Julie Walker

DIFFERENT NARRATIVE STRUCTURES IN CHOPIN'S LAST STYLE: A METHODOLOGY FOR THE NARRATIVE AND THYMIC ANALYSIS

Introduction

Our researches want to understand the stylistic evolution of Chopin in the last years of his life. After making a corpus of 22 works1 , we investigate them with a double analytical level. The first, traditional and the more frequent, is a formal analysis, which reveals the different themes, keys, modulations, phrases, sections and musical parts with the examination of the score (analysis of the "signifier"). The second wants to take into account the expressivity of the work and its semantic level, in other words the "significate2 ". To do this, the narrative level of a musical work of art can be expressed by the concept of topic3 , which is, according to Leonard Ratner, a "reservoir of characteristics figures 4 " coming from the historical musical knowledge. Those historical "references", sorts of expressive unities, are the subjects of the musical discourse and play the role of actors 5 , which take part to the evolving process of the musical piece and highlight new elements important for the work's comprehension (noted NA for "narrative actor" in our analyses).

The interaction and relation between these topics create different links and events, which are not identifiable with only a formal analysis. The thymic states 6 report the moments of tension and detente hearable by the auditors but not visible as such in the score. The notion of "tension" was develop by Claude Zilberberg and Jacques Fontanille (1996,1998) and considered like the combination of two "valences": intensity (the "quantity" of energy which makes the perception more or less lively) and extensity (the "scope", spatial or temporal etc.) into a "tensive scheme". Raphaël Baroni studies the different kinds of presence of the narrative tension in literature between suspense, curiosity and surprise 7 . Concerning the thymic analysis, Greimas and Courtés mostly developed it in term of "axiological analysis" between two main states: euphoria 8 and dysphoria 9 . This kind of analysis is possible in music because of the historical legacy of the use of musical styles or genres into society and culture. The musical themes refer to this knowledge and are forwarded thanks to the topics. The global aim is to represent in a scientific way the evolution and the concatenation of the topics and Other terminologies exist concerning these unities: Ujfalussy (intonations), Hatten (Topic fields), Tarasti (semes), Grabócz (semes, semanteme, classeme, semantic isotopy). 4 Leonard [START_REF] Ratner | Intonation, Charakterbildung und Typengestaltung in Mozarts Werken[END_REF], Classic Music, New York, Editions Schirmer, p. 9 5 This is a concrete and individualized unity which is registered in the « surface discourse » and not in the deep structure like the « actant ». It « takes part to the process ». See A.J. Greimas and J. Courtès (1986), Sémiotique : dictionnaire raisonné de la théorie du langage, Paris, Hachette. In musicology, Robert Hatten, one of the first which talk about "actors" or "agent" into music, differentiates the lyrical and the dramatic modes, in others word, the "enacted" and the "enacting". 6 The word « thymia » comes from Greek « thumos » which means « passion ». It is used in psychology to relate the humors.

Greimas defined it like a notion which « serves to articulate the semanticism directly linked to the perception of the man and is own body » (Sémiotique : dictionnaire raisonné de la théorie du langage, op. cit, p. 396). Denis Bertrand defines it like a « affective basis disposition -determining the relation that a sensible body maintains with his environment: relation positive or negative » (Précis de sémiotique littéraire (2000), Paris, Nathan University, p. 225). 7 Raphaël Baroni (2007), La tension narrative, Paris, Seuil 8 I.e. « optimistic », (positive, pleasure). 9 I.e. « pessimistic » or « dramatic », (negative, un-pleasure). See Louis Herbert (2006), « L'analyse thymique », in Louis Hébert (ed.), Signo, online, Rimouski (Québec), consulted the 4/08/14, http://www.signosemio.com/greimas/analysethymique.asp, and "Thymie", Centre National de Ressources Textuelles et lexicales, online, consulted the 4/09/2014, http://www.cnrtl.fr/definition/thymie including the moments of tension and detente. For each musical work, the thymic evolution, revealing these interactions, can take different shapes and there are different redundant "macro-structures" in Chopin's last style. We want to explain this new methodology in a more theoretical way and then illustrate it with three examples of musical works.

A new methodology: from the narrativity to the detection of the thymic states in the musical works of art

Theory of the methodology

First, the musical piece, independently from its genre, form or size, is analyzed in a formal and classical way. We come to several musical phrases grouped in sections, themselves put together into parts. The keys and the main themes are highlighted and the musical language investigated, particularly the new stylistic elements, which can help us to understand the last style 10 .

Then, the different sections, which have some thematic unity, will be studied by examining their "topic" or musical style: a march (military, funerary, etc.), a dance (waltz, minuet, mazurka, polonaise, oberek, kujawiak, etc.), a virtuoso or brilliant style, a contrapuntal writing, a lamentation (doloroso), a hymn (homophonic), an improvised style, etc. So, their name are linked to their stylistic characteristics and also to their atmosphere and ambiance. Like Eero Tarasti showed it for the Polonaise Fantaisie 11 , it's about to define "the main actors from their "genres" or "topics" (…). The musician-semiotician takes in consideration the dynamic of their actions or the character of their situation 12 ". For example, even if some styles have intrinsically a specific "color" (a doloroso theme induced obligatory a characteristic ambiance of malaise and pain and consequently of dysphoria in the context of the 19th century), it is not the case for all of the topics (per example the "march" or other type of texture like the hymn, the counterpoint, or the accompanied melody, etc.). That's why the description of those unities will be specific to each case and take the "character of their situation". Their name has no importance in fact, the important thing is the relation and the interactions between them. For example, the mazurka can be very lively (Mazurka op. 59 n°3) or melancholic (Mazurka op. 56 n°3) and a polonaise can insist of a very lyrical or military aspect (both themes are present in Polonaise op. 44). So, all these topics become finally specific of the musical universe of Chopin and that can be compared to the papers of other researchers who revealed the different significates for several composers 13 . Once the list of the topics is established for each work, their concatenation will be studied to create the thymic scale. In this purpose, we have to take into account the moments of tension and detente hearable when we are listening to the piece and transmitted by our cultural and social musical legacy (for example, the "funeral march" is something considered as dysphoric in the 19 th century, a "pastoral theme" like euphoric, etc.). We also take into account the other directives of the score (dynamics and other indications from the composer). The elements that mainly determinate the thymic changes are: ruptures and contrasts of texture and musical density (passage from a "lyrical melody" to an "appassionato" style), modulations (major/minor colorations and others), dynamics (theme piano then fortissimo, crescendo etc.) and the character (indications like dolce or agitato for example).

Case Studies

Thanks to this methodology and the realization of the thymic curves, different structures could be highlighted. Indeed, different narrative behaviors were revealed and permit to identify three courses, illustrated by the three following works.

Impromptu op. 36 in F-sharp major

The Impromptu op. 36, composed in 1840, is the oldest work of our corpus. At the formal level, it presents a clearly juxtaposition of different themes, included in an ABA form. The topic of the "pastoral lyricism" (NA1) 10 In the last years, we find different stylistic redundant characteristics thanks to our analyses like contrapuntal writing, use of emancipated genres, complication of the harmonic language, chromaticism or conclusions with secondary themes. This is a main part of study of your thesis. 11 Eero Tarasti (1996), Sémiotique musicale, "La narrativité dans l'OEuvre de Chopin", Presses universitaires de Limoges, p.

