Manon Lelandais
email: manon.lelandais@univ-nantes.fr

Gaëlle Ferré
email: gaelle.ferre@univ-nantes.fr

Prosodic boundaries in subordinate syntactic constructions

Keywords: subordination, embedding, demarcation, boundaries

Based on a video recording of conversational British English, this paper tests whether several different subordinate syntactic structures are evenly vocally integrated to their environment. "Secondary constructions" have been described in linguistics as dependent, subordinate forms elaborating on primary elements of discourse. Although their verbal and vocal characteristics have been deeply analysed, few studies have provided a qualified picture of their vocal integration. Beyond showing that secondary constructions are not evenly dependent on their environment, the results suggest that prosody demarcates secondary constructions more than it integrates them. The creation of a break preferentially takes place retrospectively, immediately after the subordinate structure through rhythmic features and/or pitch upsteps.

Introduction

This article deals with subordination in spontaneous speech, more specifically on the sequences containing "secondary constructions" in English, operating at the syntactic level of modification.

In syntactic and discourse studies, "secondary material" refers to elements modifying or specifying some primary features, often described as additions associated to another propositional content in the host or embedding structure [START_REF] Huddleston | The Cambridge Grammar of the English Language[END_REF]. This paper focuses on adverbial clauses, appositive clauses, and restrictive relative clauses, as illustrated in examples [START_REF] Huddleston | The Cambridge Grammar of the English Language[END_REF][START_REF] Lehmann | Towards a Typology of Clause Linkage[END_REF][START_REF] Tao | Understanding Non-restrictive Which-clauses in Spoken English, Which is Not an Easy Thing[END_REF]. In [START_REF] Huddleston | The Cambridge Grammar of the English Language[END_REF], the adverbial clause specifies the circumstances in which I tried driving once in her car is valid as an utterance. The adverbial clause restricts the spatial scope in which the referential elements must be understood.

(1) Adverbial clause (see appendix at the end of the paper for transcription conventions) Rhianna L i tried driving once in her car SC when we were on a # little road in the countryside # R and hem (swallows) she said turn left #

In [START_REF] Lehmann | Towards a Typology of Clause Linkage[END_REF], the appositive relative clause qualitatively evaluates a place called Tropicana, which can however be identified independently.

(2) Appositive relative clause Beth L and then we went into # a place called Tropicana # SC which was horrible (laughs) # R it's on Saint Mary's street near the castle Lastly, in [START_REF] Tao | Understanding Non-restrictive Which-clauses in Spoken English, Which is Not an Easy Thing[END_REF], the restrictive relative clause increases the relevance of the Spanish girls, creating a subcategory for this referent.

(3) Restrictive relative clause Joey L the Spanish girls SC that were there # R on our second one Secondary constructions are generally defined as dependent on another predication [START_REF] Lehmann | Towards a Typology of Clause Linkage[END_REF]. However, the literature shows little consensus in defining clear scopes and boundaries for these forms [START_REF] Tao | Understanding Non-restrictive Which-clauses in Spoken English, Which is Not an Easy Thing[END_REF], [START_REF] Couper-Kuhlen | Intonation and Grammar[END_REF]. This study therefore questions whether they all express the same degree of dependence upon their cotext.

We investigate the production process of secondary constructions in English, focusing on demarcation at the segmental and suprasegmental levels. The main hypothesis, arising from the consensus in the previous results, is based on the capacity of these constructions to show distinct forms of autonomy in function of their syntactic type. Different degrees of prosodic demarcation are consequently identified from this new perspective, providing a qualified picture of their insertion in discourse.

Theoretical background

Syntactic subordination

In the traditional categorial division of clause complexes into two uneven and complementary syntactic subgroups, i.e. a main clause and a subordinate, modifiers are viewed as optional constituents functioning at a phrasal or clausal level [START_REF] Halliday | An Introduction to Functional Grammar[END_REF], arising from the concept of minimal utterances: some elements of the message are deemed semantically useful without standing as constitutive elements. However, this acceptation has been reproved by a number of linguists (e.g. [START_REF] Haiman | Subordination' in Universal Grammar[END_REF], [START_REF] Chafe | Linking Intonation Units in Spoken English[END_REF]), described as imprecise for analysing spontaneous speech, especially regarding the nature of introductory elements.

Prosodic subordination

Prosodic subordination is essentially achieved through intonation [START_REF] Bolinger | Intonational Signals of Subordination[END_REF]. Throughout a vocal paragraph, pitch height naturally declines in a progressive manner [START_REF] Wennerstrom | The Music of Everyday Speech. Prosody and Discourse Analysis[END_REF]. A subordinate unit is signalled through downwards changes in key (i.e. major levels in a speaker's pitch range) or in pitch height. Intonation can convey subordinating information that is not marked through verbal means. Syntactic units can prosodically be made autonomous or grouped depending on the speaker's semantic and/or pragmatic strategies [START_REF] Mertens | Intonational Grouping, Boundaries, and Syntactic Structure in French[END_REF], [START_REF] Lee | On the Role of Intonation and Pauses in the Interpretation of English Parenthetical Adverbs[END_REF].

To integrate a prosodic unit to an adjacent segment, F0 (i.e. Fundamental Frequency = pitch) generally rises on the final syllable of the inserted segment, which indexes it as prefacing further speech, continuing the paragraph and the ongoing point [START_REF] Wells | English Intonation: An Introduction[END_REF]. A downstepped tone compared to a preceding high tone corresponds to the general neutral relationship between two prosodic groups, often used to express seamless continuity [START_REF] Wennerstrom | The Music of Everyday Speech. Prosody and Discourse Analysis[END_REF].

On the contrary, a variation on the initial syllable signals a boundary. Likewise, a discourse segment featuring a low final syllable (termination contour) does not embed the following segment, and is autonomous regarding what ensues. Boundaries in speech can also be created with a variation of tempo [START_REF]Prosodic Systems and Intonation in English[END_REF]. Silent pauses conjointly participate in the segmentation of discourse [START_REF]Continuing and Restarting[END_REF].

Method

Corpus recording

The corpus used for this study, ENVID, is a collection of dialogues in British English. This collaborative corpus gathers video recordings realised between 2000 and 2012. Five dialogues were selected, making up a total of 2 hours and 10 minutes of interaction. Each interaction was recorded in a soundproof studio, guaranteeing its prosodic treatment. The participants are British people aged 20 to 23. Each participant had a lavalier microphone, providing two separate audio tracks. Two audio files corresponding to each microphone were created in a WAV format, facilitating the analysis of overlapping speech.

Corpus transcription

The corpus was first edited in Praat [START_REF] Boersma | Praat: Doing Phonetics by Computer[END_REF] for a standard orthographic transcription, in which secondary constructions were localised and coded on a separate track as SC. All the annotations made in Praat where then exported into Elan [START_REF] Sloetjes | Annotation by Category: ELAN and ISO DCR[END_REF], an annotation tool facilitating data pooling and extraction.

Syntactic annotation

A total of 228 forms were annotated in the corpus, representing 4.82% of the total speaking time (i.e. 2.01 form/min): 88 adverbial clauses (1.88% of speaking time -0.78 form/min), 81 restrictive relative clauses (1.71% of speaking time -0.71 form/min), and 58 appositive relative clauses (1.23% of speaking time -0.52 form/min)

The selection targeted forms without an interruption, surrounded with immediate left and right co-texts other than a single silent pause yielding the speaking turn.

The selected forms were classified according to their syntactic type in Praat (adverbial clause, appositive clause, restrictive relative clause). A second track delimitates their environment: the preceding tone-unit as described below was labelled L (left co-text), the subsequent one labelled R (right co-text).

Prosodic annotation

The corpus was segmented into tone-units, according to the British school of intonation [START_REF] Wells | English Intonation: An Introduction[END_REF], [START_REF]Prosodic Systems and Intonation in English[END_REF], based on dynamic pitch contours.

The Momel-Intsint algorithm [START_REF] Hirst | A Praat Plugin for Momel and INTSINT with Improved Algorithms for Modelling and Coding Intonation[END_REF], [START_REF] Bigi | SPPAS: a Tool for the Phonetic Segmentation of Speech[END_REF] was used for the automatic annotation of the F0 target points in the signal. Annotations are made in two respects: the algorithm notes pitch height (in Hz) on target syllables, which then allowed us to calculate mean F0 values for specific segments. The algorithm also codes symbolic (relative) values of intonation, in which each measured F0 value is compared to preceding ones, i.e. significant changes in the F0 curve either regarding the speaker's pitch range (Top, Bottom) or regarding the neighbouring tones or sequences of tones (Upstep, Downstep, Same, Low, High).We are here particularly interested in values which indicate a significant pitch reset (Top, Bottom), or a significant change in pitch key (Upstep -change towards higher pitch range, Downstep -towards lower pitch range). We are also interested in the value "Same" which, if found in greater number in our sequences, would indicate that there is no break in between the different elements of the sequence.

Within each segment of the sequences under study, the nature of each nuclear contour was also coded manually (fall; fall-rise; rise; rise-fall; flat). Pitch key was then annotated in regards to each speaker's specific range (high; mid; low) on both the whole segments (L, SC, R) and the boundary (initial and final) syllables in these segments.

Working hypotheses

Based on the theoretical background defined by the literature and on our observations from the annotation, a specific list of vocal cues (including rhythmical and intonational patterns) is taken into account to survey different types of discourse boundaries. We mainly expect forms showing little autonomy.

If the constructions are not autonomous, they are expected to be integrated in the same tone-unit as the main clause, or to show continuation contours (final rises). They should be uttered in a low or mid-key, the usual declination line of the paragraph being followed without any break. These forms should not cause any important change in rhythm, featuring few pauses.

Results

This paper evaluates the prosodic autonomy of secondary constructions (integration vs. demarcation). We test whether these forms mainly create a break or whether they are preferentially integrated to their left and/or right co-text. After identifying and measuring the most relevant prosodic disjunctive cues drawing on our hypotheses, the three syntactic types can be placed on a continuum from dependence to autonomy. The particularities are detailed for each type from the most integrated form to the most disruptive.

Restrictive relative clauses

Restrictive relative clauses feature only two disruptive prosodic cues. They show a distinct duration pattern: SC is significantly the longest segment of the sequence in which it is inscribed (L: F(54, 54) = 3.01, p < .0001; R: F(54, 54) = 4.48, p < .0001). This type also features the highest number of intraconstituent silent pauses (44.6%, making a total of 10.26 sec). Most of these pauses are not accompanied with any significant increase or decrease in speech rate, and do not coincide with the syntactic boundaries; they are linked with displaced demarcation (i.e. a displaced pause making it possible for the speaker to keep the conversational turn while processing). Example (4) below illustrates these tendencies: SC is the longest segment in the sequence. The conjunction that is separated from the clause it introduces with a silent pause, which is not correlated with any hesitation marker. While indexing a close link between the end of L and the beginning of SC, Michelle pauses outside syntactic boundaries to indicate she has not finished delivering the ongoing informational unit; her speaking turn is then secured.

Restrictive relative clauses are also the only type in which a majority of occurrences (55.6%) are directly bound to the left co-text under a same intonational unit, through flat contours on the left co-text's lexical items. In example (5) associated with Figure 1 below, SC shares a common prosodic and predicative unit with its left co-text, whose prosodic nucleus is in SC. An important break is however marked with R, through substantial extra-constituent silent pauses: While L and SC display final rising contours, SC's and R's initial syllables are downstepped (Intsint "D" value). Although realised in distinct tone-units, these three segments are intonationally linked.

