Jérôme Chevalier

Laurent Gremillard

Rachid Zenati

Yves Jorand

Christian Olagnon

Gilbert Fantozzi

Slow Crack Growth in Zirconia Ceramics with Different Microstructures

Keywords: Crack, Stress Corrosion, Toughness, Transformation, R-Curve

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INTRODUCTION

The concept of stress induced phase transformation in zirconia ceramics represents one of the most remarkable innovations in the ceramic field. Indeed, it was shown in the 70's first by Garvie et al. [l] then by Gupta et al. [2] that zirconia exhibited a transformation toughening mechanism acting to resist to crack propagation. The stress induced phase transformation involves the transformation of metastable tetragonal grains towards the monoclinic phase at the crack tip which, accompanied by a volumetric expansion, leads to compressive stresses acting to reduce the driving force for crack propagation. Zirconia ceramics can be found in the form of tetragonal zirconia polycrystals (TZP), stabilised in the tetragonal symmetry by an addition of different oxides, for example Cerium (Ce-TZP) or Yttrium (Y-TZP) oxides. Zirconia ceramics can also exist in the form of tetragonal precipitates in a cubic matrix. An example is the Magnesia Partially Stabilised Zirconia, as referred to as Mg -PSZ. Zr02 can also be used as one of the components in a composite material, as Zr02 dispersed ceramics. An example of such a composite would be the Ah03 -Zr02 composites (ZTA). Zirconia can also be found in the form of single crystals for optical properties. In these later, the cubic structure is stabilised by 10 mol.% Yttria and no phase transformation occurs. An extensive number of studies has led to optimal processing routes for increasing the toughness of zirconia based ceramics. It is well recognised that zirconia ceramics can exhibit a toughness from 6 MPavm for the 3Y-TZP or the ZTA composites to more than 15 MPavm for the Ce-TZP or some Mg-PSZ which demonstrate a high toughening capability [START_REF] Green | Transformation Toughening of Ceramics[END_REF]. These fracture toughness values are much higher than the 1.5 MPavm measured for zirconia single crystal [START_REF] Pajares | [END_REF].

Ceramics are generally susceptible to slow crack growth (SCG) under static loading which can cause delayed fa ilure. Theoretical and experimental studies in glass or in single crystals (especially sapphire) have shown that SCG was a consequence of corrosion assisted by high stresses at the crack tip, as referred as to stress corrosion (5)(6)(7)(8)[START_REF] Lawn | Fracture of Brittle Solids[END_REF][START_REF] Michalske | [END_REF](11)(12). Such a stress corrosion is believed to occur in zirconia based ceramics (13)(14)[START_REF] Nag | Fracture Mechanics of Ceramics[END_REF]. The analysis of SCG in these materials is however difficult due to the effect of transformation toughening on crack velocities. This is the aim of this paper to compare the SCG behaviour of different zirconia materials, with different microstructures, thus with different toughening capabilities. This paper therefore summ arises different SCG laws obtained in the same laboratory, mainly with the same Double Torsion (DT) technique. This work is an attempt to show that the SCG behaviour of all zirconia based ceramics is basically the same, i.e. stress corrosion by water molecules at the crack tip, the observed difference lying on the toughening capability of a given zirconia.

MATERIALS AND EXPERIMENTAL PROCEDURE

Experimental procedure: the Double Torsion method

The Double Torsion (DT) was used as a direct method to study Slow Crack Growth in the different zirconia ceramics. The geometry of the DT specimen, the loading configuration and the analysis of experimental results are given in details in refs [START_REF] Chevalier | [END_REF][START_REF] Chevalier | Fracture Mechanics of Ceramics[END_REF] and also in ref [START_REF] Ebrahimi | [END_REF] in the same proceedings. The experimental results were obtained from either relaxation tests (for the determination of crack velocities higher than 10• 7 m/s) or constant loading method (for crack velocities lower than 10• 7 m/s). The method has been compared with indirect methods on micro-cracks in 3Y-TZP and gives an excellent agreement with the latter (19]. The advantage of the DT configuration is to visualise directly the crack extension at the surface of the specimen, and the overall shape of the V K1 (crack velocity versus applied stress intensity factor) can be obtained without any assumption. Some results are presented for laminate zirconia based composites. They were obtained with the SENB method.

Materials Cubic Zirconia Single Crys tal:

A 10%mol. Yttria stabilised zirconia single crystal (Marketec inc.) was chosen as a model brittle material, without any reinforcement due to transformation toughening. Only limited literature has been devoted to the study of mechanical properties of zirconia single crystal, because the properties of interest are optical. However, it was found by Pajares et al. [START_REF] Pajares | [END_REF] that zirconia single crystal fracture characteristics were anisotropic. The toughness measured by the SENB (Single Edge Notched Beam) method varied from 1.5 to 1.9 MPavm along <110> or <100> direction respectively. Well defined indentation cracks were observed along <110> direction. Thus, it is believed that this latter is a privileged cracking direction. DT specimens were therefore oriented along the <110> direction, the { 100} plane corresponding to the specimen plane.

3Y-TZP Ceramics:

3%mol. Yttria stabilised zirconia ceramics present outstanding mechanical properties, especially strength which can reach more than one GPa. These very important strength values obtained for 3Y-TZP are a combination of high toughness (about 6-7 MPavm) and very fine grain size (0,5 µm for most commercial products). Thus, it was considered in this study as another model material, with a small grain size (no or little bridging) and a moderate amount of transformation toughening.

Two commercial powders were used as starting materials. Tosoh co-precipitated powder TZ3Y (Tosoh inc.) was chosen as a highly pure zirconia, with less than 100 ppm impurity content. Daiichi powder was chosen because it exhibits naturally a significant amount of alumina and Silica content (total 1500 ppm) that are beneficial for ageing properties. The powders were elaborated by a slip casting method, all with a 80 wt. % solid content. 2.5 wt. % Colloidal silica was added in one slurry of Tosoh powder in order to achieve a zirconia ceramic with a significant amount of silica phase. The different slurries were put in plaster moulds to eliminate water, then dried at 25°C during 7 days, and fired at different temperatures (heating rate 100°C/h) during 5 hours in air in order to achieve different grain sizes. Table 1 summar ises the different zirconia ceramics that were elaborated in this work and the firing conditions. By choosing two different powders, with or without silica addition and by firing at different temperatures, it was possible to process 3Y-TZP with different microstructures, thus different mechanical properties. fracture toughness could be increased, if the length of the transformation zone was restricted to approximately the zone width, which is not the case for monolithic Ce-TZP. They proposed to change the frontal zone in Ce-TZP by introducing laminar composites, containing layers of Ce-Zr02 and either Ah03 or mixture of Ah03 and Ce-Zr02. They used a colloidal method that allowed formation of layers with thickness as small as l 0 µm. In such conditions, a fracture toughness of more than 30 MPa.m 11 2 was predicted. Thus our aim was to process some laminar composites by slip casting method to investigate the extend of crack resistance of this new generation of materials in comparison to the monolith. Slip casting by pressure filtration route was used to elaborate: {i)-Dense Ceria Stabilised Zirconia (l 0%mol. Ce-TZP). (ii)-Multilayer composites based on Ce-TZP. Multilayer composites with alternate layers of dense Ce-TZP and mixture of 20 wt. % Ah03 and 80 wt. % Ce-Zr02 were fabricated by sequential slip casting.