195-217

12 Márta Grabócz (2009), Musique narrativité, signification, "Bref apercu sur l'utilisation des concepts de narrativité", Paris, Editions l'Harmattan, p. 40 13 Jozsef Ujfalussy listed the « intonations » in Mozart's works (1961), Robert Hatten reveals the « Topic fields » in Beethoven's sonatas and quartets (1994 and 2004), Márta Grabócz studied the « semes », « classemes/semantemes » and « semantic isotopies » in Liszt's music (1996), Eero Tarasti works about the semes in Stravinski, Sibelius and Wagner's music linked with myths (1978). is important and is declined into three different textures: a lyrical accompanied melody (NA1), a "pastoral hymn" (NA2) and a "pastoral barcarolle" (NA4) because of its changes of density and rhythm. The first is very calm, in 4 th notes and 2 th notes, indicated legatissimo. The first bars are like an introduction and the melody enters after 6 measures, lyrical. The left hand is characterized by an alternation between the note c (second and fourth beat of the measure, like some internal pedal) and others notes of the harmony. The simplicity of the melody and the harmony, the slow harmonic rhythm, the pedal, the rocking accompaniment, the major mode, the quiet dynamics are all characteristics of this passage which permit us, according to Robert Hatten, to notice the topic of pastoral14 . The second texture keeps the same atmosphere and global characteristics. However, the rhythm, totally homorythmic, creates a higher density typical of a hymn (NA2). The third texture develops a ternary rhythm in 8 th notes at the left hand with the return of the pastoral melody of NA1. Gradually, this rhythmic cell will be incorporated in the right hand too, accompanying the melody (graphically marked by the 8 th notes in the score). With the lyrical color and this ternary rhythm, we thought that this could be a topic of barcarolle, Venetian song in 6/8 or 12/8 measures. The other themes refer to the "Victorious march" (NA3), with an obstinate pointed rhythm and a strong atmosphere, and also to the "Virtuoso and brilliant style" (NA5) with an incessant and fast pattern of 32th notes. We reach to the table below, which list all these topics. The thymic states are symbolized by "+" for euphoric and "-" for dysphoric. From a stylistic point of view, the piece presents some topics, which are emancipated genres (march, barcarolle, hymn). The piece also develops a more contrapuntal writing, even if this is reserved for the moment (bars 44-46, with a complex harmonic enchainment and during the topic "virtuoso" and "brilliant style" with the left hand which tries to follow some melodic line). Moreover, the conclusion uses a secondary theme (the NA2 "Pastoral hymn"), which seems at first glance subordinated to the others. With the comparison with our other analyses, we realized that Chopin often chooses to conclude with a pattern or theme previously subsidiary or a hymnic texture more or less developed 15 . With this list, we tried then to "calibrate" the thymic scale showing the relations and interaction between the topics. We also added the general thymic category, which considers the main and global thymic state of the actor. After our analysis, we came to this table: The different topics are not differentiated concerning their thymic states and the different modulations, only real modifications in the musical discourse, won't contrast with the general atmosphere of euphoria. So, the amplitude is not very important (6 unities) and Chopin stays almost always in a euphoric zone.

Fantasia op. 49 in F sharp minor

This piece, much more longer that the previous one, was composed in 1841 16 . It presents a general tripartite form (ABA type) with a lot of different themes and motives, which complicates the musical discourse. Our analysis reach to this list of topics, disposed with their thymic states and the bars in the score. There are three main actors with the topics of the "agitato" style, of the "heroic style" and the "military march". These three topics constitute the principal thematic material of the piece. Between their occurrences, the topic of an "improvised style" related them, but with different roles. After an introduction around the "march" first "funeral" and then more "nocturnal" which present some alternation between major and minor keys, (NA1 in figure 13, indicated Tempo di Marcia by Chopin), the first passage of the "improvised style" (NA2, figure 14) in arpeggios, with fermatas and an important rubato (indicated poco a poco doppio movimento, with also the same type of alternation between major and minor), increases the tension until the first main theme with the topic of "ballade" and indicated agitato (NA3). It is characterized by a complex and dense material in an epic tone, specific to the Ballades of Chopin. After the passage of the topic of the "heroic" style (NA4, figure 11) distinguishable by the pointed rhythm and the victorious color, the "military march" (NA5, figure 12) sounds in a cheerful and bouncy atmosphere with a detached rhythm in 4/4.

After that, the "improvised style 2" (NA6) comes again and brings, like before, the "agitato 2" (NA7). But, this one will be truncated to return to the arpeggios, whose role will be modified by the composer ("Improvised style 3", NA8). Indeed, after they plunged the piece into dysphoria and brought the "agitato" themes (NA2 to NA3 and NA6 to NA7, figure 15), Chopin changes it color to make it peaceful and calm, in a real atmosphere of plenitude (NA8). They reach to the central part represented by a hymnic and homophonic texture of a deep contemplation (NA9 topic of the "hymn", figure 14). After this oneiric parenthesis, the "improvised style 4" (NA10) comes back but in its first role, followed by the entire thematic unit (NA11 "agitato 3", NA12 "heroic style 2" and NA13 "military march 2"). To conclude, Chopin chooses to use the "improvised style 5" (NA14) again, and this can be surprising because this theme seems at first glance secondary (because of its role of "transition") and this remain us the type of conclusion of the Impromptu op. 36. This is a subterfuge established by Chopin and revealed by the narrative analysis (hence its necessity in complementarity with the traditional analysis), because he uses the same material in two different and opposite roles: NA2, NA6 and NA10 bring dysphoria but NA8 and NA14 brings calm and euphoria. At the moment of the conclusion, it is the second role that is used, the one, which brings calm and serenity and becomes a twirling and peaceful theme. So, this motive is a sort of keystone of the whole piece. The second important thing is the topic of the "agitato" style. At each return, the piece is more and more sinking into dysphoria with transposed and varied repetitions. Gradually, it draws a global evolution of tension, which increased more and more. After the last occurrence of the "ballade", the global descending phase begins until the euphoric climax, which conclude the piece. This is a typical romantic form because the "setting abyss" of the beginning (represented by the "gloomy march" referring to the spleen of this period) comes after main adventures (and many passages between the two thymic categories) to the situation of final equilibrium (euphoria), which is, furthermore a "repented" theme from its first dysphoric role.

So, this work has the particularity to finish in its positive climax. The negative climax is represented by the genre of the "agitato" style.

For the general thymic categories, there are much more changing that in the Impromptu op. 36. We can also noticed that the topic of the "improvised style" belongs generally to both thymic categories because of the continuous major/minor colorations, but like we explain it before, before the topic of the "hymn" and in the coda it changes it role to become globally euphoric.

Nocturne op. 55 n°2 in E-flat major

This piece was composed between 1842 and 1844. The progression is totally different from the Fantaisie op. 49. Very little thematic material is used and there are no numerous contrasts in this ABA form. The whole piece stays in a lyrical texture of accompanied melody, with the insertion of a writing sometimes more contrapuntal (question/answer, voices of alto and soprano independent) while the left hand keep a uniformed accompaniment of 8 th notes. Our analyses reach to these topics: So, only two topics are present and decline the lyrical style. The first (NA1) noted "Oniric Nocturne", have a really lyrical character. In this middle section, the thematic material is modified to become more melancholic (minor key, weak dynamics). Moreover, the melody is more static and gives the impression of running in circles in this passage, with a rhythmic ostinato. However, the global atmosphere isn't modified. After a return of the topic of "oniric nocturne A' " (NA3), the end of the piece reveals in a concise way a passage of more homophonic writing suggesting a hymnic texture again and sketching some third topic ("Hymn", NA4). The relation and interactions between these topics, symbolized by the thymic states, are little changing (only two phases and four values). The euphoric state is mainly present.

Identity of the narratives structures in the last style

It is interesting to see which interactions and which links are present between the topics in the musical work, but it is even more interesting to compare them one with each other to see if some identities emerge: does certain narrative structures resemble, and if so, what are their characteristics? These kinds of questions are a central aspect for many narratologists of music. For example, Robert Hatten showed in Musical Meaning in Beethoven 17 and Interpreting Musical Gestures, Topics, and Tropes18 , how the relation between the topics and expressive genres and the logic enchainment of it will serve to express and translate some dramatics itinerary 19 . In the same way, Márta Grabócz talks about the "capacity to define the relations between them (the topics), to determinate their power relations, their "action", the denouement of their plot, and their cathartic issue, if there is any20 ".