Adverbial clauses

Given their mobility in the macrostructure (i.e. they can be preposed to the main nuclear syntactic configuration or postponed), adverbial clauses show a diverse prosodic configuration with respect to boundedness and internal variation. However, some disjunctive markers are common to all forms.

Displaying three disruptive cues, adverbial clauses are demarcated thanks to intra-constituent intonational resources. These forms are both produced and followed with a greater pitch height variation within the neighbouring tones of the same unit than the other types. Their immediate right co-text (i.e. R) features more demarcative tone values (hence pitch movement) than the other segments, as shown in Figure 3, associated with example ( 7): [START_REF] Chafe | Linking Intonation Units in Spoken English[END_REF] Alex L (h) you know SC when you're not allowed to laugh R and then there's like a massive silence While sharing a single prosodic contour with L, SC's final syllable is higher than the initial one (284 Hz vs. 219 Hz) and does not match R's beginning, which is downstepped (Intsint "D" value). This preposed SC forms a landmark for what follows, embedding R, which nonetheless comprises more demarcative tone values.

Adverbial clauses are also demarcated thanks to rhythmic strategies. Although they do not feature any distinctive duration or speech rate, they display the highest distribution of filled pauses (19.7%), mostly occurring immediately after their production, i.e. in the right co-text. The most frequent hesitation marker is hem, as shown in R in (8) further below.

While L and SC are indexed as in a common cognitive unit through their fluent delivery and a structuring silent pause at the end of SC, R is indexed as requiring a greater processing load. Kate combines the discourse marker like, both suggesting approximation and functioning as a filler, with the hesitation marker hem.

Appositive clauses

The prosodic signals indexing independence in appositive clauses are more numerous and varied than in the other forms we treated so far. They display the greatest number of vocal boundary cues with eight segmental and suprasegmental markers.

Appositive clauses are the shortest and fastest forms; both in their embedding sequence (L: F(54, 54) = 2, p < .05; R: F(54, 54) = 1.9, p < .05) and compared to the other types (duration restrictives: F(54, 54) = 3.5, p < .0001, adverbials: p > .05; speech rate: p > .05). This host sequence also reveals the highest distribution of extra-constituent silent pauses (36.5%, making up a total of 30.8 sec; restrictives: F(57, 44) = 3.8, p < .0001; adverbials: F(57, 36) = 2.40, p < .005), 61% of which are immediately following the subordinate segment (i.e. in between SC and R).

In addition, appositive clauses show a relatively high distribution of filled pauses (15.8%), most of them located after their production. They then show a very irregular rhythm. Example (9) illustrates these tendencies: From the point of view of intonation, 98.3% of occurrences are uttered in a separate tone-unit. These forms are then vastly made autonomous. They also show less integrative rising contours: 79 % of the sequences containing them do not feature any rising tone, as shown by example [START_REF] Mertens | Intonational Grouping, Boundaries, and Syntactic Structure in French[END_REF] and Figure 5: If the absence of a pause between L and SC suggests their proximity at a propositional level, the definitive falling contours indicate they make two distinct discursive moves. While L supports the main theme (the staff is very rude), SC turns this new argument into a concession, with a change in point of view. While the pitch key in SC is lower, the similar contour signals similarity to L. This lower key is another prototypical means for achieving disruption in these forms. While pitch discontinuity marks the segment following the subordinate form in the other types, 60% of speakers produce appositive clauses with a simultaneous change in pitch height (>20 Hz; p < .05 for these speakers). Some forms are then realised with a genuine "parenthetical" intonation (i.e. lower F0 and no modulation [START_REF] Hirschberg | Intonational Features of Local and Global Discourse Structure[END_REF]- [START_REF] Lelandais | Multimodal Analysis of Parentheticals in Conversational Speech[END_REF]). The following segment is globally uttered with a higher key (1.48 in L; 1.31 in Sc; 1.40 in R -pitch key is normalised according to each speaker's pitch range; however, p > .05). An initial upstep on the first syllable of the right cotext corroborates a non-neutral interval between SC and R.

Discussion

Our analysis confirms that the different syntactic types can be distinguished in their degree of autonomy. They can be positioned on a continuum, from integration (restrictive relative clauses) to autonomy (appositive clauses), going through intermediate combinatory strategies (adverbial clauses).

Restrictive relative clauses are the most intonationally integrated forms to their (left) co-text. However, they rhythmically stand out (duration, filled pauses).

Adverbial clauses are freer, reflecting their pragmatic ability to project an interpretative frame for several consecutive segments or to close a unit. Their intonation conveys change, indicating an alteration in the interpretative/informational networks rather than segmenting adverbial clauses as isolated units. The postponed forms present variations from the prototypical afterthought realisation [START_REF] Chafe | How People Use Adverbial Clauses[END_REF]. Speakers also index more processing difficulties after the production of this type. However, these subsequent segments are more characterised with hem, signalling a local interruption, than with uh, signalling a strong boundary [START_REF] Shriberg | Intonation of Clause-internal Filled Pauses[END_REF]- [START_REF] Clark | Using 'uh' and 'um' in Spontaneous Speaking[END_REF].

Appositive clauses are finally the most independent forms, mainly showing a total prosodic autonomy. They are the only forms displaying such an array of disjunctive cues, both at the segmental and suprasegmental levels. Disruption is then more perceptible, with the highest significant gap in pitch, and the weakest distribution of continuation contours.

Disjunction is mostly expressed by speakers through rhythmical resources, in all types of secondary constructions. Duration and extra-constituent silent pauses are especially drawn on to create a break. As far as intonation is concerned, initial pitch upsteps are widely used. The presence of these resources suggests that prosody demarcates secondary constructions more than it integrates them; their location indicates that prosodic boundaries are preferentially marked retrospectively, immediately after the secondary construction.

Conclusions

When analysing how semiotic units form larger sequences of action in discourse and conversation, spontaneous speech presents both complex chains of structures embedded in one another, and disruptions in which the discourse parts no longer follow one another. Secondary constructions introduce a break when they establish a different assertive position from the preceding utterance [START_REF] Lelandais | Multimodal Analysis of Parentheticals in Conversational Speech[END_REF]. While this break can directly be created through syntactic or discursive means, prosody creates a break immediately afterwards through rhythmic features or pitch upsteps, signalling the previous elements have to be recontextualised.

This study aimed at demonstrating that a wide survey of prosodic resources for demarcation sheds new light on subordination, deriving from numerous interactions between segmental and suprasegmental cues. One way to widen the picture of prosodic boundaries would be to take final syllable lengthening into account. However, there is little consensus on the array of thresholds to be taken into account regarding the sampler types and weights of syllables; we are currently working to provide the most qualified and reliable criteria.

Appendix: transcription conventions

Figure 1 :

 1 Figure 1: Praat pitch contour of example 5 (the transcription track shows segments -L, SC, R -in the sequence).

  s had all her life # R the # cos there's the black one Besides, they feature the highest distribution of integrative continuation contours (58% in L only; 16% both in L and SC as shown in (6) below; 9% in SC only).

Figure 2 :

 2 Figure 2: Two continuation rises in example 6 illustrated in the Praat curve and the Intsint codes (U = upstep, D = downstep).

( 6 )

 6 Rhianna it's basically # L uh the school # SC hem which is right # below R that uh rents # the flat out

Figure 3 :

 3 Figure 3: Final rise in example 7 illustrated in the Praat curve and the Intsint codes (H = high), followed by more pitch movement in R (L = low).

( 8 )Figure 4

 84 Figure4shows once again more demarcative tone values in R. This postponed SC is realised in a distinct tone-unit. Yet, contrary to what could be expected, SC shows an initial upstep (Intsint "U" value) despite its lack of internal variation.

Figure 4 :

 4 Figure 4: Initial upstep in example 8 followed by more pitch movement in R, illustrated in the Praat curve and the Intsint codes (U = upstep, T = top).

Figure 5 :

 5 Figure 5: Two similar contours in example 10 with a lower pitch key in SC illustrated in the Praat curve.

( 10 )

 10 Rhianna L even compared to Easyjet SC which is another low cost company # R yeah i hate Ryanair


Sébastien Martin
email: sebastien.martin@insa-lyon.fr

First order quasilinear equations with boundary conditions in the L ∞ framework

Keywords: Semi Kružkov entropy-flux pair, Maximum principle, BV estimates, Vanishing viscosity method, Doubling variable method Mathematics Subject Classifications: 35K65, 35L65, 35L60

We study a class of first order quasilinear equations on bounded domains in the L ∞ framework. Using the "semi Kružkov entropy-flux pairs", we define a weak-entropy solution, state an existence and uniqueness result, and a maximum principle.

Introduction

In this paper, Ω ⊂ R d , d ≥ 1, is a bounded smooth domain. Let us denote by ∂Ω the boundary of Ω and by n the outer normal vector to ∂Ω. We denote Q T ≡ (0, T ) × Ω and Σ T ≡ (0, T ) × ∂Ω. Let us consider this set of equations:

∂u ∂t + ∇ • f (t, x, u) + g(t, x, u) = 0 on Q T (1) u(0, •) = u 0 on Ω (2) "u = u D " on Σ T (3)
where the sense of the boundary condition will be precised further. We consider the following assumption:

Assumption 1 (i) f and g are two functions defined on [0, T ] × Ω × R such that

f ∈ C 2 ([0, T ] × Ω × [a, b]) d , g ∈ C 2 ([0, T ] × Ω × [a, b])
(ii) f , ∇ • f and g are Lipschitz continuous w.r.t. u, uniformly in (t, x), the constants of Lipschitz continuity being respectively denoted

L [f ] , L [∇•f ] , L [g] . (iii) (u 0 , u D ) ∈ L ∞ (Ω; [a, b]) × L ∞ (Σ T ; [a, b]), (iv) (∇ • f + g) (•, •, a) ≤ 0 and (∇ • f + g) (•, •, b) ≥ 0 uniformly in (t, x).
From a mathematical point of view, numerous works have approached or investigated this field. On unbounded domains, existence and uniqueness of a solution for quasilinear first order equations domains has been solved in the pioneering works of Oleȋnik [START_REF] Oleȋnik | Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation, Uspehi Mat[END_REF], Volpert [START_REF] Volpert | Spaces BV and quasilinear equations[END_REF] and Kružkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] who introduced the concept of weak entropy solutions and related "Kružkov entropy-flux pairs" u -k , sgn(u -k)(f (t, x, u) -f (t, x, k)) .

When dealing with bounded domains, under some regularity assumptions on the data, Bardos, Le Roux and Nédélec [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF] also proved existence and uniqueness of a weak entropy solution satisfying a "Kružkov entropy-flux pair" formulation including boundary terms; for this, they introduced an appropriate mathematical boundary condition that must be understood in a particular way. Nevertheless, when considering L ∞ data, the lack of regularity prevents from using the result of Bardos, Le Roux and Nédélec. This difficulty was overcome, at least in the case of autonomous scalar conservation laws on bounded domains, by Otto [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF][START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF] who introduced "boundary entropy-flux pairs"

H(u, k), Q [f ] (u, k)
satisfying particular properties (to be recalled further), which enable to state existence and uniqueness of a so-called weak entropy solution and a maximum principle for this solution. Finally, using a lemma proposed by Vovelle [START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF], it appears that a formulation using "semi Kružkov entropy-flux pairs"

(u -k) ± , sgn ± (u -k)(f (t, x, u) -f (t, x, k))
is equivalent to a formulation based on "boundary entropy-flux pairs". Here, u → (u -κ) ± are the so-called "semi Kružkov entropies" [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF][START_REF] Serre | structures géométriques, oscillation et problèmes mixtes[END_REF][START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF] defined by

(u -κ) + =      u -κ, if u ≥ κ, 0, otherwise.
and (u -κ) -= (κ -u) + .