The starting materials were: (i) Ce-TZP powder (10 mol% Ce02) with a mean particle size d50 = 0.5µm (Mandoval I Zirconia Sales Ltd CEZ-10, U.K.), and a specific surface area 20 m 2 /g (BET), (ii) a-Ah03 powder with d50 = 0.3 µm (Baikowski SM8, France), and a specific surface area 10 m 2 /g (BET). Two different aqueous slips containing either Ce-TZP or mixed composition of 80 wt% Ce-TZP and 20 wt% Ah03 with a solid content of 70 wt% were prepared after homogenisation in zirconia or alumina ball mill for 24h at pH 9.

Those suspensions were casted in a filtration device with rectangular aluminium mould. The green compacts were dried during one week at 25°C and fired at 1450°C for lh in air, at a heating rate of 100°C/h.

Full density (> 99% theoretical density) and 1-2 µm grain size were obtained.

The following mechanisms have been identified as active in the toughening of Ce-TZP systems: transformation shielding for the monolith, and interfacial mechanisms (i.e. crack deflection and delamination) for composites. The enlargement of the process zone is also efficient to shield the crack. The crack propagation behaviour of multilayer ceramics were determined at room temperature using SENB experiments, because DT is not adapted in this last case, due to the curve crack front which would complicate the analysis.

RESULTS AND DISCUSSION

Slow Crack Growth by stress corrosion in zirconia ceramics

Different systematic studies on Slow Crack Growth have been conducted on glass, alumina single crystals and mica [5][6][7][8][START_REF] Lawn | Fracture of Brittle Solids[END_REF][START_REF] Michalske | [END_REF][11][12]. As a general trend, they have shown that an unique relation could be established, for a given environment, between the velocity (V) and the stress intensity factor (K I ) with three different stages attributed to distinct mechanisms. In the first stage (Region I, for low velocities), crack propagation is limited by the thermally activated reaction kinetic between the ceramic and corrosive molecules. These molecules were clearly identified in the case of silica as Hp [5]. Region II, at intermediate velocities, depends on environment, but is much less sensitive to K1. This region is often attributed to gaseous diffusion along the crack interface. Region III, for high velocities represents the fracture of ceramic bonds without chemical reaction with environmental species. Although mentioned by Wiederhom [7] in glass, the occurrence of a threshold is still under debate. Recently, Wan et al. [23][24] considered this point and the presence of a threshold was definitely shown in mica. These authors and Maugis [25] considered that the threshold corresponds to an equilibrium with a null crack velocity. In this respect, crack healing should occur below the threshold, which was partly shown in the case of mica [24] .

A large number of studies have focused on SCG in zirconia ceramics. However, there is only a sparse collection of data where the crack rate versus the applied stress intensity factor (V-K1) has been obtained for a large range of crack rates. Knechtel et al. [13] have obtained V-K1 curves for Y-TZP and A)z03 I Y-TZP ceramics in the crack velocity regime from 10-12 m/s to 10-2 m/s. Their measurements have been confirmed and completed by Chevalier et al. [ 19] for a 3Y-TZP in different atmospheres with more experimental data. The V-K1 diagram for a commercial 3Y-TZP zirconia ceramic (Supplied by Norton Desmarquest Fine Ceramics, France), with a grain size of 0.5 µm is presented in Figure I for different environments. The overall curve in air presents three different stages and a threshold value K10 below which no propagation takes place. In water, only one stage is observed and the slope of the curve is similar to that of the first stage in air. It is important to point out the results in silicon oil or under vacuum where no water molecules are present: the V-K1 curve presents a much higher slope and SCG operates only for stress intensity factors near K1c. These results suggest that a stress corrosion mechanism operates, as in glass or in monocrystalline ceramics. This is supported by the increase of crack velocities in water when the temperature increases. The first stage in air can be fitted by a power law:

V == A•K/ (1)
where A and n are constants dependent on the material properties. Chevalier et al. [ 19] have found a value for n of about 32 for 3Y-TZP ceramics with different grain sizes and Knechtel et al. [13] have found 29 and 44 for their materials. The value for K10 was determined to 3.5 MPa--im in air, 3.1 MPa--im in water and 4.9 MPa--im under vacuum conditions. A systematic study on the effect of temperature on crack velocities has lead to an activation energy of 70 KJ/mol.

Effect of transformation toughening and th e presence of a glas sy ph as e on Slow

Crack Growth in 3Y-TZP Figure 2 represents the V-K1 laws obtained for the different 3Y-TZP ceramics and for single crystal. The data for silica glass obtained with the same DT technique are also plotted, they will be discussed further. Regarding the results for the 3Y-TZP, it is clear that an increase in the grain size leads to a shift towards the high stress intensity factors, expressing an increase of crack resistance due to higher transformability. In comparison with zirconia single crystal (and silica), V-K1 curves for polycrystalline 3Y-TZP are shifted towards much higher stress intensity factors. It is obviously a consequence of transformation toughening that act to shield the crack from the external driving force. It is however interesting to note that all the curve, even for the single crystal ceramic, exhibit three distinct stages corresponding to the three stress corrosion propagation stages discussed above. The second stage, which corresponds to the gaseous diffusion of water molecules inside the crack path occurs for the same crack rates (�10•4 m/s). In terms of stress intensity factor, the stress -induced transformation leads to a shielding Kish according to:

K r = K rtip + K rsh (2)
where K itip is the local crack tip stress intensity factor and Ki the external applied stress intensity factor. The shielding stress intensity factor in toughened ceramics with a regular zone shape has been evaluated as [26]:

with C h = 0.214•E• Vce T -(l+v) • ( ./3) s (l-v)•cr� l27t (3) ( 4
)
where E is the Young modulus, V r the volume fraction of transformed particles, er is the volume dilatation associated to the transformation, v the Poisson ratio, crmc is the local stress for phase transformation and K i the applied stress intensity factor. Eqn (3), which suggests that the shielding stress intensity factor is proportional to the applied one, has been verified experimentally by Chevalier et al.

