Silvana Condemi

Stéphane Mazières
email: stephane.mazieres@univ-amu.fr

Pierre Faux

Costedoat Caroline

Andres Ruiz-Linares

Pascal Bailly

Jacques Chiaroni

Mazières Stéphane

Faux Pierre

Ruiz-Linares Andres

Bailly Pascal

Chiaroni Jacques

Blood groups of Neandertals and Denisova decrypted

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

o

  High coverage o Low contamina^on o Unadmixed o Few uncalled snps o Presence of the 3 ABO phenotypes o Non-secretor alleles o DIII type 4, *ceEK à sub-Saharan background, à par;al D, par;al ce (RH:-18) o Neandertal-specific RHD haplotype o Denisova : unknown RHD (c.929A>G, Pro310Arg) o Homozygous at 6 loci: K-k+, Fy(a-b+), Jk(a+b-), S-s+, Di(a-b+) o Ancestral alleles: DARC (rs12075 "Fy b ", c.125A, p.42Asp), SLC4A1 (rs5036 "Memphis", c.166G, p.56Glu), Kel (rs8176038 "Js b ", c.1790T>C, p. 597Pro) 1. Consolida4on of an African origin Ancestral forms for RH, Kell, Duffy, Kidd, MNS and Diego African clusters of RHD, RHCE variants Absence of the an^gen combina^on RhC, RhE, K, Fya, Jkb and S 2. Admixture before expansion towards Oceania RHD*DIII type4 with c.602G and DUC2 : only 2 cases in Australia & Papua New Guinea 3. Gene4c signature of environmental pressure Polymorphism at the ABO level à balanced selec^on Non-secretor FUT2 snps à exposi^on to « Norovirus-like » 4. Neandertal demography at risk High level of homozygosity Par^al RhD, Rhc and Rhe an^gens à risk of hemoly^c disease of the foetus and the new born Evolutionary outcomes Vindija 33.19 Archaic popula+ons References • Bokelman et al. 2019. A gene2c analysis of the Gibraltar Neanderthals. PNAS, 116 (31) 15610-15615 • Meyer et al. A high-coverage genome sequence from an archaic Denisovan individual. Science. 2012; 338(6104):222-6. • Mafessoni et al. A high-coverage Neandertal genome from Chagyrskaya Cave. Proc Natl Acad Sci U S A. 2020. • Möller et al. Erythrogene: a database for in-depth analysis of the extensive varia2on in 36 blood group systems in the 1000 Genomes Project. Blood Advances. 2016; 1(3):240-9. • Prüfer et al. The complete genome sequence of a Neanderthal from the Altai Mountains. Nature. 2014; 505(7481):43-9. • Prüfer et al. A high-coverage Neandertal genome from Vindija Cave in Croa2a. Science. 2017; 358(6363):655-8. • Schoeman EM, Roulis EV, Perry MA, Flower RL, Hyland CA. Comprehensive blood group an2gen pro-file predic2ons for Western Desert Indigenous Australians from whole exome sequence data. Transfu-sion. 2019; 59(2):768-78. • Thorvaldsdodr et al. Integra2ve Genomics Viewer (IGV): high-performance genomics data visualiza2on and explora2on. Brief Bioinform. 2013; 14(2):178-92. • Wagner & Flegel. The Rhesus Site. Transfus Med Hemother. 2014; 41(5):357-63. • Yamamoto et al. Molecular gene2c basis of the histo-blood group ABO system. Nature. 1990; 345(6272):229-33. • Yamamoto & Hakomori S. Sugar-nucleo2de donor specificity of histo-blood group A and B transferases is based on amino acid subs2tu2ons. J Biol Chem. 1990; 265(31):19257-62. strand and alignement with blood group Locus Genomic References o ABO genotype calls : 5 key snps à enzyme ac^vity o RHD and RHCE haplotypes : The Human RhesusBase, o Key indels: ABO c.261delG, c.1061delC, and RHCE 209bp o Low-mapped variants : IGV à Mapping Quality threshold


est destinée au dépôt

  Mesozooplankton structure and functioning on shelf and oceanic waters of during the Kerguelen Bloom during the Keops2 survey (15 October -20 November 2011) Francois Carlotti, Marie-Paule Jouandet, Antoine Nowaczyk, Mireille Harmelin-Vivien To cite this version: Francois Carlotti, Marie-Paule Jouandet, Antoine Nowaczyk, Mireille Harmelin-Vivien. Mesozooplankton structure and functioning on shelf and oceanic waters of during the Kerguelen Bloom during the Keops2 survey (15 October -20 November 2011). ICES/PICES 6th Zooplankton Production Symposium, May 2016, Bergen, Norway. hal-03006174


Walid Baaziz

Georgian Melinte

Ovidiu Ersen

Cuong Pham-Huu

Izabela Janowska
email: janowskai@unistra.fr

Effect of nitriding/nanostructuration of few layer graphene supported iron -based particles; catalyst in graphene etching and carbon nanofilaments growth

Stable, highly faceted and dispersed iron nitride particles supported on few layer graphene are obtained by ammonia decomposition on iron-based particles at the temperature commonly used for the synthesis of N-doped CNTs and graphene etching. The TEM/EELS analysis reveals nitrogen diffusion in a bulk of the particles. The resulted facet FeN x catalyst exhibits high activity in the etching of graphene, which is assisted by catalyst reorganization. The ammonia decomposition is used for the first time for graphene etching, while the highly facet catalyst has an impact on the etched channels structures. According to the shape of the active planes of the catalyst, the etching results in sharp "V" channels' ends and often "step-like" edges. The FeN x morphology proves previously reported triangularisation of arches in highly doped by N carbon nanotubes. The conditioning of the catalyst by its shaping and nitrogen incorporation is investigated additionally in the carbon nanostructures formation, for decomposition of ethane. The herringbond CNFs, "hollow" bamboo-like CNFs/CNTs or CNTs are observed effectively.

Introduction

Design of carbon nanostructures (CNTs, CNFs, graphene) with specific chemical/physical properties in view of particular applications is of high interest. Since the CCVD method allows high yield synthesis of carbon nanostructures, a tailoring of the final structures by controlling the operating conditions constitutes a large area of research, due to the impact of many parameters such as catalyst, support, carbon precursor... [1][2][3] In general, Ni is considered as an efficient catalyst for CNFs production, while Fe is for CNTs formation. 4 However even for a given metal the family of CNFs and CNTs is large and diverse as long as shape and diameter of catalyst vary. This is related to different local kinetics of carbon diffusion/precipitation and orientation of formed carbon layers -parallel to the metal catalyst surface. 5,6 The round shape, graphite supported Ni catalyst particles lead to CNTs, whereas the sharp edges ones to fishbone CNFs formation. 7 A desirable structural conditioning of catalyst remains difficult to achieve, control and observe due to the particles' easy shape reconstruction. 8,9 On the other hand, a chemical doping of CNTs with heteroatoms, such as N, results in bamboo-like structure; 3,10 where for high N content, the orientation of graphitic walls is closed to the "fishbone" CNFs 11 and arches reveal triangular cross-section 12 -suggesting the presence of faceted catalyst. Ndoped CNTs demonstrate high performance particularly in catalysis. [13][14][15][16] The most common way to synthesize nanostructures with N incorporated into the carbon matrix is "in-situ" doping/CVD growth on Fe with ammonia as nitrogen precursor. [17][18][19] It involves ammonia decomposition 20 , which is also an important reaction in energy production; first: by a nitridation 21 -activation of metals (alternative to Pt in ORR reaction), [22][23][24][25][26] second: for the production of CO x -free H 2 for PEMFCs [27][28][29] . The nitriding of iron or iron oxides particles aims also at materials with high magnetic performance. 30,31 High reactivity of Fe with carbon renders also this metal, together with Ni, a useful catalyst for etching of graphene, allowing access to planar carbon nanostructures with specific properties related to their planarity, defined size and shape (quantum dots, nanorribons). [32][33][34] Contrary to the carbon nanofilaments growth, the impact of catalyst shape in the etching of graphene was never investigated.

In the present work we report on the nitridation and faceting of few layer graphene (FLG) supported iron based nanoparticles (Fe 3 O 4 ) with ammonia at the temperature commonly applied for the etching of graphene and CVD synthesis/ N-doping of carbon nanostructures. The morphology of the treated iron-based particles is investigated by TEM/EELS technique. The impact of the nanostructurisation by faceting/nitrogen incorporation of/into the catalyst is investigated in the etching of graphene; and, by decomposition of ethane, in the growth of carbon nanostructures.

Results and discussion

Morphology of ammonia treated iron based particles

The FLG supported spherical Fe 3 O 4 particles with the average diameter of (8 2 nm) 35 , (Fig. 1A), are treated by NH 3 /Ar at 800°C for 2h. Despite high temperature, the treatment results in coalescence of the particles with maximum final size of 50 nm, while an amount of particles remains at their initial size (Fig. 1B, Fig. 5). The etching of FLG is additionally systematically observed (Fig. 1B). According to the TEM analysis the overall dispersion of particles remains very high, while the spherical shape of particles changes into the polygonal (hexagonal, rectangular and, punctually triangular) with more or less visible sharp edges. TEM analysis depicts the existence of three main types of particles; i.e. core-shell in majority, homogeneous and semi-core-shell particles located at the edges of graphene (SI). The strong variation in the particles morphology is the result of partial coalescence which depends on localization on the FLG surface, distance between particles, ramping and local temperature. The similar morphology distribution was observed in our previous study for the iron based particles after treatment with hydrogen, although the particles remained spherical.