The functions sgn ± (u -κ)(f (•, •, u) -f (•, •, κ)) are the corresponding "semi Kružkov fluxes", where u → sgn ± (u) is the derivative of the function u → u ± with value 0 at point 0. Notice that the "semi Kružkov entropy-flux pairs" formulation is very similar to the initial one of Kružkov. But:

What is the appropriate definition of a weak entropy solution for first order quasilinear equations (i.e. including non-autonomous fluxes and source terms) on bounded domains with L ∞ data ? Answering this question would draw a complete parallel with the results of Bardos, Le Roux and Nédélec [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF] and those of Otto [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF] and Vovelle [START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF]: indeed, the analysis of scalar conservation laws with L ∞ data, initiated by Otto, would be extended to quasilinear first order equations, studied by Bardos, Le Roux and Nédélec. What sufficient conditions lead to a maximum principle ? Indeed, such a property is crucial when studying some physical problems.

Thus, it is the purpose of this paper to give a general framework which is valid for first order quasilinear equations on bounded domains with L ∞ data. Among the difficulties, we can observe that, when dealing with non autonomous fluxes and source terms, a formulation with "boundary entropy-flux pairs" is not possible anymore. Fortunately, the concept of "semi Kružkov entropy-flux pairs" allows to overcome difficulties. This work is organized as follows: in Section 1, we state the definitions and establish a maximum principle; in Section 2, we prove the existence result; in Section 3, we prove the uniqueness result. Existence and uniqueness theorems are based on techniques that have been widely used in [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF][START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF][START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF]. But we point out the fact that these arguments have never been gathered with the appropriate definition of a weak entropy solution in this general framework in order to establish an existence and uniqueness theorem along with a maximum principle: in fact, we deeply use the results detailed in [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF], up to the following modifications: proofs for existence and uniqueness are adapted to the "semi Kružkov entropy-flux pairs", dealing with additional terms induced by the source term and non-autonomous property of the flux.

1 Definition, initial / boundary conditions, maximum principle

Definition 1 Suppose that Assumption 1 holds. A function u ∈ L ∞ (Q T , [a, b]
) is said to be a weak entropy solution of problem ( 1)-( 2)-(3) if it satisfies

(P SK )                                Q T (u -k) ± ∂ϕ ∂t + sgn ± (u -k)(f (t, x, u) -f (t, x, k)) ∇ϕ -sgn ± (u -k) ∇ • f (t, x, k) + g(t, x, u) ϕ dx dt + Ω (u 0 -k) ± ϕ(0, x) dx + L [f ] Σ T (u D -k) ± ϕ(t, r) dγ(r) dt ≥ 0 ∀φ ∈ D((-∞, T ) × R d ), φ ≥ 0, ∀k ∈ R
Let us explain the way the boundary / initial conditions are satisfied for this problem. Interestingly, the concept of "boundary entropy-flux pairs" defined by Otto is still the key point. Thus, let us recall their definition:

Definition 2 Let (H, Q [f ] ) be in C 1 (R 2 ) × C 1 ((0, T ) × Ω × R 2 ) d . The pair (H, Q [f ]
) is said to be a "boundary entropy-flux pair" (for the flux f ) if:

1. for all w ∈ R, s → H(s, w) is a convex function, 2. ∀w ∈ R, ∂ s Q [f ] (•, •, s, w) = ∂ s H(s, w) ∂f ∂s (•, •, s), 3. ∀w ∈ R, H(w, w) = 0, Q [f ] (•, •, w, w) = 0, ∂ s H(w, w) = 0.
Let us recall the lemma provided by Vovelle [START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF], which gives the link between "semi Kružkov entropy-flux pairs" and "boundary entropy-flux pairs":

Lemma 3 (i) Let η ∈ C 1 (R; R)
s -→ p 1 α i (s -κ i ) -+ q 1 β j (s -κj ) +
where α i ≥ 0,

β j ≥ 0, κ i ∈ [a
̺→0 + Σ T Q [f ] (t, r, u(t, r -̺ n(r)), u D (t, r)) • n(r) β(t, r) dγ(r) dt ≥ 0, (4)
PROOF. We directly use the proof of Lemma 7.12 in [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF], adapted to the particular case of the "semi Kružkov entropy-flux pairs". Thus, we easily state that if u ∈ L ∞ (Q T ) satisfies (P SK ), then, defining the quantity ess lim

̺→0 + Σ T sgn ± (u(t, r -̺ n(r)) -v D (t, r)) f (t, r, u(t, r -̺ n(r))) -f (t, r, v D (t, r))) • n(r) β(t, r) dγ(r) dt (5)
exists for all β ∈ L 1 ((0, T )×R d-1 ), β ≥ 0 a. e., and all v D ∈ L ∞ ((0, T )×R d-1 ). Moreover, we have:

ess lim ̺→0 + Σ T sgn ± (u(t, r -̺ n(r)) -v D (t, r)) f (t, r, u(t, r -̺ n(r))) -f (t, r, v D (t, r))) • n(r) β(t, r) dγ(r) dt ≥ -L [f ] Σ T (u D (t, r) -v D (t, r)) ± β(t, r) dγ(r) dt, for all β ∈ L 1 ((0, T )×R d-1
), β ≥ 0 a. e., and all v D ∈ L ∞ ((0, T )×R d-1 ). Then, taking v D = u D , every "boundary flux" Q [f ] is uniformly approximated by a linear combination of "semi Kružkov fluxes" (see Lemma 3), every coefficient being non-negative, which preserves the inequality and concludes the proof.

To complete the scope of boundary / initial conditions, we recall the following result, which is proved with the same arguments as in Lemma 7.41 of [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF]:

Lemma 5 (Initial condition) Let u ∈ L ∞ (Q T ) satisfying (P SK ). Then, ess lim t→0 + Ω u(t, x) -u 0 (x) dx = 0 (6)
Now we give some details on the way the boundary condition is satisfied:

Remark 6
The boundary condition 4 is nothing less than the one obtained in [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF][START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF], up to a generalization to non-autonomous fluxes and taking account of a source-term which does not interfere in the boundary condition. We have proved that it is satisfied, although working only with the "semi Kružkov entropy-flux pairs" formulation (let us recall that a "boundary entropy-flux pairs" formulation is not possible anymore). However the way to understand the boundary condition is given in [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF][START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF][START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF]: generally speaking, the problem should be overdetermined and the boundary equality cannot be required to be assumed at each point of the boundary, even if the solution is a regular function. But, with additional assumptions, the more comprehensive "BLN" condition is recovered: if u admits a trace, i.e. there exists

u |Σ T ∈ L ∞ (Σ T ) such that ess lim ̺→0 + Σ T |u(τ, r -̺ n(r)) -u |Σ T (τ, r)| dγ(r) dτ = 0
then Eq. ( 4) is equivalent to the following equation (see [START_REF] Dubois | Boundary conditions for nonlinear hyperbolic systems of conservation laws[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF])

Q [f ] (•, •, u |Σ T , u D ) • n ≥ 0, a.e. on Σ T .
Considering the particular boundary entropy / flux pairs

H + δ (z, κ) = {(z -κ) + } 2 + δ 2 -δ, H - δ (z, κ) = {(κ -z) -} 2 + δ 2 -δ, Q ± [f ],δ (•, •, z, κ) = z κ ∂ 1 H ± δ (λ, k) ∂f ∂u (•, •, λ) dλ,
and letting δ → 0, we obtain the following uniform convergences:

Q ± [f ],δ (•, •, z, κ) → sgn ± (z -κ)(f (•, •, z) -f (•, •, κ)).
Finally taking the boundary flux

Q [f ] (•, •, s, w) = sgn + (s -max(w, k))(f (•, •, s) -f (•, •, max(w, k))) +sgn -(s -min(w, k))(f (•, •, s) -f (•, •, min(w, k)))
yields the classical condition given by Bardos, Le Roux and Nédélec [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF]:

for a.e. (t, r) ∈ Σ T , ∀k ∈ [min(u |Σ T , u D ), max(u |Σ T , u D )], sgn(u |Σ T (t, r) -u D (t, r))(f (t, r, u |Σ T (t, r)) -f (t, r, k)) • n(r) ≥ 0. ( 7
)
Before stating existence and uniqueness results in next sections, we prove:

Theorem 7 (Maximum principle) Under Assumption 1, if u satisfies (P SK ), then a ≤ u ≤ b a.e. on Q T .

PROOF. Set k = a in (P SK ). Since we have by Assumption 1 (iii) and (iv), (u 0 -a) -= 0, (u D -a) -= 0, the boundary / initial terms vanish. Then if we choose a particular testfunction which only depends on time t, we obtain:

Q T (u -a) -φ ′ (t) -sgn -(u -a) ∇ • f (t, x, a) + g(t, x, u) φ(t) dx dt ≥ 0 for all φ ∈ D([0, T [), φ ≥ 0. Now, using ∇ • f (t, x, a) + g(t, x, u) = ∇ • f (t, x, a) + g(t, x, a) + g(t, x, u) -g(t, x, a)
and Assumption 1 (iv), we get

Q T (u -a) -φ ′ (t) -sgn -(u -a) (g(t, x, u) -g(t, x, a)) φ(t) dx dt ≥ 0, (8)
for all φ ∈ D([0, T [), φ ≥ 0. Furthermore, we can check that:

-L [g] (u -a) -≤ sgn -(u -a) (g(t, x, u) -g(t, x, a)) ≤ L [g] (u -a) -
and Inequality (8) implies

Q T (u -a) -φ ′ + L [g] φ ≥ 0. Defining the function q a (t) = e -L [g] t Ω (u -a) -(t, x) dx, (9)
the above inequality gives

T 0 q a (t)e L [g] t φ ′ (t) + L [g] φ(t) dt ≥ 0. Denoting ψ(t) = e L [g] t φ(t), we infer that for all ψ ∈ D([0, T [), ψ ≥ 0, T 0 q a (t) ψ ′ (t) dx dt ≥ 0, ( 10
) Let τ < T , δ τ = T -τ and r ∈ D([0, T [) be such that: r is non-increasing, r ≡ 1 on [0, τ ], r ≡ 0 on [τ + δ τ /2, T [. Choosing ψ(t) = r(t) (T -t)/T in Inequality (10) gives - 1 T T 0 q a (t)r(t) dt + T 0 q a (t) T -t T r ′ (t) dt ≥ 0
Since r ′ ≤ 0, the second term of the left-hand side is negative. Since r(t) = 1, ∀t ∈ (0, τ ) and r ≥ 0, the first term is upper bounded by

- 1 T τ 0 q a (t) dt
which is consequently non-negative. But, q a is obviously a non-negative function, so that q a ≡ 0, on (0, τ ).Therefore, we deduce from the definition of q a (see Eq. ( 9)) that (u -a) -= 0 on Ω × (0, τ ). Letting τ → T , we have u ≥ a a.e. Similarly, by choosing k = b in (P SK ) (with the "semi Kružkov entropy" u → (u -b) + ), we prove u ≤ b a.e.

Remark 8 Under Assumption 1 (iv), we check that only the restriction to the set [a, b] of functions s → f (t, x, s) and s → g(t, x, s) plays an active role. Therefore, it is sufficient to consider functions f (t, x, •) and g(t, x, •) defined on [a, b] instead of R, as proposed in Assumption 1 (i).