[19] and Alcala et al. [27]. Reporting Eqn (3) in Eqn (2), the stress intensity factor at the crack tip, Kitip, can be expressed as:

(5)

Csh increases with the material transformability. If we assume that the intrinsic mechanism of SCG at the crack tip is the same for all zirconia ceramics, i.e the fracture of the zirconia bonds (or the silica bonds present in the intergranular vitreous phase) by water molecules, the crack rate must always been given by:

(6)
Therefore, the experimentally determined V-Ki law is:

(7)
From Eqn (5), it can be argued that the intrinsic toughness K1ctip (the toughness we would measure if there was no transformation toughening) is related to the experimental toughness K ie by: (

By reporting Eqn (8) in Eqn (7), one obtains:

(9)
Therefore the rationalisation of the V-K1 curves by the experimentally determined toughness Kie must lead to the same 'master curve', whatever the microstructure. This plot should depend only on the intrinsic V-K1tip law of a given ceramic. This V-(K1IK1c) plot is presented in Figure 3. K1 (MPa'1m) The different zirconia ceramics exhibit nearly the same rationalised V-K1/K.1c curves. However, there is a significant difference between ultra pure 3Y-TZP ceramics (To. Jµm, Tosµm, T1oµm) and 3Y�TZP exhibiting silica (and alumina) in the starting powder (DoJµm and T-So 3µm). Of importance is the coincidence between zirconia single crysr}l and the fonner and between silica and the larer. This could traduce a fracture of zirconia bounds in the case of pure 3 Y-TZP ceramics and a significant amount of fracture of silica bounds for the less pure 3Y-TZP. Transmission Electron Microscopy (TEM) analysis was achieved on the different 3 Y TZP in order to investigate the presence of an amorphous silica rich grain boundary phase, which should appear in zirconia ceramics as mentioned earlier by several authors. All pre zirconia ceramics (processed with the ultrapure Tosoh powder) exhibit any grain boundary phase, neither at the grain boundary nor at triple points, as shown in Figure 4.a and 4.b in To Sµm• In contrast, Do. 3µm erbited a thin amorphous layer of about 1 nm or less ( cf Figure 5.a), which is in agreement with the value predicted by Clark (28). The presence of amorphous triple points was also observed for this ceramic (Figure 5.b). Energy Dispersive Spectroscopy revealed the presence of silica and alumina in the grain boundaries. At last, T-So3µm, which was processed from Tosoh powder and an addition of colloidal silica, exhibited a important number of large glassy triple points (Figure 6). However, the presence of silica rich glassy phase at the grain boundary was not clear, and most of the observed grain bordaries were clean. Thus, re effect of grain size and of the presence of a grain boundary glassy phase is obvious from the theoretical analysis developed above and from the different results :

(i) an increase of the grain size leads to increased crack resistance according to Equations (3) and ( 4) where O'mc decreases when the grain size increases.

(ii)

The presence (even partial) of a glassy phase at the grain boundary and/or at the triple junctions bereen grains leads to a combination of fracre of silica bords and zirconia bords, instead of zirconia bounds only for pure zirconia ceramics. The inrsic propagation law is changed in the presence of silica, and rus the V-K1/K1c law is not the same. However, the difference between 'pure' and Si02 doped 3Y-TZP is not very important because of rsfonnation shielding that act to increase crack resistance in all cases. It is to note that such rationalisation has been conducted for other results on different types of zirconia ceramics from the literare in zirconia by Chevalier et al. [29].They have concluded that all zirconia based ceramics exhibit approximately the same 'master' V K1IK1c law under both static and cyclic fatigue. However, the presence of silica rich glassy phase at the grain boundary was not clear.

Modelling of Slow Crack Growth in zirconia ceramics

The treatment of environment-sensitive crack velocity in ceramics by stress corrosion models has been widely documented for single crystal ceramics or glass [START_REF] Lawn | Fracture of Brittle Solids[END_REF]. Formulations for the crack velocity in the three regions should be written :

Region I: ( £01 ) [a1 •(G * -2 r) ] V = 2 • v. •a •exp --• sinh -----'------'- 1 ° 0 kT kT (10)
Region II:

V - 64. G * . a/ . PE 11 - ( I) �(2 • tr • m • kT) 3 • tr • E • a 0 • In - a o
Region ill :

( E olll) . [all l • {G* -2 y.y)l V =2•� • a •exp ---•smh ---'----"'""" Ill 0 0 kT kT (11) ( 12
)
where G* is the net crack tip driving force for crack propagation, i.e. G-Gsh, Gsh being the effect of reinforcement on crack tip driving force. In these equations, Uo is the Boltzmann frequency, ao the bond spacing, k the Boltzmann constant, y and ys the surface energy in the presence or absence of water molecule respectively, T the absolute temperature. The bond spacing for zirconia can be taken as ao=a/2v2 where a is the lattice parameter, about 0.5 nm, for a crack propagation through <11 O> direction. Em and Emu are the activation energy of stress corrosion in region I and region ill. a.1 and am traduce the stress dependence of crack velocity. Region II is related to the diffusion controlled stage of crack propagation, m is therefore the mass of water molecules and PE the water pressure. The first term of equation ( 11) incorporates the pressure differential between the mouth (source) and tip (sink) which regulates the flow rate, so l represents the crack opening displacement, on the order of 1 µm. E is the Young modulus, 220 GPa for zirconia ceramics.

G* can be calculated directly from G. Indeed, in the case of zirconia single crystal, G*=G because no microstructural reinforcement occurs. In the case of zirconia polycrystals, the contribution of shielding can be estimated from Equation ( 5) by a comparison of the V-K1 law for a given polycrystal and for the single crystal. For all the velocities measured, the applied K1 and the intrinsic Kuip were proportional. The constant Csh has been calculated for the three grain sizes investigated in this work. This gives a proportionality factor between the applied driving force G and the crack tip driving force G*by:

I I ----• G * = --• G * (1-c sh ) 2 C G sh
Table 2 gives this proportionality factor as a function of the grain size.

Table 2.

Grain size 0.3 m 0.5 m 1.0 µm Cash 0,128 0,104 0.0865

The fit of experimentally determined V-G* laws by the three stress corrosion equations are shown in Figure 7. In this fit, it was considered that region I and II act in series and that region II and region ill are parallel mechanisms.