A

The element mapping performed, by HR-TEM/EELS technique, on the core shell particles of  50 and 35 nm, (Fig. 2) reveals that oxygen appears in the same region than iron -suggesting the existence of iron oxide species (Fig. 2A); or only in the shell (Fig. 2 B). The detected oxygen can partially or totally originate from the oxygenation once the particle is exposed to air 37 (highly reductive conditions) and induces observed disorder in the shell structure. The analysis reveal the presence of nitrogen, and contrary to oxygen the nitrogen appears exclusively in the core of particles with average diameter of c.a. 25 and 22 nm, respectively (Fig. 2D). In general, a detection of carbon by TEM/EELS is difficult due to the carbon background from the support, although a very thin carbon layer at the surface can be detected for the core-shell particles suspended over transparent etched channels (Fig. 3). This indicates very low dissolution of carbon in iron particles; carbon which could originate from FLG support or from CH 4 formed during the FLG etching. The low dissolution of carbon from FLG is at first linked up with high graphitization degree of FLG. The second group of particles exhibits highly defined shape with sharp edges and more homogenous redistribution of elements, including nitrogen (Fig. 4). Rarely, these particles exhibit very weak signals from nitrogen (oxygen) which could suggest quasi purely iron composition (Fig. 4B). This might be an effect of lower local ammonia concentration; 38 or the analytical noise insufficiency due to the low nitrogen concentration caused by its high distribution within the particles. The presence of nitrogen in Fe particles reveals adsorption and bulk diffusion of nitrogen originating from ammonia decomposition:

(1) Since the XRD pattern confirms the formation of FeN x (SI), a detailed crystallographic structure will not be defined in this work due to its diversity, which can be related to the existence of different morphology/composition of particles. However, the preliminary data from electron diffraction analysis suggest the contribution of Fe 16 N 2 (SI), which is in agreement with earlier reported single phase ''-Fe 16 N 2 , observed for ammonia treated spherical core-shell nanoparticles. 31 The presence or absence of oxygen-rich disordered shells in the formed here particles could be linked to different oxidation potential of nitrides and pure iron, and related formation of passive layer. The oxidation potential of nitrides increases also with a percentage of N. 39

Stability of FeN x /FLG system

The shape retransformation of particles, when NH 3 /Ar is switched to Ar during the cooling step, does not happen. Curiously, the spherical like particles can be punctually observed, but the FeN x core of particles is highly faceted. Highly stable and well dispersed FeN x phase containing particles is a combined effect of nitrogen diffusion, faceting and interaction with support. The additional experiments reveal that the faceting is less pronounced when the particles are treated under H 2 /Ar gas (Fig. 5); and in turn, the faceted but highly coalesced big particles are observed when NH 3 /Ar treatment is applied for the particles supported on Al 2 O 3 (Fig. 6).

The relatively high dispersion of the particles treated with ammonia is prevented by faceting of particles, where higher content of atoms is exposed in the basal plane to interact with the support. The additional stabilizing factors are the edges of graphene. Due to the faceting of particles, a dispersion of particles remains quite high over the FLG surface despite a treatment in highly reductive atmosphere (2h at 800°C) and etching of the support. Similar stabilization of metal on the FLG support by faceting of particles and the edges of graphene we have observed in our previous study for in-situ TEM heated Pt particles, but only up to 700°C. 40,41

FLG etching

Apart from nitrogen, a decomposition of ammonia (1) produces hydrogen. The presence of the latter can result into a gasification of carbon to methane (2) and, in the case of highly graphitized carbon support, the etching of support by Fe and the formation of channels in a direction defined by the crystallographic orientation of graphene.

(

) 2
Such etching of the FLG support is presently observed. The ammonia decomposition over iron based particles does a nitridation of metal and etches the graphene surface. Here, due to the faceted catalyst, the "ends" of the graphene channels are sharp. The figure 7 presents the carbon maps of the particles described above, i.e. of the core-shell particle from fig. 2A (Fig. 7A) and of homogenous particles from Fig. 4A (Fig. 7B). It is evident that the shape of etched channels' ends corresponds to the edge shape of the homogenous particles or to the faceted core of core-shell particles with disordered shells. Moreover, the edges' ends of the channels always depict a "V" shape, with apexes localized symmetrically or not (Fig. 7A,B).

Figure 7. The carbon maps corresponding to the: (A) core-shell particle from fig. 2A and (B) to the homogenous particle from Fig. 4A. "V" shape of etched channels' ends with symmetrically (A) and non-symmetrically localized apex (B).

Statistical TEM/EELS analyses display the highest concentration of nitrogen in the active part of particle: near the "V" ends of channels, which indicates the high catalytic activity of the FeN x phase and reorganization of the catalyst with displacement of FeN x phase/core (Fig. 2A, SI).

The edges of the etched channels reveal often "steps-like" shape (Fig. 8), however many edges are straight lines as long as the particle does not turn, which suggests the equally active facets of catalyst particle (SI). A direction of cutting, which starts from the graphene edges, follows crystallographic directions of graphene, similarly as it was the case of HOPG etched with Ni nanoparticles (SI). 33 Fig. 8A features a 2D-projection TEM image of the etched channel by the active core-shell particle with hardly exhibited facets. The 3D mode analysis (Fig. 8B) performed on the same zone at the graphene surface level, in longitudinal xy direction, clearly evidences three sharp edges in the particle.

The observed "step-like" edges shape of channel suggests a 2D rotation and "creep" movement of particle, which may results from competition between the most active facet and the preferential orientation of etching -defined by graphene lattice. The final turning of particles might be caused by

Carbon nanofilaments structure

The sharp edges of FeN x catalyst have not only an impact on the structurization of graphene channels.

The polygonal and punctually triangle shape of FeN x core/particles (SI), observed here, together with the "V" shape of the channels "ends", confirms a contribution of FeN x phase in the earlier reported triangularisation of nitrogen rich arches in bamboo-like CNTs. 12 Presently, the impact of catalyst shape and nitrogen incorporation on the growth of carbon nanofilaments is investigated by performing a decomposition of ethane. A decomposition of the hydrocarbon is preluded by faceting (and reduction) step of the FLG supported Fe 3 O 4 with H 2 /Ar, at first, and NH 3 /Ar in the second experiment.

The C 2 H 6 decomposition on the H 2 /Ar faceted catalyst results in the growth of CNFs with specific herringbone-type structure, in which graphene sheets are packed perpendicularly to the longitudinal fibers axis, and suggests quasi same diffusion/precipitation of carbon at base/slanting planes of catalyst (Fig. 9A). A similar structure was observed exclusively for the small Ni particles ( 15 nm), as a secondary product of acetylene decomposition. 5 Despite highly faceted catalyst, a decomposition of ethane on iron nitride results in "hollow" bamboolike CNFs/CNTs structures with arches, and conical or parallel to longitudinal axis orientation of external walls (Fig. 9 B). (The impact of catalyst diameter on the formation of "hollow" structures can be here omitted, as curiously, a diameter of ammonia treated particles is slightly lower (20-35 nm) than the one of hydrogen treated particles (35-45 nm)). According to the literature, the "hollow" structure suggests that carbon diffusion hardly happens in the bulk (or basal plane). This indicates the contribution of FeN x species, where the introduction of nitrogen causes the increase of bulk density. 3 We suggest also that the observed above high activity of the FeN x phase in the etching of graphitized carbon may be considered in the growth of doped carbon nanostructures. Depending on the applied conditions during the growth of N-doped carbon nanostructures, such etching of the precipitated carbon layers might occur. The conical shape of the walls, including external ones, in bamboo-likes CNFs depicts faster precipitation of carbon at the slanting planes of catalyst compare to the herringbond CNFs originated from H 2 /Ar treated particles. This indicates that the sharp edges faceted catalyst acts. 5 In some bamboo-likes filaments, however, the orientation of the external walls is parallel to a longitudinal axis, which uncovers rather CNTs morphology and suggests different faceting /nitrogen content in the bulk and at the surface of catalyst (the parallel external walls were found also for triangular aches CNTs 12 ). We have assigned this carbon structure to the punctually observed spherical like particles with highly facet FeN x core, detected even by standard TEM analysis (Fig. 10). For this reason we call them bamboo-like CNTs and not CNFs. To complete the investigation on the catalystgrown nanofilaments structure relation, a decomposition of ethane is performed additionally on the reduced at 400°C (not faceted), spherical iron catalyst. Adequately, it results in the formation of typical hollow carbon nanotube structure (SI).

Experimental

Ammonia treatment of Fe 3 O 4 /FLG was performed in a tubular quartz reactor at 800°C for 2 h, using the gas mixture of NH 3 :Ar (50:50 cc). The cooling step was performed under NH 3 :Ar (50:50 cc) mixture or pure Ar.

B

The synthesis of carbon nanostructures (CNFs, bamboo-like CNFs/CNTs and CNTs) were performed under ethane decomposition at 800°C for 1h in the presence of H 2 / Ar with the ethane: H 2 :Ar ratio of 100:10:100 cc, respectively. The synthesis of bamboo-like CNFs and CNFs were preluded by faceting of FLG-supported Fe 3 O 4 particles with NH 3 /Ar (above) and H 2 /Ar (same procedure) respectively. The synthesis of CNTs was preluded by 2h reduction step with H 2 /Ar, 50:50 cc at 400°C. For the TEM analyses, a drop of suspension containing the FLG (Al 2 O 3 ) supported iron-based NPs nanoaggregates was deposited on a TEM grid covered by a holey carbon membrane. The energy filtered images of the tilt series were obtained using a JEOL 2100F (FEG) TEM/STEM electron microscope operating at 200 kV, equipped with a TRIDIEM post-column imaging filter of the Gatan Company. The images acquired on a 2048*2048 pixel cooled CCD detector were hardware-binned to 512*512 pixels. The three-window method was used to acquire EFTEM images with a monotonic dependence of the chemical signal as a function of concentration.

