domain: sdv shs

text:
Elsa Faugère

Monica Castro

Guillaume Ollivier

Taxonomy's rebirth triggered by the international biodiversity policies : An historical and anthropological approach

Introduction

In 1985, the neologism « biodiversity » was invented by Walter G. Rosen during the preparation of the National Forum on Biological Diversity held in 1986 in Washington D.C. (Le Guyader 2008 : 7). For Marris 2006 and other authors, it is during this National Forum that the biologists gave the first important alert concerning the crisis of biodiversity which had a political and mediatised impact. For these biologists, biological diversity was disappearing so quickly that some of them do not hesitate any longer to call this crisis the 6 th extinction, even though such an expression is disputed by other biologists (Le Guyader 2008: 1718). A few years before this alert and the invention of the concept of biodiversity, a fogging experiment was conducted in 1982 by Terry Erwin on an isolated tree in the Amazonian forest. This fogging experiment had an unexpected result. It showed that the magnitude of species biodiversity was much higher than what was suspected before (Le Guyader 2008 : 8). This result was very instructive, Le Guyader goes on to say: « The naturalists' dream -to describe exhaustively all the living species, to identify and to classify them vanished. The discrepancy in the order of magnitude (from 10 to 100 million species) shows that we are incapable of quantifying the number of species. And we realized then that to study species biodiversity, it will be necessary to use a much bigger strike force than the work of some isolated naturalists who work alone, without resources, during short period of time in rich environments" (Le Guyader 2008)1

In the 80's, the biologists discovered simultaneously how very ignorant they were about species biodiversity and how much this species biodiversity was threatened. These discoveries would be crucial in driving several international initiatives which, during the 90's and the 2000's would try to provide some solutions to the following two problems:

The clear lack of scientific knowledge on species biodiversity The necessity and the urgency of elaborating conservationists' policies on biodiversity before its disappearance.

The scientific discipline whose aim is to collect, to identify and to classify new species is taxonomy. But taxonomy is also in crisis, said all the specialists. At the beginning of the 90's the biologists considered that there was a "taxonomy impediment". This taxonomy impediment would considerably restrict the urgent production of new scientific knowledge on biodiversity and also the urgent need for new policies of conservation.

"The taxonomic impediment is a term that describes the gaps of knowledge in our taxonomic system (including knowledge gaps associated with genetic systems), the shortage of trained taxonomists and curators, and the impact these deficiencies have on our ability to manage and use our biological diversity". (Darwin Declaration, 1998, UNEP/CBD/COP/4/Inf.28, 2 may 1998).

During the 1990's and the 2000's many international answers have tried to overcome this taxonomy impediment. We can classify those answers in two groups. The first group of answers are international initiatives, governmental and non governmental, trying to reinvent (or to modernize) taxonomy by using informatics, Internet and genetics in order to create new virtual and global networks of taxonomists all around the world or in order to change the way by which taxonomists identify new species.

Some examples of multigovernmental initiatives are :

Since the first meeting (1993) of the Conference of the Parties of the Convention on Biological Diversity, the "taxonomy impediment" was highlighted. The experts said that it could prevent the implementation of the CBD. Such considerations would lead to the creation, in 2002, of the Global Taxonomy Initiative which was created to remove or reduce the "taxonomic impediment". "The GTI has been established to address the lack of taxonomic information and expertise available in many parts of the world, and thereby to improve decisionmaking in conservation, sustainable use and equitable sharing of the benefits derived from genetic resources". This is the first time in history that taxonomy has such recognition at this high level in international policy. (http://www.cbd.int/gti/)

In 2001, in another governmental international organization, the Organization for Economic Cooperation and Development (OECD) (OCDE in French), the Biodiversity Informatics Subgroup of the Megascience Forum created the Global Biodiversity Information Facility (GBIF). The aim of the GBIF is to create a virtual portal on the Internet in order to make access to primary biodiversity data, free and open, for every one (individual, scientists, managers, politics, etc.). It is a meta data base on biodiversity. "Facilitating digitisation and global dissemination of primary biodiversity data, so that people from all countries can benefit from the use of the information, is the mission of the Global Biodiversity Information Facility" http://www.gbif.org/GBIF_org/bg10#whatdo) Some examples of non governmental initiatives are :

In 1993, creation of the Global Network for Taxonomy, called BionetInternational, dedicated to promoting taxonomy in the southern countries (http://www.bionet intl.org/opencms/opencms/whoWeAre/history.jsp

In 2002, creation of the Consortium for the Barcode of Life (CBOL), devoted to developing DNA barcoding (le codebarre de l'ADN) as a global standard for the identification of biological species. DNA barcode sequences can be obtained reasonably quickly and cheaply". http://barcoding.si.edu/index.html, etc.

The second kind of answer giving to overcome the taxonomic impediment is the organization of a new kind of naturalist expedition. The first expedition of this kind was organized by some French biologists in 2006, and was led on the island of Santo (Vanuatu, South Pacific). Its aim was to product new scientific knowledge on biodiversity before its extinction. This list is not at all exhaustive. But it shows the existence of new dynamics, carried by different kinds of social actors at different levels, all dedicated to the same goal: to overcome the taxonomy impediment. Their common aim is to "modernize" taxonomy by using new technologies of information and communication (informatics and Internet) and genetics, so that taxonomy can be able to take on the new challenges identified by the biologists that is : to accelerate the production of new scientific knowledge on biodiversity ; and to contribute to a better management (or conservation) of biodiversity . Between march 2005 and november 2006, I studied, as an ethnographer, the preparation and the implementation of the biggest naturalist expedition of all time, according to the media. It had gathered, during 4 months, more than 150 scientists coming from 20 different countries, for the same goal: to make an inventory of all the biodiversity of Santo's Island. For the three organizers (Philippe Bouchet, from the National Museum of Natural History of Paris, Hervé Le Guyader (Institut de recherche pour le développement (IRD) and Olivier Pascal (PronaturaInternational, an NGO), the challenge and the main goal of this expedition was to document and to record primary data on all of the biodiversity of Santo before its disappearance. It was then to produce new scientific knowledge on the unknown biodiversity of this island. The goal of this expedition was clearly within the tradition of Natural History as it was created and developed since the 18 th century. If this expedition seems to look like its predecessors, a more precise analysis shows that it tried to renew this style by being embedded in contemporary geopolitical and scientific context and challenges, trying to overcome the taxonomy impediment. If we can argue that, it is because of several of its specificities. The first one is the size of the expedition. By gathering 150 scientists for the same goal on the same island during 4 months, the organizers thought it will help to produce a great amount of new scientific taxonomical knowledge on biodiversity. The second one is the use of a new engine, called "Canopy Glider" (arboglisseur) which was very important to attract the media and private sponsors. Because of the size of this expedition and of the use of the Canopy Glider, the global cost (1,1 millions Euros) was high. But it served to attract both the media and private sponsors (mécènes). The privatization of the funding of this expedition and its media coverage were both two important features of Santo 2006. The media coverage was a real success for the organizers. Many journalists from radio, television, newspapers, well covered the event. From the point of view of the organizers, the media coverage is one crucial goal for changing public and political perceptions of the crisis of biodiversity.

We can read on the website of Pronatura International: "With Santos2006, we were able to show:

• That the necessary change of scale for the knowledge of biodiversity is possible and brings quantifiable results. Thanks to the human and technical resources available in this unprecedented operation, more than 10,000 species were inventoried, hundreds of which are new to science. While the field of investigation in biodiversity is overwhelming, we confirmed that the challenge can be met with meticulous and exhaustive exploration techniques.

• That it is possible to share the results of a difficult subject with a large audience by integrating the different domains of science, technology and communication in the same operation. The diffusion of the operation through different media channels (Internet, daily publications, films and books) reached a large audience" http://www.pronatura.org/?lang=en&page=biodiversity, page consultée le 15 mai 2008)

The third specificity of the expedition is more ethical and political. From the beginning, the organizers announced the crucial importance for them to respect the CBD's spirit. What does this mean? It means first to obtain the Prior Informed Consent of local populations. It means also to share, with the inhabitants of Santo, some benefits of the access and uses of biodiversity. Concretely these benefits consisted of:

the restoration of a local ship funded by the expedition (the cost was 10% of the total budget of the expedition) ; the implication of some NiVanuatu students, researchers and governmental officers in the expedition ; some forms of local and national restitution : communication, reports and a final collective book (in progress) called the Natural History of Santo.

For the organizers, it was essential to respect the "CBD's spirit". It was a matter of ethics, of politics but also of science. They know that it is a sine qua non condition if they want to organize others such expeditions in southern countries, which means being able to pursue their work of documenting and recording biodiversity before its loss.

The fourth specificity of the expedition was the use, for marine biodiversity, of DNA barcoding, along with morphological identifications. DNA barcoding is a new method used for the identification of species:

"A molecular barcode is a sequence of DNA present in all living organisms. And this sequence of DNA is the very same in all specimens of the same species. It then allows determining which species a specimen belongs to even if we only know its DNA's sequence. It can be used as a taxonomical identification tool for living organisms, in the same way as barcodes for commercial products" (Nicolas Puillandre, http://acces.inrp.fr/santo/donnees/biodivmarine/barcoding/questcequelebarcode, page consultée le 21 mai 2008).

In Expedition Santo 2006, the presence of both "barcoders" and morphological taxonomists leaded to some disputes and controversies. The barcoders tended to qualify the morphological taxonomists as "collectors of rare poststamps". And the morphological taxonomists tended to see the molecularists as young "idiots" who where only interested by small pieces of flesh of animals, and who were incapable of recognizing any specimen with their own eyes. One challenge of this expedition was to gather and to conciliate those two different methods of identifying species.

Our Planet Reviewed. Biodiversity Hotspots : Taking a closer look at the 'rich and famous'.

For the organizers, this expedition was such a success that they have implemented a new strategic plan "for filling key gaps in our knowledge of the world's major biodiversity areas". This plan is called "Our Planet Reviewed. Biodiversity 'hotspots' : taking a closer look at the 'rich and famous". It consists on "expeditions over the next decade that will focus on littleknown sites and on neglected animal and plant groups (invertebrates, fungi…), especially those whose study requires special organizational skills and logistical resources". On the website of PronaturaInternational, they explain their new strategy in the following way:

"Despite significant efforts made over the last several decades, the biodiversity of our planet remains largely unknown and is disappearing many times faster than we discover it. ProNatura International and its partner, The National Museum of Natural History (MNHN) in Paris, are now developing a new initiative to help address this issue, aimed at significantly boosting our knowledge of Earth's biodiversity by filling gaps through the exploration and description of a carefully selected set of key sites.

The new challenge for our project teams is to apply their scientific and technical knowledge in regions of the world where the most pressing issues of Biodiversity and Conservation converge. We will thus focus on areas that are recognized as global hotspots for biodiversity, but where numerous knowledge gaps constitute a serious impediment to sound conservation and resource management". http://www.pronatura.org/projects/planet_reviewed_en.pdf, page consultée le 15 mai 2008.

Conclusion

To conclude, we can say that the international taxonomical landscape (or network) seemed to have change since the 80's, thanks to the politicization (mise en politique) of the "taxonomy impediment". One of the most important innovations brought by the Expedition Santo 2006 and the program Our Planet Reviewed is maybe their focus on invertebrates. Indeed, invertebrates constitute the most important part of biodiversity, in quantitative terms, but maybe also in terms of ecosystem services. And it is the less studied and the less known part of biodiversity.

It is also the part of biodiversity that interests the least "public opinion", media, politics, but also the international conservationist community and even the biologists, as some members of Santo 2006's expedition told me. For all of them (except invertebrate specialists), it is much more prestigious to work on mammals or birds than on arthropods or molluscs. We can then consider that terrestrial and marine invertebrates have found new "spokesmen" and a new "apparatus" (dispositif) to exist, thanks to this new kind of expedition. By succeeding in interesting the media and a large sector of the public in those little neglected animals, some of which are even invisible to the human eye, this new kind of naturalist expedition may contribute to changing the occidental relationship to nature.

2

  Towards a new kind of naturalist expedition on biodiversity : the case of Santo 2006 2 Organized by : Philippe Bouchet (National Museum of Natural History, Paris), Olivier Pascal (Pronatura International), Hervé le Guyader (Institute for Research and Development, IRD). Contributing Foundations were the following: Stavros S. Niarchos Foundation, Total Foundation for Marine Biodiversity, Veolia Environnement Foundation, The Alfred P. Sloan Foundation, European Distributed Institute of Taxonomy, Fonds Pacifique, the French Ministry of Research, and the French Embassy in Vanuatu. Partner corporations were: Universal Sodexho, Ricoh, Solvay, Océan Vert, National Geographic, VRAI d'agriculture biologique, Telecom Vanuatu Limited.

« Le rêve des naturalistes -décrire exhaustivement l'ensemble des espèces, les répertorier et les classer, s'éloigne. Ensuite, le désaccord sur l'ordre de grandeur (de 10 à 100 millions d'espèces) indique que l'on est incapable de quantifier le nombre d'espèces. Et enfin, étudier la « biodiversité spécifique » nécessite une toute autre force de frappe que celle de quelques naturalistes isolés réalisant, sans trop de moyens, des missions de courte durée dans des environnements riches ».


Changes in carbon metabolism

Gas exchanges : photosynthesis ; respiration

Glutathione Glutathione Reductase

-this metabolite acts to reduce Asc through glutathione reductase activity.

Reducing power (NADPH & NADH)

-these nucleotides are the driving force of the detoxifying system (for regeneration of ascorbate pool).