All along our analyses, we noticed three omnipresent templates and illustrate it with the three precedent examples. This means that there are three sorts of concatenation of the topics in Chopin's late works.

Non contrasted narrative structure

The thymic curve of this first model has the following characteristics: the thematic material (the topics), doesn't present strong contrasts and consequently, there is one only global thymic state into the whole work . It can be one same topic used and declined (Berceuse op. 57, with a form of variation around the same topic of the "berceuse", into one atmosphere) or different topics, which stay close concerning their thymic power, in other words in their atmosphere, density, texture, etc. The amplitude is weak (generally under 6 unities). Here is the list of the works belonging to this category of structure: the Impromptu op. 36 (amplitude of 6, 8 phases), the Prelude op. 45 (amplitude of 5, 18 phases), the Mazurka op. 56 n°1 (2, 11 phases), the Berceuse op. 57 (3,14), the Barcarolle op. 60 (6, 12) and the largo of the Sonata for piano and cello op. 65 (4, 6).

Complex narrative structure

The second type of narrative structure is the most represented into our corpus. It regroups pieces with a lot of different thematic material and topics, important changes of atmosphere, of ruptures and contrasts (and consequently a lot of phases in the curve) and a strong amplitude between the actors (more or equal to 7). Here is the list of the works concerned: the Polonaise op. 44 (amplitude of 12, 25 phases), the Fantasia op. 49 (amplitude of 16, 31 phases), the Ballade op. 52 (17, 31 phases), the Polonaise op. 53 (12, 15), the Mazurka op. 56 n°3 (7, 9), the Mazurka op. 59 n°1 (7, 21), the Mazurka op. 59 n°3 (8, 11), the Polonaise Fantaisie op. 61 (20,42), the Nocturne op. 62 n°1 (9, 7), the Waltz op. 64 n°3 (8, 12), the allegro (13, 27), scherzo (8, 17) and finale (10, 17) of the Sonata for piano and cello op. 65 and the Mazurka op. 68 n°4 (7, 12).

Simple narrative structure

To finish, the last identity proposes a structure with little changes represented by just a few phases into the curve. The amplitude is weak or moderate and the thematic material not very numerous: the topics are almost 2 or 3 in the whole piece. So, there is less ruptures because of the few topics present, which are more global and include a larger amount of quantity of musical material then before (number of phases under 4). The works concerned are: the Nocturne op. 55 n°2 (amplitude of 3, 2 phases), the Nocturne op. 62 n°2 (amplitude of 6, 4 phases), the Mazurka op. 63 n°3 (5, 4 phases) and the Waltz op. 64 n°2 (4, 3).

Synthesis of the results

To synthetize all this results, the figure 26 proposes a recapitulation of the narrative structure and thymic course of each work. The use of these structures is varied: 58 % are complex, 25% are less contrasted and 17% are simple.

We added a second column, which takes into account the general thymic evolution, from the beginning to the end the piece. Indeed, this is also an important dimension of Chopin's dramaturgy. When we compare these two categories, several elements can be highlighted. For example, the non-contrasted narrative structures always present the same thymic state at the beginning and at the end of the work: mostly the euphoric one (++, 6 times of 7) or dysphoric (--, for the Prelude op. 45). The simple narrative structure presents the same thymic state at the beginning and the end, but it can be from the two sides (++ or --). The complex narrative structure proposes mostly a different state at the beginning and at the end but even so the same state (about 60% and 40 %).

There are different evolutions concerning the correlation between the size and the narrative structures used by Chopin. Till 1844, the complex structures are reserved to big pieces (polonaise, ballade, fantasia etc.) but from the Mazurka op. 56 n°3, these structures will also characterized little forms (Mazurkas op. 59, Waltz op. 64 n°3, Mazurka op. 68 n°4). The "non contrasted structure" is regularly present until 1846 and then discarded, except for the largo of the Sonata for cello and piano op. 65. This identity touches big and little forms as well (Barcarolle and Nocturnes). The "simple structures" are reserved to little forms (nocturne, mazurka, and waltz) and after some try with the Nocturne op. 55 n°2, it was neglected until a concentrate use in the years 1846/47 with the Nocturne op. 62 n°2, Mazurka 63 n°3 and Waltz 64 n°2.

Method's limit

Of course, one problem resists in this methodology: the elaboration of the thymic scale can be criticized and considered as subjective, even more when the material is numerous and changing. However, we are thinking that even if there will be certainly changes into the scale with other people, the number of contrasts and the climax will be very similar, if we are basing on the same musical criteria (changes of texture, density modulations, dynamics, cultural and social musical knowledge in history, etc.). So, even if the "calibrating" could be different, the amplitude (more or less important and representing the biggest difference between two thematic materials) and the number of phases (number of ruptures and contrasts) will globally stay the same. More, we are focusing on the emotion expressed by the composer in his music and language (according to historian and social elements) and not specifically on the emotion felt by the auditor, which is maybe another step with psychoacoustics studies. So, our study takes into account the score and the musical material and not a specific recording, which add in addition a subjective variable. These could be the next point of our analyses, which wants to integrate the performance of the pianist. So, the topics and their thymic category and values could be compared with the empirical values extracted from the performance with an audio descriptor (tempo, dynamic, etc.). In fact, this variable of the pianist will be interesting to see how the schools and generations of artists modify the narrative structure with their musical choices.

We are too remaining that the values are not data themselves; the main thing is the relation and connections with the others into the whole scale. The details about the curve and the numbers chosen are only a way to reach to the chart, symbolizing the topics and their interaction. It is the general behavior of the curve, which determines the narrative structure. It doesn't matter if one theme has this or another value: it is the general trend and evolution of the curve, which is sufficient to reveal the narrative structure. Some types of relations symbolize typical behaviors: closer are the values, closer are the characteristics of the musical material (the topics) and conversely; the balancing values between two opposite numbers reveals an alternation of major/minor colorations for the same melody. So, the chart symbolizes the characteristics of the different topics basing on the score. We are considering the ruptures, modulations, changes of texture and density, dynamics, etc. and the different identities could be highlighted without the chart. If the number of ruptures is very important and the material very different (between a oneiric hymn and at a appassionato theme for example), it is a complex narrative structure; if the material is very similar (between two lyrical themes for example) it is a non-contrasted structure, and if it is analogous in addition of just a few ruptures, this will be a simple structure (only two similar main themes for example).

Conclusion

This paper is about a methodology which wants to be innovative and useful to represent in another way the musical work's narrativity and emotion, according to the moments of tension and detente, created by the relations between the different topics and represented by the thymic categories euphoria and dysphoria. The evolution of the different topics reveals different macro-structures called "narrative structures". The "complex", "simple" and "non-contrasted" structures characterized all the 21 pieces of our corpus, composed between 1841 and his death, linked to the last style. Of course, this methodology will continue to be developed and try to be improved. We can also imagine that this methodology could be used for other composers, periods or musical styles and reveals other types of structures.

Figure 1 ,

 1 Figure 1, Beginning of the topic of the "Pastoral lyrcisim", Impromptu op. 36, bars 1-8

Figure 2

 2 Figure 2 Beginning of the topic "Pastoral hymn" (bar 30), Impromptu op. 36, bars 29-33

Figure 3

 3 Figure 3 a. Beginning the topic of the « Pastoral barcarolle », Impromptu op. 36, bars 61-63 ______ b. Part of the topic of the « Pastoral barcarolle », Impromptu op. 36, bars 76-78

Figure 4 Figure 5

 45 Figure 4 Beginning of the topic of the "Victorious march", Impromptu op. 36, bars 39-43

Figure 6

 6 Figure 6 List of the topics of the Impromptu op. 36

  the Fantaisie op. 49, the Nocturne op. 55 n°2, the Mazurka op. 59 n°1, the Mazurka op. 59 n°3, the Barcarolle op. 60, the Nocturne op. 62 n°1 and the Nocturne op. 62 n°2.