Conclusions

The new important aspects linked up with the conditioning of iron based catalyst by nitridation/faceting; and its reactivity with carbon are reported:

-A decomposition of ammonia with the formed active FeN x catalyst may be a way to etches graphene with required nanostructuration of graphene tranches and potential decoration of etches by nitrogen; -An observed bulk diffusion of nitrogen into the iron-based particles and interaction of the particles with FLG condition the particles' faceting and stability; -The punctually observed facet triangle FeN x catalyst explains a triangularization of arches in highly CNTs highly doped by N; -The influence of the shape/nitridation on the variable carbon layers orientation and carbon diffusion/precipitation in the grown carbon nanofilaments is presented for the first time within a given catalytic system.

-The FeN x morphological investigations contribute to the knowledge on the catalyst for ammonia decomposition, an important reaction in energy production.

Notes and references †

Electronic Supplementary Information (ESI) available: Additional data dealing with the crystallographic structure and reorganization of FeN x particles (XRD, HR-TEM, TEM) complementary description of carbon nanotubes growth.

See DOI: 10.1039/b000000x/

Figure 1 .

 1 Figure 1. The TEM images of Fe 3 O 4 /FLG hybrid before (A) and after ammonia treatment (B).

36

 36

Figure 2 .

 2 Figure 2. Representative TEM micrographs and elemental mapping (TEM/EELS) of iron base coreshell particles with nitrogen rich core and oxygen contained core-shell (A) or shell (B).

Figure 3 .

 3 Figure 3. TEM and carbon/nitrogen elements micrographs of core-shell nanoparticle obtained after ammonia treatments.

Figure 4 .

 4 Figure 4. TEM micrographs and C/Fe/N mapping of two homogenous-type particle (A) and (B)

Figure 5 .

 5 Figure 5. TEM micrographs showing dispersion of NH 3 /Ar (A) and H 2 /Ar (B) treated Fe 3 O 4 particles supported on FLG.

Figure 6 .

 6 Figure 6. TEM micrographs showing extremely low dispersion of NH 3 /Ar treated Fe 3 O 4 particles supported on Al 2 O 3 .

  the graphene surface, e.g. higher local defects concentration. The later is also responsible for the formation of the observed large void in the graphene surface due to the uncontrolled gasification reaction.

Figure 8 .

 8 Figure 8. Micrographs of the active in carbon etching, iron nitride particle with hardly exhibited facets by TEM projection (A) and visible facets by longitudinal xy direction at the graphene surface level analysis (B). The step-like edges of channel are visible on the (A, B) micrographs.

Figure 9 .

 9 Figure 9. (A) Herringbone-type CNFs grown on H 2 /Ar faceted catalyst. (B) "Hollow" bamboo-like CNFs/CNTs grown on NH 3 /Ar faceted catalyst.

Figure 10 .

 10 Figure 10. TEM micrographs of hollow bamboo-like CNFs (A) and CNTs (B) and related to the CNTs catalyst particle.

Acknowledgment

The authors would like to acknowledge the project FreeCats (NMP3-SL-2012-280658) for financial support.


est destinée au dépôt

Faculté des Sciences et Technologies

CrCr

  Carbide transformations occurring between 1000 and 1100°C in the sub-surface of a Ni-25Cr-0.50C alloy induced by hot oxidation Patrice Berthod, Elodie Conrath To cite this version: Patrice Berthod, Elodie Conrath. Carbide transformations occurring between 1000 and 1100°C in the sub-surface of a Ni-25Cr-0.50C alloy induced by hot oxidation. Eurocorr 2014, Sep 2014, Pise, Italy. wt.% in the carbide free-zone → less th an 0.03w t.% C Horizontal enlargment of the BCC matrix domain: mo re carbon possible in th e carbide free-zone at 1200°C wt.% in the carbide free-zone → less than 0.05wt.%C Horizontal enlargment of the FCC matrix domain: more carbon possible in the carbide free-zone at 1200°C , Faculty of Sciences and Technologies, University of Lorraine, BP 70239, Bd des Aiguillettes, 54506 Vandoeuvre-lès-Nancy, France patrice.berthod@univ-lorraine.fr Alloys and superalloys elaborated by foundry practice are often strengthened by carbides. They constitute an important family of refractory alloys for applications at high temperature [1]. The metallic carbide-forming elements present in the reinforcing carbides are easily oxidable [2]. When they are used in hot conditions in aggressive environments, such alloys are subjected to changes of the carbides initially present in the sub-surface [3,4]. These changes are induced by high temperature oxidation occurring on surface. Such microstructure modifications may possibly modify the local refractoriness [5] as well as the local mechanical properties in sub-surface. These phenomena can be of different types: (eventually) development of a carbide-free zone (case of alloys reinforced by chromium carbides and/or TaC carbides) or not (case of HfC-reinforced alloys), local coarsening/nature change of the carbides initially present, precipitation of new ones ... This study focuses on the oxidation-induced phenomena encountered in the sub-surfaces by ternary Ni-Cr-C alloys, with additionally the study of the effect of the temperature and duration of oxidation. The phenomena are interpreted using thermodynamic calculations [6,7]. Oxidation thus induces curious microstructure phenomena in the sub-surface of the carbide-strengthened alloys. Some of them (new carbides zone) are much more difficult to interpret/explain than the other (carbide-free zone notably). Obviously, in addition to temperature-dependent, these later phenomena are also time-dependent, as sugge sted by the additional tests with longer durations. Further work may consi st in new duration increasing to follow further the evolution with time of this particular intermediate zone. In parallel one can also wonder how the oxidizing atmosphere may influence these phenomena (e.g. presence of water vapour in air).

Example 3 :CrExample 4 : 2 : 1 :

 3421 external and internal oxidation with dev elopment of a carbide-f ree zone f or three M-30Cr-1C-15Ta cast alloy s oxidized during 46 h at 1150°C; outwards dif f usion of Cr and Ta atoms wt.% in the carbide free-zo ne → less than 0. 04wt.%C Horizontal enlargment of the FCC m atrix dom ain: more carbon possible in the carbide free-zone at 1200°C internal (and external) oxidation only (no dev elopment of carbide-f ree zone) f or three {Ni-25Cr}-based {Hf C carbides}-containing cast alloy s oxidized during 46 hours at 1150°C; no Hf dif fusion Example Dev elopment of a carbide-f ree zone but with coarsening of the intermediate carbides and precipitation of new ones Example Surf ace and sub-surf ace states of the M-30Cr-0.4C alloy s af ter oxidation: carbides-f ree zones in all cases and phenomena concerning carbides in the case of the Ni-30Cr-0.4C af ter oxidation f or 50h at 1000°C Isothermal ternary Co-Cr-C phase diagrams (f or 1000 and 1200°C) computed with Thermo-Calc showing the maximal carbon content compatible with the absence of carbides in the carbide-f ree zone, taking into account the Cr-prof ile across this zone Isothermal ternary Ni-Cr-C phase diagrams (f or 1000 and 1200°C) computed with Thermo-Calc showing the maximal carbon content compatible with the absence of carbides in the carbide-f ree zone, taking into account the Cr-prof ile across this zone Isothermal ternary Fe-Cr-C phase diagrams (f or 1000 and 1200°C) computed with Thermo-Calc showing the maximal carbon content compatible with the absence of carbides in the carbide-f ree zone, taking into account the Cr-prof ile across this zone Ev olution with time of the intermediate zone where carbides are modif ied (coarsened f or example) at 1000°C and 1050°C f or a cast Ni-25Cr-0.25C alloy Use of thermodynamic calculations for interpreting the phenomena In most cases, the carbides disappear during oxidation, progressively from the {external oxide, alloy} interface. As demonstrated by thermodynamic calculations these growing carbide-free zones are much poorer in carbon that the initial alloy (here 0.4 or 0.5 wt.%) since the maximal carbon content value is only the tenth part of the initial one. Furthermore it is probably almost inexistent in this area. The carbon atoms initially present as carbides have probably diffused towards the oxidation front to be oxidized into gases. However, in some cases, it seems that a part of this carbon quantity has, on the contrary, diffused inwards to promote the appearance of new quantities of carbide deeper, either by coarsening the present ones, or by precipitating new ones finer. The reason of this inversion of diffusion direction remains to explain, but one can guess that the resulting local microstructure modifications may have consequences on the mechanical, thermal expansion and refractoriness properties of the concerned alloys.


Konstantin Avrachenkov
email: k.avrachenkov@inria.fr

Vladimir Gaitsgory
email: vladimir.gaitsgory@mq.edu.au

Lucas

Lucas Gamertsfelder
email: lucas.gamertsfelder@hdr.mq.edu.au

LP Based Bounds for Cesàro and Abel Limits of the Optimal Values in Non-ergodic Stochastic Systems

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

In this paper, we study asymptotic properties of problems of control of stochastic discrete time systems with time averaging and time discounting optimality criteria, and we establish that the Cesàro and Abel limits of the optimal values in such problems can be evaluated with the help of a certain infinite-dimensional (ID) linear programming (LP) problem and its dual. Note that matters related to the existence and the equality of such limits have been investigated by many; see, e.g., [START_REF] Arisawa | On ergodic stochastic control[END_REF], [START_REF] Arapostathis | Ergodic Control of Diffusion Processes[END_REF], [START_REF] Bardi | Capuzzo-Dolcetta I Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems and Control: Foundations and Applications[END_REF], [START_REF] Bishop | Examples concerning Abel and Cesaro limits[END_REF], [START_REF] Buckdahn | On representation formulas for long run averaging optimal control problem[END_REF], [START_REF] Gaitsgory | Linear programming approach to deterministic infinite horizon optimal control problems with discounting[END_REF], [START_REF] Grüne | Asymptotic controllability and exponential stabilization of nonlinear control systems at singular points[END_REF], [START_REF] Grüne | On the relation between discounted and average optimal value functions[END_REF], [START_REF] Khlopin | Tauberian theorem for value functions[END_REF], [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF], [START_REF] Oliu-Barton | A uniform Tauberian theorem in optimal control[END_REF], [START_REF] Quincampoix | On existence of a limit value in some non-expansive optimal control problems[END_REF], [START_REF] Sznajder | Some comments on a theorem of Hardy and Littlewood[END_REF]. A distinct feature of the present paper is that the Cesàro and Abel limits of the optimal values are evaluated with the help of LP tools.

