u(τ ) L ∞ dτ . (70)
Therefore, we deduce,

1 0 diam(V R,s (s)) 4π 3 R 3 dR ≤ 8π √ 3 exp s 0 ∇u(τ ) L ∞ dτ 1 0 R 4 dR = 8π 5 √ 3 exp s 0 ∇u(τ ) L ∞ dτ . (71)
Thanks to (71), from (68), we obtain,

π 3 ω(s) L ∞ ≤ π 3 ω 0 L ∞ + 8π 5 √ 3 exp s 0 ∇u(τ ) L ∞ dτ ∇ω(s) L ∞ + s 0 B(α(s),1) u(X(α, t), t) ω 0 (α) • n(α) dα dt. (72)
For all t ∈ [0, T ], we introduce the set V s (t) = X(B(α(s), 1), t). Then, thanks to (44), we have,

|V s (t)| = |B(α(s), 1)| = 4π 3 . ( 73
)
We introduce the inverse A(y, t) = X -1 (y, t) called the back to label map which satisfies A(X(α, t), t) = α and X(A(y, t), t) = y. Thanks to (44), for all t ∈ [0, s] , using the change of variable y = X(α, t) with α ∈ B(α(s), 1), we deduce,

Vs(t) u(y, t) ω 0 (A(y, t)) • n(A(y, t)) dy = B(α(s),1) u(X(α, t), t) ω 0 (α) • n(α) dα. (74)
Then, from (72), using (74), we deduce,

π 3 ω(s) L ∞ ≤ π 3 ω 0 L ∞ + 8π 5 √ 3 exp s 0 ∇u(τ ) L ∞ dτ ∇ω(s) L ∞ + s 0 Vs(t) u(y, t) ω 0 (A(y, t)) • n(A(y, t)) dy dt. (75)
Let us prove now, that there exists a constant C > 0 such that for all t ∈ [0, s]

Vs(t) u(y, t) ω 0 (A(y, t)) • n(A(y, t)) dy ≤ C ω(t) L ∞ ω 0 L ∞ .
For all t ∈ [0, s], we introduce the function f t defined from R 3 to R by,

f t (y) = ω 0 (A(y, t)) • n(A(y, t))χ Vs(t) (y) for all y ∈ R 3 . (76)
Notice that,

R 3 f t (y) dy = 0, (77) indeed, R 3 f t (y) dy = Vs(t) ω 0 (A(y, t)) • n(A(y, t)) dy = B(α(s),1) ω 0 (α) • n(α) dα,
by using the change of variable y = X(α, t) with α ∈ B(α(s), 1). Since, we have,

B(α(s),1) ω 0 (α) • n(α) dα = 1 0 ∂ ∂r B(α(s),r) ω 0 (α) • n(α) dα dr = 1 0 ∂B(α(s),r) ω 0 (α) • n(α) dγ(α) dr.
On one hand, for all r > 0, we have,

∂B(α(s),r) ω 0 (α) • n(α) dγ(α) = B(α(s),r) ∇ • ω 0 (α)dα.
On the other, we have ∇ • ω 0 = 0, then, we deduce,

B(α(s),1) ω 0 (α) • n(α) dα = 0,
and therefore, we get R 3 f t (y) dy = 0. From (76), we get f t L ∞ ≤ ω 0 L ∞ , moreover since supp f t ⊂ V s (t) and thanks to (73), we get for all q 0 > 1, f t L q 0 ≤ ω 0 L ∞ ( 4π 3 ) 1 q 0 . Therefore for all q 0 > 1, f t ∈ L q 0 (R 3 ) and R 3 f t (y) dy = 0, then, we deduce that f t ∈ H 1 (R 3 ) (see [START_REF] Stein | Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals[END_REF],

chapter III, §1, 1.2.4). For any t ∈ [0, s], since the space BM O r (V s (t)) is the dual of H 1 z (V s (t)) (see Section 3), then we obtain,