Figure 7

 7 Figure 7 Table of the thymic values of the Impromptu op. 36

Figure 9

 9 Figure 9 List of the topics of the Fantaisie op. 49

Figure 8 Figure 9

 89 Figure 8 Beginning of the topic of the "agitao" style, Fantaisie op. 49, bars 68-70

Figure 11 Figure 12

 1112 Figure 11 Beginning of the topic of the "march", Fantaisie op. 49, bars 1-4

Figure 13

 13 Figure 13 End of the topic « Improvised style 2 » to the « ballade 2 » (to dysphoria), Fantaisie op. 49, bars 150-158

Figure 14 ,

 14 Figure 14, Passage from the topic "improvised style 3", (second role, to euphoria) to the topic of the "Hymn", Fantaisie op. 49, bars 192-200

Figure 15

 15 Figure 15 Last bars of the topic "improvised style 5", conclusion of the piece, Fantaisie op. 49, bars 322-332

Figure 17

 17 Figure 17 List of the topics of the Nocturne op. 55 n°2

Figure 18 Figure 19

 1819 Figure 18 Beginning of the "Lyrical reverie", Nocturne op. 55 n°2, bars 1-3

Figure 20

 20 Figure 20 Topic of the "hymn", Nocturne op. 55 n°2, bars 62-67

Figure 21

 21 Figure 21 Thymic values of the Nocturne op. 55 n°2

Figure 16 Table of the thymic values of the Fantaisie op. 49

		188
		192
	4. Polonaise 1	199	+
	5. Jerky march 1	223	+
	6. Improvised style 2	227
		235	-+
		244
		252
	7. Ballade 2	261	-
		276
		294
	8. Improvised style 3	310 312	-+
		320
	9. Hymn	322	+
	10. Improvised style 4		-+
	11. Ballade 3		-
	12. Polonaise 2		+
	13. Jerky march 2		+
	14. Improvised style 5
	and conclusion		+
	Narrative actors	Bars	General thymic category
	1. Gloomy and nocturnal	1	-
	march	21
		42
		93
		109
		127
	2. Improvised style 1	143 147	-+
		149
		155
		164
		172
	3. Ballade 1	180	-

Based on the studies of M-P, Rambeau, T. Zielinski, J.Kallberg and M. Tomaszewski.

Márta Grabócz evokes the long supremacy of the signifier over the signified, long ignored in musical analysis: The musicologists quoted above noticed that the traditional musicology developed many kind of description of the signifier, while ignoring the signified. In french in the text : Les musicologues cités ci-dessus font remarquer que la musicologie traditionnelle a développé des moyens de description du « signifiant » tout en ignorant le « signifié ». Márta Grabócz (2009), Musique, narrativité, signification, Paris, Editions l'Harmattan, p. 21

This concept was elaborated by L.[START_REF] Ratner | Intonation, Charakterbildung und Typengestaltung in Mozarts Werken[END_REF] and developed byAllanbrook (1983), Agawu (1991) andMonelle (2000).

Robert Hatten (1994), Musical Meaning in Beethoven Markedness, Correlation, and Interpretation, Bloomington, Indiana UP, p. 97-99

This piece has 332 bars for 13 minutes music on average.

Robert S. Hatten (1994), Musical Meaning in Beethoven Markedness, Correlation, and Interpretation, op. cit

Robert S. Hatten (2004a), Interpreting Musical Gestures, Topics, and Tropes, Indina University Press

He also talk about this point in his forthcoming paper : Robert S. Hatten, « Les degrés de narrativité en musique », to appear in Márta Grabócz (ed.), Narratologie musicale: théorie et analyse (Topiques et strategies narratives en musique), Paris, Editions des Archives contemporaines

Márta Grabócz (2009),Musique narrativité, signification, op. cit, p. 18, in french in the text : à définir les relations qui existent entre eux, à déterminer leurs rapports de force, leur « action », le dénouement de leur intrigue, puis leur issue cathartique, s'il y en a.

Bibliography

 BARONI, Raphaël (2007), La tension narrative, Suspense, curiosité et suspense, Paris, Seuil BERTRAND, Denis (2000), Précis de sémiotique littéraire, Paris, Nathan University COURTÉS, Joseph (1991)


Pierre Cantin

Well-posedness of the scalar and the vector advection-reaction problems in Banach graph spaces

come

Well-posedness of the scalar and the vector advection-reaction problems in Banach graph spaces

Introduction.

Let Ω be a domain of R 3 with Lipschitz-continuous boundary ∂Ω and consider u : Ω → R and u : Ω → R 3 solving the following first-order homogeneous boundary valued problems

β•∇u + µu = s a.e. in Ω, (1a)
u = 0 a.e. on ∂Ω -,

and

∇(β•u) + (∇×u)×β + µu = s a.e. in Ω, (2a)
u = 0 a.e. on ∂Ω -.

In this paper, β denotes a Lipschitz-continuous R 3 -valued vector field on Ω, and µ and µ denote two bounded reaction coefficients taking R and R 3×3 values, respectively. The inflow boundary ∂Ω -is defined as ∂Ω -= {x ∈ ∂Ω | β(x)•n(x) < 0}, with n the exterior unit normal of ∂Ω. These problems (or variants) have been studied several times in the literature. We mention in particular the pioneering work of [START_REF] Bardos | Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport[END_REF] and Beirão da [START_REF] Beirão Da Veiga | Boundary-value problems for a class of first order partial differential equations in Sobolev spaces and applications to the Euler flow[END_REF] for the well-posedness analysis in smooth domains with regular model parameters and also the work of [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF] when the problem is expressed in unbounded domains with irregular model parameters. More recently, [START_REF] Girault | L p and W 1,p regularity of the solution of a steady transport equation[END_REF] proved (using a viscous and a Yosida regularization) the well-posedness of these problem in L p (Ω) for all p > 2 under the assumption β ∈ W 1,2 (Ω), and also the W 1,p (Ω)-regularity of solution of (1) if s ∈ W 1,p (Ω) and if β ∈ W 1,∞ (Ω) is sufficiently small.

In this paper, we analyze the well-posedness of problems ( 1) and (2) in Banach graph spaces of power p ∈ (1, ∞). Observing that these problems define two Friedrichs systems, it is well-known (see Ern & Guermond (2006a) or [START_REF] Ern | An intrinsic criterion for the bijectivity of Hilbert operators related to Friedrichs' systems[END_REF]) that the well-posedness is a consequence of the positivity of the R-valued Friedrichs tensor σ β,µ;p := µ -

1 p ∇•β, (3)
for the first problem (1) and of the positivity of the lowest eigenvalue of the R 3×3 -valued Friedrichs tensor

σ β,µ;p := µ + µ T 2 + ∇β + ∇β T 2 - 1 p (∇•β) Id, (4)
for the second problem (2). The first contribution of this work concerns the well-posedness in Banach graph spaces. Following the analysis of Friedrichs system in Hilbert space proposed in the aforementioned works, we establish the well-posedness of these two problems for positive (in the above sense) Friedrichs tensors (3) and ( 4). The second part of this work is devoted to the analysis when these assumptions are not satisfied. Introducing a so-called potential (whose existence follows from the regularity and the trajectory of the vector field in Ω, see [START_REF] Devinatz | The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives[END_REF], we prove that one may extend the Friedrichs positivity assumptions so as to consider null or reasonably negative tensors. This paper is organized as follows. First, some notations are introduced and we recall the classical statement of the Banach-Nečas-Babuška (BNB) theorem. Section 2 is concerned with the scalar problem (1); we prove that this problem is well-posed in the Banach graph space of power p ∈ (1, ∞) if the infimum of the Friedrichs tensor (3) takes positive values and we extend this result to consider null or reasonably negative values. In Section 3, we extend these results to prove the well-posedness of the vector problem (2) under similar assumptions on the Friedrichs tensor (4).