Allowing one to use convex duality theory, the LP approach to various classes of optimal control problems have been studied extensively in the literature. For example, LP formulations of problems of optimal control of stochastic systems evolving in continuous time have been considered in [START_REF] Arapostathis | Ergodic Control of Diffusion Processes[END_REF], [START_REF] Bhatt | Occupation Measures for Controlled Markov Processes: Characterization and Optimality[END_REF], [START_REF] Buckdahn | Stochastic optimal control and linear programming approach[END_REF], [START_REF] Fleming | Convex duality approach to the optimal control of diffusions[END_REF], [START_REF] Kurtz | Existence of Markov Controls and Characterization of Optimal Markov Controls[END_REF], [START_REF] Stockbridge | Time-Average Control of a Martingale Problem: A Linear Programming Formulation[END_REF]. Various aspects of the LP approach to problems of optimization of discrete time stochastic systems (controlled Markov chains) or continuous time systems with random jumps have been discussed in [START_REF] Borkar | A convex analytic approach to Markov Decision Processes[END_REF], [START_REF] Hernandez-Lerma | Infinite Linear Programming and Multichain Markov Control Processes in Uncountable Spaces[END_REF]- [START_REF] Hordijk | Constrained undiscounted stochastic dynamic programming[END_REF], [START_REF] Piunovskiy | Optimal Control of Random Sequences in Problems with Constraints[END_REF], [START_REF] Piunovskiy | Discounted Continuous-Time Markov Decision Processes with Unbounded Rates: The Convex Analytic Approach[END_REF]. In the deterministic setting, the LP approach has been developed/applied in [START_REF] Goreac | Linearization techniques for L ∞ -control problems and dynamic programming principles in classical and L ∞ control problems[END_REF], [START_REF] Hernandez-Hernandez | The Linear Programming Approach to Deterministic Optimal Control Problems[END_REF], [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI-relaxations[END_REF], [START_REF] Rubio | Control and Optimization. The Linear Treatment of Nonlinear Problems[END_REF], [START_REF] Vinter | Convex Duality and Nonlinear Optimal Control[END_REF] for systems evolving in continuous time considered on a finite time interval. The applicability of the LP approach to deterministic continuous and discrete time systems considered on the infinite time horizon has been explored in [START_REF] Borkar | Linear programming formulation of long run average optimal control problem[END_REF], [START_REF] Borkar | LP Formulations of Discrete Time Long-Run Average Optimal Control Problems: the Nonergodic Case[END_REF], [START_REF] Gaitsgory | Linear Programming Formulations of Deterministic Infinite Horizon Optimal Control Problems in Discrete Time, Discrete and Continuous Dynamical Systems[END_REF]- [START_REF] Gaitsgory | On sets of occupational measures generated by a deterministic control system on an infinite time horizon[END_REF].

In [START_REF] Borkar | Linear programming formulation of long run average optimal control problem[END_REF] and [START_REF] Borkar | LP Formulations of Discrete Time Long-Run Average Optimal Control Problems: the Nonergodic Case[END_REF], in particular, it was shown (for deterministic, continuous time systems in [START_REF] Borkar | Linear programming formulation of long run average optimal control problem[END_REF], and for deterministic, discrete time systems in [START_REF] Borkar | LP Formulations of Discrete Time Long-Run Average Optimal Control Problems: the Nonergodic Case[END_REF]) that the Cesàro and Abel limits of the optimal values are bounded from above by optimal values of certain infinite-dimensional linear programming (IDLP) problems and that these limits are bounded from below by the optimal values of the corresponding dual problems. This paper extends this line of research to stochastic systems evolving in discrete time. In more detail, we consider the discrete time stochastic control system y(t + 1) = f (y(t), u(t), s(t)), t = 0, 1, 2, . . . , [START_REF] Arisawa | On ergodic stochastic control[END_REF] and we assume that the following conditions are satisfied:

• The function f (y, u, s) : Y × Û ×S → R m is continuous in (y, u) on Y × Û and is Borel measurable in s on S,
where Y is a compact subset of R n , Û is a compact metric space, and S is a Polish space. • s(t) ∈ S, t = 0, 1, ..., is a sequence of independent, identically distributed random elements. • The controls u(t), t = 0, 1, ..., are defined by a sequence of functions π def = {π t (y), t = 0, 1, ...} that are Borel measurable selections of a multivalued map

U (•) : Y Û so that u(t) = π t (y(t)) ∈ U (y(t)), t = 0, 1, ..., (2)
where U (•) is upper semicontinuous and compactvalued (that is, U (y) is compact for any y ∈ Y ). • f (y, u, s) ∈ Y for any y ∈ Y , any u ∈ U (y), and any s ∈ S (that is, the set Y is forward invariant with respect to system (1)). Let Π stand for the set of sequences of measurable selections of U (•):

Π def = {π = {π t (•), t = 0, 1, ...} | π t (y) ∈ U (y) ∀ y ∈ Y, π t (•) are Borel measurable}.
For any sequence π ∈ Π (for convenience, such sequences will be referred to as control plans) and any initial condition y(0) = y 0 ∈ Y , let (y π,y0 (•), u π,y0 (•)) stand for the statecontrol trajectory obtained in accordance with (1) and [START_REF] Arapostathis | Ergodic Control of Diffusion Processes[END_REF].

Consider the following optimal control problems

1 T min π∈Π E T -1 t=0 k(y π,y0 (t), u π,y0 (t)) def = v T (y 0 ), (3)
min π∈Π ∞ t=0 ( 1
-) t k(y π,y0 (t), u π,y0 (t)) def = h (y 0 ), (4)
where k(y, u)

: Y × Û → R is a continuous function with |k(y, u)| ≤ M ∀(y, u) ∈ Y × Û , M = const,
and ∈ (0, 1) (that is, (1 -) is a discount factor).

Proposition 1.1: Under the assumptions made above, the optimal value functions v T (•) and h (•) are lower semicontinuous and satisfy the equations:

T v T (y) = min u∈U (y) {k(y, u) + (T -1)E[v T -1 (f (y, u, s))]}, (5) h (y) = min u∈U (y) { k(y, u)+(1-)E[h (f (y, u, s))]} ∀ y ∈ Y.
(6) Also, the minima in (3) and (4) are achieved. Proof. The proof follows Theorems 2.4.6 and 7.2.1 in [START_REF] Bäuerle | Markov Decision Processes with Applications to Finance[END_REF].

We will be interested in evaluating lim T →∞ v T (y 0 ) and lim →0 h (y 0 ) (these limits are commonly referred to as Cesàro and Abel limits). More specifically, we will establish that lim sup T →∞ v T (y 0 ) and lim sup →0 h (y 0 ) are bounded from above by the optimal value of a certain IDLP problem, and that lim inf T →∞ v T (y 0 ) and lim inf →0 h (y 0 ) are bounded from below by the optimal value of the corresponding dual problem (see Theorem 3.5). An immediate consequence of this result is the statement that the Cesàro and Abel limits of the optimal values exist and are equal to each other if there is no duality gap (see Corollary 3.6).

Note that many results obtained in this paper are stated and proved similarly to their deterministic counterparts obtained in [START_REF] Borkar | LP Formulations of Discrete Time Long-Run Average Optimal Control Problems: the Nonergodic Case[END_REF]. However, the part of Theorem 3.5 that establishes the upper bound for the Cesàro and Abel limits of the optimal values (this being the most important result of the paper) is stronger than the corresponding statement (Theorem 3.1) in [START_REF] Borkar | LP Formulations of Discrete Time Long-Run Average Optimal Control Problems: the Nonergodic Case[END_REF] since, in contrast to the latter, it is not assumed that v T (•) and h (•) are continuous.

For the convenience of the reader, we leave out most of the proofs from the following sections. These proofs may be found in the preprint this work is based on (see [START_REF] Avrachenkov | LP Based Upper and Lower Bounds for Cesaro and Abel Limits of the Optimal Values in Problems of Control of Stochastic Discrete Time Systems[END_REF]).

II. MODEL

Let us introduce some notations and definitions that will be used in the subsequent sections. Let G stand for the graph of U (•),

G def = graph(U ) = {(y, u) : u ∈ U (y), y ∈ Y } ,
and let P(G) stand for the set of probability measures defined on Borel subsets of G. (Note that, due to upper semicontinuity of U (•), the graph G is a compact subset of Y × Û .) Given a control plan π ∈ Π and an initial condition y(0) = y 0 ∈ Y , denote by γ π,y0,T ∈ P(G) and γ π,y0, d ∈ P(G) the probability measures defined as follows:

for any Borel Q ⊂ G, γ π,y0,T (Q) = 1 T E T -1 t=0 1 Q (y π,y0 (t), u π,y0 (t)) , (7)
γ π,y0, d (Q) = E ∞ t=0 ( 1
-) t 1 Q (y π,y0 (t), u π,y0 (t)) , (8)
where 1 Q (•) is the indicator function of Q. The measures defined by ( 7) and ( 8) will be referred to as occupational measure and, respectively, discounted occupational measure generated by the control plan π. Note that from ( 7) and ( 8)

it follows that G q(y, u)γ π,y0,T (dy, du) = 1 T E T -1 t=0 q(y π,y0 (t), u π,y0 (t)) (9)
and

G q(y, u)γ π,y0, d (dy, du) = E ∞ t=0 ( 1
-) t q(y π,y0 (t), u π,y0 (t)) (10)
for any Borel measurable function q on G. In fact, the definitions ( 7) and ( 8) are equivalent to that the equality ( 9) and ( 10) are valid if q(•) is an indicator function of Q.