The different parameters a.i, a.ui. y, ys, Em and Eom deduced from this plot are summarised in Table 3. They can be used to predict V-G* function of zirconia in any environmental condition. Figure 8 shows the V-G* laws of the different zirconia ceramics, with or without glassy phase, in comparison with silica glass and zirconia single crystal. Even for D0.3µm and T-S0.3µm, the V-G* law is much nearer from the single crystal zirconia than for silica glass. This should indicate that all the grain boundaries are not wetted by the glassy phase and that a significant proportion of fracture occurs also for zirconia bounds. As fracture occurs more easily for silica than for zirconia bounds, the lowest step in the chain should rule the overall crack velocity. It is also important to note that the real composition of the glassy phase is not known exactly and can differ from that of the silica glass tested in the present work. Crack tip energy release rate G* Modelling of SCG in 3Y-TZP with stress corrosion and transformation toughening models was possible, because stress-assisted phase transformation zones were limited. Indeed, it was shown by several authors that the zone with in 3Y-TZP was less than 10 µm [START_REF] Nag | Fracture Mechanics of Ceramics[END_REF]. Thus, the R-Curve of 3Y-TZP ceramics is rising steeply over the first tens of microns then becomes constant [30). This means that DT results obtained in 3Y-TZP were obtained on the plateau of the R-Curve. Equation ( 4) is only valid on this plateau, which is not the case for Ce-TZP with an R-Curve rising over several millimetres. Figure 9 represents the transformation zone observed in the Ce-TZP ceramic, on the surface of a DT specimen. The zone width is about I 00 µm and the zone length reaches several hundreds of microns (�1500 µm). 22,00 ,------------------� 18,00. This leads to extensive crack resistance, as represented in Figure 10 for a measurement with the SENB method: the toughness reaches 18 MPavm after several hundreds of microns extension, and we remember that the intrinsic toughness of zirconia is of 2 MPavm. Of importance is also the load-displacement curve obtained by DT for the Ce-TZP material on Figure 11 which shows an extensive non-linear behaviour, in comparison with more common brittle ceramics.

1 0 •2 ,-., 1 0 °" "' 6 '--' 0 1 0 .a "<3 0 Q) 1 0 -8
Acoustic Emission can be monitored during the loading of a specimen. It was shown by Gogotsi et al. [31] that Acoustic Emission was generally observed before crack propagation, traducing extensive transformation even before crack extension. This was verified in our experiments both by SENB and DT methods. At last, a number of secondary cracks were observed that act also to reduce the crack tip driving force. The DT was used to identify the Slow Crack Growth behaviour of the Ce-TZP ceramic. Two successive relaxation tests were conducted on the same specimen. The analysis of the tests was made with a compliance analysis, as described in [START_REF] Ebrahimi | [END_REF] in this series of proceedings. As shown in Figure 12, The V-K1 law depends drastically on the crack length, as a consequence of increasing crack shielding. The V-K1 law is shifted towards higher stress intensity factor values as the crack advances. This effect has been analysed in details for alumina in ref [START_REF] Ebrahimi | [END_REF]. It however more pronounced for Ce-TZP as a consequence of a much stronger R-Curve behaviour. The second relaxation corresponds to the plateau of the R-Curve and Slow Crack Growth occurs for applied stress intensity factors near 18 MPavm. In this latter experiment, for high applied loads, a spasmodic crack propagation was observed, which was traduced by a complex V-K1 law, with slow down -speed up sequences. This effect which has not been described elsewhere has been attributed to the generation of a secondary crack outside the transformation zone, when the main crack was stopped by the transformation zone. The sudden drop of the load during relaxation tests was always connected to a crack branching: the main crack was slowed down by the transformation, and a secondary crack started at high speed, then again slowed down by another branch of transformation zone.

Crack Velocity (mls), (log.) 10-2 ---------------...,,. Further work is needed to analyse precisely SCG in Ce-TZP ceramics. It would be interesting to suppress the R-Curve behaviour of the material from the experimental SCG results to describe the real intrinsic V-Kiti p of this material in comparison with 3Y-TZP or single crystal. However, it is clear that the strong reinforcement in this material, as compared to the single crystal or even to the 3Y-TZP ceramics shows the potential of such a ceramic for load bearing applications, where SCG is of prime importance. Figure 14 represents the V-K1 laws that were obtained in this work for the single crystal, 3Y-TZP ceramic with a 0.5 µm grain size and for the Ce-TZP ceramic. The SCG data spread from 2 to 18 MPa.../m for the same basic material, as a consequence of transformation toughening capability.

Crack propagation in Ce-TZP based composites

High fracture toughness has been achieved for Ce-TZP that undergoes extensive martensitic transformation from tetragonal to monoclinic phase. However, the transformation zones surrounding cracks in this material, tend to extend far ahead of the crack a distance of 10 to 20 times the zone width ( cf. Figure 9). The extra transformed material ahead of the crack degrades the toughening. For a steady-state cracking (crack growth at constant K1), with a zone containing a uniform volume fraction, f, of transformation product over a distance w from the crack surface, the shielding stress intensity factor is:

A•f•er •w 11 2 K ------- sh - (1-v) (14)
Where, Eis the Young's modulus, vis the Poisson's ration, eT is the net dilatation and A is a dimensionless constant dependent on the zone shape ahead of the crack tip. In order to assess the role of the extended frontal zone on crack tip shielding, the shape parameter <<A» can be evaluated for any specified zone shape using the weight function method of Mc Meeking et al. [26].

Marshall et al [22) estimated that the fracture toughness could be increased, if the length of the transformation zone was restricted to approximately the zone width, which is the case for magnesia-partially-stabilised zirconia (Mg-PSZ) or 3Y-TZP. They proposed to change the frontal zone in Ce-TZP by introducing laminar composites, containing layers of Ce-Zr02 and either Ah03 or mixture of Alz03 and Ce-ZrOz. They used a colloidal method that allowed formation of layers with thickness as small as 10 µm. In such conditions, a fracture toughness of more than 30 MPa.../m was predicted. Our purpose was therefore to still increase the crack resistance of the present Ce-TZP by using this concept. Laminar materials with alternate layers of dense Ce-TZP (A) and a mixture of 80 wt. % Ce-TZP and 20 wt. % Ah03 (B) were fabricated using colloidal technique. The technique involved sequential slip casting by pressure filtration of solution containing suspended particles to form the layered green body followed by drying and natural sintering at 1450°C for lh. The technique is described in more details in ref [32]. The final materials consisted of different layers (from 3 to 9) in sequences A/BIA/BIA with an overall thickness of 3 mm. Notched beam crack growth experiments were conducted at room temperature. The dimensions of the beams were 30 mm * 5 mm * 3 mm with an initial notch of 1 mm depth.

The apparent toughness (apparent because the behaviour is far from linear elastic) of the composites can reach values up to 50 MPa.../m, which is exceptionally high for ceramic based materials. Figure 15 shows a micrograph of a notched composite loaded in flexion.

An enlargement of the transformation zone at the first interface is apparent on the micrograph, as well as autocatalic transformation in the last zirconia layer. Strong

Figure 1 .

 1 Figure 1. V-K1 law for a commercial 3Y-TZP ceramic with a grain size of 0.5 µm (from ref [19)). • : air, 25°C ; A: distilled water, 25°C ; /:J. : distilled water, 75°C ; 0 : silicon oil, 25°C ; 0: secondary vacuum, 25°C.