1.1. Notations. In this paper, p denotes any real number in (1, ∞) with p its conjuguate number such that 1 p + 1 p = 1. The inner, the cross and the tensor products in R 3 are denoted by •, × and ⊗ respectively. To alleviate the notation, |•| denotes either the Lebesgue measure of a set, the absolute value of a real number, the Euclidean norm of a vector or the Frobenius norm of a tensor. As usual, the Banach space L p (Ω) collects all measurable functions v : Ω → R whose the absolute value raised to the power p is Lebesgue integrable, i.e., ||v||

L p (Ω) = ( Ω |v| p ) 1 p < ∞.
Similarly, the Banach space L p (Ω) collects all measurable functions v : Ω → R 3 whose the Euclidean norm raised to the power p is Lebesgue integrable, i.e., ||v||

L p (Ω) = ( Ω |v| p ) 1 p < ∞. We denote C ∞ (Ω) (resp. C ∞ (Ω)) the space of infinitely differentiable R-valued functions (resp. R 3 -valued) on Ω and C ∞ c (Ω) (resp. C ∞ c (Ω)
) the subspace of those that are compactly supported in Ω.

1.2. Banach-Nečas-Babuška (BNB) theorem. Consider the following abstract variation- nal problem Find u ∈ U s.t. a(u, v) = f, v V ,V , ∀v ∈ V, (5)
where U and V are two Banach spaces equipped with ||•|| U and ||•|| V , respectively, V is reflexive, a ∈ L(U ×V ; R), f ∈ V and •, • V ,V is the duality pairing between V ≡ L(V ; R) and V . A necessary and sufficient condition for (5) to be well-posed is given by the (BNB) theorem, see e.g., [START_REF] Ern | Theory and practice of finite elements[END_REF].

Theorem 1 (Banach-Nečas-Babuška). The abstract problem (5) is well-posed if and only if:

(BNB1) There exists C bnb > 0 such that

C bnb ||v|| U ≤ sup w∈V \{0} a(v, w) ||w|| V , ∀v ∈ U. (BNB2) For all w ∈ V , (∀v ∈ U, a(v, w) = 0) =⇒ (w = 0).
2. Scalar advection-reaction problem. This section analyzes the well-posedness of the continuous problem (1) in Banach graph spaces and generalized the sign condition on the Friedrichs tensor σ β,µ;p defined by (3).

2.1. The graph space. The Banach graph space of power p associated with (1) is defined by

V β;p (Ω) := {v ∈ L p (Ω) | β•∇v ∈ L p (Ω)} , (6)
and is equipped with the norm ||v||

V β;p (Ω) := (||v|| p L p (Ω) + ||β•∇v|| p L p (Ω) )
1 p for all v ∈ V β;p (Ω). This space defines a reflexive Banach space owing to the first and the second Clarkson inequalities (see [START_REF] Brezis | Analyse fonctionelle; théorie et applications[END_REF]) where for all v ∈ V β;p (Ω), β•∇v ∈ L p (Ω) means that the linear form

C ∞ c (Ω) ϕ → - Ω v ∇•(βϕ), (7)
is bounded in L p (Ω), so that β•∇v is the Riesz representative of (7) in L p (Ω). To specify the meaning of the trace of a function in V β;p (Ω), we introduce the space L p (|β•n| ; ∂Ω) given by

L p (|β•n| ; ∂Ω) := v : ∂Ω → R is Lebesgue measurable on ∂Ω | ∂Ω |β•n| |v| p < ∞ , (8)
which is a Banach space when equipped with the norm

||v|| L p (|β•n|;∂Ω) = ( ∂Ω |β•n| |v| p )
1 p for all v ∈ L p (|β•n| ; ∂Ω). As observed by Ern & Guermond (2006a), the existence of traces in L 2 (|β•n| ; ∂Ω) for function in V β;2 (Ω) is not always guaranteed. A necessary and sufficient condition is the wellseparation of the boundary ∂Ω with respect to the vector field β, i.e., dist(∂Ω -, ∂Ω + ) > 0 with

∂Ω ± = {x ∈ ∂Ω | ± β(x)•n(x) > 0} . (9)
In this following, we always assume that this condition is satisfied. Let us adapt the proof of (Ern & Guermond, 2006a, Lemma 3.1) to the general case p ∈ (1, ∞) to prove the existence of such traces.

Lemma 1 (Trace in L p (|β•n| ; ∂Ω)). The trace map γ : C ∞ (Ω) → L p (|β•n| ; ∂Ω) with γ(ϕ) = ϕ |∂Ω for all ϕ ∈ C ∞ (Ω), extends continuously to V β;p (Ω), i.e., there exists C γ > 0 such that ||γ(v)|| L p (|β•n|;∂Ω) ≤ C γ ||v|| V β;p (Ω) , ∀v ∈ V β;p (Ω).
Proof. Owing to the separation of the boundary from assumption ( 9), there exist

ψ + , ψ -∈ C ∞ (Ω) such that ψ + + ψ -≡ 1 on ∂Ω, ψ ± ≥ 0, ψ + |∂Ω -≡ 0 and ψ - |∂Ω + ≡ 0. Proceeding as in Ern & Guermond (2006a), we infer that ∂Ω |β•n| |ϕ| p = ∂Ω (β•n)|ϕ| p (ψ + -ψ -) = Ω ∇•(β|ϕ| p (ψ + -ψ -)), ∀ϕ ∈ C ∞ (Ω),
where we have used the partition of the unity on the boundary and the Stokes formula. Applying now the Leibniz product rule and recalling that ∇|ϕ| p = pϕ|ϕ| p-2 ∇ϕ, we obtain

∂Ω |β•n| |ϕ| p = p Ω (ψ + -ψ -)(β•∇ϕ)ϕ|ϕ| p-2 + Ω |ϕ| p ∇•(β(ψ + -ψ -)).
Next, Hölder's and Young's inequalities together with the identity

||ϕ|ϕ| p-2 || p L p (Ω) = ||ϕ|| p L p (Ω) yield Ω (β•∇ϕ)ϕ|ϕ| p-2 ≤ ||β•∇ϕ|| L p (Ω) ||ϕ|| p/p L p (Ω) ≤ 1 p ||β•∇ϕ|| p L p (Ω) + 1 p ||ϕ|| p L p (Ω) . It follows that ||ϕ|| L p (|β•n|;∂Ω) ≤ C (||β•∇ϕ|| p L p (Ω) + p||ϕ|| p L p (Ω) ) 1 p with the constant C = 2 1 p (||ψ + - ψ -|| L ∞ (Ω) + ||∇•(β(ψ + -ψ -))|| L ∞ (Ω) ) 1 p . Then, observing that p 1 p ≤ e p-1 p
≤ e, we obtain [START_REF] Jensen | Discontinuous Galerkin methods for Friedrichs Systems with irregular solutions[END_REF], Theorem 2)), this inequality holds as well for any function in V β;p (Ω).

||ϕ|| L p (|β•n|;∂Ω) ≤ C γ ||ϕ|| V β;p (Ω) , ∀ϕ ∈ C ∞ (Ω), with C γ = eC . Finally, recalling that C ∞ (Ω) is dense in V β;p (Ω) for all p ∈ (1, ∞) (see
Owing to the existence of traces in L p (|β•n| ; ∂Ω), the following integration by parts formulae hold.

Lemma 2 (Integration by parts). For all v ∈ V β;p (Ω) and for all w ∈ V β;p (Ω),

Ω (β•∇v) w + Ω (β•∇w) v + Ω (∇•β) vw = ∂Ω (β•n) vw. ( 10a
)
In addition, for all v ∈ V β;p (Ω) and for all z ∈ W 1,∞ (Ω),

Ω (β•∇v) v|v| p-2 z + 1 p Ω (∇•β) |v| p z + 1 p Ω β•∇z |v| p = 1 p ∂Ω (β•n)|v| p z. (10b)
Proof. These formulae follow from the density of C ∞ (Ω) in V β;p (Ω) for all p ∈ (1, ∞). The first one results from the Leibniz product rule while the second one is a consequence of the identity

β•∇(ϕ|ϕ| p-2 z) = ϕ|ϕ| p-2 β•∇z + (p -1)|ϕ| p-2 z β•∇ϕ, (11)
for all ϕ ∈ C ∞ (Ω) and for all z ∈ W 1,∞ (Ω).