Therefore, these equalities are valid for linear combinations of indicator functions. The validity of ( 9) and ( 10) for any Borel function follows from the fact that any such function can be presented as uniform limit of linear combinations of indicator functions.

Let us denote by Γ T (y 0 ) the set of occupational measures and by Θ (y 0 ) the set of discounted occupational measures:

Γ T (y 0 ) def = π∈Π {γ π,y0,T }, Θ (y 0 ) def = π∈Π {γ π,y0, d }.
Note that, due to ( 9) and [START_REF] Borkar | A convex analytic approach to Markov Decision Processes[END_REF], problems (3) and ( 4) can be rewritten in the form

min γ∈Γ T (y0) G k(y, u)γ(dy, du) = v T (y 0 ) (11)
and min

γ∈Θ (y0) G k(y, u)γ(dy, du) = h (y 0 ), (12)
respectively.

To describe convergence properties of occupational measures, the following metric on P(G) will be used:

ρ(γ , γ ) := ∞ j=1 1 2 j | G q j (y, u)γ (dy, du) - G q j (y, u)γ (dy, du)|
for γ , γ ∈ P(G), where q j (•), j = 1, 2, . . . , is a sequence of Lipschitz continuous functions dense in the unit ball of the space of continuous functions C(G) from G to R. This metric is consistent with the weak * convergence topology on P(G), that is, a sequence γ k ∈ P(G) converges to γ ∈ P(G) in this metric if and only if

lim k→∞ G q(y, u)γ k (dy, du) = G q(y, u)γ(dy, du)
for any q ∈ C(G). Using the metric ρ, we can define the "distance" ρ(γ, Γ) between γ ∈ P(G) and Γ ⊂ P(G) and the Hausdorff metric ρ H (Γ 1 , Γ 2 ) between Γ 1 ⊂ P(G) and Γ 2 ⊂ P(G) as follows:

ρ(γ, Γ) def = inf γ ∈Γ ρ(γ, γ ), ρ H (Γ 1 , Γ 2 ) def = max{ sup γ∈Γ1 ρ(γ, Γ 2 ), sup γ∈Γ2 ρ(γ, Γ 1 )}.
Note that, although, by some abuse of terminology, we refer to ρ H (•, •) as a metric on the set of subsets of P(G), it is, in fact, a semi-metric on this set (since ρ H (Γ 1 , Γ 2 ) = 0 implies Γ 1 = Γ 2 if Γ 1 and Γ 2 are closed, but the equality may not be true if at least one of these sets is not closed).

III. MAIN RESULTS

Consider the IDLP problem

min γ∈W G k(y, u)γ(dy, du) def = k * ( 13
)
where W be defined by the equation

W def = {γ ∈ P(G)| G E ϕ(f (y, u, s)) -ϕ(y) γ(dy, du) = 0 ∀ϕ ∈ C(Y )}.
Proposition 3.1: The following equalities are valid:

lim →0 ρ H ( coΘ , W ) = 0, where Θ def = y0∈Y {Θ (y 0 )}, lim T →∞ ρ H ( coΓ T , W ) = 0, where Γ T def = y0∈Y {Γ T (y 0 )}.
where co stands for the closed convex hulls of the corresponding sets.

Proof. The proof is given in [START_REF] Avrachenkov | LP Based Upper and Lower Bounds for Cesaro and Abel Limits of the Optimal Values in Problems of Control of Stochastic Discrete Time Systems[END_REF].

Corollary 3.2: Due to ( 11) and ( 12), from Proposition 3.1 it follows that:

lim →0 min y0∈Y h (y 0 ) = k * , (14)
lim

T →∞ min y0∈Y v T (y 0 ) = k * . ( 15
)
Proposition 3.3: The set W allows the following (equivalent) representation:

W = γ ∈ P(G) γ 1 (Q) = G P (Q|y, u)γ(dy, du) (16) ∀ Borel Q ⊂ Y ,
where γ 1 is the marginal of γ, that is,

γ 1 (Q) = G 1 Q (y)γ(dy, du),
and P (dy|y, u) is the transition law associated with system (1), that is,

P (Q|y, u) = E[1 Q (f (y, u, s))] ∀ (y, u) ∈ G.
Proof. The proof is given in [START_REF] Avrachenkov | LP Based Upper and Lower Bounds for Cesaro and Abel Limits of the Optimal Values in Problems of Control of Stochastic Discrete Time Systems[END_REF].

REMARK. The validity of the representation ( 16) makes the results established by Proposition 3.1 and Corollary 3.2 consistent with well known results in Markov control processes theory; see [START_REF] Hernandez-Lerma | Infinite Linear Programming and Multichain Markov Control Processes in Uncountable Spaces[END_REF]- [START_REF] Hernandez-Lerma | Further Topics on Discrete-Time Markov Control Processes[END_REF], [START_REF] Piunovskiy | Optimal Control of Random Sequences in Problems with Constraints[END_REF], [START_REF] Piunovskiy | Discounted Continuous-Time Markov Decision Processes with Unbounded Rates: The Convex Analytic Approach[END_REF] and references therein. Many of the latter are obtained under assumptions that are lighter than the assumptions we are using in this paper. Note that some of our assumptions can be relaxed too. For example, the assumption about compactness of the state space Y can be replaced by the assumption about the tightness of the set of occupational measures that make the results of Proposition 3.1 valid. However, to make the presentation more expository, we stick to using simpler (albeit more restrictive) assumptions.

Let us consider the following IDLP problem

inf (γ,ξ)∈Ω(y0) G k(y, u)γ(dy, du) def = k * (y 0 ), (17)
where the feasible set Ω(y 0 ) is defined by (γ, ξ)

∈ P(G) × M + (G) such that Ω(y 0 ) def = {(γ, ξ) ∈ P(G) × M + (G)| γ ∈ W, G (ϕ(y 0 ) -ϕ(y))γ(dy, du)+ G E ϕ(f (y, u, s)) -ϕ(y) ξ(dy, du) = 0 ∀ϕ ∈ C(Y )},
where M + (G) stands for the space of nonnegative measures defined on Borel subsets of G. One way to obtain the above problem is to follow formally the construction of [START_REF] Avrachenkov | Constraint augmentation in pseudo-singularly perturbed linear programs[END_REF] and [START_REF] Avrachenkov | Singularly perturbed linear programs and Markov decision processes[END_REF]; it is obtained, in fact, by augmenting problem [START_REF] Buckdahn | Stochastic optimal control and linear programming approach[END_REF] with additional constraints and an additional "decision variable" ξ.

The problem dual to the augmented IDLP problem (17) can be written in the following form sup (µ,ψ,η)∈D(y0)

µ def = d * (y 0 ), (18)
where D(y 0 ) is the set of triplets (µ,

ψ(•), η(•)) ∈ IR × C(Y ) × C(Y ) that for all (y, u) ∈ G satisfy the inequalities k(y, u) + (ψ(y 0 ) -ψ(y)) + E[η(f (y, u, s))] -η(y) -µ ≥ 0, E[ψ(f (y, u, s))] -ψ(y) ≥ 0. ( 19
)
Note that the optimal value of problem ( 18) can be equivalently represented as

d * (y 0 ) = sup (ψ,η)∈C(Y )×C(Y ) min (y,u)∈G {k(y, u)+ (ψ(y 0 ) -ψ(y)) + E[η(f (y, u, s))] -η(y)}, (20)
where ψ satisfies the second inequality in [START_REF] Gaitsgory | On sets of occupational measures generated by a deterministic control system on an infinite time horizon[END_REF].

The following proposition establishes the validity of the weak duality inequality.

Proposition 3.4: The optimal values of ( 17) and ( 18) are related by the inequality d * (y 0 ) ≤ k * (y 0 ).

(21) Proof. Take any (γ, ξ) ∈ Ω(y 0 ) and (µ, ψ, η) ∈ D(y 0 ). Integrating the first inequality in [START_REF] Gaitsgory | On sets of occupational measures generated by a deterministic control system on an infinite time horizon[END_REF] with respect to γ and taking into account that γ ∈ W , we conclude that G k(y, u)γ(dy, du) + G (ψ(y 0 ) -ψ(y))γ(dy, du) ≥ µ.

Since (γ, ξ) ∈ Ω(y 0 ), from the second inequality in [START_REF] Gaitsgory | On sets of occupational measures generated by a deterministic control system on an infinite time horizon[END_REF] it follows that

G (ψ(y 0 ) -ψ(y))γ(dy, du) = - G E[ψ(f (y, u, s))] -ψ(y) ξ(dy, du) ≤ 0. Therefore, G k(y, u)γ(dy, du) ≥ µ.
Taking first inf over all (γ, ξ) ∈ Ω(y 0 ) in the left-hand-side and then sup over all (µ, ψ, η) ∈ D(y 0 ) in the right-handside, one establishes the validity of [START_REF] Grüne | Asymptotic controllability and exponential stabilization of nonlinear control systems at singular points[END_REF].

Our main result is the following theorem.

Theorem 3.5: The lower and upper Cesàro/Abel limits of the optimal value functions in problems ( 3) and ( 4) satisfy the inequalities, ∀ y 0 ∈ Y ,

d * (y 0 ) ≤ lim inf T →∞ v T (y 0 ) ≤ lim sup T →∞ v T (y 0 ) ≤ k * (y 0 ), d * (y 0 ) ≤ lim inf →0 h (y 0 ) ≤ lim sup →0 h (y 0 ) ≤ k * (y 0 ),
where k * (y 0 ) is the optimal value of the augmented IDLP problem [START_REF] Gaitsgory | Linear Programming Formulations of Deterministic Infinite Horizon Optimal Control Problems in Discrete Time, Discrete and Continuous Dynamical Systems[END_REF] and d * (y 0 ) is the optimal value of its dual [START_REF] Gaitsgory | Linear programming approach to deterministic infinite horizon optimal control problems with discounting[END_REF]. Proof. The proof is given in [START_REF] Avrachenkov | LP Based Upper and Lower Bounds for Cesaro and Abel Limits of the Optimal Values in Problems of Control of Stochastic Discrete Time Systems[END_REF].