Existence

Existence is obtained from the vanishing viscosity method. We consider the following set of equations:

∂u ε ∂t + ∇ • f (t, x, u ε ) + g(t, x, u ε ) = ε∆u ε on Q T (11) u ε (0, •) = u 0 ε on Ω (12) u = u D ε on Σ T ( 13
)
where the following assumption holds:

Assumption 2 (i) u D
ε and u 0 ε satisfy compatibility conditions on Σ T ∩ Q T : in particular, u 0 ε and u D ε should be a restriction, on the sets {0} × Ω and Σ T respectively, of a smooth function ψ ε defined on Q T satisfying

∂ψ ε ∂t + ∇ • f (t, x, ψ ε ) + g(t, x, ψ ε ) = ε∆ψ ε , on {0} × ∂Ω (ii) u D ε and u 0 ε are smooth functions: u D ε ∈ C 2 (Σ T ; [a, b]), u 0 ε ∈ C 2 (Ω; [a, b]).
Under Assumption 2, the parabolic problem ( 11)-( 13) admits a unique solution [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF]). We study the convergence of {u ε } when ε tends to 0. As in [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF], we introduce the following tools:

u ε ∈ C 2 (Q T ) (see Chapter V, §6 in
Definition 9 Let us consider µ > 0 small enough. We define the functions:

s(x) =      min (dist(x, ∂Ω), µ) , if x ∈ Ω -min (dist(x, ∂Ω), µ) , if x ∈ R d \ Ω ξ ε (x) = 1 -exp - L [f ] + εR ε s(x) , with R = sup 0<s(x)<µ |∆s(x)|.
Notice that s is Lipschitz continuous in R d and smooth on the closure of the set {x ∈ R d , |s(x)| < µ}. Moreover, it can be proved (see [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF]):

Proposition 10 ξ ε being defined in Definition 9, for all ϕ ∈ D(R d ), ϕ ≥ 0, L [f ] Ω ∇ξ ε ϕ ≤ ε Ω ∇ξ ε ∇ϕ + (L [f ] + εR) ∂Ω ϕ (14)
Lemma 11 Let (u, u D , u 0 ) satisfy equations ( 11)-( 13), the data satisfying Assumption 2 (subscripts are dropped for convenience). Then,

(i) for all ϕ ∈ D(] -∞, T [×R d ), for all k ∈ R, Q T (u -k) ± ∂ϕ ∂t + sgn ± (u -k) f (t, x, u) -f (t, x, k) ∇ϕ (15) -sgn ± (u -k) ∇ • f (t, x, k) + g(t, x, u) ϕ + ε (u -k) ± ∆ϕ ξ ε + Ω (u 0 -k) ± ϕ(0, •) ξ ε ≥ -2ε Q T (u -k) ± ∇ϕ∇ξ ε -(L [f ] + Rε) Σ T (u D -k) ± ϕ,
(ii) the following maximum principle holds: a ≤ u ≤ b.

PROOF.

Proof of (i): Let us define the functions:

sgn η ± (z) =      H η (z), if z ∈ R ± -H η (-z), if z ∈ R ∓ , I ± η (z) = z 0 sgn η ± (t) dt
where the function H η is a classical approximation of the Heaviside graph:

H η (z) = z/ηχ [0,η[ (z) + χ [η,+∞[ (z).
Obviously, the pairs

I ± η (z, k), sgn η ± (z -k) f (t, x, z) -f (t, x, k)
mimick the behaviour of the "semi Kružkov entropy-flux pairs". Notice that

I ± η (•, k) ∈ C 1 (R) is piecewise convex. Multiplying Eq. (11) by sgn η ± (u-k) ϕ ξ ε , with ϕ ∈ D(] -∞, T [×R d )
, we obtain (after integration by parts):

Q T I ± η (u, k) ∂ϕ ∂t + sgn η ± (u -k) f (t, x, u) -f (t, x, k) ∇ϕ -sgn η ± (u -k) ∇ • f (t, x, k) + g(t, x, u) ϕ + ε I ± η (u, k)∆ϕ ξ ε + Q T sgn η ± (u -k) f (t, x, u) -f (t, x, k) ϕ ∇ξ ε + Ω u 0 k sgn η ± (v -k) dv ϕ(0, •) ξ ε + Q T f (t, x, u) -f (t, x, k) • ∇u sgn η ± ′ (u -k) ϕ ξ ε ≥ ε Q T ∇ I ± η (u, k) ϕ ∇ξ ε -2I ± η (u, k)∇ϕ∇ξ ε
After some computation, we state that:

sgn η ± (u -k) f (t, x, u) -f (t, x, k) ≤ L [f ] I ± η (u, k) + L [f ] η
Moreover, using Proposition 10 with I ± η (u, k)ϕ instead of ϕ, we get:

L [f ] Q T I ± η (u, k)ϕ ∇ξ ε ≤ ε Q T ∇ I ± η (u, k)ϕ ∇ξ ε + (L [f ] + Rε) Σ T I ± η (u D , k)ϕ
Using these two results in the previous inequality gives:

Q T I ± η (u, k) ∂ϕ ∂t + sgn η ± (u -k) f (t, x, u) -f (t, x, k) ∇ϕ -sgn η ± (u -k) ∇ • f (t, x, k) + g(t, x, u) ϕ + ε I ± η (u, k)∆ϕ ξ ε + Ω u 0 k sgn η ± (v -k) dv ϕ(0, •)ξ ε + Q T f (t, x, u) -f (t, x, k) • ∇u sgn η ± ′ (u -k)ϕξ ε ≥ -2ε Q T I ± η (u, k)∇ϕ∇ξ ε -(L [f ] + Rε) Σ T I ± η (u D , k)ϕ -L [f ] η Q T ϕ ∇ξ ε
Now, let η tend to 0. The first and second terms of the left-hand side give:

Q T (u -k) ± ∂ϕ ∂t + sgn ± (u -k) f (t, x, u) -f (t, x, k) ∇ϕ -sgn ± (u -k) ∇ • f (t, x, k) + g(t, x, u) ϕ + ε (u -k) ± ∆ϕ ξ ε + Ω (u 0 -k) ± ϕ(0, •) ξ ε
The last term of the left-hand side tends to 0 by Lemma 2 in [4]1 . Finally, the right-hand side tends to:

-2ε

Q T (u -k) ± ∇ϕ∇ξ ε -(L [f ] + Rε) Σ T (u D -k) ± ϕ
and the proof is concluded.

Proof of (ii): The result is obtained as in the proof of Theorem [START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF], by working with Eq. ( 15) instead of (P SK ). Now we propose the following L 1 -stability result:

Lemma 12 Let (u 1 , u D 1 , u 0 1 ), (u 2 , u D 2 , u 0
2 ), satisfy equations ( 11)-( 13), the corresponding data satisfying Assumption 2. Then, for all t ∈ (0, T ),

Ω u 1 (t, •) -u 2 (t, •) ξ ε ≤ Ω u 0 1 -u 0 2 ξ ε + (L [f ] + Rε) Σ T u D 1 -u D 2 e L [g] T (16)
PROOF. The proof follows the idea used in [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF] and only needs to be adapted to our problem: let us denote

w = u 1 -u 2 , w D = u D 1 -u D 2 , w 0 = u 0 1 -u 0 2 and let us introduce ϕ δ (z) = (z 2 + δ 2 ) 1/2 . Multiplying ∂w ∂t + ∇ • f (t, x, u 1 ) -f (t, x, u 2 ) + g(t, x, u 1 ) -g(t, x, u 2 ) -ε∆w = 0 by ϕ δ ′ (w)ξ ε and integrating over (0, t) × Ω, we get Ω ϕ δ (w(t, •)) ξ ε - Ω ϕ δ (w 0 ) ξ ε (17) - t 0 Ω f (τ, x, u 1 ) -f (τ, x, u 2 ) ϕ δ ′′ (w) ∇w ξ ε + ϕ δ ′ (w) ∇ξ ε + g(τ, x, u 1 ) -g(τ, x, u 2 ) ϕ δ ′ (w)ξ ε +ε ∇w 2 ϕ δ ′′ (w)ξ ε + ∇ϕ δ (w)∇ξ ε = 0
Now we study the behaviour of each term w.r.t δ: using the uniform Lipschitz continuity of f , Young's inequality and the fact that

z 2 ϕ δ ′′ (z) = z 2 δ 2 (z 2 + δ 2 ) -3/2 < δ, we get -f (τ, x, u 1 ) -f (τ, x, u 2 ) ∇w ϕ δ ′′ (w)ξ ε + ε ∇w 2 ϕ δ ′′ (w)ξ ε ≥ -L [f ] |w| |∇w| + ε ∇w 2 ϕ δ ′′ (w)ξ ε ≥ - L 2 [f ] 4ε w 2 ϕ δ ′′ (w)ξ ε ≥ - L 2 [f ] δ 4ε ξ ε Moreover, observing that |z|ϕ δ ′ (z) ≤ ϕ δ (z), we obtain -f (τ, x, u 1 ) -f (τ, x, u 2 ) ϕ δ ′ (w) ∇ξ ε ≥ -L [f ] |w| |ϕ δ ′ (w)| |∇ξ ε | ≥ -L [f ] ϕ δ (w)|∇ξ ε |
Following the same idea, we get

g(τ, x, u 1 ) -g(τ, x, u 2 ) ϕ δ ′ (w)ξ ε ≥ -L [g] |w| |ϕ δ ′ (w)| ξ ε ≥ -L [g] ϕ δ (w)ξ ε
Finally, using the previous inequalities, we state that

Ω ϕ δ (w(t, •))ξ ε - Ω ϕ δ (w 0 )ξ ε - L 2 [f ] δ T 4ε Ω ξ ε -L [f ] t 0 Ω ϕ δ (w)|∇ξ ε | -L [g] t 0 Ω ϕ δ (w)ξ ε + ε t 0 Ω ∇ (ϕ δ (w)) ∇(ξ ε ) ≤ 0.
Then, putting these inequalities in Eq. ( 17) and using Inequality ( 14), we get

Ω ϕ δ (w(t, •))ξ ε Ω ϕ δ (w 0 )ξ ε ≤ L [g] t 0 Ω ϕ δ (w)ξ ε + (L [f ] + Rε) t 0 Ω ϕ δ (w D ) + L 2 [f ] δT 4ε Ω ξ ε
Now let δ tend to 0. We obtain

Ω |w(t, •)| ξ ε ≤ Ω |w 0 | ξ ε + (L [f ] + Rε) Σ T |w D | + L [g] t 0 Ω |w| ξ ε .
Applying Gronwall's lemma concludes the proof.