Weak formulation.

To examine the well-posedness of (1), we introduce the bilinear form a β,µ;p ∈ L(V 0 β;p (Ω)×L p (Ω); R), where V 0 β;p (Ω) := {w ∈ V β;p (Ω) | w |∂Ω -= 0}, and such that for all v ∈ V 0 β;p (Ω) and all w ∈ L p (Ω),

a β,µ;p (v, w) := Ω (β•∇v) w + Ω µ v w. ( 12
)
Observe that, for all p ∈ (1, ∞), V 0 β;p (Ω) is a closed subspace of V β;p (Ω) owing to Lemma 1. Assuming that s ∈ L p (Ω), the weak formulation of (1) in the graph space V 0 β;p (Ω) is:

Find u ∈ V 0 β;p (Ω) s.t. a β,µ;p (u, v) = Ω s v, ∀v ∈ L p (Ω). ( 13
)
It is readily seen that if u ∈ V 0 β;p (Ω) solves ( 13), the equation (1a) holds in L p (Ω) and the boundary condition (1b) holds in L p (|β•n| ; ∂Ω). Note that the boundary conditions are strongly enforced in (13).

2.3. Well-posedness for positive Friedrichs tensor. To examine the uniqueness of the weak solution u of (13) in the graph space V β;p (Ω), we recall the R-valued Friedrichs tensor

σ β,µ;p := µ - 1 p ∇•β. (14)
Hereafter, we assume that this tensor satisfies the so-called Friedrichs positivity assumption (H p ):

(H p ) ess inf Ω σ β,µ;p > 0. We define the reference time τ = (ess inf Ω σ β,µ;p ) -1 .

Proposition 1 (Uniqueness under (H p )). Assume that (H p ) holds. Then,

a β,µ;p (v, v|v| p-2 ) ≥ τ -1 ||v|| p L p (Ω) , ∀v ∈ V 0 β;p (Ω). (15)
Proof. Let v ∈ V 0 β;p (Ω). Observing that v|v| p-2 ∈ L p (Ω), the quantity a β,µ;p (v, v|v| p-2 ) is welldefined. Owing to the integration by parts formula (10b) with z ≡ 1 on Ω (so that β•∇z ≡ 0), we infer that

a β,µ;p (v, v|v| p-2 ) = Ω µ - 1 p ∇•β |v| p + 1 p ∂Ω (β•n)|v| p ,
whence, using the definition ( 14) of the Friedrichs tensor σ β,µ;p and the fact that

v |∂Ω -= 0, we obtain a β,µ;p (v, v|v| p-2 ) = Ω σ β,µ;p |v| p + 1 p ∂Ω + (β•n)|v| p .
The desired bound then follows from (H p ) and the definition of ∂Ω + .

In the proof of the well-posedness of (13) under the assumption (H p ), we need to introduce the two following Lipschitz spaces

Lip 0 (Ω) := v : Ω → R | v ∈ Lip(Ω) and v |∂Ω -≡ 0 , (16)
and

Lip c (∂Ω + ) := v : ∂Ω → R | v ∈ Lip(∂Ω) and v is compactly supported on ∂Ω + , (17)
which satisfy the following Proposition.

Proposition 2 (Surjectivity of traces). For all v ∈ Lip c (∂Ω + ), there is w ∈ Lip 0 (Ω) such that w |∂Ω = v.

Proof. In order to stay general, we denote d = 3 the dimension of Ω. Let v ∈ Lip c (∂Ω + ) and denote K its compact support on ∂Ω + . Owing to the Borel-Lebesgue property, we define

{B i } 1≤i≤N the finite family of open sets in R d covering K, i.e., K ⊂ 1≤i≤N (B i ∩ ∂Ω + ) ∂Ω + , (18)
and we denote {θ i } 1≤i≤N the partition of the unity subordinate to this covering, i.e., for all

1 ≤ i ≤ N , we have 0 ≤ θ i ≤ 1, θ i ∈ C ∞ c (B i
) and 1≤i≤N θ i|K ≡ 1. Recalling that the boundary ∂Ω is assumed to be Lipschitz-continuous, we introduce, for all

1 ≤ i ≤ N , the local bi-Lipschitz charts ψ i : Q → B i where Q := {(x , x d ) ∈ R d-1 ×R | |x | < 1 and |x d | < 1}, such that ψ i (Q + ) = B i ∩ Ω with Q + = {(x , x d ) ∈ R d-1 ×R | |x | < 1 and 0 < x d < 1} and ψ i (Q 0 ) = B i ∩ ∂Ω + with Q 0 = {(x , 0) ∈ R d-1 ×R | |x | < 1}. Denoting v i = v |B i ∩∂Ω + , we introduce the function ṽi : Q + → R defined as the extrusion of v i • ψ i in Q + , i.e., for all (x , x d ) ∈ Q + , ṽi (x , x d ) = v i • ψ i (x , 0). Next, mapping back to Ω, we consider w i : Ω → R such that w i (x) = ṽi • ψ -1 i (x) for all x ∈ B i ∩ Ω and w i (x) = 0 for all x ∈ Ω\B i ∩ Ω.
Finally, collecting these functions {w i } 1≤i≤N , we observe that the function w = 1≤i≤N θ i w i satisfies the desired conditions.

Theorem 2 (Well-posedness under (H p )). Assume that (H p ) holds. Then the problem (13) is well-posed.

Proof. We apply Theorem 1. Adapting the proof of [START_REF] Ern | Theory and practice of finite elements[END_REF], Theorem 5.7), the condition (BNB1) follows from Proposition 1 with the constant

C bnb = (τ p + (1 + ||µ|| L ∞ (Ω) τ ) p ) 1 p .
Let us prove the second condition (BNB2). Consider w ∈ L p (Ω) such that a β,µ;p (v, w) = 0 for all v ∈ V 0 β;p (Ω). Owing to the inclusion C ∞ c (Ω) ⊂ V 0 β;p (Ω), it follows that µw -∇•(βw) = 0 a.e. in Ω, so that the dense inclusion C ∞ c (Ω) ⊂ L p (Ω) implies that β•∇w = (µ -∇•β)w ∈ L p (Ω), whence w ∈ V β;p (Ω). Applying now the integration by part formula (10a), we observe that

∂Ω (β•n) vw = a β,µ;p (v, w) - Ω (µ -∇•β) vw + Ω (β•∇w) v = 0, for all v ∈ V 0 β;p (Ω).
In particular, observing that Lip 0 (Ω) ⊂ V 0 β;p (Ω) for all p ∈ (1, ∞), and owing to Proposition 2, we have

∂Ω + (β•n) vw = 0, ∀v ∈ Lip c (∂Ω + ).
Owing to the vanishing integral property (see [START_REF] Brezis | Analyse fonctionelle; théorie et applications[END_REF], Lemma IV.2)), this identity yields w |∂Ω + = 0. Now, testing the identity µw -∇•(βw) = 0 by an arbitrary y ∈ L p (Ω) and using the chain rule ∇

•(βw) = β•∇w + (∇•β)w, we infer that 0 = Ω (µw -∇•(βw)) y = Ω (µ -∇•β)wy - Ω β•∇w y.
Hence, the particular choice y = w |w| p -2 along with the identity (10b) with p replaced by p and with z ≡ 1 yields

0 = Ω (µ -∇•β) |w| p + 1 p Ω (∇•β) |w| p - 1 p ∂Ω (β•n) |w| p = Ω σ β,µ;p |w| p - 1 p ∂Ω (β•n) |w| p ≥ τ -1 ||w|| p L p (Ω)
, where we have used that w |∂Ω + = 0 and the assumption (H p ). As a result, w = 0 a.e. in Ω, so that the condition (BNB2) is satisfied. Owing to Theorem 1, there exists a unique solution solving the problem (13).