It is necessary to

O 3 O 3
The sensitivity of a leaf to ozone lies on the stomatal conductance, on the existing scavenging system and the detoxifying capacity (regeneration).

The aim is to determine a subcellular model taking into account these aspects. Increasing tolerance to ozone by elevating foliar ascorbic acid confers greater protection against ozone that increasing avoidance (Chen and Gallie, 2005, Plant Physiology, 138, 1673-1689) Increasing the level of ascorbate through enhanced ascorbate recycling provided greater protection against oxidative damage than reducing stomatal area aperture.

Our data lead to hypothesize that O3-tolerance in bean depends more on a superior potential cultivar-specific ability more than stress-induced physiological and biochemical adjustments to avoiding and countering stressinduced oxidative damage. (Guidi et al., Environmental pollution, 2010, 158, 3164-3171) Good correlation between tolerance to ozone and high endogenous levels of antioxidant metabolites such as AA and GSH in tobacco (Pasqualini et al., Plant cell Environment, 2001, 24, 245-252) Ozone flux was best explained by stomatal conductance and symplastic (rather than apoplastic) ascorbate explaining 66% of the total variation in leaf flux. It is important to measure processes other than stomatal conductance to explain steady-state leaf-level fluxes of pollutant gases. (Eller and Parks, 2006, Plant Cell Environment, 29, 1742-1750).

Other factors than constitutive total apoplastic ascorbate contribute to the differential ozone tolerance of two clones of trifolium repens. The symplastic redox status could play a role (D'Haese et al., 2005, Plant Cell Environment, 28, 623-632).

The increase in foliar ascorbate under ozone stress in the sensitive poplar clone Eridano, with a greater stomatal conductance, was insufficient to counteract ROS accumulation and the consequent oxidative stress. The higher influx of ozone into Eridano leaves compared with the tolerant clone I-214 resulted in a lower potential detoxification capacity per unit of ozone influx. (Di Baccio et al., 2008, Tree Physiol., 28, 1761-1772).

Recent developments highlight the complexity of redox-dependent defence reactions and the importance of interactions between the reduction state of soluble redox couples and their concentration in mediating dynamic signalling in response to stress -i.e. reactive oxygen species (Noctor, 2006, Plant cell Environment, 29, 409-425).

COST

  confirm that ozone stimulates the activity of the cytosolic enzymes providing NADPH.

FP0903

  indirect radiative forcing due to O 3 increase (part of nonassimilated CO 2 ) For comparison, estimates of the direct radiative forcing due to O3 increase only are shown by the square and the triangle Increasing ozone concentration in the atmosphere decreases plant productivity Ozone will lower the capacity of plants to sequester the increasing level of carbon (rise in atmospheric CO 2 ) Consequence: indirect radiative forcing of ozone, due to additional level of CO 2 in the atmosphere, which is added to the direct radiative forcing

2 / 12 CO 2 WUE Cuticular non-stomatal deposition

	CO 2
	Extracellular space	Apoplast	Plasmalemma	Symplast		PAR	Chloroplast	CO 2
		Cell wall		Cytosol			PSII	Photochemistry	Rubisco
								ATP	Calvin
								NADPH	cycle
					Detoxification
						Asc	GSSG	NADPH
								Trioses Phosphate
						DHA	GSH	NADP	NTR/Trx
								Catabolism
		Asc		Asc	GSSG	NAD(P)H	NAD(P)H
		DHA		DHA	GSH	NAD(P)	Krebs cycle
								NTR/Trx	Mitochondria

Δ 13 CO 4.Regeneration of antioxidants Dizengremel et al., 2008, Env. Pollution COST FP0903, Rome, October 2010

An An overview overview of of cell cell metabolism metabolism PAR Calvin cycle Photochemistry Rubisco NADPH ATP PSII CO 2 Trioses Phosphate Catabolism NAD(P)H

	CO 2
			Detoxification
				Glycolysis	HMP
	Asc	Asc	GSSG
	DHA	DHA	GSH	NAD(P)

Apoplast Extracellular space Cytosol Cell wall Symplast ROS ROS O 3 1st existing detoxifying barrier ROS O 3 Asc DHA Detoxification Detoxification PEP PEPc Mitochondria Krebs cycle How ozone impacts How ozone impacts cell cell metabolism metabolism

	Plasmalemma	Rubisco C O 3	PEPc C O 3	Rubisco / PEPc ratio C O 3
			Asc	GSSG	NADPH
			DHA	GSH	NADP	NTR/Trx
			NAD(P)H
					NTR/Trx
					COST FP0903, Rome, October 2010

Poplar 60 ppb O 3 , 2 weeks 9.1 17.0 1.35 0.36 6.74 47.2 Effect Effect of ozone on of ozone on carboxylation carboxylation Rubisco & PEPc Norway spruce 200 ppb O 3 , 12 weeks

	CO 2
	Specific activity RubisCo		Calvin	Chloroplast α-cetoglutarate
	(nkat.mg prot Malate -1 )		cycle			cycle GS-GOGAT
	AOA	NADPH		Trioses P	AA
	6.2 Glucose 6P 0.45 Gal3PDH 13.2	2.55 G6PDH	2.43 NADPH 29.3
	Aleppo pine AOA 200 ppb O 3 , 5 weeks CO 2 NADPH PEPc 13.6	PEP 7.5	0.55	1.96 HMP	24.7	α-cetoglutarate 3.82
	NADPH	Pyruvate	NADPH
	Malate		ME	Pyruvate		IDH
	Malate	Krebs cycle	citrate /isocitrate	isocitrate
					Mitochondrion	Dizengremel et al., 2009, Plant Biology

Fontaine et al. (1999), Physiologia Plantarum Pelloux et al. (2001), Plant, Cell & Environment Dizengremel (2001), Plant Physiol. Biochem. Fontaine et al. (2003), Physiologia Plantarum COST FP0903, Rome, October 2010

Effect Effect of ozone on enzymes of ozone on enzymes linked linked to to catabolism catabolism Glycolysis, Krebs cycle and anaplerotic pathway

			Which metabolites/enzymes are relevant to measure?
			Ascorbate	-total Asc (symplastic + apoplastic) could be a good indicator
	Norway spruce Ascorbate peroxydases of ozone resistance.		Poplar
			200 ppb O 3 , 12 weeks			100 ppb O 3 , 24 days
					PEPc		5
	Relative activity (O 3 /C ratio)	5 10 1			G6PDH Fumarase PFK	Relative activity (O 3 /C ratio)	1 2 3 4	control ozone
		0					0
		0	2 0 Time of fumigation (days) 4 0 6 0	8 0		PEPc	PK Fum NAD EM	NADP EM
	Sehmer et al. (1998), Physiologia Plantarum; Dizengremel et al. (2009), Plant Biology
							COST FP0903, Rome, October 2010

Robusta (sensitive +++) Cima (tolerant) Carpaccio (tolerant) Parameters : Gas exchanges + Chlorophyll Growth + Leaf surface Ascorbate + glutathione NADP(H) + NADPH generating enzymes PEPc + RubisCo Guard cells transcriptomic Metabolomic

  Poplar Experiment in phytotronic chambers• 3 genotypes Pop. deltoïdes × Pop. nigra (Carpaccio, Cima, Robusta).

	Towards Towards a model for Ozone a model for Ozone Risk Risk Assessment Assessment : :
	from from atmosphere atmosphere to to cell cell
	• Ozone 120ppb
	Stomatal conductance
	Robusta
	Cima
	Carpaccio

The ozone group

The ozone group at at Nancy Nancy University University (4)


Etat et Marché dans le contexte de la globalisation économique

Jacques Fontanel

In

Evaluation des politiques publiques

Université Pierre Mendes France Grenoble, Fragments de cours Grenoble, 2004Grenoble, .2004 Résumé : Dans le cadre du processus de globalisation économique, l'Etat devient un agent économique déclinant, contrôlé par des organisations internationales qui fixent des règles qu'il ne peut pas souverainement contrevenir, sans subir des effets négatifs dans ses accords internationaux. En revanche, l'Etat reste le garant de l'intérêt national général, même si parfois il défend principalement des intérêts privés. Il doit cependant toujours assurer ses droits régaliens, soutenir son économie et organiser la sécurité interne et externe de ses citoyens.

In the process of economic globalization, the State becomes a declining economic agent, controlled by international organizations that set rules that it cannot sovereignly contravene, without suffering negative effects in its international agreements. On the other hand, the State remains the guarantor of the general national interest, even if sometimes it mainly defends private interests. However, it must always guarantee its sovereign rights, support its economy and organise the internal and external security of its citizens.

Etat, marché, globalisation, capitalisme, conflits économiques. State, market, globalisation, capitalism, economic conflicts


domain: sdv sde

text:
J P Thermoz
email: thermoz@corse.inra.fr

Many major threats around the world can cause important losses of yield in Citrus production, due to viruses, phytoplasmas, bacteria and fungi. When acceding new varieties and when propagating trees we have to take care not to diffuse pathogens. Within the French certification scheme for fruit trees, partners from research, development, official services and nurseries cooperate to propagate Citrus trees free of diseases. The French Citrus production is located in Corsica and in overseas departments. Corsica is free of airborne and vector transmitted diseases. For 50 years INRA-GEQA manages an important germplasm of about 5,000 trees grown outdoor and is able to provide seeds of rootstocks and budwoods of most Citrus varieties. INRA-GEQA is moving the scheme of sanitary control to a quality insurance system in order to secure the quality of the budwoods.

The European regulation for fruit trees Two groups of texts are giving obligation for fruit trees :

The measures for Plant Protection The marketing rules. The European regulations for plant protection list the prohibited pests and origins of plants in the long directive 2000/29/EU and the additive texts. They are transcribed in the national laws of each European country. The marketing rules (directive 93/48/EU) give for the whole Union the limitations for fruit trees, mainly phytosanitary measures. They define the minimum level for marketing fruit scions, called Conformitas Agraria Communitatis (CAC level). We can consider these requirements as mandatory certification. A member state is not allowed to take national prohibitions out of these texts. The French fruit tree plant certification is a voluntary procedure.

The French certification for fruit trees

Beyond the commitment of the European regulations for fruit tree plants, the French nurserymen grow certified scions of a superior quality. The French fruit tree plant certification is a voluntary procedure. This certification is supervised by the French Ministry of Agriculture and carried out by the CTIFL (Interprofessional Center for Fruit and Vegetables). It is an official procedure. Specifications are given for :

The varietal identity by filiation and reference to an official catalogue The sanitary status Some cultivation conditions such as distances, rotations Criteria for quality of the scion such as the height of grafting and the diameter.

Along the thirty last year, the growers have earn benefit from trees of the best quality. The French Citrus certification scheme


Prélèvement ressources Consommation d'espace Rejets et pollution

Impacts Activities

Lagoon activities

Conchyliculture

•

  Existence of a lot of « unmonetarised » activities (out of the market goods & services) • Small size of the Thau's territory (smaller than institutionnal level of data & stats collection) • Caractère artisanal et pour partie informel de la conchyliculture et la pêche • Caractère diffus et décentralisés des flux économiques du tourisme (plusieurs secteurs)

  of permanent population (réf : recensement de 1999)Prise en compte des cinq sous zones et des coûts fonciers Prise en compte de la réglementation (SCOT) Prise en compte des tendances généralesCalculation of the monetary output of water sellings

Introduction Espèces et biodiversité Maritime Espace côtier Activités liées à l'exploitation du plan d'eau (ressources, espace, paysage)

	Pêche	Ressources	Zones de	Aménagements		Fréquentation
		halieutiques	pêche, filets	portuaires		bateaux
			calés, stock
			coquillers
	Conchyliculture	Ressources	Lotissements	Mas et ports	Déchets conchylicoles	collecteurs naissains
		trophiques	Conchylicoles	conchylicoles		et retrempage
		primaires				produits
	Activités portuaires		Circulation	Aménagements	Risque de rejets de	Fréquentation
	et de navigation		maritime	portuaires	carburants et d'eaux	bateaux
					usées
	Tourisme littoral	Pêche	Baignade	Urbanisation	Rejets station
		sportive	nautisme		épuration
		Consommation
		d'eau
	Thermalisme	Consommation		Urbanisation	Risques rejets
		eau sources			spécifiques
		thermales			Rejets station
					épuration
	Activités liées à l'

exploitation des ressources naturelles autres que plan d'eau

	Agriculture	Eau pour	Exploitation	Pollution hydraulique et	Impact biodiversité
		l'iIrrigation	des terres	phréatique en fonction	selon semences et
				des traitements	traitements

Activités ou aménagements liés à la population

	Construction de	Modification des	Bord francs	Urbanisation	Risques hydrauliques
	bâtiments et	paysages	comblement	et réseaux routiers	liés à l'impact des
	travaux publics				aménagements sur
					l'infiltration
	Activités	Consommation	Usages	Urbanisation	Rejets station	Développement
	résidentielles liées	d'eau	récréatifs		épuration	plaisance
	à la population		et plaisance		Pollution
			Autres

activités situées en zone côtière

	Industries	Consommation	Urbanisation	Rejets
		d'eau		Eaux usées
		Modification
		paysages
	Caves	Consommation	Urbanisation	Rejets
	coopératives	d'eau		Eaux usées

  Mesure du poids économique du tourisme

	Fréquentation * budget = flux financiers liés au tourisme
		Souvenirs
	Produits terroir	5%
	5%
		CA des structures
	Loisirs 8%	Hébergement
		accueil (34%)
		34%
	Transport
	11%
		Surplus
		Activités
		économie
		récréatives
		résidentielle
		produits
	14%
	Restaurant	terroirs
		23%
		souvenirs (18%)
		Alimentation
		Flux directs tourisme