Vs(t) u(y, t) ω 0 (A(y, t)) • n(A(y, t)) dy = Vs(t) u(y, t)f t (y) dy ≤ u(t) BM Or (Vs(t)) f t H 1 z (Vs(t)) = u(t) BM Or (Vs(t)) f t H 1 (R 3 ) . (78)
Due to the equivalence of norms in Hardy space H 1 (R 3 ) and thanks to Lemma 3.1, there exists an absolute constant ̺ > 0 such that,

f t H 1 (R 3 ) ≤ ̺ ( f t L 1 + 2|V s (t)| f t L ∞ ) ≤ 3̺|V s (t)| f t L ∞ . (79)
From ( 76), we obtain,

f t H 1 (R 3 ) ≤ 3̺|V s (t)| ω 0 L ∞ .
Thanks to (73), we deduce,

f t H 1 (R 3 ) ≤ 4π̺ ω 0 L ∞ . (80)
From ( 9), we have, u(t) BM Or (Vs(t)) = sup

Q⊂Vs(t) 1 |Q| Q |u(x, t) -u Q (t)| dx , (81)
where u Q (t) = 1 |Q| Q u(y, t)dy, and the supremum is taken over all cubes Q in the domain V s (t)

whose sides are parallel to the coordinate axes. For all x ∈ Q ⊂ V s (t), using Lemma 3.2, we get,

|u(x, t) -u Q (t)| ≤ 1 |Q| Q |u(x, t) -u(y, t)| dy ≤ ω(t) L ∞ diamQ.
Then, from (81), we deduce,

u(t) BM Or (Vs(t)) ≤ ω(t) L ∞ sup Q⊂Vs(t) diamQ. (82)
Since, for all Q cube of R 3 , the measure of Q is given by,

|Q| = diam(Q) √ 3 3 .
For all Q ⊂ V s (t), cube of R 3 , we have, |Q| ≤ |V s (t)|, and we deduce,

diamQ ≤ √ 3|V s (t)| 1 3 . (83)
Then, using (82), ( 83) and ( 73), we deduce, u(t) BM Or (Vs(t)) ≤ √ 3 4π 3

1 3 ω(t) L ∞ . (84)
Then, using ( 80) and ( 84), from (78), we deduce that there exists an absolute constant C > 0 such that,

Vs(t)
u(y, t) ω 0 (A(y, t)) • n(A(y, t)) dy ≤ C ω(t)

L ∞ ω 0 L ∞ . (85)
Using (85), from (75), we deduce that there exists an absolute constant C 1 > 0 such that,

ω(s) L ∞ ≤ ω 0 L ∞ + C 1 exp s 0 ∇u(τ ) L ∞ dτ ∇ω(s) L ∞ +C 1 s 0 ω(t) L ∞ ω 0 L ∞ dt, (86)
which is valid for all s ∈]0, T ] and therefore valid for all s ∈]0, T * [. Now, we use the scaling property of Euler equations (37) and the vorticity equations (39), in other words, for all λ > 0 and β > 0, the couple of vector-valued functions (u λ,β , ω λ,β ) defined for all (z, σ) on R 3 × 0, T * λβ by, u λ,β (z, σ) = λu(βz, λβσ) ω λ,β (z, σ) = λβω(βz, λβσ)

are respectively solution of Euler equations (37) and vorticity equations (39) for the initial data λu 0 (β •) and λβω(β •).

Then, Inequality (86) holds for the couple (u λ,β , ω λ,β ) and we get for all σ ∈ 0, T * λβ ,

ω λ,β (σ) L ∞ ≤ ω λ,β 0 L ∞ + C 1 ∇ω λ,β (σ) L ∞ exp σ 0 ∇u λ,β (θ) L ∞ dθ +C 1 ω λ,β 0 L ∞ σ 0 ω λ,β (θ) L ∞ dθ. (88)
For all s ∈ [0, T * [, setting σ = s λβ in (88) and using (87), we deduce,

ω(s) L ∞ ≤ ω 0 L ∞ + C 1 β ∇ω(s) L ∞ exp s 0 ∇u(τ ) L ∞ dτ +C 1 ω 0 L ∞ s 0 ω(t) L ∞ dt, (89)
then taking the limit as β → 0 in (89), we obtain for all s ∈ [0, T * [,

ω(s) L ∞ ≤ ω 0 L ∞ + C 1 ω 0 L ∞ s 0 ω(t) L ∞ dt.
Then, thanks to Gronwall Lemma, we deduce for all s ∈ [0, T * [,

ω(s) L ∞ ≤ ω 0 L ∞ exp(C 1 ω 0 L ∞ s). (90)
Then, we have

T * 0 ω(s) L ∞ ds ≤ 1 C 1 exp(C 1 ω 0 L ∞ T * )
< +∞, which leads to a contradiction with (41), therefore T * = +∞ and Inequality (90) holds for all s ≥ 0, which conclude the proof.