We have also the following immediate corollary. Corollary 3.6: Let, for a given y 0 ∈ Y , the strong duality equality be valid:

k * (y 0 ) = d * (y 0 ). ( 22
)
Then the Cesàro and Abel limits of the optimal values exist and are equal:

lim T →∞ v T (y 0 ) = lim →0 h (y 0 ) = k * (y 0 ) = d * (y 0 ). (23)

IV. EXAMPLE

Let the dynamics be one-dimensional and be described by the equation (compare with (1))

y(t + 1) = y(t)u(t)s(t) ∀ t = 0, 1, ... , (24)
where Y = [-1, 1] and U (y) = {-1, 1} (that is, the control can be either equal to 1 or to -1). Assume that s(t) takes only two values: s(t) = 1 with probability 3 4 and s(t) = -1 with probability 1 4 . Consider problems (3) and (4) with k(y, u) = y. It can be readily understood, that, in this example, the plan π * = {π * t (y), t = 0, 1, ...}, where, for any t = 0, 1, ...., π * t (y) = +1 for y ∈ [-1, 0] and π * t (y) = -1 for y ∈ (0, 1], is optimal in both problem (3) and problem (4) (as this is the plan that maximizes the probability for the state variable to be negative). The optimal values of problems ( 3) and ( 4) can be evaluated to be as follows

v T (y 0 ) = - 1 2 |y 0 | + 1 T y 0 + 1 2 |y 0 | ∀ y 0 ∈ Y, (25)
h (y 0 ) = - 1 2 |y 0 | + y 0 + 1 2 |y 0 | ∀ y 0 ∈ Y. (26)
(By a direct substitution, one can verify that v T (y 0 ) and h (y 0 ), defined in accordance with ( 25) and ( 26), satisfy the dynamic programming equations ( 5) and ( 6), respectively.) From ( 25) and ( 26) it follows that

lim →0 h (y 0 ) = lim T →∞ v T (y 0 ) = - 1 2 |y 0 |. (27)
The augmented IDLP problem [START_REF] Gaitsgory | Linear Programming Formulations of Deterministic Infinite Horizon Optimal Control Problems in Discrete Time, Discrete and Continuous Dynamical Systems[END_REF] takes the form

inf (γ,ξ)∈Ω(y0) G yγ(dy, du) = k * (y 0 ), (28)
where Ω(y 0 ) is the set of pairs (γ, ξ) ∈ P(G) × M + (G) that satisfy the equations

G 3 4 ϕ(yu) + 1 4 ϕ(-yu) -ϕ(y) γ(dy, du) = 0, (29)
∀ ϕ ∈ C([-1, 1]), and
G (ϕ(y 0 ) -ϕ(y))γ(dy, du) = (30)
- G 3 4 ϕ(yu) + 1 4 ϕ(-yu) -ϕ(y) ξ(dy, du), ∀ ϕ ∈ C([-1, 1]
), and where

G = Y ×U = [-1, 1]×{-1, 1}
in this case. The corresponding dual problem (see [START_REF] Goreac | Linearization techniques for L ∞ -control problems and dynamic programming principles in classical and L ∞ control problems[END_REF]) is

sup (ψ,η)∈C([-1,1])×C([-1,1]) min (y,u)∈G {y + (ψ(y 0 ) -ψ(y)) (31)
+ 3 4 η(yu)) + 1 4 η(-yu) -η(y)} = d * (y 0 ),
where the ψ functions are assumed to satisfy the inequality

3 4 ψ(yu) + 1 4 ψ(-yu) -ψ(y) ≥ 0, (32)
∀ y ∈ [-1, 1], and ∀ u ∈ {-1, 1}. If a function ϕ is even, then 3 4 ϕ(yu) + 1 4 ϕ(-yu) -ϕ(y) ≡ 0 (since u is either equal to 1 or to -1). Therefore, [START_REF] Hordijk | Constrained undiscounted stochastic dynamic programming[END_REF] is satisfied for all γ ∈ P(G), while [START_REF] Khlopin | Tauberian theorem for value functions[END_REF] Thus, the constraints [START_REF] Khlopin | Tauberian theorem for value functions[END_REF] ensure that the occupational measures γ generated by the state-control trajectories satisfy the property γ(Y \ Y y0 ) = 0. This is consistent with the system's dynamics (see [START_REF] Hernandez-Hernandez | The Linear Programming Approach to Deterministic Optimal Control Problems[END_REF]), according to which the only states attended by the state trajectories are y 0 and -y 0 . Let γ(dy, du) where δ a stands for the Dirac measure concentrated at a, and where κ(y) is equal to 1 for y ∈ [-1, 0] and equal to -1 for y ∈ (0, 1] (that is, for an arbitrary function q(u) on U , U q(u)δ κ(y) (du) = q(1) ∀ y ∈ [-1, 0] and U q(u)δ κ(y) (du) = q(-1) ∀ y ∈ (0, 1]).

Via a direct substitution into ( 29) and ( 30), it can be verified that (γ, ξ) ∈ Ω(y 0 ) (note that it is sufficient to verify the validity of ( 29) and ( 30) only for the even and odd test functions ϕ(•)). Therefore,

k * (y 0 ) ≤ G yγ(dy, du) = - 1 2 |y 0 |. (33)
On the other hand, it can also be verified that the pair of functions ( ψ(y), η(y)), This inequality, along with ( 21) and [START_REF] Lehrer | A uniform Tauberian theorem in dynamic programming[END_REF], allows one to conclude that the optimal value of the IDLP problem [START_REF] Hordijk | Linear programming and Markov decision chains[END_REF] and the optimal value of the dual problem [START_REF] Kurtz | Existence of Markov Controls and Characterization of Optimal Markov Controls[END_REF] are equal (that is, the strong duality equality is valid) and also that (γ, ξ) is an optimal solution of the former and ( ψ(y), η(y)) is an optimal solution of the latter. Note that the common optimal value of problems ( 28) and ( 31) coincides with the Cesàro and Abel limits [START_REF] Hernandez-Lerma | Further Topics on Discrete-Time Markov Control Processes[END_REF].

ψ(y) def = - 1 2 |y|, η (y)
V. IMPROVEMENT OF THE ESTIMATES FROM BELOW Proposition 5.1: The feasible set Ω(y 0 ) allows another (equivalent) representation in the form

Ω(y 0 ) = {(γ, ξ) ∈ P(G) × M + (G)| γ 1 (Q) - G P (Q|y, u)γ(dy, du) = 0, ξ 1 (Q) - G P (Q|y, u)ξ(dy, du) + γ 1 (Q) = 1 Q (y 0 ) ∀ Borel Q ⊂ Y }, (34)
where P (Q|y, u) is the transition probability kernel corresponding to system (1), and where γ 1 and ξ 1 are marginals of γ and ξ, respectively.

Proof. The proof is given in [START_REF] Avrachenkov | LP Based Upper and Lower Bounds for Cesaro and Abel Limits of the Optimal Values in Problems of Control of Stochastic Discrete Time Systems[END_REF].

If the feasible set Ω(y 0 ) is presented in the form [START_REF] Oliu-Barton | A uniform Tauberian theorem in optimal control[END_REF], then the dual problem to [START_REF] Gaitsgory | Linear Programming Formulations of Deterministic Infinite Horizon Optimal Control Problems in Discrete Time, Discrete and Continuous Dynamical Systems[END_REF] takes the form (see [START_REF] Hernandez-Lerma | Infinite Linear Programming and Multichain Markov Control Processes in Uncountable Spaces[END_REF]):

sup (µ,ψ,η)∈ D(y0) µ def = d * (y 0 ), (35)
where D(y 0 ) is the set of triplets (µ,

ψ(•), η(•)) ∈ IR × B(Y ) × B(Y ) that for all (y, u) ∈ G satisfy the inequalities k(y, u) + (ψ(y 0 ) -ψ(y)) + E[η(f (y, u, s))] -η(y) -µ ≥ 0, E[ψ(f (y, u, s))] -ψ(y) ≥ 0,
where B(Y ) stands for the space of bounded Borel functions on Y . Note that the following inequality holds

d * (y 0 ) ≤ d * (y 0 ) ≤ k * (y 0 ), (36)
where the second inequality in ( 36) is just a version of the weak duality inequality, and it may be established similarly to Proposition 3.4.

The following proposition improves upon the estimates from below in Theorem 3.5.

Then the Cesàro and Abel limits of the optimal values exist and are equal:

(as in Example 1), then, by [START_REF] Piunovskiy | Discounted Continuous-Time Markov Decision Processes with Unbounded Rates: The Convex Analytic Approach[END_REF], d * (y 0 ) = d * (y 0 ) = k * (y 0 ). That is, (37) is valid, with [START_REF] Rubio | Control and Optimization. The Linear Treatment of Nonlinear Problems[END_REF] taking a form identical to [START_REF] Hernandez-Lerma | Infinite Linear Programming and Multichain Markov Control Processes in Uncountable Spaces[END_REF]. REMARK: Sufficient conditions for the equality (37) (the strong duality) to be valid have been studied in [START_REF] Hernandez-Lerma | Infinite Linear Programming and Multichain Markov Control Processes in Uncountable Spaces[END_REF]. Note that the strong duality may not be true in the general case. An example, in which the Cesàro and Abel limits of the optimal values are not equal to each other, and, therefore, by Theorem 3.5, there is a duality gap, is given in [START_REF] Shvartsman | Lack of Equality between Abel and Cesaro Limits in Discrete Optimal Control and the Implied Duality Gap[END_REF].