Lemma 13 Let (u, u D , u 0 ) satisfy equations ( 11)-( 13), the data satisfying Assumption 2. We suppose furthermore that u D has a smooth extension to Q T , denoted u D . Then, there exists a constant λ which only depends on u 0 Ω , u D Σ T , T , Ω, f and g such that

sup t∈(0,T ) Ω ∂u ∂t (t, •) + ∇u(t, •) ≤ λ (18)
Here, we used the notation

u 0 Ω = Ω ∆u 0 + ∇u 0 + u 0 u D Σ T = sup Q T ∆u D + ∂u D ∂t + ∇u D + u D + Q T ∇ 2 ∂u D ∂t + ∇ 3 u D + ∂ 2 u D ∂t 2 + ∇ ∂u D ∂t + ∇ 2 u D PROOF.
In this proof, we will say that a constant "does not depend on ε" if it only depends on u 0 Ω , u D Σ T , T , Ω, f and g. Moreover, for the sake of simplicity, u D will be identified to u D . The proof follows the idea developped in [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF] (up to the source term g and non-autonomous property of f ): in particular, we may observe the influence of the derivatives ∂f /∂t, ∂f /∂x, ∂g/∂t, ∂g/∂x and ∂g/∂u in this statement. Indeed, this leads to the main differences with the proof stated in [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF], since additional terms have to be treated in order to get a BV estimate. The proof is organized in two steps:

Step 1: Boundness of

Ω ∂u ∂t (t, •)
Let us still denote u D the smooth extension of u D onto Q T . We introduce

v = u -u D , e = ∂ 2 u D ∂t 2 + ∇ • ∂f ∂u (•, •, u) ∂u D ∂t -ε∆ ∂u D ∂t + ∇ • ∂f ∂t (•, •, u) + ∂g ∂u (•, •, u) ∂u D ∂t + ∂g ∂t (•, •, u),
so that we easily get

∂ 2 v ∂t 2 + ∇ • ∂f ∂u (•, •, u) ∂v ∂t + ∂g ∂u (•, •, u) ∂v ∂t -ε∆ ∂v ∂t = -e. ( 19
)
Multiplying Eq. ( 19) by ϕ δ ′ (∂v/∂t), with ϕ δ (z) = √ z 2 + δ 2 , and integrating over (0, t) × Ω, we obtain

Ω ϕ δ ∂v ∂t (t, •) - Ω ϕ δ ∂v ∂t (0, •) - t 0 Ω ∂f ∂u (τ, x, u) • ∇ ∂v ∂t ∂v ∂t ϕ δ ′′ ∂v ∂t + t 0 Ω ∂g ∂u (τ, x, u) ∂v ∂t ϕ δ ′ ∂v ∂t + t 0 Ω ε ∇ ∂v ∂t 2 ϕ δ ′′ ∂v ∂t + e ϕ δ ′ ∂v ∂t = 0, (20)
by using the property ϕ δ ′ (∂v/∂t) = 0 on Σ T . Further, we have

- ∂f ∂u (τ, x, u) • ∇ ∂v ∂t ∂v ∂t ϕ δ ′′ ∂v ∂t + ε ∇ ∂v ∂t 2 ϕ δ ′′ ∂v ∂t ≥ - 1 4ε ∂f ∂u (τ, x, u) 2 ∂v ∂t 2 ϕ δ ′′ ∂v ∂t ≥ - L 2 [f ] δ 4ε .
Thus, letting δ → 0 in Eq. (20) implies the following inequality

Ω ∂u ∂t (t, •) ≤ Ω ∂u D ∂t (t, •) + Ω ∂v ∂t (0, •) + t 0 Ω |e| + L [g] t 0 Ω ∂v ∂t . (21)
Now, let us analyse each term of the right-hand side in the previous inequality:

◮ (step 1) Analysis of Ω ∂u D ∂t (t, •) . It is obviously bounded by c 1 = u D Σ T . ◮ (step 1) Analysis of Ω ∂v ∂t (0, •)
. We obtain from Eq. ( 11)

Ω ∂v ∂t (0, •) = Ω -∇ • f (0, •, u 0 ) -g(0, •, u 0 ) + ε∆u 0 - ∂u D ∂t (0, •) .
So far, we have:

Ω -∇ • f (0, •, u 0 ) = Ω (∇ • f ) (0, •, u 0 ) + ∂f ∂u (0, •, u 0 ) ∇u 0 ≤ L [∇•f ] Ω |u 0 | + L [f ] Ω |∇u 0 | ≤ c (1) 2
where c

(1)

2 only depends on f and u 0 Ω . Moreover,

Ω g(0, •, u 0 ) ≤ |Ω| sup g(t, x, s) , (t, x, s) ∈ Q T × [a, b] ≤ c (2) 2 ,
where c

(2) 2

only depends on g and Ω. Further, for ε bounded (which can be assumed, for instance ε ≤ 1), we get

Ω ε∆u 0 - ∂u D ∂t (0, •) ≤ Ω ∆u 0 + |Ω| sup Q T ∂u D ∂t ≤ c (3) 2 ,
where c

2 only depends on u 0 Ω , u D Σ T and Ω. Thus, the sum satisfies:

Ω ∂v ∂t (0, •) ≤ c 2 := c (1) 2 + c (2) 2 + c (3) 2 .
◮ (step 1) Analysis of t 0 Ω

|e|. Let us recall that, from the definition of e:

t 0 Ω |e| ≤ t 0 Ω ∂ 2 u D ∂t 2 + ∇ • ∂f ∂u (•, •, u) ∂u D ∂t + ∇ • ∂f ∂t (•, •, u) + ∂g ∂u (•, •, u) ∂u D ∂t + ∂g ∂t (•, •, u) + ε∆ ∂u D ∂t .
Now, we have

t 0 Ω ∂ 2 u D ∂t 2 ≤ Q T ∂ 2 u D ∂t 2 ≤ c (1)
3 with c

(1)

3 = u D Σ T . Moreover, ∇ • ∂f ∂u (•, •, u) ∂u D ∂t = ∂f ∂u (•, •, u) • ∇ ∂u D ∂t + ∂ 2 f ∂u 2 (•, •, u) • ∇u ∂u D ∂t + ∇ • ∂f ∂u (•, •, u) ∂u D ∂t .
Thus, each term can be controlled in the following way (the last term comes from the non-autonomous property of f ):

t 0 Ω ∂ 2 f ∂u 2 (•, •, u) • ∇u ∂u D ∂t ≤ c (2) 3 t 0 Ω ∇u , with c (2)
3 := sup

Q T ×[a,b] ∂ 2 f ∂u 2 sup Q T ∂u D ∂t , t 0 Ω ∂f ∂u (•, •, u) • ∇ ∂u D ∂t ≤ c (3) 3
:= L [f ] Q T ∇ ∂u D ∂t t 0 Ω ∇ • ∂f ∂u (•, •, u) ∂u D ∂t ≤ c (4)
3 := sup

Q T ×[a,b] ∇ • ∂f ∂u sup Q T ∂u D ∂t T |Ω|,
Further again, as f may explicitely depend on (t, x) and g may be a nonzero function, we state the additional estimates

t 0 Ω ∇ • ∂f ∂t (•, •, u) ≤ t 0 Ω ∇ • ∂f ∂t (•, •, u) + ∂ 2 f ∂t ∂u (•, •, u) • ∇u ≤ c (5) 3 + c (6) 3 Ω t 0 ∇u with c (5) 3 = sup Q T ×[a,b] ∇ • ∂f ∂t T |Ω| and c (6) 3 = sup Q T ×[a,b] ∂ 2 f ∂u 2 sup Q T ∂u D ∂t and also t 0 Ω ∂g ∂u (•, •, u) ∂u D ∂t + ∂g ∂t (•, •, u) + ε∆ ∂u D ∂t ≤ c (7) 3 ,
with c

(7) 3 = L [g] sup Q T ∂u D ∂t + sup Q T ×[a,b] ∂g ∂t + sup Q T ∆ ∂u D ∂t T |Ω|. Taking c 3 = max c (1) 3 + c (2) 3 + c (4) 3 + c (5) 3 + c (7) 3 , c (3)
3 + c (6) 3

and using the previous inequalities gives:

t 0 Ω |e| ≤ c 3   1 + Ω t 0 ∇u   . ◮ (step 1) Analysis of L [g] t 0 Ω ∂v ∂t .
We have, obviously, the property:

L [g] t 0 Ω ∂v ∂t ≤ L [g]   t 0 Ω ∂u ∂t + t 0 Ω ∂u D ∂t   ≤ c 4   1 + t 0 Ω ∂u ∂t   , with c 4 = L [g] max 1, sup Q T ∂u D ∂t |Ω| T .
Thus, recalling Inequality (21) along with the previous results, we obtain

Ω ∂u ∂t (t, •) ≤ c 5   1 + t 0 Ω |∇u| + t 0 Ω ∂u ∂t   (22)
by taking, for instance, c 5 = 4 i=1 c i which does not depend on ε.

Step 2: Boundness of

Ω ∇u(t, •)
For this, we proceed in two steps, namely Steps 2 (a) and 2 (b) , which will be gathered in order to conclude Step 2. Let us first proceed to Step 2 (a) . Recalling that v = u -u D and denoting

h 1 = ∂u D ∂t + ∇ • (f (t, x, u D )) + g(t, x, u D ) -ε∆u D , (23)
we have

∂v ∂t + ∇ • (f (t, x, u) -f (t, x, u D )) + g(t, x, u) -g(t, x, u D ) -ε∆v = -h 1 .(24)
We multiply Eq. ( 24) by ϕ δ ′ (v) β, where β ∈ D(R), β ≥ 0, depends only on the space variable and ϕ δ (z) = (z 2 + δ 2 ) 1/2 -δ.. After integration over (0, t) × Ω, and since ϕ δ

′ (v) = 0, ϕ δ (v) = 0, ∇ϕ δ (v) • n = 0 on Σ T , we obtain Ω ϕ δ (v(t, •)) β - Ω ϕ δ (v(0, •)) β -ε t 0 Ω ϕ δ (v) ∆β - t 0 Ω ϕ δ ′ (v)(f (τ, x, u) -f (τ, x, u D ))∇β -(f (τ, x, u) -f (τ, x, u D ))∇v ϕ δ ′′ (v) β + t 0 Ω ϕ δ ′ (v) (g(τ, x, u) -g(τ, x, u D ))β + ε|∇v| 2 ϕ δ ′′ (v) β = - t 0 Ω ϕ δ ′ (v) h 1 β.
We let δ → 0 and thus

Ω |v(t, •)| β - Ω |v(0, •)| β -ε t 0 Ω |v| ∆β - t 0 Ω sgn(u -u D )(f (τ, x, u) -f (τ, x, u D ))∇β (25)
+ t 0 Ω sgn(u -u D )(g(τ, x, u) -g(τ, x, u D ))β ≤ - t 0 Ω sgn(u -u D ) h 1 β.

Now we choose

β(x) = γ s(x) ρ ,
where s(x) is defined as before, ρ is a strictly positive number and γ ∈ D(R) is a fixed non-negative function such that γ(0) = 0 and γ(σ) = 1, for σ ≥ 1.

Let us study the behaviour with respect to ρ of each term.