Remark 4.1 Theorem 4.1 have been established for all 1 < q < ∞ and r > 2 + 3 q , in fact, the result holds also for any r > 1 + 3 q , indeed since C ∞ 0 (R 3 ) is dense in W r,q (R 3 ) and thanks to Theorem 3.6 in [KP], then Theorem 4.1 follows for any r > 1 + 3 q .

  Lebesgue space L p , normed by the corresponding Lebesgue norm. The operator (I -∆) α 2 = G -α , which for α > 0 is most easily defined by means of the Fourier transform. It corresponds, in fact, to multiplication of the Fourier transform f of by (1 + |ξ|) α 2 .


Prof. Dr Gert Hein
email: gert.hein@med.uni-jena.de

Thorsten Eidner

Peter Oelzner

Michael Rose

Alexander Wilke

Gunter Wolf

Sybille Franke

Influence of Rituximab on markers of bone remodeling in patients with rheumatoid arthritisa prospective open label pilot study

Keywords: rituximab, rheumatoid arthritis, bone remodeling, inflammation, B-cells

Immune system and bone are interacting in a complex way. Rheumatoid arthritis is characterized not only by joint destruction, but also by development of systemic osteopenia and osteoporosis. The CD20-depleting antibody Rituximab (Rtx) is a novel therapeutic option able significantly to slow the destructive joint process of rheumatoid arthritis. However, there are little data whether Rtx influences systemic bone remodeling. In the present prospective study we evaluated the influence of Rtx on markers of bone metabolism with a follow up of 3-15 months after Rtx therapy (2 dose of each 1000 mg) in 13 patients with rheumatoid arthritis. There was no significant change of the bone formation markers bone alkaline phosphatase and c-terminal propeptide of collagen I. However, a non-significant tendency of decrease of RANKL (with no chance of osteoprotegerin) and a significant decrease of the bone degradation marker desoxypyridinolin crosslinked collagen I was observed 15 months after Rtx application. These initial results provide no evidence of a negative systemic influence of Rtx on bone remodeling. In contrast, it appears that Rtx lowered osteoclast activity often found increased in active rheumatoid arthritis contributing to osteoporosis in this disease.

Introduction

Rheumatoid arthritis (RA) is a chronic inflammatory disease driven by autoimmune processes that consequently results in progressive degradation and destruction of the affected joint cartilage and juxtaarticular bone [START_REF] Lee | Rheumatoid Arthritis[END_REF]. A common finding of RA is a systemic osteopenia/osteoporosis caused by multiple factors including chronic inflammation [START_REF] Gough | Generalized bone loss in patients with early rheumatoid arthritis[END_REF][START_REF] Solomon | The relatioship between focal erosions and generalized bone loss in patients with early rheumatoid arthritis[END_REF][START_REF] Haugeberg | Bone loss in patients with rheumatoid arthritis: results from a population-based cohort of 366 patients followed up for two years[END_REF].The interaction between various inflammatory cells and bone is mediated by cytokines, chemokines but also by direct cell-cell-contact [START_REF] Caetano-Lopes | Osteoimmunologythe hidden immune regulation of bone[END_REF]. Two cell types are critically involved in bone remodeling, the bone matrix generating osteoblasts and the bone resorbing osteoclasts. The development and function of osteoblasts and osteoclasts is fundamentally influenced by inflammatory and immunological processes [START_REF] Nakashima | The dynamic interplay between osteoclasts and the immune system[END_REF][START_REF] Horowitz | Immunologic regulation of bone development[END_REF][START_REF] Takaynagi | Osteoimmunology: shared mechanisms and crosstalk between the immune and bone systems[END_REF].