VI. CONCLUSIONS

We have studied the problems of control of stochastic discrete-time systems with time averaging and time discounting optimality criteria. In particular, we have established that the Cesàro and Abel limits of the optimal values in such problems can be estimated with the help of a certain infinitedimensional linear programming problem and its dual.

One possible future research direction is to find easily verifiable sufficient conditions for strong duality. Another interesting research direction is to relax some of the assumptions, e.g., the assumption about the compactness of the state space.

  is converted to G (ϕ(y 0 )ϕ(y))γ(dy, du) = 0 in this case. The latter equality implies that G |y 0 | l γ(dy, du) = G |y| l γ(dy, du) for any l = 1, 2, ..., which, in turn, implies that γ(Y y0 ) = 1, where Y y0 def = {y : |y| = |y 0 |}.

  dy) δ κ(y) (du), ξ(dy, du) def = δ y0 (dy)δ κ(y) (du),

  yu) -ψ(y) = 0, ∀ y ∈ [-1, 1], and ∀ u ∈ {-1, 1}. Therefore (compare the latter with (31) and (32)), -1 2 |y 0 | ≤ d * (y 0 ).

Proposition 5 . 2 :

 52 The following inequalities hold.lim infT →∞ v T (y 0 ) ≥ d * (y 0 ) ∀ y 0 ∈ Y, lim inf →0 h (y 0 ) ≥ d * (y 0 ) ∀ y 0 ∈ Y.Proof. The proof is given in[START_REF] Avrachenkov | LP Based Upper and Lower Bounds for Cesaro and Abel Limits of the Optimal Values in Problems of Control of Stochastic Discrete Time Systems[END_REF].Corollary 5.3: Let, for a given y 0 ∈ Y , the strong duality equality be valid: k * (y 0 ) = d * (y 0 ).

h

  (y 0 ) = k * (y 0 ) = d * (y 0 ). (38) Proof. Follows directly from Proposition 5.2 and Theorem 3.5. Note that, if k * (y 0 ) = d * (y 0 )

ACKNOWLEDGEMENT

We would like to thank Yi Zhang from the University of Liverpool for useful comments on the present work.


Joffrey Mianné

Amel Nasri

Chloé Nguyen Van

Chloé Bourguignon

Mathieu Fieldès

Engi Ahmed

Christine Duthoit

Nicolas Martin

Hugues Parrinello

Anaïs Louis

CRISPR/Cas9-mediated gene knockout and interallelic gene conversion in human induced pluripotent stem cells using non-integrative bacteriophage-chimeric retrovirus-like particles

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


María Anguiano
email: anguiano@us.es

Renata Bunoiu
email: renata.bunoiu@univ-loraine.fr

On the flow of a viscoplastic fluid in a thinperiodic domain

, where a stationary incompressible Navier-Stokes flow was studied. Recently, the model of the thin porous medium under consideration in this paper was introduced in [FaEtAl16], where the flow of an incompressible viscous fluid described by the stationary Navier-Stokes equations was studied by the multiscale asymptotic expansion method, which is a formal but powerful tool to analyse homogenization problems. These results were rigorously proved in [AS18] using an adaptation (introduced in [AS17]), of the unfolding method from [CiEtAl08]. This adaptation consists of a combination of the unfolding method with a rescaling in the height variable, in order to work with a domain of fixed height, and to use monotonicity arguments to pass to the limit. In [AS17], in particular, the flow of an incompressible stationary Stokes system with a nonlinear viscosity, being a power law, was studied. For nonstationary incompressible viscous fluid flow in a thin porous medium we refer to [An17], where a nonstationary Stokes system is considered, and [An217], where a nonstationary non-Newtonian Stokes system, where the viscosity obeys the power law, is studied. For the unfolding method applied to the study of problems stated in other type of thin periodic domains we refer for instance to [Gr04] for crane type structures and to [GrEtAl17] for thin layers with thin beams structures, where elasticity problems are studied.

Viscoplastic fluids are quite often encountered in real life. We mention oils, polymer solutions, vocanic lavas, muds and clays, avalanches, liquid chocolate. The theory of the fluid mechanics of such materials has several different applications, as for instance in the oil and gas industry, which can be found in the ground, which is a porous medium. The most commonly studied viscoplastic fluid is the Bingham fluid. In our thin porous medium, we consider the flow of a nonlinear viscoplastic Bingham flow, whose yield stress itself depends on the small parameter characterizing the geometry of the domain, denoted e. The first study of this type of problem is due to [START_REF] Lions | Écoulement d'un fluide viscoplastique de Bingham dans un milieu poreux[END_REF], where the problem was studied in a classical porous medium, by using the multiscale asymptotic expansion method. A nonlinear Darcy law was obtained after the passage to the limit e ! 0. The corresponding convergence result was proved in [START_REF] Bourgeat | A note on homogenization of Bingham flow through a porous medium[END_REF] with the two-scale convergence method and then recovered in [START_REF] Bunoiu | Unfolding Method for the Homogenization of Bingham flow[END_REF] with the periodic unfolding method from [START_REF] Cioranescu | The periodic Unfolding Method in Homogenization[END_REF]. For the study in a porous medium with a doubly periodic structure we refer to [START_REF] Bunoiu | Bingham Flow in Porous Media with Obstacles of Different Size[END_REF], where a more involved nonlinear Darcy law was derived. The flow of a Bingham fluid was also studied in thin domains of small height, denoted e. We refer the reader to [START_REF] Bunoiu | Fluide de Bingham dans une couche mince[END_REF], [START_REF] Bunoiu | Asymptotic behaviour of a Bingham fluid in thin layers[END_REF], and [START_REF] Bunoiu | Asymptotic Analysis of a Bingham Fluid in a Thin T-like Shaped Structure[END_REF] for these studies, where a lower dimensional Bingham-like law was exhibited from the limit problem, after the passage to the limit e ! 0. This law was already used in the engineering (see [START_REF] Liu | Approximate equations for the slow spreading of a thin sheet of Bingham plastic fluid[END_REF]), but no rigourous mathematical derivation was previously known.

The paper is organized as follows. In Section 1.2 we state the problem: we define in (1.1) the thin porous medium (see also Fig. 1.3), in which we consider the flow of a viscoplastic Bingham fluid with velocity verifying the nonlinear variational inequality (1.5). In Section 1.3 we state and prove the main result of our paper, Theorem 1. We then give in Section 1.4 some conclusions and perspectives and we end the paper with a list of References.

Statement of the Problem

The domain: the periodic porous medium is defined by a domain w and an associated microstructure, or periodic cell Y 0 = [ 1/2, 1/2] 2 , which is made of two complementary parts: the fluid part Y 0 f , and the solid part

Y 0 s (Y 0 f S Y 0 s = Y 0 and Y 0 f T Y 0 s = ?).
More precisely, we assume that w is a smooth, bounded, connected set in R 2 , and that Y 0

s is an open connected subset of Y 0 with a smooth boundary

∂Y 0 s , such that Y 0 s is strictly included in Y 0 .
The microscale of the porous medium is a small positive number e. The domain w is covered by a regular mesh of square of size e: for k 0 2 Z 2 , each cell

Y 0 k 0 ,e = ek 0 + eY 0 is divided in a fluid part Y 0 f k 0 ,e
and a solid part Y 0 s k 0 ,e , i.e. is similar to the unit cell Y 0 rescaled to size e. We define Y = Y 0 ⇥ (0, 1) ⇢ R 3 , which is divided in a fluid part Y f = Y 0 f ⇥ (0, 1) and a solid part Y s = Y 0 s ⇥ (0, 1), and consequently Y k 0 ,e = Y 0 k 0 ,e ⇥ (0, 1) ⇢ R 3 , which is divided in a fluid part Y f k 0 ,e and a solid part Y s k 0 ,e .

We define L e (see Fig.

W e = {(x 1 , x 2 , x 3 ) 2 w e ⇥ R : 0 < x 3 < e}. (1.1)
We make the assumption that the solids t(Y 0 s k 0 ,e ) do not intersect the boundary ∂ w (see Fig. 1.2): The problem: in the domain W e defined in (1.1), we consider the stationary flow of an incompressible Bingham fluid. Following [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF], the problem is formulated in terms of a nonlinear variational inequality.

Y f k , Y s k ,
For a vectorial function v, we define (1  i, j  3)

(D(v)) i, j = 1 2 ∂ x j v i + ∂ x i v j , |D(v)| 2 = D(v) : D(v),
where : denotes the full contraction of two matrices: for A = (a i, j ) 1i, j3 and B = (b i, j ) 1i, j3 , we have A : B = Â 3 i, j=1 a i j b i j . We consider the space

V (W e ) = {v 2 (H 1 0 (W e )) 3 | div v = 0 in W e },
and for u, v 2 (H 1 0 (W e )) 3 , we introduce

a(u, v) = 2µ Z W e D(u) : D(v)dx, (1.3) j(v) = p 2g e Z W e |D(v)|dx, (1.4) (u, v) W e = Z W e u • vdx,
where the positive real µ is the viscosity of the Bingham fluid and the positive real g is related to the yield stress of the Bingham fluid. More precisely, the yield stress of the Bingham fluid under consideration in this work is of the form ge, where e is the parameter related to the geometry of the domain. This yield stress is exactly the one considered by [START_REF] Lions | Écoulement d'un fluide viscoplastique de Bingham dans un milieu poreux[END_REF], where the flow of a Bingham fluid in a classical porous medium was studied.

Let f 2 (L 2 (W )) 3 be given such that f = ( f 0 , 0) and f e 2 (L 2 (W e )) 3 be defined by f e (x) = f (x 0 , x 3 /e), a.e. x 2 W e . The model of the flow is described by the following nonlinear variational inequality:

Find u e 2 V (W e ) such that a(u e , v u e ) + j(v) j(u e ) ( f e , v u e ) W e , 8v 2 V (W e ).