Locations de bateaux et bases nautiques 27% bateaux promenades sur l'étang 5% Plongée sous-marine 18% Randonnée 5% Vélo 8% Découverte de la nature 16% Club équestre 9% Pêche amateurs 5% Chasse aux gibiers d'eau 1% Aviron 6% Sales turnover of activities in the Thau lagoon's (2002/2004)

	Distribution des activités récréatives
	(en nombre de structures : total : 94 (2003)
		Estimation du
		chiffre d'affaires	Total
	Tourisme traditionnel	38 281
	Tourisme de santé	62 500	101 455 (*)
	Tourisme de plaisance	674
	Activités sportives	1 975
	Balades Protection nature	1 040 282	3 351
	Chasse/pêche	53
	Conchyliculture Pêche lagunaire	36 710 9 540	46 250
	Total	151 056

(*)

ensemble des flux dont hébergement (34%) et activités récréatives, produits du terroir, souvenirs (18 %) Tourisme (Commune d'Agde exclue) 67% Activités récréatives 2% Activités traditionnelles 31% Autres activités récréatives 0% Activités Récréatives balades 1% Activités récréatives sportives 1% Tourisme traditionnel 26% Pêche lagunaire 6% Tourisme de santé 42% Conchyliculture 25% Tourisme nautique 0%

	Activités récréatives 145
	Tourisme	1 172
	(emplois hébergements)

DIRECT EMPLOYMENT's ANALYSIS

	3675	Activités portuaires 1 300
				+ matelots
				(entre 60 et 200)
		Conchyliculture 2079
		Pêche	296	3692
				emplois
				lagune
				+ 660 bénévoles
		Agriculture	1 924

6 916 36 232 19% 10% 10% + ouvriers agricoles Structuration spatio fonctionnelle et types de populations Analysis of « urban pressures » Thau basin aera A9 A75 Agde Mèze Urban pressure due to residential expansion ; industrial and port activites Urban pressure due to tourism Maritime activities related to lagoon exploitation Low urban pressure Important agricultural activity Urban pressure due to residential expansion -agricultural activity Sète Montpellier attraction area Coastline aera Back country rural aera Progression démographique + 27% Migrations journalières : 18 231 sorties et 14 438 entrées 44 % du territoire consacré à l'agriculture (80% viticulture) Béziers Montpellier A9 A75 Agde Zone rurale d'arrière pays Mèze Pression urbaine résidentielle Activité industrielle et portuaire Pression urbaine touristique Activités maritimes liées à la lagune Faible pression urbaine Activité agricole importante Pézénas 915 7155 Pression urbaine résidentielle Activité agricole Sète 1169 Zone d'attraction (desserrement) de Montpellier Zone littorale 129 505 habitants in 20 communes Zone Bassin de Thau Evolution

  of the price of non agricultural land (achats par les non agriculteurs)

	80
	70
	60
	50
	40
	30
	20
	10
	0
	d'

après données SAFER Hérault

		Total : 33 130 075
	Types of population « Only » 26% of worn water rejected into the lagoon 11 598 226 15 782 001 Water demand analysis Eau potable Eau brute Communes Type de réseau d'assainissement Lieu de rejet Balaruc les Bains séparatif mer Nb abonnés EU Volumes collectés ou facturés (m3) * Réseau EU (km) Source
	Population résidente Population Résidente 9 534 129 505 Résidents locaux Agde Résidents migrants 102 309 27 196 Touristes venant dans la résidence secondaire de la famille ou d'amis 8% Résidents temporaires 13% Touristes 2 419 484 Résidents Population touristique 3 430 883 154,476 Lyonnaise des eaux, données 2003 Total 80 159 (29 258 193 nuitées) 209 754 Temporaires Touristes Total 27 007 53 153 209 754 Résidents nés dans le département 31% Résidents nés hors département 18% Agde Sète Frontignan Balaruc les Bains Balaruc le Vieux Marseillan Mèze/Loupian Cournonsec Gigean Montagnac Montbazin Pinet/Pomerols Poussan/ Bouzigues Villeveyrac Aumelas Castelnau de Guers Florensac mer Pertes Pertes Synthèse des flux S Fabrègues 8 641 139 1 549 659 1 839 940 957 699 Mer Etang Cours d'eau étang Cours d'eau Mer 67% Etang 12% Cours d'eau étang 14% Cours d 'eau et mer 7% Balaruc le Vieux pseudo-séparatif mer 1. Equipement et organisation de la gestion de l'eau Logements collectifs Usages domestiques Aumelas 169 36 666 Mairie 2 . Composantes et déterminants de la demande en eau Demande marchande eau potable Demande en eau brute Ménages résidents Touristes Collectivités Activités économiques Agriculture 3. Volumes d'eaux T E spécifiques Villeveyrac séparatif cours d'eau rejoignant la lagune S Activités Usage industriel spécifique E 2 458 000 (en gras italique données estimées à partir de la moyenne de la zone) S Sète séparatif mer Total zone 58 907 12 766 137 usées collectées Entreprises diverses de biens et services T V I technique courant R I S 3 623 200 Poussan séparatif cours d'eau rejoignant la lagune Villeveyrac 1 070 214 467 SDEI, données 2003 -Ecosite 4 055 722 Sète 11 829 3 257 114 162 SDEI Sète, données 2003 I Usage P Pomerols séparatif cours d'eau rejoignant la lagune Pomerols 1150 90 000 Mairie, données 2003 Industrie Réseaux de distribution d'eau potable Réseaux de collecte des eaux usées Collectivités publiques (communes) Exploitations secteur primaire C O L L E C Usage agricole et espaces verts colelctifs E N T E Pinet 557 96 030 SDEI, données 2003 432 522 Pinet séparatif cours d'eau rejoignant la lagune R 7 842 900 1 297 566 Mèze unitaire lagune de Thau Montbazin pseudo-séparatif cours d'eau rejoignant la lagune Loupian 1 035 Mèze 4 727 Montagnac 1 520 289 157 Mairie, données 2003 Montbazin 879 214 758 SDEI, données 2003 -Ecosite Prospective SDEI, données 2003 -Ecosite 1 186 646 SDEI, données 2003 -Ecosite d'assainissement Marseillan Plage pseudo-séparatif canal dans les marais longeant la lagune Marseillan 7 730 363 013 SDEI, données 2003 et techniques M E N A G E S Population touristique Population résidente Logements individuels Résidences secondaire Usages d'arrosage Usage Arrosage individuel 1 297 326 3 303 197 1 715 183 952 879 Bouzigues séparatif cours d'eau rejoignant la lagune Balaruc les Bains 2 129 639 632 38,2 CGE Frontignan, données 2003 8 390 743 Marseillan unitaire canal dans les marais longeant la lagune Gigean 1588 344 544 Ecosite Université Montpellier 3 (2) 3 335 536 894 515 1 143 455 Cournonsec séparatif cours d'eau rejoignant la lagune Frontignan pseudo-séparatif mer Gigean séparatif cours d'eau rejoignant la lagune Loupian séparatif cours d'eau rejoignant la lagune Balaruc le Vieux 706 125 330 15,4 CGE Frontignan, données 2003 Frontignan 8 407 1 188 180 Elected officials Scientists (15) 119,7 CGE Frontignan, données 2003 Bouzigues 836 and inter-commune services (20) SDEI, données 2003 -Ecosite Poussan 1 628 SDEI, données 2003 -Ecosite Castelnau de Guers 550 86 233 Ifremer (3), USTL (5) Investigations on Thau's EID, Commission Intercommunalité (CCNBT, CABT) Mairie, données 2002 Cournonsec 692 100 923 SDEI, données 2003 Fabrègues 2 171 612 500 SDEI, données 2003 Scientifique des étangs Maires (16) « common representations » Université Montpellier 1 (3) Autres : Député
	collecti vités	3 314 220	2 435 628	30%

Professionnal Org. and Associations (10)

		Investigations on Thau's « representations » among institutionnal et scientific stakeholders (58) QUESTIONNAIRE (PRINCIPLES) Main questions about representations Résults : frequentation (Montpellier) Résults : frequentation (Mèze et Sète) Representations of the lagoon itself
	(examples -free format answers) 1. Enquêtes de rue, situations diverses, marchés, centres 34 30% commerciaux, rues passantes en centre ville 32
	51% 2. Lieux : Sète, Mèze et Montpellier 60% 20 25% 3. Questionnaires adaptés : -un pour Sète et Mèze 10 9 50% 1 2	4
	20% 15% 40% Conchyliculture et What represents for you the Thau Lagoon ? -un pour Montpellier Environnement et Travail Pollution Usage, récréatif, Conchyliculture, récréatif, agréable Conchyliculture, paysage environnement, pêche paysage agréable pêche, usage, pêche,	Travail, usage, récréatif, agréable
	Which have been the problems encountered for 15 years ? 19% 21% 30% 10% NOMBRE D 'ENQUÊTES RÉALISÉES 35
	Constraints and assets of the shells of Thau ? 5% 30 20% 30	9%
	25 20 15 10	Administrations (13) CG (DARE + Ports), Région CEPRALMAR (2) SMNLR Aff. Mar. Services Vétérinaires DDAF (Mise), DDASS, Sète 67 10% 0% ne fréquentent pas l'étang 1à 3 fois / mois 12 0% Ne fréquentent pas 3 l'étang Préfecture, DIREN, Conchyliculture et pêche Environnement et paysage Pollution 4 à 5 fois / 1 fois par an Mèze 56 mois à 8 fois / mois 8 6 Usage, récréatif, agréable Agence de l'eau 218	Prud'homie, SRC, CCI, CRPM, Ch. Agri, Fédération plaisance, Fédération hôtellerie, SPN, Tour du Vallat, Ass. Plaisanciers Montpellier 95 pas jours sais les ne 2 à 4 fois / an à 15 fois /mois 11 10 plus de 4 fois 1 2 tous Conchyliculture, pêche, usage, Conchyliculture, pêche, Environnement, paysage, usage, paysage récréatif, agréable environnement, récréatif, agréable	2

Thermes Montpellier 95 enquêtes Mèze et Sète 123 enquêtes Results in terms of structures and logics of institutions & stakeholders Dispositifs de gestion intégrée

	Approche structurelle et analytique
	Mesures sectorielles	Analyse socio-politique
	• Bilan des équipements et des	des acteurs et des dispositifs
	infrastructures environnementales	• SMVM
	• Mesures sectorielles (CTE, Normes de	• 1°contrat d'étang
	pollution industrielle et navigation)	• 2°contrat d'étang
	• Inventaires des sites et ressources classés	• Préparation du 3°contrat
	et protégés
	Les conflits d'usages
	Les risques conchylicoles
	Construction d'indicateurs d'apprentissage
	institutionnel

Données spécifiques au modèle (collectées et calculées) Coefficients impacts environnementaux (utilisation des ressources, rejets…) Détail des activités liées au littoral Coefficients impacts sociaux Vecteur des niveaux de résultats (entrée des simulations) Indicateurs classiques d'impact (PIB, Export,Emploi) Indicateurs d'impact spécifiques développement durable du littoral

	I C EP pop Optimisation of RR (reuse rate) according to various criteria I C ENP ind St ENP max PU St ENP PU pompe Cap ppge Opt. % REU Principles Données collectées et calculées Données comptabilité nationale ACTIVITES Resultats : bilans (balances) Realisations Présentation de la matrice Structure de la matrice Employement (computed) Environnement (selected results) Results Origin of waste (déchets ménagers)
	I C EP entr Px EP 71 biens sont consommés par les Ventes EP entr Ventes EP pop Ventes tot Ventes tot ENP Vente ENP ind Px ENP agr Px ENP ind I C ENP agr Vente ENP agr Ventes tot EP for water management 1 2 3 4 5 6 7 8 9 10 11 différentes activités Biens en ligne DECHARGE CLIMATISATION Calculation of the investments in equipment Inv St EP PU St EP Surf coll pluie Inv St ENP PU surf col Inv St EU Cap dessal EdM Cap potab ENP St EP max St EU max PU St EU PU Dessal EdM PU Potab ENP PU Epur EU Inv Pppge Inv Dessal Inv Epur Inv Coll pluie Inv Potab Inv Equip Inv St Cap epur EU Critère micro Série3 Série5 Macro Economic • Permet d'envisager une variété quasi infinie d'applications, et donc de scénarios étudié • Exploitation du modèle sur divers scénarios Level of activity Productions Consumptions Non-relation (zero) (negative coef) (positive coef) SURPLUS 71 Total 177 ARTISANAT TOURISME STN.EPURAT COMPOSTAGE AUTR.INDUST EXPORTATIONS 20 • Les productions de déchets… IND.AGRO.AL MOULE.MER Modelling • S'appuie sur le modèle « Input-Output » (permet une représentation détaillée des structures et du fonctionnement de systèmes productifs) Matrice des consommations intermédiaires " TAP " Matrice des aménagements, technologies et projets/scénarios Surplus B I • Economiques; en comptabilisant des Groupement d'activités Nombre de variables AUTR.INDUST RES.ROUTES PS. NON CONST. AMNGT.RTES ARTISANAT • Réutilisation partielle de la modélisation appliquée SOL et ROUTES SERV.DISTRIB AUTOBUS HABIT CONC IND.AGRO.AL VARIABLES LOCAUX et CONSTRUCTIONS E N S échanges monétarisés • Sociaux; en indiquant la création , le maintien ou la disparition d'emplois • Environnementaux; en déterminant divers facteurs de qualité des milieux au Languedoc Roussillon • Adaptation des nomenclatures d'activité • Correction (pondérations, actualisation) des • Calage du modèle sur les données du territoire DECHETS SERVICES IMPORTATIONS 20 HUITRE MOULE.ETANG PECHE.ETANG P.HAB.ISOL P.HAB.CLIM consommations d'eau, d'énergie P.RES.SEC PECHE.MER coefficients techniques GOODS Activities Overprods Exchanges VEHICULES POPULATION AGRICULTURE ET PËCHE 13 INDUSTRIE EAU 11 ENERGIE Activités en colonne MAR&FRUIT VITICULTURE SERV.FINANC P.LOC.PROF. CAMS&UTS SERV.ENVIR • La répartition de l'emploi entre les TR.AGRIC SERV.ADMIN ESP.VERT /EP CULT.DIV P.LOC.TECH HABIT LOT. différentes activités • Les répartition en matière de P.HEB.TOUR AQUACULT CAMPING ELEVAGES Origin of CO2 emissions	Assessments

LAGUNAGE ENTRET.BAT ENTRET.MEC COMB.SLG INCIN.D.MEN. INCIN.D.IND.