2.4. Well-posedness for non-positive Friedrichs tensor. Summarizing the results obtained so far, we have proved under assumption (H p ) the well-posed of (1) in the graph space V 0 β;p (Ω). This section aims to extend this result under the new assumption (H p ) so as to include the situation where the infimum of the Friedrichs tensor σ β,µ;p takes null or slightly negative values.

(H p ) ess inf Ω σ β,µ;p ≤ 0 and there exists a non-dimensional function ζ ∈ Lip(Ω) such that

ess inf Ω e ζ σ β,µ;p - 1 p β•∇ζ > 0. ( 19
)
We define the reference time τ = ess inf Ω e ζ σ β,µ;p -1 p β•∇ζ -1

.

Proposition 3 (Uniqueness under (H p )). Assume that (H p ) holds. Then,

a β,µ;p (v, e ζ v|v| p-2 ) ≥ τ -1 ||v|| p L p (Ω) , ∀v ∈ V 0 β;p (Ω). ( 20
) Proof. Let v ∈ V 0 β;p (Ω). Observing that a β,µ;p (v, e ζ v|v| p-2 ) = a β, µ;p (v, v|v| p-2 ),
where we have denoted β = e ζ β and µ = e ζ µ, the identity (20) follows from Proposition 1 if ess inf Ω σ β, µ;p > 0, i.e., if (H p ) holds since we have

σ β, µ;p = e ζ σ β,µ;p - 1 p β•∇ζ .
Remark 1 (Using integration by parts). Instead of using Proposition 1 in the proof of Proposition 3, it is also possible to obtain this result by applying the general integration by part formula (10b) with z = e ζ v|v| p-2 .

Example 1. Assumption (H p ) indeed generalizes the assumption (H p ) since it is now possible to consider situations that cannot be handled under (H p ). For example, considering the rotating field β = (y, -x, z + 1) expressed in the Cartesian coordinates of R 3 and a reaction coefficient µ ∈ R, we have σ β,µ;p = µ -1 p . If µ ≤ 1 p , this Friedrichs tensor does not satisfy (H p ), whereas (H p ) does, for example with the potential ζ(x) = α(1 + z) 2 for all α ∈ R such that 8α < µp -1.

Remark 2 (Existence of ζ). Following [START_REF] Devinatz | The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives[END_REF] and considering a continuously differentiable field β ∈ C 1 (R d ), the existence of the potential ζ relies on the assumption that every solution of the Cauchy problem d t x(t) = β(x(t)), x(0) = x 0 ∈ Ω remains in the domain Ω for a finite time only. Observing that the proof in this reference is based on the flow box theorem, the extension to a less regular field (e.g. β ∈ Lip(Ω)) is a priori not obvious.

We are now in a position to state the well-posedness of (13) under assumption (H p ).

Theorem 3 (Well-posedness under (H p )). Assume that (H p ) holds. Then the problem (13) is well-posed.

Proof. We follow the same ideas as in the proof of Theorem 2. The condition (BNB1) is inferred from Proposition 3 with

C bnb = (||e ζ || p L ∞ (Ω) τ p + (1 + ||µ|| L ∞ (Ω) τ ||e ζ || L ∞ (Ω) ) p ) 1 p .
Turning to the second condition (BNB2), we consider w ∈ L p (Ω) such that a 0 β,µ;p (v, w) = 0 for all v ∈ V 0 β;p (Ω). Proceeding as in the proof of Theorem 2, this implies that w belongs to the graph space V β;p (Ω) and that it satisfies µw -∇•(βw) = 0 a.e. in Ω and w |∂Ω + ≡ 0. Let us prove that w ≡ 0 a.e. in Ω. First, we observe that replacing p by p and choosing z = e ζ(1-p ) in (10b) yields

Ω (β•∇w) w |w| p -2 e ζ(1-p ) = 1 p ∂Ω (β•n) |w| p e ζ(1-p ) - 1 p Ω (∇•β) |w| p e ζ(1-p ) - 1 -p p Ω (β•∇ζ) |w| p e ζ(1-p ) .
Hence, testing the identity µw -∇•(βw) = 0 with the function w |w| p -2 e ζ(1-p ) , we infer that

0 = Ω (µ -∇•β) |w| p e ζ(1-p ) - Ω (β•∇w) w |w| p -2 e ζ(1-p ) = Ω (µ -∇•β) |w| p e ζ(1-p ) + 1 p Ω (∇•β) |w| p e ζ(1-p ) - 1 p Ω (β•∇ζ) |w| p e ζ(1-p ) - 1 p ∂Ω (β•n) |w| p e ζ(1-p ) .
Then, collecting these terms and using the fact that w |∂Ω + = 0, we obtain

0 = Ω e ζ(1-p ) σ β,µ;p - 1 p β•∇ζ |w| p - 1 p ∂Ω (β•n) |w| p e ζ(1-p ) ≥ Ω e ζ(1-p ) σ β,µ;p - 1 p β•∇ζ |w| p .
As a result, owing to (H p ) and the fact that w |∂Ω + ≡ 0, it follows that w ≡ 0 a.e. in Ω. Owing to Theorem 1, we conclude that ( 13) is well-posed under assumption (H p ).

3. Vector advection-reaction problem. In this section, we apply similar ideas to analyze the well-posedness of the vector-valued problem (2) in Banach graph spaces where we generalize the assumption of the sign of the Friedrichs tensor σ β,µ;p defined by (4). For the sake of brevity, the proofs are omitted if they are straightforwardly adapted from those of the scalar case in Section 2.

3.1. The graph space. Let us introduce the graph space

V β;p (Ω) := {v ∈ L p (Ω) | (β•∇)v ∈ L p (Ω)} , (21)
where the i-th component in the Cartesian basis of (β•∇)v is given by β j ∂ j v i (where repeated indices are summed) and where (β•∇)v ∈ L p (Ω) means that the linear form

C ∞ c (Ω) ϕ → - Ω ∇•(β⊗ϕ)•v, (22)
is bounded in L p (Ω), so that (β•∇)v is the Riesz representative of (22) in L p (Ω). Equipped with the norm ||v|| V β;p (Ω) := (||v|| p L p (Ω) + ||(β•∇)v|| p L p (Ω) )
1 p for all v ∈ V β;p (Ω), this space defines a reflexive Banach space. The following proposition states that the problem (2) is well-defined in the graph space V β;p (Ω).

Proposition 4 (Equivalent definition of V β;p (Ω)).

V β;p (Ω) = {v ∈ L p (Ω) | ∇(β•v) + (∇×v)×β ∈ L p (Ω)} , where ∇(β•v) + (∇×v)×β ∈ L p (Ω) means that the linear form C ∞ c (Ω) ϕ → - Ω (β ∇•ϕ + ∇×(ϕ×β)) •v, (23)
is bounded in L p (Ω).

Proof. Let v ∈ V β;p (Ω). By definition, we have

Ω (β•∇)v•ϕ = - Ω ∇•(β⊗ϕ)•v, ∀ϕ ∈ C ∞ c (Ω). Now, recalling the identity ∇•(β⊗ϕ) = β ∇•ϕ + ∇×(ϕ×β) + (ϕ•∇)β for all β ∈ Lip(Ω) and for all ϕ ∈ C ∞ c (Ω), it follows that Ω (β•∇)v•ϕ = - Ω (β ∇•ϕ + ∇×(ϕ×β)) •v - Ω ((ϕ•∇)β) •v, ∀ϕ ∈ C ∞ c (Ω).
Hence, observing that

((ϕ•∇)β)•v = ((∇β)ϕ)•v, we obtain Ω ((β•∇)v + (∇β) T v) •ϕ = - Ω (β ∇•ϕ + ∇×(ϕ×β)) •v, ∀ϕ ∈ C ∞ c (Ω).
Hence, the linear form ( 23) is bounded, yielding ∇(β•v) + (∇×v)×β ∈ L p (Ω), so that the inclusion holds. Note that we have the identity

(β•∇)v + (∇β) T v = ∇(β•v) + (∇×v)×β a.e. in Ω.
Since the proof of the converse inclusion is similar, the proof is completed.