◮ (step 2 (a) ) Behaviour w.r.t. ρ of t 0 Ω sgn(u-u D ) (f (t, x, u)-f (t, x, u D ))∇β:
Obviously, one has

∇β = γ ′ s(x) ρ 1 ρ ∇s(x)
and ∇s(x) = 0, on Ω \ K µ with K µ = {x ∈ Ω, dist(x, ∂Ω) < µ}. Thus, each point x ∈ K µ (for µ small enough) can be described as x = r(x) -s(x) n(r), where r(x) is the nearest boundary point to x, and n(r) is the outer vector to ∂Ω at point r(x). Let us notice that ∇s(x) = -n(r), if x ∈ K µ . From the previous observations, we deduce the following equality (for the sake of simplicity, F (u, u D )(τ, x) denotes the value of the function

sgn(u -u D ) (f (•, •, u) -f (•, •, u D )) at point (τ, x) ∈ Q T ): t 0 Ω F (u, u D )(τ, x) ∇β(x) dx dτ = t 0 Kµ F (u, u D )(τ, x) γ ′ s(x) ρ 1 ρ ∇s(x) dx dτ = t 0 µ 0 ∂Ω F (u, u D )(τ, r -sn(r)) γ ′ s ρ 1 ρ (-n(r)) dγ(r) ds dτ = - t 0 µ/ρ 0 ∂Ω F (u, u D )(τ, r -σρ n(r)) γ ′ (σ) n(r) dγ(r) dσ dτ = - µ/ρ 0 γ ′ (σ)   t 0 ∂Ω F (u, u D )(τ, r -σρ n(r)) n(r) dγ(r) dτ   dσ.
Thus, letting ρ → 0, we obtain (recalling that F (u, u D ) = 0 on Σ T ):

lim ρ→0 t 0 Ω F (u, u D )(τ, x)∇β(x) dx dτ = - +∞ 0 γ ′ (σ)   t 0 ∂Ω F (u, u D )(τ, r) n(r) dγ(r) dτ   dσ = - +∞ 0 γ ′ (σ) dσ   t 0 ∂Ω F (u, u D )(τ, r) n(r) dγ(r) dτ   = 0. ◮ (step 2 (a) ) Behaviour with respect to ρ of t 0 Ω |v| ∆β:
The particular choice of β gives ∆β(x) = ∇ • 1 ρ γ ′ s(x) ρ ∇s(x) and thus:

∆β(x) = 1 ρ d i=1 1 ρ γ ′′ s(x) ρ ∂s(x) ∂x i 2 + γ ′ s(x) ρ ∂ 2 s(x) ∂x 2 i . Now, if x ∈ K µ , then ∂s(x) ∂x i = -n i (r), n i being the ith component of n, so that 1 ρ 2 γ ′′ s(x) ρ d i=1 ∂s(x) ∂x i 2 = 1 ρ 2 γ ′′ s(x) ρ n(r) 2 = 1 ρ 2 γ ′′ s(x) ρ
and, as a consequence,

∆β(x) =        1 ρ 2 γ ′′ s(x) ρ + 1 ρ γ ′ s(x) ρ ∆s(x), on K µ , 0, elsewhere.
Thus, since v(τ, r(x)) = 0 (r(x) being a boundary point) and using the previous expression of ∆β, we have

t 0 Ω v ∆β = t 0 Ω v(τ, x) -v(τ, r(x)) ∆β = t 0 Kµ v(τ, x) -v(τ, r(x)) γ ′′ s(x) ρ 1 ρ 2 + 1 ρ γ ′ s(x) ρ ∆s(x) .
Let us focus on the first right-hand side term of the previous equality:

t 0 µ 0 ∂Ω v(τ, r -s n(r)) -v(τ,
≤ R on K µ , t 0 Kµ v(τ, x) -v(τ, r(x)) 1 ρ γ ′ s(x) ρ ∆s(x) dx dτ ≤ R t 0 µ 0 ∂Ω v(τ, r -s n(r)) -v(τ, r) 1 ρ γ ′ s ρ dr ds dτ ≤ R t 0 µ/ρ 0 ∂Ω v(τ, r -σρ n(r)) -v(τ, r) γ ′ (σ) dr dσ dτ = R ρ µ/ρ 0 σ γ ′ (σ)   t 0 ∂Ω v(τ, r -σρ n(r)) -v(τ, r) σρ dr dτ   dσ.
Letting ρ → 0 (notice that the second right-hand side term tends to 0) gives

lim ρ→0 t 0 Ω v ∆β = +∞ 0 σ γ ′′ (σ)   t 0 ∂Ω ∇v(τ, r) • n(r) dr dτ   dσ = - t 0 ∂Ω ∇v(τ, r) • n(r) dr dτ
As a consequence, Inequality (25) becomes

Ω |v(t, •)| + ε ∂Ω ∇v • n ≤ Ω |v(0, •)| + L [g] t 0 Ω |v| + t 0 Ω |h 1 |. ( 26
)
Now, we proceed to Step 2 (b) . Let us denote

z i = ∂v ∂x i , z = ∇v.
Then we have

∂z i ∂t + ∇ • ∂f ∂u (t, x, u) z i -ε∆z i = -h (i) 2 ,
with

h (i) 2 = ∂ 2 u D ∂x i ∂t + ∇ • ∂f ∂u (t, x, u) ∂u D ∂x i + ∇ • ∂f ∂x i (t, x, u) + ∂ ∂x i g(t, x, u) -ε∆ ∂u D ∂x i .
Multiplying the previous equation by ∂φ δ /∂ξ i (z), with φ δ (ξ) = (|ξ| 2 + δ 2 ) 1/2 , adding the terms (i = 1, d), we have, using the usual Einstein summation convention (i.e. whenever an index appears twice in one expression, the summation over this index is performed):

t 0 Ω ∂z i ∂t ∂φ δ ∂ξ i (z) = Ω φ δ (v(t, •)) - Ω φ δ (v(0, •)), t 0 Ω ∆z i ∂φ δ ∂ξ i (z) = - t 0 Ω ∂z i ∂x j ∂ 2 φ δ ∂ξ i ∂ξ k (z) ∂z k ∂x j + t 0 ∂Ω ∂z i ∂x j n j ∂φ δ ∂ξ i (z), t 0 Ω ∇ • ∂f ∂u (•, •, u) z i ∂φ δ ∂ξ i (z) = - t 0 Ω ∂f j ∂u (•, •, u) z i ∂ 2 φ δ ∂ξ i ∂ξ k (z) ∂z k ∂x j + t 0 ∂Ω ∂f j ∂u (•, •, u)n j z i ∂φ δ ∂ξ i (z).
Due to the estimate (obtained exactly as in [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF])

ε ∂z i ∂x j ∂ 2 φ δ ∂ξ i ∂ξ k (z) ∂z k ∂x j - ∂f j ∂u (•, •, u) z i ∂ 2 φ δ ∂ξ i ∂ξ k (z) ∂z k ∂x j = δ 2 (|z| 2 + δ 2 ) 3/2 ε|∇z| 2 - ∂f j ∂u (•, •, u) z i ∂z i ∂x j ≥ - 1 4ε ∂f ∂u (•, •, u) 2 δ 2 |z| 2 (|z| 2 + δ 2 ) 3/2 ≥ - L [f ] δ 4ε , we obtain for δ → 0 Ω z(t, •) -z(0, •) ≤ t 0 Ω |h 2 | + lim sup δ→0 t 0 ∂Ω ∂f ∂u (•, •, u) • n z j ∂φ δ ∂ξ j (z) -ε∇z i • n ∂φ δ ∂ξ i (z) .
Due to z = 0 on Σ T , we have on Σ

T z = ∇v = (∇v • n) n, ∆v = D 2 v(n, n) + ∆s∇v • n,
where D 2 v is the bilinear form of the second differential of v. Therefore, the integrand can be rewritten as

∂f ∂u (•, •, u) • n z j ∂φ δ ∂ξ j (z) -ε∇z i • n ∂φ δ ∂ξ i (z) = ∂f ∂u (•, •, u) • n |∇v| 2 (|∇v| 2 + δ 2 ) 1/2 -εD 2 v n, ∇v (|∇v| 2 + δ 2 ) 1/2 = ∂f ∂u (•, •, u) • ∇v -εD 2 v(n, n) ∇v • n (|∇v| 2 + δ 2 ) 1/2 . Moreover, ∇ • f (t, x, u) -f (t, x, u D ) = (∇ • f )(t, x, u) -(∇ • f )(t, x, u D ) + ∂f ∂u • ∇u - ∂f ∂u • ∇u D = (∇ • f )(t, x, u) -(∇ • f )(t, x, u D ) + ∂f ∂u (t, x, u) • ∇v - ∂f ∂u (t, x, u D ) - ∂f ∂u (t, x, u) • ∇u D .
Thus, for (t, x) ∈ Σ T , ∂f ∂u (t, x, u D ) -∂f ∂u (t, x, u) = 0, and we obtain

∂f ∂u (t, x, u) • ∇v = ∇ • f (t, x, u) -f (t, x, u D ) -(∇ • f )(t, x, u) -(∇ • f )(t, x, u D ) ,
in which a non classical contribution appears, due to the fact that f may depend on x. Since ∂v/∂t = 0 on Σ T , we have for (t, x)

∈ Σ T ∂f ∂u (t, x, u) • n z j ∂φ δ ∂ξ j (z) -ε∇z i • n ∂φ δ ∂ξ i (z) = ∂v ∂t + ∇ • f (t, x, u) -f (t, x, u D ) -ε∆v + ε∆s∇v • n ∇v • n (|∇v| 2 + δ 2 ) 1/2 -∇ • f (t, x, u) -∇ • f (t, x, u D ) ∇v • n (|∇v| 2 + δ 2 ) 1/2 = (-h 1 + ε∆s∇v • n) ∇v • n (|∇v| 2 + δ 2 ) 1/2 -∇ • f (t, x, u) -∇ • f (t, x, u D ) + g(t, x, u) -g(t, x, u D ) ∇v • n (|∇v| 2 + δ 2 ) 1/2 -g(t, x, u) -g(t, x, u D ) ∇v • n (|∇v| 2 + δ 2 ) 1/2 . Since (t, x) ∈ Σ T , we obtain (u = u D ): ∂f ∂u (t, x, u)•n z j ∂φ δ ∂ξ j (z)-ε∇z i •n ∂φ δ ∂ξ i (z) = (-h 1 + ε∆s∇v • n) ∇v • n (|∇v| 2 + δ 2 ) 1/2 .
Putting this in the last inequality gives:

Ω z(t, •) ≤ Ω z(0, •) + t 0 Ω |h 2 | + t 0 ∂Ω h 1 + εR ∇v • n which, together with Inequality (26) implies Ω ∇v(t, •) + R v(t, •) ≤ Ω ∇v(0, •) + R v(0, •) + t 0 Ω |h 2 | + t 0 ∂Ω h 1 and, as a consequence, Ω ∇u(t, •) + R u(t, •) ≤ Ω ∇u D (t, •) + R u D (t, •) + Ω ∇v(0, •) + R v(0, •) + t 0 Ω |h 2 | + t 0 ∂Ω h 1 . (27)
Let us analyse each term of Inequality (27).

◮ (step 2 (b) ) Analysis of Ω ∇u D (t, •) + R u D (t, •) . We easily state that Ω ∇u D (t, •) + R u D (t, •) ≤ c 6 := sup Q T ∇u D + R sup Q T u D |Ω| T. ◮ (step 2 (b) ) Analysis of Ω ∇v(0, •) + R v(0, •) .
Clearly, one has:

Ω ∇v(0, •) + R v(0, •) ≤ Ω ∇u 0 + ∇u D (0, •) + R u 0 + u D (0, •) ≤ c 7 := Ω ∇u 0 + R u 0 + sup Q T ∇u D + R sup Q T u D . ◮ (step 2 (b) ) Analysis of t 0 Ω |h 1 |. Recalling the expression of h 1 , t 0 Ω |h 1 | = t 0 Ω ∂u D ∂t + ∇ • (f (t, x, u D )) + g(t, x, u D ) -ε∆u D ≤ sup Q T ∂u D ∂t + ∆u D + sup Q T ×[a,b] |g| |Ω| T + t 0 Ω ∇ • (f (t, x, u D )). Since ∇ • (f (t, x, u D )) = ∇ • f (t, x, u D ) + ∂f ∂u (t, x, u D )∇u D , we have t 0 Ω |h 1 | ≤ c 8 , with c 8 = sup Q T ∂u D ∂t + ∆u D + L [f ] ∇u D + sup Q T ×[a,b] |g + ∇ • f | |Ω| T . ◮ (step 2 (b) ) Analysis of t 0 Ω |h 2 |.
First, let us develop the expression of h 2 :

h (i) 2 = ∂ 2 u D ∂x i ∂t +∇ • ∂f ∂u (t, x, u) ∂u D ∂x i + ∂ 2 f ∂u 2 (t, x, u) ∇u ∂u D ∂x i + ∂f ∂u (t, x, u) ∇ ∂u D ∂x i +∇ • ∂f ∂x i (t, x, u) + ∂ 2 f ∂x i ∂u (t, x, u) ∇u + ∂g ∂x i (t, x, u) + ∂g ∂u (t, x, u) ∂u ∂x i -ε∆ ∂u D ∂x i .
in which non classical contributions appear (in comparison with [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF]), due to the fact that f (resp. g) depends on (t, x) (resp. (t, x) and u). From this equality, we deduce the following estimate:

t 0 Ω |h 2 | ≤ c (1) 9 + c
(2) 9

t 0 Ω ∇u ≤ c 9   1 + t 0 Ω ∇u   with c 9 = max c (1) 9 , c (2) 9
and the following constants:

c (1) 9 = sup Q T ×[a,b] ∇ • ∂f ∂u sup Q T ∇u D + sup Q T ×[a,b] ∇ 2 f + sup Q T ×[a,b] ∇g |Ω| T +L [f ] Q T ∇ 2 u D + Q T ∇ 3 u D , c (2) 9 = sup
Q T ×[a,b] ∂ 2 f ∂u 2 sup Q T ∇u D + L [g] + sup Q T ×[a,b] ∇ • ∂f ∂u .