Calculated indicators

(from model's results on each scenario)

Population Emploi

RESULTS

(examples)

• Le développement du tourisme dans le souci de valeurs environnementales génère :

-Une forte augmentation du nombre de résidences secondaires ;

-Une hausse de l'emploi ;

-Une hausse du surplus de monnaie retiré des échanges avec l'extérieur. • La baisse des productions liées à la lagune :


Annabelle Le

Study of casein aggregation into micelle in mammary epithelial cell

est destinée au dépôt

Study of casein aggregation into micelle in mammary epithelial cell

Annabelle Le Parc 1 , Joëlle Léonil 2 and Eric Chanat 1 1 INRA, UR 1196, Génomique et Physiologie de la Lactation, F-78352 Jouy-en-Josas, France.

2 INRA, URM 1253, Science et Technologie du Lait et de l'oeuf, F-35042 Rennes, France.

Caseins, a family of acidic phosphoproteins which represent 78% of bovine milk proteins, interact with colloidal calcium phosphate (CCP) and aggregate to organize into a supramolecular structure: the casein micelle. The structure of the micelle determines the functional characteristics of the milk protein fraction and has an impact on food processing. Despite the importance of the nutritional and functional value of casein micelle, the intrinsic organisation and the mechanisms of formation of these structures in the mammary epithelial cell (MEC) has not been fully established.

The aim of this work is to obtain information about the specific arrangement of the caseins in the structure and to characterize the interactions between the various components of the micelle. To understand the structural complexity of the casein micelle, we believe it is essential to gain new knowledge about the formation of this structure within the MEC in order to elucidate the role of the individual casein and of the CCP in the aggregation process. Since the aggregation of the caseins appears to be initiated in the endoplasmic reticulum (ER), and proceeds during their transport to the apical surface, the dynamic of elaboration of the micelle will be studied in vivo. One project is to characterize the initial micellar states formed in the ER of rat MEC. We will then follow the evolution of this basic structure in the secretory pathway. The possible role of disulfide bridges in the formation and the stabilization of the micellar states will be investigated. In addition, we will try to precise the role of the calcium in the first steps of casein aggregation process.

With this aim, we have purified rough microsomes using differential centrifugation techniques. Morphological observations by electron microscopy indicate that the rough ER derived membrane fraction was almost pure. Permeabilisation of rough microsomes with saponine, at pH 6.8, induces the release of soluble resident proteins (PDI, BIP and calreticuline) of the ER but not that of the membranous chaperone calnexine. This approach is currently applied to characterize the native micelles in the ER. The conditions allowing the conservation of the aggregated state of the caseins within permeabilized rough microsomes or their release from that compartment will be determined. The micelles obtained from rough microsomes will be isolated by density sucrose gradient and their structure compared to that of the casein micelle found in rat milk. The size, homogeneity, stoechiometry of the caseins and type of interactions within the structure, as well as the occurrence of disulfide bounds will be investigated.

4th

  International Dairy Federation dairy Science and Technology week" 20-24 avril 2009 à Rennes


Virginie Moreaux
email: virginie.moreaux@bordeaux.inra.fr

E Lamaud

J M Bonnefond

A Bosc

D Loustau

Energy Energy , water and , water and carbon carbon exchanges exchanges in in young young coniferous coniferous stands stands in South West of France in South West of France Effect of the presence of a layer of gorse

Intensification of the forest practices

Sustainibility of the forest ecosystem towards the soil fertility,

Background around the Landes forest (France).

• The process of cultivation and growth follows a production cycle that extends over a period of 45-50 years (Lesgourgues et al. 1997).

Introduction

Results

Soil water content (0-80cm) SWC higher in the weeded plot due to:

• lesser ET -weaker rainfall interception -lesser transpiration Partitionning between GPP and Re

Results

GPP and Re higher in the control plot, with a earlier decreasing in GPP Is this earlier decrease in GPP during the drought linked to the layer vegetation?

Effects of weeding and thinning on radiative balance:

Slight decreasing of the available energy (4%) Reducing of latent heat flux but few change in sensible heat on hydrological processes and carbon fluxes:

No attenuation of the summer drought since effects on transpiration on tree growth look identical. Dynamic of the ET and GPP on the control plot is more important and react faster to dry conditions: attributed to the dynamic a the gorse layer

Next step:

Modelling: using a 3D-model (Maestra, Wang, 1989, Medlyn, 1997): better representation of the understorey layer in young stand.

  closed on the CONTROL with a slope of 0.78 compared to a slope of 0.85 on WEEDED. • Literature: Wilson et al 2002 : slope: 0.79 ± 0.01, intercept: 3.7 ± 2.0 Berbigier et al 2001 : slope: 0.86, intercept: 0.007 -aRg + LW -LW↑ Effect of albedo is compensated by the effect of LW ↑ ↑ Vegetation layer is more sensitive to drought Partitionning between evapotranspiration (ET) and transpiration (Tr)

  fluxes more pronounced on the control plot, with higher values Dry and warm events: high reduction of the fluxes on the control plot Dry and cold events : high reduction of the fluxes, for both realesed and fixed carbon Temporal evolution of NEE (filtered data)

Context

Introduction

	Introduction	Material & method		Results & discussion
	Context
	Renewable energy production (Mtep)	2006	2020	2006/2020
		Heat	9.7	19.7	+10.1
		Biomass	8.8	15.0	+6.2
		Geothermal energy Solar energy Waste	0.4 0.00 0.4	2.3 0.9 0.9	+1.9 +0.9 +0.5	© INRA
		Biogaz	0.00	0.6	+0.5
		Electricity	5.6	12.9	+7.2
		Hydraulic energy	5.2	5.8	+0.6
		Land Wind energy	0.2	3.6	+3.4
		Sea Wind energy	0.0	1.4	+1.4
		Biomass	0.2	1.4	+1.2
		Photovoltaic energy	0.0	0.5	+0.5
	Other (geothermal energy, marine energy...)	0.0	0.1	+0.1
		+

12 Mm 3 in 2012 (ie. 30%) + 20 Mm 3 in 2020 (ie. 1 Mm 3 /yr)

Introduction Environmental round table, 2007

Acknowledgment: Denis Loustau and Eric Lamaud Michel Sartore, Didier Garrigou, Pierre Trichet, Fréderic Bernier, Jean-Luc Denou, Jean Michel Beau, Salles commune

Conclusion and prospects

Results

Introduction

Material & method

Meteorological characteristics: June 2009 -May 2010

Annually:

• Mean annual T= 12.4 °C


Gloria Aguilar
email: gaguilar@unizar.es

Laurent Lévi
email: laurent.levi@univ-pau.fr

Monique Madaune-Tort
email: monique.madaune-tort@univ-pau.fr

On the well-posedness for the coupling of multidimensional quasilinear diffusion-transport equations

Keywords: Coupling problem, degenerate parabolic-hyperbolic equation, entropy solution

This paper deals with the coupling of a quasilinear parabolic problem with a first order hyperbolic one in a multidimensional bounded domain Ω. In a region Ωp a diffusion-advection-reaction type equation is set while in the complementary Ω h ≡ Ω\Ωp, only advection-reaction terms are taken into account. Suitable transmission conditions along the interface ∂Ωp ∩ ∂Ω h are required. We select a weak solution characterized by an entropy inequality on the whole domain. This solution is given by a vanishing viscosity method.

Introduction

We are interested in a coupling of a quasilinear parabolic equation with an hyperbolic first-order one in a bounded domain Ω of R n ; n ≥ 1. The main motivation for considering this problem is the study of infiltration processes in an heterogeneous porous media. For instance, in a stratified subsoil made up of layers with different geological characteristics, the effects of diffusivity may be negligible in some layers. Such a coupled problem occurs also in fluid-dynamical theory for viscous-compressible flows around a rigid profile so that near this profile the viscosity effects have to be taken into account while at a distance they can be neglected. Another example arises in heat transfer studies as mentioned in [START_REF] Gastaldi | Coupling of two-dimensional hyperbolic and elliptic equations[END_REF].

We consider the case of two layers, that is sufficient. Then, the geometrical configuration is such that: Ω = Ω h ∪ Ω p ; Ω h and Ω p are two disjoint bounded domains with Lipschitz boundaries denoted by Γ l = ∂Ω l , l ∈ {h, p} and Γ hp = Γ h ∩ Γ p is such that H n-1 (Γ hp ∩ (Γ l \Γ hp )) = 0 where for q in [0, n + 1], H q is the q-dimensional Hausdorff measure over R n+1 . For l in {h, p}, ν l is the outward normal unit vector defined H n-1 -a.e. on Γ l .

The interface is denoted by Σ hp =]0, T [×Γ hp . At last, Q =]0, T [×Ω and for l in {h, p}, Q l =]0, T [×Ω l , Σ l =]0, T [×Γ l . Now, due to a combination of conservation laws and Darcy's law, the physical model is described as follows: For any positive and finite real T , find a measurable and bounded function u on Q such that,

∂ t u - n i=1 ∂ xi (f (u)∂ xi P ) + g(t, x, u) = 0 in Q h , ∂ t u - n i=1 ∂ xi (f (u)∂ xi P ) + g(t, x, u) = ∆φ(u) in Q p , u = 0 on ]0, T [×∂Ω, u(0, .) = u 0 on Ω. (1)
Then, suitable conditions on u across the interface Σ hp must be added. As for the linear problem studied by F. Gastaldi and al. in [START_REF] Gastaldi | Coupling of two-dimensional hyperbolic and elliptic equations[END_REF] or for the one dimensional nonlinear problem studied by G. Aguilar and al. in [START_REF] Aguilar | Analysis of a nonlinear parabolic-hyperbolic problem[END_REF], these transmission conditions include the continuity property of the flux through the interface formally written here as:

-f (u)∇P.ν h = (∇φ(u) + f (u)∇P ).ν p on Σ hp .

(2)

Moreover, the transmission conditions involve a property on the (dis)continuity of the function u via an entropy condition.

Let us mention that this problem has already been studied by the authors for a nondecreasing flux function f when ∇P.ν h ≤ 0, H n-1 -a.e. on Γ hp in [START_REF] Aguilar | Coupling of Multidimensional Parabolic and Hyperbolic Equations[END_REF] and when ∇P.ν h has a constant sign H n-1 a.e. on Γ hp in [START_REF] Aguilar | Nonlinear Multidimensional Parabolic-Hyperbolic Equations[END_REF].

Assumptions and notations

• The datum P is a known stationary function that belongs to W 2,∞ (Ω) and such that ∆P = 0 which is not restrictive as soon as [START_REF] Aguilar | Coupling of Multidimensional Parabolic and Hyperbolic Equations[END_REF] includes some reaction terms.

• The reaction function g belongs to W 1,∞ (]0, T [×Ω × R) and we set • The initial data u 0 belong to L ∞ (Ω). Thus we can define the nondecreasing time-depending function

M ′ g = ess sup
M : t ∈ [0, T ] → M (t) = u 0 L ∞ (Ω) e M ′ g t + M 0 e M ′ g t -1 M ′ g .
To simplify we write M = M (T ). Now, we assume local hypotheses on f and φ.

-The flux function f is a Lipschitzian function on [-M, M ] with constant M ′ f and such that f (0) = 0. To express the boundary conditions on the frontier of the hyperbolic area, we introduce as in [START_REF] Otto | Conservation Laws in Bounded Domains, Uniqueness and Existence via Parabolic Approximation[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF] the function F defined on [-M, M ] 3 by:

F(a, b, c) = 1 2 {sgn(a -b)(f (a) -f (b)) -sgn(c -b)(f (c) -f (b)) + sgn(a -c)(f (a) -f (c))}.
φ is a increasing Lipschitzian function on [-M, M ] such that φ(0) = 0.

• For any positive real µ, sgn µ is the Lipschitzian approximation of the function "sgn" defined by:

∀τ ∈ [0, +∞[ , sgn µ (τ ) = min τ µ , 1 and sgn µ (-τ ) = -sgn µ (τ ).
• Throughout this work, σ (resp. σ) denotes the variable on Σ l (resp. Γ l ), l ∈ {h, hp, p}. This way, for any t of [0, T ], σ = (t, σ).

• For any real a and b,

I(a, b) = [min(a, b); max(a, b)].