Figure 2 .Figure 3 .

 23 Figure 2. V-K1 laws obtained for different 3Y-TZP zirconia cerainics compared to single crystal and silica. + : T1.oµm; • : To. Sµm; e : ToJµm ; 0 : T-So. Jµm ;!:;,. : Do. Jµm; * : zirconia single crystal; + : silica glass.

Figure 4 .

 4 Figure 4. Grain boundaries and triple points for T0•5pm without any glassy phase.

Figure 5 .

 5 Figure 5. Presence of a thin amorphous glassy layer in a grain boundary and at a triple point in D0.3µm.

Figure 6 .

 6 Figure 6. Nwnerous glassy phase pockets at triple points in T-So.Jµm . Note the rounded shape of grains.

Figure 7 .

 7 Figure 7. Intrinsic V-G* law of pure zirconia ceramics. + : T 1oµm ; •: T o.sµm ; e : T o.Jµm.

Figure 8 .

 8 Figure 8. V-G* laws for all 3Y-T ZP as compared to silica glass. +: T 1.oµm ; • : To. Sµm ; e: T o. Jµm; O: T-So. Jµm ; !:!; Do. Jµm; * : zirconia single crystal ; + : silica glass.

  200µm

Figure 9 .

 9 Figure 9. Transformation zone in IOwt.% mol Ce-TZP.

Figure 10 .

 10 Figure 10. Crack resistance (KR) versus relative crack length (a!W, with W: specimen height of 6 mm) of Ce TZP measured by SENB method.

Figure 11 .

 11 Figure 11. Load-displacement curve for Ce-TZP measured with the DT method.

18 Figure 12 .

 1812 Figure 12. V-K1 law of a Ce-TZP for two successive DT relaxation tests.

Figure 13 .

 13 Figure 13. Crack branching observed at each acceleration of the crack rate in the Ce-TZP ceramic.

Table 1 .

 1 Nomenclature of the different processed 3Y-TZP ceramics.

	Desi1mation To . 3um To . sum T1.oum T-So.3um Do.3um	Powder Tosoh Tosoh Tosoh Tosoh + 2.5 wt.% Si02 1400°c Firing T0 Grain size Density 1400°C 0.3 µm >9 8% 1450°C 0.5 µm >9 8% 1550°C 1.0 µm >98% 0.3 µm >9 8% Daiichi 1350°C 0.3 µm >9 8%

Table 3 .

 3

	3	s 6	850	am/k 1200	E OI 70 KJ/mol.	E ou

**

*

  3.4. Crack Growth in Ce-TZP: importance of th e R-Curve on Slow Crack Growth.

Ce-TZP Ceramics:Ce-TZP ceramics present a large amount of tranformation toughening, and fracture toughness up to 15-20 MPavm were reported. Thus, it was interesting to study the SCG in this crack resistant zirconia ceramic in comparison to 3Y-TZP with a much more limited transformation capability, or to the single crystal without any transformation. Multilayer zirconia based composites were introduced by Marshall et al[20][21][22] who estimated that the

delamination and crack deviation was seen at all the interfaces and act for the high toughness of this materials.


Luigi Ambrosio
email: ambrosio@sns.it

Aymeric Baradat
email: baradat@mis.mpg.de

Yann Brenier
email: brenier@dma.ens.fr

Monge-Ampère gravitation as a Γ-limit of good rate functions

. The main step in this derivation is the Γ-convergence of the good rate functions corresponding to a one-parameter family of large deviation principles. Surprisingly, the derived model includes dissipative phenomena. As an illustration, we show that it leads to sticky collisions in one space dimension.

Introduction

On a periodic domain such as T d = (R/Z) d , Newtonian gravitation is commonly described in terms of the density of probability f (t, x, ξ) to find gravitating matter at time t, position x ∈ T d and velocity ξ ∈ R d , subject to the Vlasov-Poisson equation ∂ t f (t, x, ξ) + div x (ξf (t, x, ξ)) -div ξ (∇ϕ(t, x)f (t, x, ξ)) = 0,

∆ϕ(t, x) = R d f (t, x, ξ) dξ -1, (t, x, ξ) ∈ R × T d × R d ,
where ϕ is the gravitational potential. Notice that the averaged density, say 1, has been subtracted out from the right-hand side of the Poisson equation, due to the periodicity of the spatial domain. This is a common feature of computational cosmology and it let the uniform density be a stationary solution. The Vlasov-Poisson system can be seen as an "approximation" to the more nonlinear Vlasov-Monge-Ampère (VMA) system ∂ t f (t, x, ξ) + div x (ξf (t, x, ξ)) -div ξ (∇ϕ(t, x)f (t, x, ξ)) = 0,

(1)

det(I + D 2 ϕ(t, x)) = R d f (t, x, ξ) dξ, (t, x, ξ) ∈ R × T d × R d , (2)
where the fully nonlinear Monge-Ampère equation substitutes for the linear Poisson equation of Newtonian gravitation. Indeed, for "weak" gravitational potential, by expanding the determinant about the identity matrix I, we get det(I + D 2 ϕ(t, x)) ∼ 1 + tr(D 2 ϕ(t, x)) = 1 + ∆ϕ(t, x)

and recover the Newtonian model approximately (and exactly as d = 1). In this paper, we will speak of "Monge-Ampère gravitation" ("MAG" in short). The Vlasov-Monge-Ampère system has been introduced and related to the Vlasov-Poisson system in [START_REF] Brenier | A geometric approximation to the Euler equations: the Vlasov-Monge-Ampère system[END_REF], and studied as an ODE on the Wasserstein space in [START_REF] Ambrosio | Hamiltonian ODEs in the Wasserstein space of probability measures[END_REF]. It can also be solved numerically thanks to efficient Monge-Ampère solvers recently designed by Mérigot [START_REF] Mérigot | A multiscale approach to optimal transport[END_REF].

It has been argued in [START_REF] Brenier | A modified least action principle allowing mass concentrations for the early universe reconstruction problem[END_REF] that the MAG may also be seen as an approximation of Newtonian gravitation for which the "Zeldovich approximation" [START_REF] Ya | Gravitational instability: An Approximate theory for large density perturbations[END_REF] (see [START_REF] Frisch | A reconstruction of the initial conditions of the universe by optimal mass transportation[END_REF][START_REF] Brenier | Reconstruction of the early Universe as a convex optimization problem[END_REF]), popular in computational cosmology, becomes exact.

In this paper we will not be directly interested in this system, but rather in its discrete version, i.e. when the number of particles is finite. As well known in optimal transport theory [START_REF] Brenier | Décomposition polaire et réarrangement monotone des champs de vecteurs[END_REF][START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF], the Monge-Ampère equation ( 2) is solved by the unique function ϕ such that the map Id + ∇ϕ realizes the optimal transport with quadratic cost from the density f dξ to the Lebesgue measure. Then, the kinetic equation ( 1) is known to be the continuous version of Newton equations of classical mechanics in a potential given by ϕ.