To conclude

Step 2, we gather Inequality (27) with all the previous bounds:

Ω ∇u(t, •) + R u(t, •) ≤ c 10  
where c 12 (= c 5 + c 11 , for instance) does not depend on ε. Applying Gronwall's lemma concludes the proof of Lemma 13.

Theorem 14 (Existence) Let us suppose that Assumption 1 holds. Let u ε be the unique solution of Eq. ( 11)-( 13) corresponding to the data (u 0 ε , u D ε ) satisfying Assumption 2 and let

lim ε→0 u D ε = u D in L 1 (Σ T ), lim ε→0 u 0 ε = u 0 in L 1 (Ω) where u D ∈ L ∞ (Σ T ; [a, b]) and u 0 ∈ L ∞ (Ω; [a, b]). Then, the sequence {u ε } ε converges to some function u ∈ L ∞ (Q T ; [a, b]) in C 0 ([0, T ], L 1 (Ω)).
Moreover u is a weak entropy solution of Eq. ( 1)-(3).

PROOF. Before entering into technical details, let us give the sketch of this proof. Our goal is to let ε tend to 0. Nevertheless, we cannot apply estimates stated in Lemma 13 on u ε because u D ε , u 0 ε satisfy compatibility conditions but do not necessarily have an extension over Q T with sufficient regularity. Thus, we introduce, by means of construction, (u D ε,h , u 0 ε,h ) which both satisfy compatibility conditions and have an extension over Q T with sufficient regularity. Moreover, (u D ε,h , u 0 ε,h ) are uniformly "close" to (u D ε , u 0 ε ) (as h → 0, uniformly w.r.t. ε), which implies that u ε,h is "close" to u ε (in a sense which will be precised further). Then, we apply Arzelà-Ascoli theorem on the sequence {u ε } in order to prove that it is relatively compact in C 0 ([0, T ]; L 1 (Ω)). Of course, we have to verify that the sequence satisfies the hypotheses of the theorem (equicontinuity and pointwise relative compactness): for this, we use the properties of u ε,h and the fact that u ε is "close" to u ε,h . In order to use Lemma 13, we need some extension of u D ε and u 0 ε to Q T , with sufficient regularity. Let us define the function u D,0 ε by

u D,0 ε (t, r + s n(r)) = u D ε (t, r), t ∈ (0, T ), r ∈ ∂Ω, |s| ≤ min(t, δ) u D,0 ε (t, x) = u 0 ε (x), -δ < t < min(dist(x, ∂Ω), δ), x ∈ Ω u D,0 ε (t, x) = 0, elsewhere.
Moreover, we mollify the above function (with a usual mollifier) which provides regularity on Q T :

u D,0 ε,h (t, x) = R d+1 u D,0 ε (t ′ , x ′ ) φ h (t -t ′ , x -x ′ ) dt ′ dx ′
Now we denote by u D ε,h (resp. u 0 ε,h ) the restriction of u D,0 ε,h to Σ T (resp. {0}×Ω). Let u ε,h be the solution of Eq. ( 11)-( 13) corresponding to the boundary and initial conditions u D ε,h and u 0 ε,h . On one hand, the uniform boundedness of u D ε , u 0 ε implies the uniform boundedness of u D ε,h , u 0 ε,h which provides (see Lemma 11) the uniform boundedness of u ε , u ε,h . Obviously, the following (strong) convergences hold:

lim h→0 u D ε,h = u D ε in L 1 (Σ T ), lim h→0 u 0 ε,h = u 0 ε in L 1 (Ω)
uniformly w.r.t. ε. This and Inequality [START_REF] Vázquez | Existence and uniqueness of solution for a lubrication problem with cavitation in a journal bearing with axial supply[END_REF] (see Lemma 12) imply

lim h→0 u ε,h = u ε in C 0 ([0, T ], L 1 (Ω)),
uniformly w.r.t. ε. On the other hand, it follows from the boundedness of

u D ε ∈ L 1 (Σ T ) and u 0 ε ∈ L 1 (Ω) that u D ε,h Σ T ≤ c h 3 , u 0 ε,h Ω ≤ c h 2 .
For fixed h > 0, it follows from Inequality (18) that the sequences

∂u ε,h ∂t , {∇u ε,h } are bounded in C 0 ([0, T ], L 1 (Ω)
). Now we propose to state that {u ε } ε is precompact in C 0 ([0, T ], L 1 (Ω)) with the Arzelà-Ascoli theorem:

(i) Equicontinuity of {u ε } ε : Let α > 0. There exists some h > 0 such that

2 Ω u ε,h (t, •) -u ε (t, •) dx < α/2, ∀t ∈ [0, T ], ∀ε > 0
and, from the uniform boundedness of ∂u ε,h /∂t, there exists δ > 0 such that

δ Ω ∂u ε,h ∂t (t, •) dx < α/2, ∀t ∈ [0, T ], ∀ε > 0 (30)
Thus, for all ε > 0 and all t 1 , t

2 ∈ [0, T ] such that |t 1 -t 2 | ≤ δ, we have Ω u ε (t 1 , •) -u ε (t 2 , •) ≤ 2 i=1 Ω u ε,h (t i , •) -u ε (t i , •) + Ω u ε,h (t 1 , •) -u ε,h (t 2 , •) ≤ 2 i=1 Ω u ε,h (t i , •) -u ε (t i , •) + t 1 -t 2 sup t∈[t 1 ,t 2 ] Ω ∂u ε,h ∂t (t, •) ≤ α Thus, the sequence u ε is equicontinuous in C 0 ([0, T ], L 1 (Ω)).
(ii) Pointwise relative compactness of {u ε } ε : For this, we use the Kolmogorov-Fréchet-Weil theorem:

⊲ Since {u ε } is uniformly bounded in L ∞ (Q T ), {u ε (t, •)} is also bounded in L 1 (Ω) (uniformly w.r.t. t ∈ [0, T ] and ε). ⊲ Let η > 0. Let us consider K η ⊂ Ω, defined by K η = {x ∈ Ω, dist(x, ∂Ω) ≥ η}. Obviously, K η is compact and sup uε(t,•) Ω\Kη u ε (t, •) ≤ max(|a|, |b|) meas(Ω \ K η ) = C(a, b, ∂Ω) η.
⊲ Recalling the existence of δ > 0 such that Inequality (30) holds, we get uniformly in t ∈ [0, T ] and ε > 0,

Ω ∆x u ε (t, • + ∆x) -u ε (t, •) ≤ 2 Ω u ε,h (t, •) -u ε (t, •) + |∆x| Ω ∇u ε,h (t, •)
which is smaller than α for |∆x| ≤ δ and Ω ∆x = {x ∈ Ω, x + ∆x ∈ Ω}. Thus, the sequence {u ε (t, •)} t∈[0,T ], ε>0 is relatively compact in L 1 (Ω). Thus, by the Arzelà-Ascoli theorem, {u ε } ε is precompact in C 0 ([0, T ], L 1 (Ω)), and since C 0 ([0, T ], L 1 (Ω)) is complete, we infer that, up to a subsequence,

lim ε→0 u ε = u in C 0 ([0, T ], L 1 (Ω)) Moreover, u ∈ L ∞ (Q T ; [a, b
]) (by passing to the limit on u ε ). Finally, u is a weak entropy solution of Eq. ( 1

( H k , Q k [f ] ) : R 2 -→ R 2 (z, w) -→ dist(z, I(w, k)), F [f ] (•, •, z, w, k) with I(w, k) = [min(w, k), max(w, k)] and F [f ] ∈ C(R × Ω × R 3
) defined as:

F [f ] (•, •, z, w, k) =                                  f (•, •, w) -f (•, •, z) for z ≤ w ≤ k 0 for k ≤ z ≤ w f (•, •, z) -f (•, •, k) for w ≤ k ≤ z f (•, •, k) -f (•, •, z) for z ≤ k ≤ w 0 for w ≤ z ≤ k f (•, •, z) -f (•, •, w) for k ≤ w ≤ z Lemma 16 Let u ∈ L ∞ (Q T ) satisfy (P SK ); then one has: for all ϕ ∈ D((0, T ) × R d ), for all k ∈ R, Q T |u -k| ∂ϕ ∂t + sgn(u -k) f (t, x, u) -f (t, x, k) ∇ϕ -sgn(u -k) (∇ • f (t, x, k) + g(t, x, u)) ϕ dx dt ≥ ess lim ̺→0 + Σ T sgn(u(t, r -̺ n(r)) -k) f (t, r -̺ n(r), u(t, r -̺ n(r))) -f (t, r -̺ n(r), k) • n(r) ϕ(t, r) dγ(r) dt,
for all β ∈ L 1 (Σ T ), β ≥ 0 a.e., and for all k ∈ R, ess lim

̺→0 + Σ T F [f ] (t, r, u(t, r -̺ n(r)), u D (t, r), k) • n(r) β(t, r) dγ(r) dt ≥ 0, for all ϕ ∈ D((0, T ) × R d ), for all k ∈ R, Q T |u -k| ∂ϕ ∂t + sgn(u -k) f (t, x, u) -f (t, x, k) ∇ϕ -sgn(u -k) (∇ • f (t, x, k) + g(t, x, u)) ϕ dx dt ≥ Σ T sgn(k -u D ) f (t, r, k) -f (t, r, u D ) • n(r) ϕ(t, r) dγ(r) dt -ess lim ̺→0 + Σ T sgn(u(t, r -̺ n(r)) -k) f (t, r, u(t, r -̺ n(r))) -f (t, r, k) • n(r) ϕ(t, r) dγ(r) dt.
with the notation F (t, r, u, k) = sgn(u -k) f (t, r, u) -f (t, r, k) , and the result is straightforward.