Functional spaces

• In the sequel, W (0, T ) is the Hilbert space

W (0, T ) ≡ {v ∈ L 2 (0, T ; H 1 0 (Ω)); ∂ t v ∈ L 2 (0, T ; H -1 (Ω))} equipped with the norm w W (0,T ) = ∂ t w 2 L 2 (0,T ;H -1 (Ω)) + ∇w 2 L 2 (Q) n 1/2
and V is the Hilbert space

V = {v ∈ H 1 (Ω p ), v = 0 a.e. on Γ p \Γ hp }
equipped with the norm v V = ∇v L 2 (Ωp) n . We denote ., . the pairing between V and V ′ .

• BV (O) with O = Ω h or O = Q h is the space of summable functions v with bounded total variation on O where the total variation is given by

T V O (v) = sup O v(x)divΦ(x)dx, Φ ∈ D(O) q , Φ L ∞ (O) q ≤ 1
where q is the dimension of the open set O. Moreover, we denote by γv the trace on Γ hp or Σ hp of a function v belonging to BV (O).

This work is organized as follows: the concept of a weak entropy solution to (1)-( 2) is defined in Section 2, 2.1 through an entropy inequality in the whole domain, the boundary conditions on the outer frontier of the hyperbolic area being expressed by referring to [START_REF] Otto | Conservation Laws in Bounded Domains, Uniqueness and Existence via Parabolic Approximation[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF]. This global formulation contains two local formulations: one on the hyperbolic area, stated in the paragraph 2.2, and one in the parabolic domain, stated in the paragraph 2.3. We also highlight some interface conditions along Σ hp in Section 3, 3.1, proper to ensure the uniqueness of a weak entropy solution to (1)-( 2), in the paragraph 3.2. The section 4 is devoted to the existence property to (1)-( 2) through the vanishing viscosity method.

The Entropy Formulation

Weak entropy solution

The definition of a weak entropy solution to (1)-( 2) has to include an entropy criterion in Q h where the quasilinear first-order hyperbolic operator is set. This way, the problem (1)-( 2) can be viewed as an evolutional problem for a quasilinear parabolic equation that strongly degenerates in a fixed subdomain Q h of Q.

As in [START_REF] Aguilar | Coupling of Multidimensional Parabolic and Hyperbolic Equations[END_REF][START_REF] Aguilar | Nonlinear Multidimensional Parabolic-Hyperbolic Equations[END_REF][START_REF] Aguilar | Analysis of a nonlinear parabolic-hyperbolic problem[END_REF], we propose a weak formulation through a global entropy inequality in the whole Q, the latter giving rise to a variational equality in the parabolic domain, to an entropy inequality in the hyperbolic one and to interface conditions along Σ hp . To establish these ones, it will be convenient to start by introducing a global entropy formulation using mollified entropy pairs. For this purpose, we set for any convex function

η in W 2,∞ loc (R) and any (w, k) in [-M, M ] 2 I (η) (w, k) = w k η ′ (φ(τ ) -φ(k)) dτ and q (η) (w, k) = φ(x) φ(k) η ′′ (τ -φ(k))(f • φ -1 )(τ ) dτ. Definition 1 A function u is a weak entropy solution to the coupling problem (1)-(2) if • u ∈ L ∞ (Q) with values in [-M, M ],
M being defined Section 1.1, φ(u) ∈ L 2 (0, T ; V ) and :

• ∀ϕ ∈ D(Q), ϕ ≥ 0, ∀k ∈ [-M, M ], ∀η ∈ W 2,∞ loc (R), Q I (η) (u, k)∂ t ϕdxdt - Qp η ′ (φ(u) -φ(k))∇φ(u).∇ϕ dx dt - Q f (u)η ′ (φ(u) -φ(k)) -q (η) (u, k) ∇P.∇ϕ dx dt - Q η ′ (φ(u) -φ(k))g(t, x, u)ϕ dx dt ≥ 0.
(3)

• ∀ζ ∈ L 1 (Σ h \Σ hp ), ζ ≥ 0, ∀k ∈ [-M, M ], ess lim τ →0 -Σ h \Σ hp F(u(σ + τ ν h ), 0, k)∇P (σ).ν h ζdH n ≤ 0, (4)
• ess lim

t→0 + Ω |u(t, x) -u 0 (x)|dx = 0. ( 5
)
Remark 1 For any u in L ∞ (Q) with values in [-M, M ], any ϕ in D(Q), ϕ ≥ 0, and k in [-M, M ], the assertion : "(3) is true for any η in W 2,∞ loc (R)" is equivalent to the entropy inequality written with the standard Kruzhkov entropy pairs:

Q |u -k|∂ t ϕ dx dt - Qp ∇|φ(u) -φ(k)|.∇ϕ dx dt - Q sgn(u -k)(f (u) -f (k))∇P.∇ϕ dx dt - Q sgn(u -k)g(t, x, u)ϕ dx dt ≥ 0. (6)

An entropy inequality on the hyperbolic zone

We first derive from (3) and ( 4) an entropy inequality in the hyperbolic domain. Indeed, Proposition 1 Let u be a weak entropy solution to (1)- [START_REF] Aguilar | Nonlinear Multidimensional Parabolic-Hyperbolic Equations[END_REF]. Then for any real k in [-M, M ] and any ϕ of

D(]0, T [×R n ), ϕ ≥ 0, Q h (|u -k|∂ t ϕ -sgn(u -k)(f (u) -f (k))∇P.∇ϕ -sgn (u -k) g(t, x, u)ϕ) dx dt ≥ -ess lim τ →0 -Σ hp sgn(u(σ + τ ν h ) -k)(f (u(σ + τ ν h )) -f (k))∇P (σ).ν h ϕ(σ)dH n - Σ h \Σ hp sgn(k)f (k)∇P (σ).ν h ϕ(σ)dH n + ess lim τ →0 -Σ h \Σ hp sgn(u(σ + τ ν h ))f (u(σ + τ ν h ))∇P (σ).ν h ϕ(σ)dH n . (7)
Proof -In (3) choose, for any positive µ and for any real τ : η ′ (τ ) = sgn µ (τ ).

By taking the limit when µ goes to 0 + thanks to the Dominated Convergence Theorem, it comes that for ϕ

in D(Q h ), ϕ ≥ 0, Q h (|u -k|∂ t ϕ -sgn(u -k)(f (u) -f (k))∇P.∇ϕ -sgn(u -k)g(t, x, u)ϕ) dx dt ≥ 0. ( 8
)
By referring to F.Otto's works in [START_REF] Otto | Conservation Laws in Bounded Domains, Uniqueness and Existence via Parabolic Approximation[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF], we deduce from (8) that for any real k in [-M, M ] and

β in L 1 (Σ h ), ess lim τ →0 -Σ h sgn(u(σ + τ ν h ) -k)(f (u(σ + τ ν h )) -f (k))∇P (σ).ν h β(σ)dH n exists. (9)
Then, it results from (8) (see [START_REF] Otto | Conservation Laws in Bounded Domains, Uniqueness and Existence via Parabolic Approximation[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF]) that, for any real k in [-M, M ] and

ϕ in D(]0, T [×R n ), ϕ ≥ 0, Q h (|u -k|∂ t ϕ -sgn(u -k)(f (u) -f (k))∇P.∇ϕ -sgn(u -k)g(t, x, u)ϕ) dx dt ≥ -ess lim τ →0 -Σ h sgn(u(σ + τ ν h ) -k)(f (u(σ + τ ν h )) -f (k))∇P (σ).ν h ϕ(σ)dH n .
To conclude we split the frontier of Ω h into Γ hp and Γ h \Γ hp and we use the boundary condition (4) on Σ h \Σ hp .

A variational equality on the parabolic zone

We give now some information on the regularity for ∂ t u in Q p and we derive from (3) a variational equality satisfied by any weak entropy solution u to the coupling problem (1)-(2).

Proposition 2 Let u be a weak entropy solution to the coupling problem (1)- [START_REF] Aguilar | Nonlinear Multidimensional Parabolic-Hyperbolic Equations[END_REF]. Then ∂ t u belongs to L 2 (0, T ; V ′ ). Furthermore, for any ϕ in L 2 (0, T ; V ),

+ ess lim τ →0 -Σ hp f (u(σ + τ ν h ))∇P (σ).ν h ϕdH n = 0. ( 10
)
Proof. This proposition is proved in [START_REF] Aguilar | Coupling of Multidimensional Parabolic and Hyperbolic Equations[END_REF] (Proposition 3.4) by starting with (3).

The Uniqueness Property

We prove the uniqueness property in the class of weak entropy solutions satisfying the strong trace property:

∃u hp ∈ L 1 (Σ hp ), ess lim τ →0 -Σ hp |u(σ + τ ν h ) -u hp (σ)|dH n = 0. ( 11
)
In this framework for any weak entropy solution u satisfying [START_REF] Peyroutet | Error estimate for a splitting method applied to convection-reaction equations[END_REF], we can establish that along the interface u satisfies two transmission conditions; the first one corresponds to (2) and the second one is an entropy-type inequality. Indeed :

About the transmission conditions along Σ hp

Proposition 3 Let u be a weak entropy solution to (1)-( 2) satisfying [START_REF] Peyroutet | Error estimate for a splitting method applied to convection-reaction equations[END_REF]. Then, for H n -a.e. σ in Σ hp ,

∀k ∈ I(u(σ), u hp (σ)), sgn(u(σ) -u hp (σ))(f (u hp (σ)) -f (k))∇P (σ).ν h (σ) ≥ 0, ( 12
)
where u(σ) = φ -1 (φ(u)(σ)) and φ(u)(σ) is the trace on Σ hp of φ(u) |Qp .

Proof. Let (ρ δ ) δ>0 be a sequence of C 1 (Ω), such that

∀δ > 0, 0 ≤ ρ δ ≤ 1, ρ δ (x) = 1 if x ∈ Γ hp ∀δ > 0, ρ δ (x) = 0 if x ∈ Ω, dist(x, Γ hp ) ≥ δ and δ∇ρ δ ∞ is bounded, ∀x ∈ Ω\Γ hp , ρ δ (x) → 0 when δ → 0 + .
This way, from [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF] and [START_REF] Peyroutet | Error estimate for a splitting method applied to convection-reaction equations[END_REF] it comes that: ∀ϕ ∈ L 2 (0, T, V ),

T 0 ∂ t u, ϕρ δ dt + Q {(∇φ(u) + f (u)∇P ).∇ϕ + +g(t, x, u)ϕ} ρ δ dx dt + Qp ∇φ(u).∇ρ δ ϕ dx dt + Qp f (u)∇P.∇ρ δ ϕ dx dt + Σ hp f (u hp (σ))∇P (σ).ν h (σ)ϕ(σ)dH n = 0. ( 13
)
Now, we take ϕ = sgn µ (φ(u) -φ(k))ψ |Qp where ψ belongs to H 1 0 (Q), ψ ≥ 0. In order to take the limit with respect to δ for the first term in the left-hand side, we use an integration by parts formula based on a convexity inequality (see e.g. [START_REF] Gagneux | Analyse mathématique de modèles non lináires de l'ingénierie pétrolière[END_REF], the Mignot-Bamberger Lemma). This way,

T 0 ∂ t u, ϕρ δ dt = - T 0 Ω I µ (u, k)ρ δ ∂ t ψ dx dt,
where

I µ (u, k) = u k sgn µ (φ(τ ) -φ(k)) dτ. (14) So clearly, lim δ→0 + T 0 ∂ t u, ϕρ δ dt = 0.
Then, for the second term in the second line of (13), as f • φ -1 is continuous, thanks to the properties of the sequence (ρ δ ) δ>0 , we can assert that for any positive µ,

lim δ→0 + Qp f (u)∇P.∇ρ δ ϕ dx dt = - Σ hp f (u(σ))∇P (σ).ν h (σ)sgn µ (φ(u)(σ) -φ(k))ψ(σ)dH n ,
and therefore it results from (13) that lim

δ→0 + Qp sgn µ (φ(u) -φ(k))ψ∇φ(u)
.∇ρ δ dx dt exists and lim

δ→0 + Qp sgn µ (φ(u) -φ(k))ψ∇φ(u).∇ρ δ dx dt = - Σ hp (f (u hp (σ)) -f (u(σ)))∇P (σ).ν h (σ)sgn µ (φ(u)(σ) -φ(k))ψ(σ)dH n . (15)
The equality (15) means in a sense that the flux through the interface is continuous. Now, let us come back to (3), written with η = sgn µ , µ > 0, and

ϕ = ψρ δ , ψ ≥ 0, ψ in H 1 0 (Q) -which is always possible thanks to a density argument. It comes that for any real k in [-M, M ], Q I µ (u, k)ρ δ ∂ t ψ dx dt - Qp sgn µ (φ(u) -φ(k))ρ δ ∇φ(u).∇ψ dx dt - Qp sgn µ (φ(u) -φ(k))ψ∇φ(u).∇ρ δ dx dt - Q (f (u)sgn µ (φ(u) -φ(k)) -q µ (u, k))∇P.∇ψ + sgn µ (φ(u) -φ(k))g(t, x, u)ψ ρ δ dx dt - Q (f (u)sgn µ (φ(u) -φ(k)) -q µ (u, k))∇P.∇ρ δ ψ dx dt ≥ 0, ( 16
)
where I µ is given by ( 14) and

q µ (u, k) = φ(u) φ(k) sgn ′ µ (τ -φ(k))(f • φ -1 )(τ ) dτ.
In order to pass to the limit when δ goes to 0 + in (16), we use (15) for the second line and we split the fourth one into an integral over Q h and an integral over Q p ; then we refer to [START_REF] Peyroutet | Error estimate for a splitting method applied to convection-reaction equations[END_REF]. It comes that:

∀k ∈ [-M, M ], ∀ψ ∈ H 1 0 (Q), ψ ≥ 0, ∀µ > 0, Σ hp (sgn µ (φ(u(σ)) -φ(k))(f (u hp (σ)) -f (u(σ)))∇P (σ).ν h (σ)ψ(σ)dH n + Σ hp (f (u(σ))sgn µ (φ(u(σ)) -φ(k)) -q µ (u(σ), k)) ∇P (σ).ν h (σ)ψ(σ)dH n - Σ hp (f (u hp (σ))sgn µ (φ(u hp (σ)) -φ(k)) -q µ (u hp (σ), k)) ∇P (σ).ν h (σ)ψ(σ)dH n ≥ 0.
So, for any positive µ and any real k in [-M, M ], and H n a.e. on Σ hp ,

(sgn µ (φ(u(σ)) -φ(k))(f (u hp (σ)) -f (u(σ)))∇P (σ).ν h (σ) + f (u(σ))sgn µ (φ(u(σ)) -φ(k)) -q µ (u(σ), k) ∇P (σ).ν h (σ) -f (u hp (σ))sgn µ (φ(u hp (σ)) -φ(k)) -q µ (u hp (σ), k) ∇P (σ).ν h (σ) ≥ 0.
And when µ goes to 0 + , q µ (w, k) goes to f (k)sgn(w -k) for all reals w and k in [-M, M ], φ being increasing. So the µ-limit provides that for H n -a.e. σ on Σ hp and for any real k in [-M, M ]:

(f (u hp (σ)) -f (u(σ)))sgn(u(σ) -k)∇P (σ).ν h (σ) + sgn(u(σ) -k)(f (u(σ)) -f (k)) -sgn(u hp (σ) -k)(f (u hp (σ)) -f (k)) ∇P (σ).ν h (σ) ≥ 0.
The desired relation is obtained by taking k in the interval I(u(σ), u hp (σ)).