In the discrete setting, the stationary Lebesgue measure is replaced by a family (a 1 , . . . , a N ) ∈ (R d ) N of N ≥ 1 points in R d (here we make the presentation in R d instead of T d for the sake of simplicity). One can for instance think of a regular lattice approximating in some region a constant density, even though in the sequel the particular choice of (a 1 , . . . , a n ) will play no role. We will consider the evolution of a cloud of N particles (x 1 , . . . , x n ) in R d whose dynamic is ruled by the discrete optimal transport problem:

σ opt = σ opt (X) := Arginf σ∈Sn N i=1 |x i -a σ(i) | 2 , X = (x 1 , . . . , x N ) ∈ (R d ) N . (3)
More precisely, the analogous of (1)(2) in this framework is easily seen to be formally:

1 ∀i = 1, . . . , N, d 2 dt 2 x i (t) = x i (t) -a σopt(i) . (4)
Following the idea of the recent paper [START_REF] Brenier | A double large deviation principle for Monge-Ampère gravitation[END_REF], we will derive this discrete dynamic from the very elementary stochastic model of a Brownian point cloud. However, in [START_REF] Brenier | A double large deviation principle for Monge-Ampère gravitation[END_REF], the derivation was obtained through a double application of the large deviation principle (LDP), through a purely formal use of the Freidlin-Wenzell theory [START_REF] Freidlin | of Grundlehren der Mathematischen Wissenschaften[END_REF]. The main purpose of the present paper is to explain how such a derivation can be made rigorous by substituting for one of the applications of the LDP a PDE method inspired by the famous concept of "onde pilote" introduced by Louis de Broglie at the early stage of Quantum Mechanics [START_REF] Broglie | La mécanique ondulatoire et la structure atomique de la matière et du rayonnement[END_REF].

The outline of the paper is the following. In Section 2 we show how to derive MAG starting from a finite number of Brownian particles. This will be done in several steps and we do not want to enter into the details now, but a key argument will be the Γ-convergence of the good rate functions associated with a family of SDEs towards an "effective" functional related to MAG. This is stated in Theorem 5, which is our main result. Section 3 is dedicated to the proof of Theorem 5. The effective functional that we obtain does not lead exactly to MAG as stated in (4) (which as already saw in footnote 1 is not well-posed in general), but also includes dissipative phenomena in those points X where the solution of the discrete optimal transport problem (3) is not unique. Even if we do not know for the moment how to treat these dissipative effects in general, the purpose of Section 4 is to show that in 1 space dimension, they lead to sticky collisions.

Notations. We will work with N particles in R d , and hence in (R d ) N . Points of (R d ) N will be denoted with capital letters, mainly X, Y or Z. Curves with values in (R d ) N will be denoted with calligraphic letters X , Y or Z. The position of X , Y and Z at time t ∈ R will be denoted by X t , Y t and Z t respectively.

2 Derivation of the discrete model

The stochastic model of a lattice with Brownian agitation

Take A = (a 1 , . . . , a N ) ∈ (R d ) N a family of N ≥ 1 points in R d . We assume each point of this lattice to be subject to Brownian agitation for times t ≥ 0. At time t, the position of point i is

a i + √ εB i t ,
where (B i ) i=1,...,d is a family of N independent normalized Brownian curves and ε monitors the (common) level of noise. As a consequence, at time t > 0, the density of probability ρ ε (t, X) for the point cloud

(a 1 + √ εB 1 t , . . . , a N + √ εB N t ) to be observed at location X = (x 1 , . . . , x d ) ∈ (R d ) N
, up to a permutation σ ∈ S N of the labels, is easy to compute. We find

ρ ε (t, X) = 1 N ! √ 2πεt dN σ∈S N N α=1 exp - |x i -a σ(i) | 2 2εt , or, in short, 1 N ! √ 2πεt N d σ∈S N exp - |X -A σ | 2 2εt ,
where | • | denotes the euclidean norm in R d or (R d ) N depending on the context, and where for all X = (x 1 , . . . , x N ) ∈ (R d ) N , X σ stands for:

X σ = (x σ(1) , . . . , x σ(N ) ).
This was the starting point of the discussion made in [START_REF] Brenier | A double large deviation principle for Monge-Ampère gravitation[END_REF], using a double large deviation principle.

In the present paper, we rather turn to a PDE viewpoint, where ρ ε is the solution of the heat equation in

(R d ) N ∂ρ ε ∂t (t, X) = ε 2 ∆ρ ε (t, X) (5)
with, as initial condition, the delta measure located at A = (a 1 , . . . , a N ) ∈ (R d ) N and symmetrized with respect to σ ∈ S N , namely:

ρ ε (0, X) = 1 N ! σ∈S N δ A σ . (6)
In some sense, we have solved the heat equation in the space of "point clouds" (R d ) N /S N , with initial position A, defined up to a permutation σ ∈ S N of the labels i = 1, . . . , N .

"Surfing" the "heat wave"

After solving the heat equation in the space of "clouds" (R d ) N /S N (5)(6), we introduce the companion ODE in the space (R d ) N :

dX ε t dt = v ε (t, X ε t ), v ε (t, X) = - ε 2 ∇ log ρ ε (t, X),
or, more explicitly

v ε (t, X) = 1 2t σ∈S N (X -A σ ) exp - |X -A σ | 2 2εt σ∈S N exp - |X -A σ | 2 2εt = 1 2t       X - σ∈S N A σ exp X • A σ εt σ∈S N exp X • A σ εt      
, where if U and V are in (R d ) N , U • V denotes the inner product between U and V . This velocity is chosen so that

∂ρ ε ∂t (t, X) + div(ρ ε (t, X)v ε (t, X)) = 0,
i.e. for the density ρ ε to be transported by the velocity field v ε . We may solve this ODE for arbitrarily chosen position X t0 ∈ (R d ) N and initial time t 0 > 0. In other words, we let the set of N "particles" X t = (x 1 (t), . . . x N (t)) ∈ (R d ) N "surf" the "heat wave" generated by the lattice subject to Brownian agitation! By doing that, we just mimic the idea of quantum particles driven by the "onde pilote", as imagined by Louis de Broglie [START_REF] Broglie | La mécanique ondulatoire et la structure atomique de la matière et du rayonnement[END_REF] at the early stage of Quantum Mechanics.