(1.5)

From [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF], we know that for every fixed e there exists a unique u e 2

V (W e ) solution of problem (1.5).

Main Convergence Result

Our aim is to study the asymptotic behavior of u e , solution of problem (1.5), when e tends to zero. For this purpose, we first use the dilatation of the domain W e in the variable x 3 , namely

y 3 = x 3 e , (1.6)
in order to have the functions defined in an open set with fixed height, denoted e W e and given by (1.2).

Namely, we define ũe 2 (H1 0 ( e W e )) 3 by ũe (x 0 , y 3 ) = u e (x 0 , ey 3 ) a.e. (x 0 , y 3 ) 2 e W e .

Let us introduce some notation which will be useful in the following: for a vectorial function v = (v 0 , v 3 ) (and, analogously, for a scalar function w), associated to the change of variables (1.6), we introduce the operators D e , D e , div e ande , defined by

D e [v] = 1 2 D e v + D t e v , |D e [v] | 2 = D e [v] : D e [v] , (D e v) i, j = ∂ x j v i for i = 1, 2, 3, j = 1, 2, (D e v) i,3 = 1 e ∂ y 3 v i for i = 1, 2, 3, div e v = div x 0 v 0 + 1 e ∂ y 3 v 3 , -e w = (-x 0 w, 1 e ∂ y 3 w) t .
We consider the space

V ( e W e ) = { ṽ 2 (H 1 0 ( e W e )) 3 | div e ṽ = 0 in e W e },
and for ũ, ṽ 2 V ( e W e ), we introduce a e ( ũ, ṽ) = 2µ Using the transformation (1.6), the variational inequality (1.5) can be rewritten as:

Find ũe 2 V ( e W e ) such that a e ( ũe , ṽ ũe ) + j e ( ṽ) j e ( ũe ) ( f , ṽ ũe ) e W e , 8 ṽ 2 V ( e W e ).

(1.7)

We start by obtaining some a priori estimates for ũe , stated in the next lemma.

Lemma 1. There exists a constant C independent of e, such that if ũe 2 (H 1 0 ( e W e )) 3

is the solution of problem (1.7), one has

k ũe k (L 2 ( e W e )) 3  Ce 2 , kD e [ ũe ]k (L 2 ( e W e )) 3⇥3  Ce, (1.8
)

kD e ũe k (L 2 ( e W e )) 3⇥3  Ce.
(1.9)

We extend the velocity ũe by zero to the W \ e W e and denote the extension by the same symbol. Obviously, estimates (1.8)-(1.9) remain valid for the extended function and the extension is divergence free too. According to this extension, problem (1.7) can be written as:

2µ Z W D e [ ũe ] : D e [ ṽ ũe ] dx 0 dy 3 + p 2g e Z W |D e [ ṽ]|dx 0 dy 3 (1.10) p 2g e Z W |D e [ ũe ]|dx 0 dy 3 Z W f • ( ṽ ũe ) dx 0 dy 3 ,
for every ṽ that is the extension by zero to the whole W of a function in (H 1 0 ( e W e )) 3 . Our main result is the following theorem: Theorem 1. Let ũe be the solution of problem (1.10). There exists û 2 L 2 (w; H 1 ] (Y ) 3 )(here "]" denotes Y 0 -periodicity), such that

ũe e 2 * Z Y 0 ûdy 0 in L 2 (W ), R Y 0 û3 dy 0 = 0, û = 0 on w ⇥Y s , û = 0 on y 3 = {0, 1}, div y û = 0 in w ⇥Y , div x 0 ✓ Z Y û0 (x 0 , y)dy ◆ = 0 in w, ✓ Z Y û0 (x 0 , y)dy ◆ • n = 0 on ∂ w,
and û is the unique solution of the limit problem

2µ Z w⇥Y D y [ û] : (D y [ ṽ] D y [ û]) dx 0 dy + p 2g Z w⇥Y D y [ ṽ] dx 0 dy p 2g Z w⇥Y D y [ û] dx 0 dy Z w⇥Y f 0 • ṽ0 û0 dx 0 dy, (1.11)
for every ṽ 2 L 2 (w; H 1 ] (Y ) 3 ) such that ṽ(x 0 , y) = 0 in w ⇥Y s , div y ṽ = 0 in w ⇥Y,

div x 0 ✓ Z Y ṽ0 (x 0 , y)dy ◆ = 0 in w, ✓ Z Y ṽ0 (x 0 , y)dy ◆ • n = 0 on ∂ w.
Proof: First step A priori estimates The change of variable (1.6) does not provide the information we need about the behavior of ũe in the microstructure associated to e W e . To solve this difficulty, we use an adaptation of the unfolding method from [START_REF] Cioranescu | The periodic Unfolding Method in Homogenization[END_REF]. In order to apply the unfolding method, we will need the following notation: for k 0 2 Z 2 , we define

k : R 2 ! Z 2 by k(x 0 ) = k 0 () x 0 2 Y 0 k 0 ,1 .
(1.12)

Remark that k is well defined up to a set of zero measure in R 2 (the set [ k 0 2Z 2 ∂Y 0 k 0 ,1 ). Moreover, for every e > 0, we have

k ✓ x 0 e ◆ = k 0 () x 0 2 Y 0 k 0 ,e .
According to the adaptation introduced in [AS17] of the unfolding method, we divide the domain W in rectangular cuboids of lateral lengths e and vertical length 1. For this purpose, given ũe 2 (H 1 0 (W )) 3 , we define ûe by ûe (x 0 , y) = ũe

✓ ek ✓ x 0 e ◆ + ey 0 , y 3 ◆ , a.e. (x 0 , y) 2 w ⇥Y, (1.13)
where the function k is defined in (1.12). For k 0 2 K e , the restriction of ûe to Y 0 k 0 ,e ⇥Y does not depend on x 0 , whereas as a function of y it is obtained from ũe by using the change of variables

y 0 = x 0 ek 0 e , which transforms Y k 0 ,e into Y .
We can obtain a priori estimates for the sequence ûe . There exists a constant C independent of e, such that ûe defined by (1.13) satisfies

D y [ ûe ] (L 2 (w⇥Y )) 3⇥3 Ce 2 , (1.14) k ûe k (L 2 (w⇥Y )) 3  Ce 2 .
(1.15)

Second step Convergence results

As in [START_REF] Anguiano | Homogenization of an incompressible non-Newtonian flow through a thin porous medium[END_REF], we can obtain compactness results for the sequences ũe and ûe satisfying the a priori estimates given before.

For a subsequence of e still denoted by e, there exist ũ 2 H 1 (0, 1; L 2 (w) 3 ), where ũ3 = 0 and ũ = 0 on y 3 = {0, 1}, û 2 L 2 (w; H 1 ] (Y ) 3 ) ("]" denotes Y 0 -periodicity), with û = 0 on w ⇥Y s and û = 0 on y

3 = {0, 1} such that R Y û(x 0 , y)dy = R 1 0 ũ(x 0 , y 3 )dy 3 with R Y û3 dy = 0, such that ũe e 2 * ( ũ0 , 0) in H 1 (0, 1; L 2 (w) 3 ), ûe e 2 * û in L 2 (w; H 1 (Y ) 3 ),
(1.16) div y û = 0 in w ⇥Y, (1.17)

div x 0 ✓ Z 1 0 ũ0 (x 0 , y 3 )dy 3 ◆ = 0 in w, ✓ Z 1 0 ũ0 (x 0 , y 3 )dy 3 ◆ • n = 0 on ∂ w, div x 0 ✓ Z Y û0 (x 0 , y)dy ◆ = 0 in w, ✓ Z Y û0 (x 0 , y)dy ◆ • n = 0 on ∂ w.
(1.18) Third Step Passage to the limit By using (1.13), we first transform the variational inequality (1.10) in a variational inequality stated in the domain w ⇥ Y. Then, by choosing suitable test functions, we pass to the limit e ! 0. By using convergences (1.16), (1.17), (1.18) we find the limit problem (1.11). The uniqueness of the solution û of problem (1.11) is proved by contradiction.

Conclusions and perspectives

By using dimension reduction and homogenization techniques, we studied the limiting behavior of the velocity for a nonlinear viscoplastic Bingham flow with small yield stress ge, in a thin porous medium of small height e and for which the relative dimension of the pores is e. We found in Theorem 1 the limit problem (1.11), in which both effects of a nonlinear Darcy law and a lower-dimensional Binghamlike law appear. Indeed, as in [START_REF] Lions | Écoulement d'un fluide viscoplastique de Bingham dans un milieu poreux[END_REF](see also [START_REF] Bunoiu | Bingham Flow in Porous Media with Obstacles of Different Size[END_REF]), problem (1.11) can be written as a nonlinear Darcy law set in the domain w. The third component of the velocity of filtration appearing in the nonlinear Darcy law equals zero and this phenomenon corresponds precisely to a two-dimensional Bingham-like law (see [START_REF] Bunoiu | Asymptotic behaviour of a Bingham fluid in thin layers[END_REF]).

In the forthcoming work [AnEtAl] we study the cases of thin porous media those periodicity parameter is a e instead of e. Different cases are analysed, following the ratio between the height e of the porous media and the relative dimension a e of the periodically distributed pores. Moreover, we consider the more involved case in which the convergence of the pressure of the flow is also studied.

  1.1) by L e = w ⇥ (0, e).

Fig. 1. 1

 1 Fig. 1.1 Views of the domain L e

Fig. 1. 2

 2 Fig. 1.2 Views of the domain w e

On the flow of a viscoplastic fluid in a thin periodic domain

Acknowledgments

María Anguiano has been supported by Junta de Andalucía (Spain), Proyecto de Excelencia P12-FQM-2466.