The uniqueness theorem

The uniqueness property of a weak entropy solution to the coupling problem ( 1)-( 2) is proved under local Hölder continuity assumption for f • φ -1 and is provided by the following statement:

Theorem 1 Assume that there exists a positive constant C and a real θ in

[ 1 2 , 1] such that ∀(v, w) ∈ [-M, M ] 2 , |(f • φ -1 )(v) -(f • φ -1 )(w)| ≤ C|v -w| θ . (17)
Let u 1 , u 2 be two weak entropy solutions to (1)-( 2) for initial data u 0,1 and u 0,2 respectively and such that (11) holds. Then, for a.e. t of [0, T ],

Ω |u 1 (t, .) -u 2 (t, .)|dx ≤ e M ′ g t Ω |u 0,1 -u 0,2 |dx.

Preliminaries

In order to use the method of doubling variables, we introduce a sequence of mollifiers (W δ ) δ>0 on R n+1 defined by

∀δ > 0, ∀r = (t, x) ∈ R n+1 , W δ (r) = ̟ δ (t) n i=1 ̟ δ (x i ),
where (̟ δ ) δ>0 is a standard sequence of mollifiers on R. We use classical results on the Lebesgue set of a summable function on Q and a similar property on the whole boundary proved in [START_REF] Peyroutet | Error estimate for a splitting method applied to convection-reaction equations[END_REF]:

Lemma 1 Let v and w be in L ∞ (Q h ) such that (8) and (11) hold. Then for any continuous function ϕ on

Q h , lim δ→0 + Q h Σ h \Σ hp sgn(v(r))f (v(r))∇P (σ).ν h (σ)ϕ( σ + r 2 )W δ (σ -r)dH n σ dr = 1 2 ess lim τ →0 -Σ h \Σ hp sgn(v(σ + τ ν h ))f (v(σ + τ ν h ))∇P (σ)ν h (σ)ϕ(σ)dH n , lim δ→0 + Q h ess lim τ →0 -Σ h \Σ hp sgn(v(σ + τ ν h ))f (v(σ + τ ν h ))∇P (σ).ν h (σ)ϕ( σ + r 2 )W δ (σ -r)dH n σ dr = 1 2 ess lim τ →0 -Σ h \Σ hp sgn(v(σ + τ ν h ))f (v(σ + τ ν h ))∇P (σ).ν h (σ)ϕ(σ)dH n ,
and

lim δ→0 + Q h Σ hp sgn(v hp (σ) -w(r))(f (v hp (σ)) -f (w(r)))∇P (σ).ν h (σ)ϕ( σ + r 2 )W δ (σ -r)dH n σ dr = 1 2 Σ hp sgn(v hp (σ) -w hp (σ))(f (v hp (σ)) -f (w hp (σ)))∇P (σ).ν h (σ)ϕ(σ)dH n
where v hp (resp.w hp ) is defined by [START_REF] Peyroutet | Error estimate for a splitting method applied to convection-reaction equations[END_REF] for v (resp. w).

Proof of Theorem 1

(i) We first compare the two solutions u 1 and u 2 in the parabolic zone. The lack of regularity of the time partial derivative of any weak entropy solution to (1)-( 2) requires a doubling of the time variable. Therefore, let χ be a nonnegative element of D(0, T ). We consider positive reals δ small enough in order that α δ : ( t, t) → α δ ( t, t) = χ((t + t)/2)̟ δ (t -t)/2 belongs to D(]0, T [×]0, T [). Then, for µ > 0, in [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF] written in variables (t, x) for u 1 we consider ϕ(t, x) = sgn µ (φ(u 1 )(t, x) -φ(u 2 )( t, x))α δ ( t, t) and in [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF] written in variables ( t, x) for u 2 , we consider ϕ( t, x) = -sgn µ (φ(u 1 )(t, x) -φ(u 2 )( t, x))α δ ( t, t) . To simplify the writing, we add a "tilde" superscript to any function in the t variable. Moreover, thanks to [START_REF] Peyroutet | Error estimate for a splitting method applied to convection-reaction equations[END_REF] we observe that in [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF], for i = 1, 2, ess lim

τ →0 -Σ hp f (u i (σ + τ ν h ))∇P (σ).ν h (σ)ϕ(σ)dH n = Σ hp f (u hp i (σ))∇P (σ).ν h (σ)ϕ(σ)dH n .
Then we integrate with respect to the corresponding time variable so that, by adding up, it comes:

T 0 T 0 ∂ t u 1 -∂ t ũ2 , sgn µ (φ(u 1 ) -φ(ũ 2 )) α δ dt d t + ]0,T [×Qp sgn ′ µ (φ(u 1 ) -φ(ũ 2 ))|∇(φ(u 1 ) -φ(ũ 2 ))| 2 α δ dx dt d t + ]0,T [×Qp sgn ′ µ (φ(u 1 ) -φ(ũ 2 ))(f (u 1 ) -f (ũ 2 ))∇P.∇(φ(u 1 ) -φ(ũ 2 ))α δ dx dt d t + ]0,T [×Qp (g(t, x, u 1 ) -g( t, x, ũ2 ))sgn µ (φ(u 1 ) -φ(ũ 2 )) α δ dx dt d t = - T 0 Σ hp f (u hp 1 (σ))∇P (σ).ν h (σ)sgn µ (φ(u 1 )(σ) -φ(u 2 )(σ))α δ dH n σ d t + T 0 Σ hp f (u hp 2 (σ))∇P (σ).ν h (σ)sgn µ (φ(u 1 )(σ) -φ(u 2 )(σ))α δ dH n σ dt. (18)
We want to pass to the limit first when µ goes to 0 + in (18). For the second and third terms in the left-hand side, we argue by using the Cauchy-Scharwz inequality in the third term,

Qp sgn ′ µ (φ(u 1 ) -φ(ũ 2 )) |∇(φ(u 1 ) -φ(ũ 2 ))| 2 α δ dx dt + Qp sgn ′ µ (φ(u 1 ) -φ(ũ 2 ))(f (u 1 ) -f (ũ 2 ))∇P.∇(φ(u 1 ) -φ(ũ 2 ))α δ dx dt ≥ - 1 2 Qp sgn ′ µ (φ(u 1 ) -φ(ũ 2 ))(f • φ -1 (φ(u 1 )) -f • φ -1 (φ(ũ 2 ))) 2 |∇P | 2 α δ dx dt + 1 2 Qp sgn ′ µ (φ(u 1 ) -φ(ũ 2 )) |∇(φ(u 1 ) -φ(ũ 2 ))| 2 α δ dx dt,
where the second term in the right-hand side is nonnegative. This way, due (17) and as P belongs to W 1,∞ (Ω), we establish that there exists a positive constant C such that

Qp sgn ′ µ (φ(u 1 ) -φ(ũ 2 )) |∇(φ(u 1 ) -φ(ũ 2 ))| 2 α δ dx dt + Qp sgn ′ µ (φ(u 1 ) -φ(ũ 2 ))(f (u 1 ) -f (ũ 2 ))∇P.∇(φ(u 1 ) -φ(ũ 2 ))α δ dx dt ≥ -C Qp |φ(u 1 ) -φ(ũ 2 )| 2θ sgn ′ µ (φ(u 1 ) -φ(ũ 2 ))α δ dx dt,
and the term in the right-hand side goes to 0 with µ as θ ≥ 1/2. For the first term in the left-hand side of (18), we use the Mignot-Bamberger Lemma (see [START_REF] Gagneux | Analyse mathématique de modèles non lináires de l'ingénierie pétrolière[END_REF]) to obtain, for a fixed t

T 0 ∂ t u 1 , sgn µ (φ(u 1 ) -φ(ũ 2 )) α δ dt = - Qp u1 ũ2 sgn µ (φ(τ ) -φ(ũ 2 ))dτ ∂ t α δ dx dt,
while for a fixed t,

- T 0 ∂ t ũ2 , sgn µ (φ(u 1 ) -φ(ũ 2 )) α δ d t = - Qp u1 ũ2 sgn µ (φ(u 1 ) -φ(τ ))dτ ∂ tα δ dx d t,
So, we are able to pass to the limit in (18) when µ goes to 0 + and it comes

- ]0,T [×Qp |u 1 -ũ2 |(∂ t α δ + ∂ tα δ ) dx dt d t ≤ ]0,T [×Qp |g(t, x, ũ2 ) -g( t, x, ũ2 )|α δ dx dt d t - T 0 Σ hp (f (u hp 1 (σ)) -f (u hp 2 (σ)))∇P (σ).ν h (σ)sgn(φ(u 1 ) -φ(u 2 (σ)))α δ dH n σ d t.
Now, we come back to the definition of α δ to express the sum ∂ t α δ + ∂ tα δ . Then we can take the limit with respect to δ through the notion of the Lebesgue's set of a summable function on ]0, T [. Therefore, as g is Lipschitzian, for any χ in D(0, T ), χ ≥ 0,

- Qp |u 1 -u 2 |χ ′ (t) dx dt ≤ M ′ g Qp |u 1 -u 2 |χ(t)dxdt - Σ hp (f (u hp 1 ) -f (u hp 2 ))∇P.ν h sgn(u 1 -u 2 )χ(t)dH n , (19)
where we remind that u i (σ) = φ -1 (φ(u i )(σ)) and φ(u i )(σ) is the trace on Σ hp of φ(u i ) |Qp .

(ii) Now we work in the hyperbolic domain. We use a doubling method for all the variables. Let ψ be such that ψ ≡ χζ where χ is a function in D(0, T ), χ ≥ 0, as in Part (i) and ζ is in D(R n ) such that: ζ ≥ 0, ζ ≡ 1 on Q h . We consider positive reals δ small enough in order that the mapping ( t, t) → χ((t+ t)/2)w δ (t -t)/2 belongs to D(]0, T [×]0, T [). Then, for such any positive δ, we define the function Ψ

δ in ]0, T [×R n ×]0, T [×R n by Ψ δ (r, r) = χ((t + t)/2)ζ((x + x) /2)W δ (r -r).
Due to the proposition 1, the inequality (7) holds for u 1 and u 2 . We choose in [START_REF] Gagneux | Analyse mathématique de modèles non lináires de l'ingénierie pétrolière[END_REF] written for u 1 in variables (t, x), k = ũ2 ≡ u 2 ( t, x) and ϕ(t, x) = Ψ δ (t, x, t, x) and in [START_REF] Gagneux | Analyse mathématique de modèles non lináires de l'ingénierie pétrolière[END_REF] written for u 2 in variables ( t, x), k = u 1 (t, x) and ϕ( t, x) = Ψ δ (t, x, t, x).