In recent years were were introduced new principles into RA therapy in with more specific mode of action concerning the influence on the autoimmune processes, the so called biologicals. TNF-blocking substances have been shown to reduce the progressive joint destruction of RA. Furthermore, there is increasing evidence that TNF-blockers are also able to stop the systemic bone loss, and may even improve the bone mineral density and systemic bone mass [START_REF] Sambrook | Tumour necrosis factor blockade and the risk of osteoporosis: back to the future[END_REF][START_REF] Lange | Increase of bone mineral density of patients with rheumatoid arthritis treated with anti-TNF alpha antibody. A prospective open -label pilot study[END_REF][START_REF] Wijbrandts | Bone mineral density in rheumatoid arthritis patients 1 year after adalimumab therapy arrest of bone loss[END_REF]. Rituximab (Rtx) is a chimeric B cell depleting anti-CD20-antibody, inducing depletion of different B-cell populations with the exception of the pro-B cells and plasma cells. This novel therapy can substantially improve signs and symptoms as well as physical function and inhibit radiological progression in RA [START_REF] Bagust | Rituximab for the tratment of rheumatoid arthritis[END_REF][START_REF] Keystone | Rituximab inhibits structural joint damage in patients with rheumatoid arthritis with an inadaequate response to tumour necrosis factor inhibitor[END_REF]. However, a potential influence of Rtx on systemic bone remodeling in RA has not been so far investigated. The crosstalk between the immune system (including the B cells) and the cells participating in bone remodeling (osteoblasts and osteoclasts) is highly complex and few experimental data indicate even possible unfavorable effects of B cell depletion on bone remodeling [START_REF] Li | B cells and T cells are critical for the preservation of bone homeostasis and attainment of peak bone mass in vivo[END_REF][START_REF] Hayer | B cell infiltrats induce endostal bone formation in inflammatory arthritis[END_REF][START_REF] Boyce | Bruton and Tec: New links in osteoimmunology[END_REF]. Therefore, the aim of our study was to evaluate the influence of Rtx therapy on markers of bone remodeling in patients with RA in a pilot study.

Patients and methods

The influence of Rtx on markers of bone markers was studied in 13 patients suffering from active RA (9 female, 4 male, mean age 54 years (32-72 years) before the first Rtx infusion, and 3, 6, 9 and 12 months after Rtx application. Rtx was given in two doses of 1000mg with an interval of 14 days. Before treatment with Rtx, all patients failed to respond to therapy with TNF-blocking substances. The study was approved by the Ethic Committee of the university.

Blood samples were taken between in the morning after overnight fasting. Serum was immediately separated, transferred into aliquots, frozen, and stored at -70° C until measurement of parameters. All parameters were determined by ELISA using the following commercial kits: Type I C-terminal collagen propeptide (CICP) and tartrateresistant acid phosphatase isoform 5b (TRAcP 5b) enzyme immunoassay were from Quidel Corporation (San Diego, CA, USA), Osteoprotegerin and total sRANKL ELISA kits were obtained from Immundiagnostik AG (Bensheim, Germany). The ELSIA for desoxypyridinolin crosslinked collagen I was from Wampole Laboratories (Princeton, NJ, USA). Bone specific alkaline phosphatase (ostase) was measured with the LIAISON BAP OSTASE Assay (DiaSorin Inc., Stillwater, MN, USA).

Statistical analysis

All data are presented as the means + standard deviation (SD). Statistical significance between different groups was first tested with the non-parametric Kruskal-Wallis test.

Individual groups were subsequently tested using the Mann-Whitney U-test.

Results

Rtx therapy had no significant effect on the measured markers of bone formation (bone specific alkaline phosphatase (ostase) and the c-terminal propeptide of collagen I (Figure 1 A+B) up to 15 months after RTX application. In addition, RTX therapy had no significant effect on the bone reabsorption parameter tartrate resistant acid phosphatase but significantly reduced serum desoxypyridinolin crosslinked collagen I concentrations after 15 months (Figure 2 A+B). Serum RANKL and osteoprotegerin concentrations were not significantly influenced after Rtx therapy (Figure 3 A+B).