Remark 1. In that case, we would use the same ODE with v = ε∇Im log ψ, ψ solving the Schrödinger equation. For instance, we could consider the free Schrödinger equation instead of the heat equation:

i∂ t + ε 2 ∆ ψ = 0, ψ(0, X) = σ∈S N exp - |X -A σ | 2 a 2 ,
with initial condition chosen according to "bosonic statistics". However, in the quantum case, the analysis gets substantially more difficult, due to the possible vanishing of the wave function ψ during the evolution.

Large deviations of the "heat wave" ODE

Let us go back to the "heat wave" ODE and add a noise of the following type:

dX ε,η t = v ε (t, X ε,η t ) dt + √ ηα(t) dB t , (7)
where η is a positive number and α is a smooth function from R * + to R * + . In other words, our "surfers" are now subject to some additional agitation, while surfing on the heat wave generated by the lattice already under Brownian agitation!

We will see that when η and ε are small, then the trajectories charged by the solution of this SDE that are in P ∈ (R d ) N at time t 0 > 0 and in Q ∈ (R d ) N at time t 1 > t 0 (up to ordering) are very close to the dynamic of MAG. Notice that the level of noise depends on time through the function α(t). It will be crucial in Subsection 2.5 since we will only recover MAG after suitable change of time.

Since, for fixed ε > 0 and t > 0, v ε is a smooth velocity field, existence of a strong solution and pathwise uniqueness for [START_REF] Brenier | Reconstruction of the early Universe as a convex optimization problem[END_REF] is standard once fixed a law for the initial position X ε,η t0 , t 0 > 0. Furthermore, we may pass to the limit η → 0, while ε > 0 is kept fixed, in the sense of large deviation: A direct application of classical Freidlin-Wentzell theory [START_REF] Freidlin | of Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Dembo | Large deviations techniques and applications[END_REF] leads to: Theorem 2. Let us fix P, Q ∈ (R d ) N the endpoints of our trajectories, up to ordering, and 0 < t 0 < t 1 two positive times. For fixed ε and as η ↓ 0, the law of the solution of (7) between times t 0 and t 1 starting from P and conditioned to arrive in Q (up to ordering) satisfies the large deviation principle on

C 0 ([t 0 , t 1 ]; (R d ) N ) of good rate function L ε defined for all X = (X t ) t∈[t0,t1] by: L ε (X ) =        t1 t0 | Ẋt -v ε (t, X t )| 2 α(t) 2 dt, if X ∈ H 1 ([t 0 , t 1 ]; (R d ) N ), X t0 ∈ {P σ } and X t1 ∈ {Q σ }, + ∞, else,
where here and in the rest of the article, we denote by {P σ } and {Q σ } the sets {P σ , σ ∈ S N } and {Q σ , σ ∈ S N } respectively.

In the rest of the article, we will call L the Freidlin-Wentzell action instead of the usual terminology "good rate function". Also, the endpoints P and Q are fixed once for all so we do not write explicitly the dependence of L ε on those. Theorem 2 asserts in particular that when ε is fixed and η is small, if X ε,η solves (7), given X ε,η t0 ∈ {P σ } and X ε,η t1 ∈ {Q σ }, X ε,η is with very high probability close to the minimizers of L ε . Now we will see that these functionals converge as ε ↓ 0 to a functional whose minimizers follow the dynamic of MAG2 , in the sense of Γ-convergence (and hence in the sense of convergence of minimizers as well). As a consequence, when both η and ε are small, given X ε,η t0 ∈ {P σ } and X ε,η t1 ∈ {Q σ }, X ε,η is close with high probability to the dynamic of MAG.

Remark 3. Here, we chose to present the result for point clouds, i.e. when the particle are still indistinguishable. However, the theorem could also be stated replacing the conditioning on X t0 ∈ {P σ } and X t1 ∈ {Q σ } by X t0 = P and X t1 = Q. Otherwise stated, reintroducing distinguishable particles at this stage would not affect the results of this section (neither Theorem 2 nor Theorem 5 below). We decided to keep on working on clouds in order to avoid crossings of particles in Section 4.

The convergence result

Define the following smooth convex function (see Lemma 9 below):

∀ε > 0, ∀t > 0, ∀X ∈ (R d ) N , f ε (t, X) := εt log 1 N ! σ∈S N exp X • A σ tε . (8)
It has the property that for all ε > 0, t > 0, and

X ∈ (R d ) N , v ε (t, X) = X -∇f ε (t, X) 2t .
As a consequence, denoting by β the smooth function 1/α 2 , we can rewrite L ε for all ε > 0 as:

L ε (X ) =        t1 t0 Ẋt - X t -∇f ε (t, X t ) 2t 2 β(t) dt, if X ∈ H 1 ([t 0 , t 1 ]; (R d ) N ), X t0 ∈ {P σ } and X t1 ∈ {Q σ }, + ∞, else.
When ε tends to zero, by virtue of the so-called Laplace's principle, we have the pointwise convergence:

lim ε→0 f ε (t, X) = max σ∈S N X • A σ =: f (X). (9)
The function f no longer depends on the time variable, and it is a convex function with finite values. As a consequence, for each X ∈ (R d ) N , the subdifferential ∂f (X) of f at X is non-empty. We will consider the extended gradient ∇f (X) of f at X defined as:

Definition 4 (Extended gradient). We call extended gradient of a real valued convex function h at X, denoted by ∇h(X), the element of ∂h(X) with minimal Euclidean norm.

Here is our Γ-convergence result:

Theorem 5. As ε tends to 0, the family of actions

(L ε ) ε>0 Γ-converges to L(X ) =          t1 t0 Ẋt - X t -∇f (X t ) 2t 2 β(t) dt, if X ∈ H 1 ([t 0 , t 1 ]; (R d ) N ), X t0 ∈ {P σ } and X t1 ∈ {Q σ }, + ∞, else.
for the topology of uniform convergence of C 0 ([t 0 , t 1 ]; (R d ) N ).

Theorem 5 can be seen as the main result of this article. In particular, it implies that any limit point as ε ↓ 0 of a sequence of minimizers of L ε is a minimizer of L. So, one can rigorously obtain an effective action to describe the double limit lim ↓0 lim η↓0 for the solution of the SDE [START_REF] Brenier | Reconstruction of the early Universe as a convex optimization problem[END_REF]. Notice that the lower semi-continuity of L is a direct corollary of the Γ-convergence. In addition, the fact that L has compact sublevels will be clear from the proof. Hence, the existence of global minimizers for L (and hence for all the forthcoming functionals) follows from the direct method of calculus of variations.

We will prove Theorem 5 in Section 3 below, but before doing so, let us show that for a specific choice of β, we recover MAG.