By integrating over Q h and adding up, it comes:

-

Q h ×Q h (|u 1 -ũ2 |(∂ t Ψ δ + ∂ tΨ δ ) -sgn(u 1 -ũ2 )(f (u 1 ) -f (ũ 2 ))(∇P.∇ x Ψ δ + ∇ P .∇ xΨ δ ) dr dr + Q h ×Q h sgn(u 1 -ũ2 )(g(t, x, u 1 ) -g( t, x, ũ2 ))Ψ δ dr dr ≤ Q h Σ h \Σ hp sgn(ũ 2 )f (ũ 2 )∇P (σ).ν h (σ)Ψ δ (σ, r) dH n σ dr + Q h Σ h \Σ hp sgn(u 1 )f (u 1 )∇ P ( σ).ν h ( σ)Ψ δ (r, σ) dH n σ dr - Q h ess lim τ →0 -Σ h \Σ hp sgn(u 1 (σ + τ ν h ))f (u 1 (σ + τ ν h ))∇P (σ).ν h (σ)Ψ δ (σ, r) dH n σ dr - Q h ess lim τ →0 -Σ h \Σ hp sgn(u 2 (σ + τ ν h ))f (u 2 (σ + τ ν h ))∇ P ( σ).ν h ( σ)Ψ δ (r, σ) dH n σ dr + Q h Σ hp sgn(u hp 1 (σ) -ũ2 )(f (u hp 1 (σ)) -f (ũ 2 ))∇P (σ).ν h (σ)Ψ δ (σ, r) dH n σ dr + Q h Σ hp sgn(u hp 2 (σ) -u 1 )(f (u hp 2 (σ)) -f (u 1 ))∇ P ( σ).ν h ( σ)Ψ δ (r, σ) dH n σ dr. (20)
Then through a classical reasoning we pass to the limit with δ on the left-hand side of (20). On right-hand side, we refer to Lemma 1. It comes:

- Q h |u 1 -u 2 |χ ′ (t)dxdt ≤ - Q h sgn(u 1 -u 2 )(g(t, x, u 1 ) -g(t, x, u 2 ))χ(t) dx dt + Σ hp sgn(u hp 1 (σ) -u hp 2 (σ))(f (u hp 1 (σ)) -f (u hp 2 (σ)))∇P (σ).ν h (σ)χ(t) dH n .
The Lipschitz condition for g provides: ∀χ ∈ D(0, T ), χ ≥ 0,

- Q h |u 1 -u 2 |χ ′ (t)dxdt ≤ Σ hp sgn(u hp 1 -u hp 2 )(f (u hp 1 ) -f (u hp 2 ))∇P.ν h χ(t) dH n +M ′ g Q h |u 1 -u 2 |χ(t) dx dt. (21)
By adding inequalities ( 19) and ( 21) we get:

- Q |u 1 -u 2 |χ ′ (t) dx dt ≤ M ′ g Q |u 1 -u 2 |χ(t) dx dt + Σ hp sgn(u hp 1 -u hp 2 )(f (u hp 1 ) -f (u hp 2 ))∇P.ν h χ(t) dH n - Σ hp sgn(u 1 -u 2 )(f (u hp 1 ) -f (u hp 2 ))∇P.ν h χ(t) dH n .
We set for H n -a.e. σ in Σ hp ,

I = sgn(u hp 1 (σ) -u hp 2 (σ)) -sgn(u 1 (σ) -u 2 (σ)) (f (u hp 1 (σ)) -f (u hp 2 )(σ))∇P (σ).ν h (σ),
and we develop a pointwize reasoning to establish that I ≤ 0, H n -a.e. on Σ hp . Indeed, for H n -a.e. σ in Σ hp , if sgn(u hp

1 (σ) -u hp 2 (σ)) = sgn(u 1 (σ) -u 2 (σ)) or sgn(u hp 1 (σ) -u hp 2 (σ)) = 0 then I = 0 while if sgn(u hp 1 (σ) -u hp 2 (σ)) = sgn(u 1 (σ) -u 2 (σ)
) and sgn(u hp 1 (σ) -u hp 2 (σ)) = 0 then sgn(I) = sgn(J) where,

J = sgn(u hp 1 (σ) -u hp 2 (σ))(f (u hp 1 (σ)) -f (u hp 2 )(σ))∇P (σ).ν h (σ).
Assume for example that u hp 1 (σ) < u hp 2 (σ), the study of the converse situation being similar. If u 2 (σ) belongs to ]u hp 1 (σ), u hp 2 (σ)[ then we may write (12) for u 1 and for u 2 and with k = u 2 (σ). By adding and taking into account that sgn(u

1 (σ) -u hp 1 (σ)) = -sgn(u 2 (σ) -u hp 2 (σ)) = 1 we obtain that J ≤ 0. If u 2 (σ) is not an element of ]u hp 1 (σ), u hp 2 (σ)[ then we take k = u hp 1 (σ) in (12) for u 2 if u 2 (σ) ≤ u hp 1 (σ) and k = u hp 2 (σ) in (12) for u 1 if u 2 (σ) ≥ u hp 2 (σ). In each case J ≤ 0. Therefore (22) becomes - Q |u 1 -u 2 |χ ′ (t)dxdt ≤ M ′ g Q |u 1 -u 2 |χ(t)dxdt,
for any nonnegative χ of D(0, T ). The inequality of the Theorem 1 follows from the initial condition ( 5) for u 1 and u 2 and thanks to the Gronwall Lemma. That completes the proof.

Existence through the vanishing viscosity method

In [START_REF] Aguilar | Nonlinear Multidimensional Parabolic-Hyperbolic Equations[END_REF] we have obtained an existence result of a weak entropy solution to ( 1)-( 2) satisfying ( 11) when along the interface all the characteristics of the first-order operator set in Q h have the same behavior: either there are all leaving the hyperbolic domain, either there are all entering in. However in the first situation, an existence property is also established in [START_REF] Aguilar | Coupling of Multidimensional Parabolic and Hyperbolic Equations[END_REF] by means of the vanishing viscosity method and thanks to the notion of process solution (note that for this special outwards characteristics framework, the uniqueness proof does not require [START_REF] Peyroutet | Error estimate for a splitting method applied to convection-reaction equations[END_REF], since data are living the hyperbolic zone along the interface). In this section we use the latter tools to provide an existence result whatever the behavior of characteristics along Σ hp but we are not able to ensure that the weak entropy solution obtained this way fulfills [START_REF] Peyroutet | Error estimate for a splitting method applied to convection-reaction equations[END_REF].

For any positive ǫ, we introduce φ ǫ = φ + ǫI R and we consider the next formal problem : find a bounded and measurable function u ǫ on Q such that

∂ t u ǫ - n i=1 ∂ xi (f (u ǫ )∂ xi P ) + g(t, x, u ǫ ) = ǫ∆φ ǫ (u ǫ ) in Q h , ∂ t u ǫ - n i=1 ∂ xi (f (u ǫ )∂ xi P ) + g(t, x, u ǫ ) = ∆φ ǫ (u ǫ ) in Q p , u ǫ = 0 on Σ, u ǫ (0, .) = u 0 in Ω, ( 22
)
subject to the transmission conditions across the interface:

-(ǫ∇φ ǫ (u ǫ ) + f (u ǫ )∇P ).ν h = (∇φ ǫ (u ǫ ) + f (u ǫ )∇P ).ν p on Σ hp , u ǫ |Q h = u ǫ |Qp on Σ hp . (23)

The viscous problem

First of all, as stated in [START_REF] Aguilar | Coupling of Multidimensional Parabolic and Hyperbolic Equations[END_REF], we remind the next existence and uniqueness result:

Theorem 2 For any positive ǫ, there exists a unique weak solution

u ǫ to (22)-(23) in W (0, T ) ∩ L ∞ (Q) with ∂ t u ǫ in L 2 loc (0, T ; L 2 (Ω))
. This solution fulfills

∂ t u ǫ -div(λ ǫ (x)∇φ ǫ (u ǫ )) + f (u ǫ )∇P ) + g(t, x, u ǫ ) = 0 a.e. on Q, (24)
u ǫ (0, .) = u 0 a.e. on Ω,

where λ ǫ (x) = ǫI Ω h (x) + I Ωp (x).
The lack of regularity of the initial data but also the fact that the diffusive term depends on the space variable through λ ǫ only allow us to establish in [START_REF] Aguilar | Coupling of Multidimensional Parabolic and Hyperbolic Equations[END_REF] the following a priori estimates: Proposition 4 There exists a constant C independent of ǫ such that:

u ǫ L ∞ (Q) ≤ M, ( 25
) (λ ǫ ) 1/2 ∇ φ(u ǫ ) 2 L 2 (Q) n + (ǫλ ǫ ) 1/2 ∇u ǫ 2 L 2 (Q) n ≤ C, (26)
∂ t u ǫ L 2 (0,T ;H -1 (Ω)) ≤ C,
where M is defined in paragraph 1.1 and φ(τ ) = τ 0 φ ′ (s)ds.

The ǫ-limit

As a consequence of the proposition 4 and by using a reasoning highlighted in [START_REF] Gagneux | Analyse mathématique de modèles non lináires de l'ingénierie pétrolière[END_REF], chapter 2:

Proposition 5 Assume that φ -1 is Hölder continuous with an exponent θ in ]0, 1[. ( 27
)
Then there exists a measurable function u of L ∞ (Q) with φ(u) in L 2 (0, T ; V ) such that up to a subsequence when ǫ goes to 0 + , the sequence (u ǫ ) ǫ>0 converges toward u in L ∞ (Q) weak * , in L q (Q p ) for any finite q and a.e. on Q p . Besides we also have

∇φ ǫ (u ǫ ) ⇀ ∇φ(u) weakly in L 2 (Q p ) n , ǫ∇φ ǫ (u ǫ ) → 0 + strongly in L 2 (Q h ) n .
To characterize the function u -that is formally to pass to the limit with respect to ǫ in ( 22)-( 23) -on the hyperbolic zone we take advantage of (25) and we use that:

Claim 1 (see [START_REF] Eymard | Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation[END_REF]) -Let O be an open bounded subset of R q (q ≥ 1) and (u n ) n>0 a sequence of measurable functions on O such that:

∃M > 0, ∀n > 0, u n L ∞ (O) ≤ M.
Then, there exist a subsequence (u ϕ(n) ) n>0 and a measurable function π in L ∞ (]0, 1[×O) such that for all continuous and bounded functions ψ on O×] -M, M [,

∀ξ ∈ L 1 (O), lim n→+∞ O ψ(x, u ϕ(n) )ξ dx = ]0,1[×O ψ(x, π(α, w)) dα ξ dx.
Such a result has first been applied to the approximation through the artificial viscosity method of the Cauchy problem in R p for conservation laws, as one can establish a uniform L ∞ -control of approximate solutions [START_REF] Diperna | Convergence of approximate solutions to conservation laws[END_REF]. It has also been applied to the numerical analysis of transport equations since "Finite-Volume" schemes only give an L ∞ -estimate uniformly with respect to the mesh length of the numerical solution (see [START_REF] Eymard | Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation[END_REF]). Here the approximating sequence is the sequence of solutions to viscous problems ( 22)-(23).

Theorem 3 If (27) holds and if f • φ -1 is Lipschitz continuous on [-M ; M ], (28)
then when ǫ goes to 0 + the sequence of solutions to viscous problems (22)-(23) ǫ>0 strongly converges in L 1 (Q) toward a weak entropy solution to the coupled parabolic-hyperbolic problem (1)-(2).

Proof. We consider the function u highlighted in the proposition 5. Since (u ǫ |Ω h ) ǫ>0 is uniformly bounded, there exist a subsequence -still labelled (u ǫ |Ω h ) ǫ>0 -and a measurable and bounded function π -called a process -on ]0, 1[×Q h such that for any continuous bounded function

ψ on Q h ×] -M, M [ and for any ξ of L 1 (Q h ) lim ǫ→0 + Q h ψ(t, x, u ǫ )ξ dx dt = ]0,1[×Q h ψ(t, x, π(α, t, x))ξ dα dx dt (29)
Our aim is first to establish that on the hyperbolic zone, the process π is reduced to u |Ω h , independently of α in ]0, 1[, and secondly to prove that u is a weak entropy solution to (1)-( 2) for initial data u 0 . To do so, for any positive µ, we take first the scalar product in L 2 (]µ, T [×Ω) between (24) and the function

∂ 1 H 1 (u ǫ , k)ζ 1 ,
where ζ 1 belongs to D(] -∞, T [×Ω), ζ 1 ≥ 0, and for any m in N * and any real k through

H 1,m (u ǫ , k) = (u ǫ -k) 2 + 1 m 2 1/2 - 1 m .
By denoting

Q 1,m (u ǫ , k) = uǫ k ∂ 1 H 1,m (τ, k)f ′ (τ )dτ, G 1,m (u ǫ , k) = g(t, x, u ǫ )∂ 1 H 1,m (u ǫ , k),
it comes after some integrations by parts

- T µ Ω (H 1,m (u ǫ , k)∂ t ζ 1 -Q 1,m (u ǫ , k)∇P.∇ζ 1 -G 1,m (u ǫ , k)ζ 1 ) dx dt - Ω H 1,m (u ǫ (µ, x), k)ζ 1 (µ, x) dx ≤ - T µ Ω λ ǫ ∂ 1 H 1,m (u ǫ , k)∇φ ǫ (u ǫ ).∇ζ 1 dx dt,
the inequality resulting from the convexity of the function ξ → H 1,m (ξ, k) for any real k.