Lemma 17 Let u ∈ L ∞ (Q T ) (resp. v ∈ L ∞ (Q T )) be a solution of (P SK ) with data (u 0 , u D ) ∈ L ∞ (Ω) × L ∞ (Σ T ) (resp. (v 0 , v D ) ∈ L ∞ (Ω) × L ∞ (Σ T )); then - Q T u -v ∂β ∂t + sgn(u -v) f (t, x, u) -f (t, x, v) ∇β -sgn(u -v) (g(t, x, u) -g(t, x, v)) β dx dt ≤ Ω u 0 (x) -v 0 (x) β(0, x) dx + L [f ] Σ T u D (t, r) -v D (t, r) β(t, r) dγ(r) dt for all β ∈ D((-∞, T ) × R d ).
PROOF. As already pointed out, any term written under the form ess lim

̺→0 + Σ T sgn(u(t, r -̺ n(r)) -v D (t, r)) f (t, r, u(t, r -̺ n(r))) -f (t, r, v D (t, r)) • n(r) β(t, r) dγ(r) dt
exists for all β ∈ L 1 (Σ T ), all v D ∈ L ∞ (Σ T ). Thus, we infer that there exists θ i,j ∈ L ∞ (Σ T ) such that: After this introduction of notations, we now apply the double variable method, initiated by Kružkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF], to the 3rd inequality stated in Lemma 16. Let ρ ε ∈ D(R d+1 ) be a symmetric regularizing sequence. We Now changing the role of (u(p),p) and (v(p ′ ),p ′ ), we get in a similar way, the inequality J ε 1 + J ε 2 + J ε 3 + J ε 4 + J ε 5 ≤ J ε 6 + J ε 7 , with Adding the two inequalities, and noticing that I ε 1 = J ε 1 and I ε 3 = -J ε 3 , we get 2I ε 1 + (I ε 2 + J ε 2 ) + (I ε 4 + J ε 4 ) + (I ε 5 + J ε 5 ) ≤ (I ε 6 + J ε 6 ) + (I ε 7 + J ε 7 ).

Σ T
I ε 1 = - 1 2 Q T Q T u(p) -v(p ′ ) ∂β ∂t p + p ′ 2 ρ ε (p -p ′ ) dp dp ′ I ε 2 = - 1 2 Q T Q T F (p, u(p), v(p ′ )) ∇β p + p ′ 2 ρ ε (p -p ′ ) dp dp ′ I ε 3 = - Q T Q T u(p) -v(p ′ ) ∂ρ ε ∂t (p -p ′ ) β p + p ′ 2 dp dp ′ I ε 4 = - Q T Q T F (
J ε 1 = - 1 2 Q T Q T v(p ′ ) -u(p) ∂β ∂t p + p ′ 2 ρ ε (p -p ′ ) dp dp ′ J ε 2 = - 1 2 Q T Q T F (p ′ , v(p ′ ), u(p)) ∇β p + p ′ 2 ρ ε (p -p ′ ) dp dp ′ J ε 3 = Q T Q T v(p ′ ) -u(p) ∂ρ ε ∂t (p -p ′ ) β p + p ′ 2 dp dp ′ J ε 4 = Q T Q T F (p ′ , v(p ′ ), u(p)) ∇ρ ε (p -p ′ ) β p + p ′ 2 dp dp ′ J ε 5 = Q T Q T sgn(v(p ′ )-u(p)) ∇•f (p ′ , u(p))+g(p ′ , v(p ′ )) β p + p ′ 2 ρ ε (p -p ′ ) dp dp ′ J ε 6 = Q T Σ T
We are now ready to let ε tend to 0. Note that this method has been widely used in the works related to hyperbolic problems [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF][START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF][START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF] but also parabolic problems [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF] or elliptic-hyperbolic problems [START_REF] Martin | Contribution à la modélisation de phénomènes de frontière libre en mécanique des films minces[END_REF][START_REF] Alvarez | A free boundary problem in theory of lubrication[END_REF][START_REF] Alvarez | On the uniqueness of the solution of an evolution free boundary problem in theory of lubrication[END_REF][START_REF] Vázquez | Existence and uniqueness of solution for a lubrication problem with cavitation in a journal bearing with axial supply[END_REF]. Thus, due to the convolution effect of ρ ε , we obviously obtain From Gronwall's lemma, we conclude that:

u(t 1 , •) -v(t 1 , •) L 1 (Ω)
≤ u(t 0 , •) -v(t 0 , •)

L 1 (Ω)
e L [g] (t 1 -t 0 ) .

As t 0 tends to 0, and using the fact that v 0 = u 0 along with the initial condition ( 6), the uniqueness is straightforward.

  be a convex function such that there exists w ∈ [a, b] with η(w) = 0 and η ′ (w) = 0. Then η can be uniformly approximated on [a, b] by applications of the kind

Lemma 4 (

 4 , b] and κj ∈ [a, b]. (ii) Conversely, there exists a sequence of "boundary entropy-flux pairs which converges to the "semi Kružkov entropy-flux pairs". Boundary condition) Let u ∈ L ∞ (Q T ) satisfying (P SK ). Then, ess lim

  , r -σρ n(r)) -v(τ, r) ρ γ ′′ (σ) dr dσ dτ = , r -σρ n(r)) -v(τ, r) σρ dr dτ   dσ. Now let us focus on the second right-hand side: since |∆s|

3 UniquenessDefinition 15

 315 limit w.r.t. ε in Inequality (15) concludes the proof. For any k ∈ R, let us denote:

θ 1 , 1

 11 (t, r)β(t, r) dγ(r) dt = ess lim̺→0 + Σ T sgn(u(t, r -̺ n) -u D ) f (t, r, u(t, r -̺ n)) -f (t, r, u D ) • n β dγ(r) dt, Σ T θ 1,2 (t, r) β(t, r) dγ(r) dt = ess lim ̺→0 + Σ T sgn(u(t, r -̺ n) -v D ) f (t, r, u(t, r -̺ n)) -f (t, r, v D ) • n β dγ(r) dt, Σ T θ 2,2 (t, r) β(t, r) dγ(r) dt = ess lim ̺→0 + Σ T sgn(v(t, r -̺ n) -v D ) f (t, r, v(t, r -̺ n)) -f (t, r, v D ) • n β dγ(r) dt, Σ T θ 2,1 (t, r) β(t, r) dγ(r) dt = ess lim ̺→0 + Σ T sgn(v(t, r -̺ n) -u D ) f (t, r, v(t, r -̺ n)) -f (t, r, u D ) • n β dγ(r) dt.

  will denote p = (t, x) ∈ Q T , p ′ = (t ′ , x ′ ) ∈ Q T , γ(p) = (t, r) ∈ Σ T , γ(p ′ ) = (t ′ , r ′ ) ∈ Σ T ,and letβ ε (p, p ′ ) = β p + p ′ 2 ρ ε (p -p ′ ) , for all p, p ′ ∈ (Q T ) 2 , for a given β ∈ D((0, T ) × R d ), β ≥ 0. Hold p ′ ∈ Q Tfixed and replace, in the 3rd inequality of Lemma 16, k by v(p ′ ) and β(p) by β ε (p, p ′ ). After integration over Q T (with respect to the variable p ′ ), and using the notationF (p, u(p), v(p ′ )) = sgn(u(p) -v(p ′ )) (f (p, u(p) -f (p, v(p ′ )))) ,we easily getI ε 1 + I ε 2 + I ε 3 + I ε 4 + I ε 5 ≤ I ε 6 + I ε 7 , with

  p, u(p), v(p ′ )) ∇ρ ε (p -p ′ ) β p + p ′ 2 dp dp ′ I ε 5 = Q T Q T sgn(u(p)-v(p ′ )) ∇•f (p, v(p ′ ))+g(p, u(p)) β p + p ′ 2 ρ ε (p -p ′ ) dp dp ′ I ε 6 = Q T Σ T θ 1,1 (γ(p)) β γ(p) + p ′ 2 ρ ε (γ(p) -p ′ ) dγ(p) dp ′ I ε 7 = -Q T Σ T F (p, v(p ′ ), u D (γ(p))) • n β γ(p) + p ′ 2 ρ ε (γ(p) -p ′ ) dγ(p) dp ′ .

θ 2 , 2

 22 (γ(p ′ )) β p + γ(p ′ ) 2 ρ ε (p -γ(p ′ )) dγ(p ′ ) dp J ε 7 = -Q T Σ T F (p ′ , u(p), v D (γ(p ′ ))) • n β p + γ(p ′ ) 2 ρ ε (p -γ(p ′ )) dγ(p ′ ) dp.

  -v) (f (t, x, u) -f (t, x, v))) ∇β,

  10 = c 6 + c 7 + c 8 + c 9 does not depend on ε. Finally, since u is a function with values in [a, b], from the previous inequality, we infer that there exists c 11 which does not depend on ε such that

									t	
									0 Ω	∇u	
	where c Ω	∇u(t, •) ≤ c 11	  1 +	t 0 Ω	∇u	 		(28)
	Now, we gather results obtained in Step 1 and Step 2: using Inequalities
	(22) and (28) gives
	Ω	∂u ∂t	(t, •) + ∇u(t, •) ≤ c 12	  1 +	t 0 Ω	∂u ∂t	+ ∇u	 

1 +

This lemma, see Saks[START_REF] Saks | Theory of the integral[END_REF], says that if v ∈ C 1 (Ω), then limη→0 Ω |∇v| 1 [|v(x)|≤η] = 0.

Acknowledgements

The author would like to thank Julien Vovelle for fruitful discussions.

PROOF.

⊲ 1st inequality -Adding the two inequalities defined by (P SK ) with each "semi Kružkov entropy-flux pair" gives the following inequality

for any ϕ ∈ D(Q T ). Thus, since u satisfies Inequality (31) along with the initial condition [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF] (see Lemma 5), the result is obtained by following the same lines of the proof of Lemma 7.12. in [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF].

⊲ 2nd inequality -The result is easily obtained by Lemma 4 applied to the particular "boundary fluxes" F [f ] (see Definition 15).

⊲ 3rd inequality -On the one hand, the function 2F

On the second hand, terms of the form ess lim

Indeed, this term is obtained by using the proof of Lemma 4: it is sufficient to add the terms of (5) corresponding to each "semi Kružkov entropy-flux pair". Therefore, each term of the following inequality exists. Thus, from the 2nd inequality, -ess lim

Now, let us focus on I ε 4 + J ε 4 + I ε 5 + J ε 5 which contains all the non classical contributions of the source term g and the non-autonomous property of the flux f . Interestingly, we will see that only the source term plays a role when passing to the limit on ε: this is because of the conservative form of the scalar conservation law. Let us reorganize the sum by rewriting it in the following form:

2 ) + K

(1)

Passing to the limit in K

(1)

3 obviously gives:

3 -K

Moreover, the limit of K

(1)

2 ) may be analysed exactly in the same manner as in the paper of Kružkov (see p. 227 in [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]) and we have:

Finally we obtain:

-

As in [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF], let us introduce the following definition:

Then, when discussing the cases, one sees that for all z 1 , z 2 , w 1 , w 2 ,

holds and using the property

one easily concludes that 1 2

The initial term is obtained by slightly modifying the proof, with test functions in the appropriate space, namely D((-∞, T ) × R d ).

Theorem 18 (Uniqueness) Under Assumption 1, problem (P SK ) admits a unique weak entropy solution.

PROOF. Considering the integral inequality of Lemma 17 with v D = u D and v 0 = u 0 and a test function which only depends on time t, we get:

for all α ∈ D(-∞, T ). Then, for an interval [t 0 , t 1 ] ⊂]0, T [, we can use in Inequality (32) the characteristic function of [t 0 , t 1 ], properly mollified, and pass to the limit on the mollifier parameter:

sgn(u -v) (g(t, x, v) -g(t, x, u)) dx dt.

Now, since we have, for all (t, x) ∈ (0, T ) × Ω: sgn(u -v) (g(t, x, v) -g(t, x, u)) ≤ L [g] u -v , we obtain:

dt.