Discussion

The interaction between the immune system and the bone is characterized by intensive crosstalk. Activated T-cells are able to produce RANKL, the down stream activator molecule of osteoclasts [START_REF] Kanamaru | Expression of membrane -bound and soluble receptor activator of NF kappa B ligand (RANKL) in human T cells[END_REF][START_REF] Nakashima | Osteoclast and the immune system[END_REF] and different immunocompetent cells also produce osteoprotegerin, the decoy receptor of RANKL [START_REF] Li | B cells and T cells are critical for the preservation of bone homeostasis and attainment of peak bone mass in vivo[END_REF]. In addition, immune cells and their mediators influence the expression of SOST and the release of sclerostin, thereby influencing Wnt-signalling with effects on generation and function of osteoblasts [START_REF] Ott | Sclerostin and Wnt Signaling -The pathway to bone strength[END_REF][START_REF] Veverka | Characterization of the structural features and interactions of sclerostin: molecular insight into a key regulator of Wnt-mediated bone formation[END_REF]. Different immunomodulating therapies such as methotrexate, azathioprine, cyclosporine, leflunomide but also biologicals (TNF-blocking substances, CTLA4antagonists or IL6 antagonists) inhibit juxtaarticulare bone resorption and joint destruction in RA. Especially for TNF-blocking substances an increase of systemic bone density and bone mass has been previously described [START_REF] Sambrook | Tumour necrosis factor blockade and the risk of osteoporosis: back to the future[END_REF][START_REF] Lange | Increase of bone mineral density of patients with rheumatoid arthritis treated with anti-TNF alpha antibody. A prospective open -label pilot study[END_REF][START_REF] Wijbrandts | Bone mineral density in rheumatoid arthritis patients 1 year after adalimumab therapy arrest of bone loss[END_REF]. Furthermore, the first report on improvement in bone turnover after application of the IL-6 receptor inhibitor tocilizumab has been published [START_REF] Garnero | Rapid and sustained improvement of bone and cartilage turnover markers with the anti-interleukin receptor inhibitor tocilizumab plus methotrexate in patients with inadaequate response to methotrexate: Results from a substudy of the multicenter double blind, placebo controlled trial of tocilizumab in inadaequate responders to methotrexate alone[END_REF]. On the other hand, methotrexate or cyclosporine may have in part unfavorable effects on bone density and quality indicating divergent effects on juxtaarticulare and systemic bone [START_REF] Zonnefeld | Methotrexate osteopathy in long-term, low-dose methotrexate treatment for psoriasis and psoriatic arthritis[END_REF][START_REF] Tannirandorn | Drug induced bone loss[END_REF].

The B cell depleting therapy with Rtx for patients with progressive RA revealed good results in the decrease of inflammatory activity and destructive potency (for review see [START_REF] Keystone | Rituximab inhibits structural joint damage in patients with rheumatoid arthritis with an inadaequate response to tumour necrosis factor inhibitor[END_REF]), but potential effects on bone remodeling have not been studied.

Experimental investigations have provided evidence that the same network of transcription factors (e.g. Ebf1 and Pax5) which regulate the B cell development is also important in the regulation of bone cells. For example, Pax5 is needed for the development of B cells [START_REF] Urbanek | Complete block of early B cell differentiation and altered pattering of the posterior midbrain in mice lacking Pax5/ BSAP[END_REF] but the loss of Pax5 resulted in significant decrease in trabecular bone in a mouse model [START_REF] Horowitz | Pax5 deficient mice exhibit early onset osteoporosis with increased osteaclast progenitors[END_REF]. Ebf1 is an essential transcription factor for B cell specification and function but also regulates the genesis of osteoblasts. Bruton´s tyrosine kinase (Btk) is not only a factor essential for B cell development, but also is involved in the regulation of terminal differentiation of osteoclasts together with Tec kinase [START_REF] Shinohara | Tyrosine kinases Btk and Tec regulate osteoclast differentiation by linking RANK and ITAM signals[END_REF]. Using isolated bone marrow cells, Li et al. [START_REF] Li | B cells and T cells are critical for the preservation of bone homeostasis and attainment of peak bone mass in vivo[END_REF] showed that cells of the B lineage in mice were responsible for 64% of total bone marrow osteoprotegerin production and mice depleted of B cells are osteoporotic.

Our study provided no evidence that Rtx therapy negatively influences bone remodeling and had no significant effect on the measured parameter of bone formation.

In contrast, the serum concentrations of the bone reabsorption marker desoxypyridinolin crosslinked collagen I was significantly reduced 15 months after treatment.

However, our study has several limitations. The patient number of this initial study was low and studies with more patients are needed. Furthermore, the measured markers are the result of ongoing or decreased juxtaarticulare bone destruction/ resorption and the systemic bone remodeling. We can also not further separate whether the observed Rtx effects are due to the efficient treatment of RA resulting in a decrease in inflammation or are due to direct effects of the antibody on the bone. However, there are currently no markers that are able to differentiate between local and systemic bone loss. In addition, bone markers alone are not sufficient to describe a decrease, a maintenance or an increase of bone density and bone mass. Imaging methods, at least the measuring of bone mineral density is needed but because of the time limitation of 12 months, a significant change in the bone mass is unlikely to occur.

Conclusion

In contrast to some experimental studies, our results give no support that a depletion of B cells by therapy with Rtx negatively influences bone remodeling.

Legends to figures

B.