Let us take the limit when µ tends to 0 + -remember that u ǫ is an element of L ∞ (Q) ∩ C([0, T ]; L 2 (Ω))and then the ǫ-limit separately on the parabolic and hyperbolic zones by using (29) and Proposition 5. We obtain:

-

Qp (H 1,m (u, k)∂ t ζ 1 -Q 1,m (u, k)∇P.∇ζ 1 -G 1,m (u, k)ζ 1 ) dxdt - ]0,1[×Q h (H 1,m (π, k)∂ t ζ 1 -Q 1,m (π, k)∇P.∇ζ 1 -G 1,m (π, k)ζ 1 ) dα dx dt ≤ Ω H 1,m (u 0 , k)ζ 1 (0, x)dx - Qp ∂ 1 H 1,m (u, k)∇φ(u).∇ζ 1 dx dt. (30)
So for any

ζ 1 in D(] -∞, T [×Ω h ), ζ 1 ≥ 0, - ]0,1[×Q h (H 1,m (π, k)∂ t ζ 1 -Q 1,m (π, k)∇P.∇ζ 1 -G 1,m (π, k)ζ 1 ) dα dx dt ≤ Ω h H 1,m (u 0 , k)ζ 1 (0, x) dx.
where the limit with respect to m provides:

- ]0,1[×Q h (|π -k|∂ t ζ 1 -sgn(π -k)(f (π) -f (k))∇P.∇ζ 1 -sgn(π -k)g(π, t, x)ζ 1 ) dα dx dt ≤ Ω h |u 0 -k|ζ 1 (0, .) dx. (31)
Let us come back to (24) and consider the L 2 (Q h )-scalar product with

H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k)) = (dist(φ ǫ (u ǫ ), I(φ ǫ (w), φ ǫ (k)))) 2 + 1 m 2 1/2 - 1 m .
Lastly ζ ǫ belongs to H 1 0 (Ω h ), ζ ǫ ≥ 0, while ξ is an element of D(]0, T [×Ω h ), ξ ≥ 0. By taking into account the convexity of the function z → H φǫ(w) 2,m (z, φ ǫ (k)) it comes:

- Q h ( uǫ k ∂ 1 H φǫ(w) 2,m (φ ǫ (τ ), φ ǫ (k))dτ )∂ t ξζ ǫ -Q w 2,m (u ǫ , k)∇P.∇ξζ ǫ -∂ 1 H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))g(t, x, u ǫ )ζ ǫ ξ dx dt ≤ - Q h Q w 2,m (u ǫ , k)∇P.∇ζ ǫ ξ dx dt -ǫ Q h ∂ 1 H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))∇φ ǫ (u ǫ ).∇(ζ ǫ ξ) dx dt, by denoting Q w 2,m (u ǫ , k) = uǫ k ∂ 1 H φǫ(w) 2,m (φ ǫ (τ ), φ ǫ (k))f ′ (τ )dτ.
Now we write:

ǫ Q h ∂ 1 H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))∇φ ǫ (u ǫ ).∇(ζ ǫ ξ) dx dt = ǫ Q h ∇(H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))).∇(ζ ǫ ξ) dx dt = ǫ Q h ∇(H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))ξ).∇ζ ǫ dx dt + ǫ Q h ∇(H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))ζ ǫ ).∇ξ dx dt -2ǫ Q h H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))∇ζ ǫ .∇ξ dx dt,
and through an integration by parts of the first line in the last right-hand side:

ǫ Q h ∂ 1 H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))∇φ ǫ (u ǫ ).∇(ζ ǫ ξ) dx dt = ǫ Q h ∇(H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))ξ).∇ζ ǫ dx dt -ǫ Q h H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))ζ ǫ ∆ξ dx dt -2ǫ Q h H φǫ(w)
2,m (φ ǫ (u ǫ ), φ ǫ (k))∇ζ ǫ .∇ξ dx dt.

Eventually,

- Q h ( uǫ k ∂ 1 H φǫ(w) 2,m (φ ǫ (τ ), φ ǫ (k))dτ )∂ t ξζ ǫ -Q w 2,m (u ǫ , k)∇P.∇ξζ ǫ -∂ 1 H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))g(t, x, u ǫ )ζ ǫ ξ dx dt ≤ - Q h Q w 2,m (u ǫ , k)∇P.∇ζ ǫ ξ dx dt -ǫ Q h ∇(H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))ξ).∇ζ ǫ dx dt +ǫ Q h H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))ζ ǫ ∆ξ dx dt + 2ǫ Q h H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))∇ζ ǫ .∇ξ dx dt. (32)
We consider now the particular choice for the function ζ ǫ ,

ζ ǫ (x) = 1 -exp - M ′ f •φ -1 ǫ ∇P ∞ + ǫL ǫ s(x) , ǫ ≥ 0, (33)
where for any positive parameter µ small enough, s(x) = min(dist(x, Γ h ), µ) for x in Ω h , with L = sup 0<s(x)<µ |∆s(x)|. That way (see [START_REF] Otto | Conservation Laws in Bounded Domains, Uniqueness and Existence via Parabolic Approximation[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF]), for any ϕ of W 1,1 (Ω h ), ϕ ≥ 0,

M ′ f •φ -1 ǫ ∇P ∞ Ω h |∇ζ ǫ |ϕdx ≤ ǫ Ω h ∇ζ ǫ .∇ϕdx + (M ′ f •φ -1 ǫ ∇P ∞ + Lǫ) Γ h ϕdH n-1 . (34)
Therefore, considering that,

|Q w 2,m (u ǫ , k)| = φǫ(uǫ) φǫ(k) ∂ 1 H φǫ(w) 2,m (τ, φ ǫ (k))(f • φ -1 ǫ ) ′ (τ )dτ ≤ M ′ f •φ -1 ǫ H φǫ(w)
2,m (φ ǫ (u ǫ ), φ ǫ (k)) a.e. on Q, and using (34) with ϕ = H φǫ(w)

2,m (φ ǫ (u ǫ ), φ ǫ (k))ξ, we argue that the first line of the right-hand side of (32) is less or equal than:

M ′ f •φ -1 ǫ ∇P ∞ Σ h H φǫ(w) 2,m (T ǫ (u ǫ ), φ(k))ξdH n + o(ǫ), with lim ǫ→0 + o(ǫ) = 0 and for H n -a.e. σ in Σ h , T ǫ (u ǫ (σ)) = 0 on Σ h \Σ hp , φ ǫ (u ǫ (σ)) on Σ hp .
Hence (32) has became:

- Q h ( uǫ k ∂ 1 H φǫ(w) 2,m (φ ǫ (τ ), φ ǫ (k))dτ )∂ t ξζ ǫ -Q w 2,m (u ǫ , k)∇P.∇ξζ ǫ -∂ 1 H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))g(t, x, u ǫ )ζ ǫ ξ dx dt ≤ M ′ f •φ -1 ǫ ∇P ∞ Σ h H φǫ(w) 2,m (T ǫ (u ǫ ), φ(k))ξdH n + o(ǫ) + ǫ Q h H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))ζ ǫ ∆ξ dx dt + 2ǫ Q h H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))∇ζ ǫ .∇ξ dx dt.
Let us interested in the ǫ-limit for this inequality: since

f • φ -1 is Lipschitz continuous, lim ǫ→0 + M f •φ -1 ǫ = M f •φ -1
and stated in [START_REF] Otto | Conservation Laws in Bounded Domains, Uniqueness and Existence via Parabolic Approximation[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF], (ζ ǫ ) ǫ>0 goes to 1 in L 1 (Ω) and (ǫ∇ζ ǫ ) ǫ>0 goes to 0 in L 1 (Ω) n . For the term over Σ h we denote

I ǫ = Σ h H φ(w) 2,m (T ǫ (u ǫ ), φ(k))ξdH n .
As soon as H φ(w) 2,m (., φ(k)) is nonlinear, the weak convergence of the traces of φ(u ǫ ) on Σ hp (observe that on Σ h \Σ hp there is no difficulty to pass to the limit on ǫ) is not sufficient. That is why we consider the sequence (H φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))ξ) ǫ>0 . Thanks to the proposition 5 and since H φǫ(w) 2,m (., φ ǫ (k)) is Lipschitz, up to a subsequence (H

φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))ξ) ǫ>0 strongly converges toward H φ(w) 2,m (φ(u), φ(k))ξ in L q (Q p ), 1 ≤ q < +∞.
Besides, thanks to a chain rule argument and estimate (26), we argue that (H

φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))ξ) ǫ>0 is uniformly bounded in L 2 (0, T ; V ) ∩ L ∞ (Q p )
and so weakly converges (up to a subsequence) towards H φ(w) 2,m (φ(u), φ(k))ξ in L 2 (0, T ; V ). The trace operator from L 2 (0, T ; V ) into L 2 (Σ p ) being linear and continuous, (H

φǫ(w) 2,m (φ ǫ (u ǫ ), φ ǫ (k))ξ) ǫ>0 weakly converges towards H φ(w)
2,m (φ(u), φ(k))ξ in L 2 (Σ p ), and so in L 2 (Σ hp ). It comes lim ǫ→0 + I ǫ = I where

I = Σ h H φ(w) 2,m (T (u), φ(k))ξdH n ,
where for H n -a.e. σ in Σ h ,

T (u(σ)) = 0 on Σ h \Σ hp , φ(u(σ)) on Σ hp .
In addition, thanks to (25) and Claim 1, we are able to pass to the ǫ-limit in the left-hand side of (32). To resume, it comes for any function

ξ of D(]0, T [×Ω h ), ξ ≥ 0: ]0,1[×Q h π k ∂ 1 H φ(w) 2,m (φ(τ ), φ(k))dτ ∂ t ξ -Q w 2,m (π, k)∇P.∇ξ -∂ 1 H φ(w) 2,m (φ(π), φ(k))g h (t, x, π)ξ dαdxdt ≥ -M ′ f •φ -1 ∇P ∞ Σ h H φ(w) 2,m (T (u), φ(k))ξdH n .
When one refers to F.Otto's reasoning in [START_REF] Otto | Conservation Laws in Bounded Domains, Uniqueness and Existence via Parabolic Approximation[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF], p. 115, lemma 7.34, this inequality implies that for any ζ of L ∞ (Σ h ) and ξ of L 1 (Σ h ), ξ ≥ 0, ess lim

τ →0 -]0,1[×Σ h Q ζ 2,m (π(α, σ + τ ν), k)∇P (σ).ν h ξdαdH n ≤ M ′ f •φ -1 ∇P ∞ Σ h H φ(ζ)
2,m (T (u), φ(k))ξdH n .

By taking for H n -a.e. σ in Σ h ζ(σ) = φ -1 (T (u(σ))) = 0 on Σ h \Σ hp , u(σ) on Σ hp , where u(σ) is defined as φ -1 (φ(u(σ))) and belongs to L ∞ (Σ hp ), and then the limit with respect to m we get the boundary conditions for π:

ess lim τ →0 -]0,1[×Σ h F(π(α, σ + τ ν), φ -1 (T (u(σ))), k)∇P (σ).ν h ξdαdH n ≤ 0. ( 35
)
To conclude, the process π fulfills (31) and (35), that means π is an entropy process solution to the quasilinear first-order hyperbolic problem set on Q h : find a measurable and bounded function w such that formally

       ∂ t w - n i=1 ∂ xi (f (w)∂ xi P ) + g(t, x, w) = 0 in Q h , w = u Γ h on Γ h , w(0, .) = u 0 on Ω h ,
where u Γ h is the element of L ∞ (Γ h ) given, for H n -a.e. σ in Σ h , by: u Γ h (σ) = 0 on Σ h \Σ hp , u(σ) on Σ hp , the trace u(σ) being defined as above. Due to [START_REF] Otto | Conservation Laws in Bounded Domains, Uniqueness and Existence via Parabolic Approximation[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF] we know that this problem has a unique solution.

Namely we may rewrite in the context of entropy process the proof provided in [START_REF] Otto | Conservation Laws in Bounded Domains, Uniqueness and Existence via Parabolic Approximation[END_REF][START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF]. It follows that if π 1 and π 2 are two process solutions for initial data u 0,1 and u 0,2 respectively, then for a.e. Classically we first deduce that when u 0,1 = u 0,2 on Ω h , there exists a measurable function u h on Q h such that a.e. on Q h , u h (.) = π 1 (α, .) = π 2 (β, .) for a.e. α and β of ]0, 1[. Another consequence of the uniqueness property is that the whole sequence (u ǫ ) ǫ>0 strongly converges to u h in L q (Q h ), 1 ≤ q < +∞. Thus u h = u |Ω h a.e. on Q h and in (30) the integrals over ]0, 1[ are performed. By taking the limit with respect to m, it follows that u fulfills (3) and [START_REF] Diperna | Convergence of approximate solutions to conservation laws[END_REF]. Boundary conditions (4) are written in (35) due to the definition of φ -1 (T (u)) that completes the proof of Theorem 3.

Conclusions

As a conclusion, let us collect the existence result stated in section 3 and the existence property of section 4.

The key point of the uniqueness theorem 1 being the existence of a strong trace of a weak entropy solution to (1)-(2), while the existence property just requires additional assumptions on f and φ. So we claim:

Corollary 1 Suppose that (27) and (28) hold. Then the coupling problem (1)-( 2) has a weak entropy solution that is the L 1 (Q)-limit of a sequence of solutions to viscous problems (22)-(23) ǫ>0 when ǫ goes to 0 + . In addition if u fulfills (11) then u is the only weak entropy solution satisfying [START_REF] Peyroutet | Error estimate for a splitting method applied to convection-reaction equations[END_REF].

Observe that in [START_REF] Aguilar | Nonlinear Multidimensional Parabolic-Hyperbolic Equations[END_REF], when the interface is included either in the set of outwards characteristics for the first order operator set in Q h , either in set of inward characteristics for the first order operator set in Q h , we highlight in each situation a weak entropy solution to (1) that fulfills [START_REF] Peyroutet | Error estimate for a splitting method applied to convection-reaction equations[END_REF]. So that, in this special framework, the coupling problem (1) has a unique weak entropy solution.

  (t,x,u)∈]0,T [×Ω×R |∂ u g(t, x, u)| and M 0 = ess sup ]0,T [×Ω |g(t, x, 0)|.

∂ 1 H

 1 φ(w) 2,m (φ(u ǫ ), φ(k))ζ ǫ ξ where for any m in N * and any real k and w in [-M, M ]

  t in ]0, T [, ]0,1[×Ω h |π 1 (α, t, x) -π 2 (β, t, x)| dα dβdx dt ≤ Ω h |u 0,1 -u 0,2 | dx e M ′ g t .

Acknowledgements. This work has been supported by CTP, the Pyrenean Work Community.