Recalling now that the boundary ∂Ω is well-separated in the sense of ( 9), functions in the graph space V β;p (Ω) have a trace in the space Proposition 5 (Integration by parts). For all v ∈ V β;p (Ω) and for all w ∈ V β;p (Ω),

L p (|β•n| ; ∂Ω) := v : ∂Ω → R 3 is Lebesgue measurable | ∂Ω |β•n| |v| p < ∞ . (24)
Ω w•(β•∇)v + Ω v•(β•∇)w + Ω (∇•β)v•w = ∂Ω (β•n) v•w. (25a)
In addition, for all v ∈ V β;p (Ω) and for all z ∈ W 1,∞ (Ω),

Ω |v| p-2 z v•(β•∇)v + 1 p Ω (∇•β)|v| p z + 1 p Ω β•∇z |v| p = 1 p ∂Ω (β•n)|v| p . ( 25b
)
3.2. Weak formulation. Similarly to Section 2, we introduce the bilinear form a β,µ;p ∈ L(V 0

β;p (Ω)×L p (Ω); R) with V 0 β;p (Ω) = {w ∈ V β;p (Ω) | w |∂Ω -= 0} such that for all v ∈ V β;p (Ω) and for all w ∈ L p (Ω), a β,µ;p (v, w) := Ω (∇(β•v) + (∇×v)×β) •w + Ω µv•w, (26)
with µ : Ω → R 3×3 the reaction tensor. Following the proof of Proposition 4, we observe that the bilinear form (26) can be reformulated as

a β,µ;p (v, w) = Ω w•(β•∇)v + Ω w•(∇β T + µ)v, (27)
and the writing µ = ∇β T + µ yields µ ∈ L ∞ (Ω) and

a β,µ;p (v, w) := Ω (β•∇)v•w + Ω µ v•w. (28)
Assuming now that s ∈ L p (Ω), the weak formulation of (2) in the graph space V 0 β;p (Ω) is:

Find u ∈ V 0 β;p (Ω) s.t. a β,µ;p (u, v) = Ω s•v, ∀v ∈ L p (Ω). (29)
We readily see that if u ∈ V 0 β;p (Ω) solves ( 29), the problem (2a) holds in L p (Ω) and the boundary condition (2b) holds in L p (|β•n| ; ∂Ω).

:= µ + µ T 2 + ∇β + ∇β T 2 - 1 p (∇•β) Id. (30)
For all x ∈ Ω, this lowest eigenvalue is denoted by ℵ p (x) and is defined as

ℵ p (x) = min (σ β,µ;p (x)y, y) | y ∈ R 3 s.t. |y| = 1 ,
where (•, •) denotes the classical Euclidean inner product in R 3 . Hereafter, we assume that this eigenvalue satisfies the following assumption.

(H p ) ess inf Ω ℵ p > 0. We define τ = (ess inf Ω ℵ p ) -1 .

Proposition 6 (Uniqueness under (H p )). Assume that (H p ) holds. Then,

a β,µ;p (v, v |v| p-2 ) ≥ τ -1 ||v|| p L p (Ω) , ∀v ∈ V 0 β;p (Ω).
Proof. Let v ∈ V 0 β;p (Ω) and consider a β,µ;p (v, v |v| p-2 ) (since v |v| p-2 ∈ L p (Ω)). Owing to the identity (27), we infer that

a β,µ;p (v, v |v| p-2 ) = Ω |v| p-2 v•(β•∇)v + Ω |v| p-2 v• (∇β T + µ) v.
Using now the integration by parts formula (25b) with z ≡ 1, we obtain

a β,µ;p (v, v |v| p-2 ) = Ω |v| p-2 v•σ β,µ;p •v + 1 p ∂Ω (β•n) |v| p , (31)
whence the result follows using (H p ) and recalling that v |∂Ω -= 0.

To take into account the situation where the smallest eigenvalue ℵ p takes null or slightly negative values in Ω, we consider the new assumption (H p ). Finally, the well-posedness of (2) holds under assumption (H p ) or (H p ). The proof follows the same ideas used to prove Theorems 2 and 3, this time using Propositions 6 and 7, respectively.

Theorem 4 (Well-posedness). Assume that (H p ) or (H p ) holds. Then the problem (29) is well-posed.

Conclusion.

In this paper, we have extended the well-posedness of problems ( 1) and ( 2), not only in Banach graph space of exponent p ∈ (1, ∞), but also under new assumptions regarding the classical Friedrichs tensor, so as to consider the situation when it takes positive, null or slightly negative values. Observing that equations (1a) and (2a) are the proxy of the Lie derivative in R 3 of a 0-and a 1-form respectively, the present analysis could be extended in a future work within the more general framework proposed by [START_REF] Heumann | Eulerian en semi-Lagrangian methods for advection-diffusion of differential forms[END_REF] to treat the advection of a differential k-form within a manifold of R d . However, the question of the existence of the potential ζ in such a more general context is still open. Another extension of this work concerns the case of non-homogeneous Dirichlet boundary condition, requiring to establish the surjectivity of the trace maps in these Banach graph spaces.

  Equipped with the norm ||v|| p L p (|β•n|;∂Ω) := ∂Ω |β•n| |v| p for all v ∈ L p (|β•n| ; ∂Ω), this space defines a Banach space. Lemma 3 (Trace in L p (|β•n| ; ∂Ω)). Let p ∈ (1, ∞). The trace map γ : C ∞ (Ω) → L p (|β•n| ; ∂Ω) with γ(ϕ) = ϕ |∂Ω for all ϕ ∈ C ∞ (Ω) extends continuously to V β;p (Ω), i.e., there exists C γ > 0 such that ||γ(v)|| L p (|β•n|;∂Ω) ≤ C γ ||v|| V β;p (Ω) , ∀v ∈ V β;p (Ω).

  3.3. Well-posedness for positive and non-positive Friedrichs tensor. The uniqueness of the solution of problem (29) relies on the sign of the lowest eigenvalue of the R 3×3 -valued Friedrichs tensor σ β,µ;p

(

  H p ) ess inf Ω ℵ p ≤ 0 and there exists a non-dimensional function ζ ∈ Lip(Ω) such thatess inf Ω e ζ ℵ p -1 p β•∇ζ > 0.We define τ -1 = ess inf Ω e ζ ℵ p -1 p β•∇ζ .Proposition 7 (Uniqueness under (H p )). Assume that (H p ) holds. Then,a β,µ;p (v, e ζ v |v| p-2 ) ≥ τ -1 ||v|| p L p (Ω) , ∀v ∈ V 0 β;p (Ω). Proof. Let v ∈ V 0 β;p (Ω). Denoting β = e ζ β, µ = e ζ µ and observing that ∇( β•v) = e ζ (∇(β•v) + (∇ζ⊗β)v) a.e. in Ω,we infer that a β,µ;p (v, e ζ v |v| p-2 ) = a β, µ;p (v, v |v| p-2 ) -a 0,∇ζ⊗β;p (v, e ζ v |v| p-2 ). Owing to (31), this identity yieldsa β,µ;p (v, e ζ v |v| p-2 ) ≥ Ω |v| p-2 v•σ β, µ;p •v -Ω e ζ |v| p-2 v• ∇ζ⊗β + β⊗∇ζ 2 •v.In addition, observing thatσ β, µ;p = e ζ σ β,µ;p -1 p (β•∇ζ)Id + e ζ ∇ζ⊗β + β⊗∇ζ 2 ,the expected result follows from assumption (H p ).

Acknowledgments. The author warmly thanks Alexandre Ern and Jean-Luc Guermond for their advice and the fruitful discussions.