A regime where Monge-Ampère gravitation arises

Let us take β(t) := t which corresponds to α(t) := 1/ √ t. Through the change of variable:

t = exp(2θ), Z θ = X exp(2θ) ,
we observe that for all X ∈ C 0 ([t 0 , t 1 ]; (R d ) N ), L(X ) = Λ(Z) with:

Λ(Z) =        θ1 θ0 Żθ -(Z θ -∇f (Z θ )) 2 dθ, if Z ∈ H 1 ([θ 0 , θ 1 ]; (R d ) N ), Z θ0 ∈ {P σ } and Z θ1 ∈ {Q σ }, + ∞, else.
(Recall the definition (9) of f .) Unexpectedly, this action is exactly the one previously suggested by the third author in [START_REF] Brenier | A modified least action principle allowing mass concentrations for the early universe reconstruction problem[END_REF] to include dissipative phenomena (such as sticky collisions in one space dimension) in the Monge-Ampère gravitational model! It turns out to be equivalent to the following one:

Λ (Z) =        θ1 θ0 | Żθ | 2 + |Z θ -∇f (Z θ )| 2 dθ, if Z ∈ H 1 ([θ 0 , θ 1 ]; (R d ) N ), Z θ0 ∈ {P σ } and Z θ1 ∈ {Q σ }, + ∞, else.
(By expanding the square and remarking that the mixed product is an exact temporal derivatives, so that its integral only involves the endpoints P and Q.)

Application of the least action principle

We observe that the points Z where f is differentiable are those for which the maximum in the definition (9) of f is reached by a unique permutation σ opt so that ∇f (Z) is nothing but A σopt . For such points Z, we get

|Z -∇f (Z)| 2 2 = |Z -A σopt | 2 2 = |Z| 2 + |A σopt | 2 2 -Z • A σopt = |Z| 2 + |A| 2 2 -f (Z)
(by definition of f and using that |A σ | = |A| for any σ ∈ S N ), while, on the set N of non-differentiability of f , we rather have

|Z -∇f (Z)| 2 2 < |Z| 2 + |A| 2 2 -f (Z).
So the action we have obtained in the previous section, namely Λ , bounds from below

Λ + (Z) =        θ1 θ0 | Żθ | 2 + |Z θ | 2 + |A| 2 2 -f (Z θ ) dθ dθ, if Z ∈ H 1 ([θ 0 , θ 1 ]; (R d ) N ), Z θ0 ∈ {P σ } and Z θ1 ∈ {Q σ }, + ∞, else.
The second action is definitely strictly larger than the first one for those curves θ → Z θ which take values in N (where f is not differentiable) on a set of times θ ∈ [θ 0 , θ 1 ] which is not negligible for the Lebesgue measure. So, the least action principle may provide different optimal curves, depending on the action we choose. However, if a curve is optimal for Λ and almost surely takes value outside of N , then it must also be optimal for Λ + . Clearly, it is much easier to get the optimality equation for such a curve, by working with Λ + rather than with Λ . By varying action Λ + , we get, as optimality equation,

d 2 Z θ dθ 2 = Z θ -A σopt , σ opt = Arginf σ∈S N Z -A σ 2 ,
which is the discrete dynamic announced in the introduction.

Of course, these equations have to be suitably modified for those curves which are optimal for action Λ but not for Λ + because they takes values in N for a non negligible amount of time. At this stage, we do not know how to do it. However, at least in the one-dimensional case d = 1, such modifications are tractable and correspond to sticky collisions as x i (t) = x j (t) occurs for different "particles" of labels i = j and during interval of times of strictly positive Lebesgue measure, see Section 4.

Proof of the Γ-convergence

The purpose of this this section is to prove Theorem 5.

The proof as a consequence of three lemmas

As we will see, Theorem 5 will be a consequence of three lemmas that we state below. Lemmas 7 and 8 both involve a family of smooth functions (g ε ) ε>0 on [θ 0 , θ 1 ] × R p for some θ 0 < θ 1 and p ∈ N, pointwise converging to a function g. On these functions, we will assume the following:

Assumptions 6. (H1) For all ε > 0 and θ ∈ [θ 0 , θ 1 ], g ε (θ, 0) = 0. (H2) For all ε > 0 and θ ∈ [θ 0 , θ 1 ], g ε (θ, •) is convex. (H3) For all θ ∈ [θ 0 , θ 1 ], g(θ, •) is convex, and the distributional derivative ∂ θ g is a L 1 loc function such that for all Y ∈ H 1 ([θ 0 , θ 1 ]; (R d ) N ), the map θ → g(θ, Y θ ) is also H 1 , and for almost all θ ∈ [θ 0 , θ 1 ], d dθ g(θ, Y θ ) = ∂ θ g(θ, Y θ ) + ∇g(θ, Y θ ) • Ẏθ . (10)
(H4) The map ∇g ε is uniformly bounded, that is:

L := sup ε>0 sup θ∈[θ0,θ1] sup Y ∈R p |∇g ε (θ, Y )| < +∞. (11)
(H5) The map ∂ θ ∇g ε is uniformly bounded, that is:

M := sup ε>0 sup θ∈[θ0,θ1] sup Y ∈R p |∂ θ ∇g ε (θ, Y )| < +∞. (12)
In order to keep the proofs simple, we did not try to optimize these assumptions for Lemmas 7 and 8, which are probably true in a far more general context. However, as we will see in the proof of Theorem 5, it suffices to check these assumptions for the family (f ε ) ε>0 after suitable change of temporal and spatial scale. This is done in Lemma 9.

Lemma 7. Let us consider θ

0 < θ 1 ∈ R, η ∈ C ∞ ([θ 0 , θ 1 ]; R * + )
and a family (g ε ) ε>0 of smooth functions from [θ 0 , θ 1 ] × R p to R pointwise converging to a function g, which satisfy (H1), (H3), (H4) and (H5) from Assumptions 6. If a family of curves

(Y ε ) ε>0 in H 1 ([θ 0 , θ 1 ]; R p ) uniformly converges to a curve Y ∈ H 1 ([θ 0 , θ 1 ]; R p ), then θ1 θ0 Ẏ ε θ • ∇g ε (θ, Y ε θ )η(θ) dθ -→ ε→0 θ1 θ0 Ẏθ • ∇g(θ, Y θ )η(θ) dθ. Lemma 8. Let us consider θ 0 < θ 1 ∈ R, η ∈ C ∞ ([θ 0 , θ 1 ]; R * + )
and a family (g ε ) ε>0 of smooth functions from [θ 0 , θ 1 ] × R p to R pointwise converging to a function g, and satisfying (H2), (H4) and (H5) from Assumptions 6. Let us fix R, S ∈ R p and define for ε > 0 and Y ∈ C 0 ([θ 0 , θ 1 ]; R p ):

K ε (Y) :=        1 2 θ1 θ0 | Ẏθ | 2 + |∇g ε (θ, Y θ )| 2 η(θ) dθ, if Y ∈ H 1 ([θ 1 , θ 1 ]; R p ) Y θ0 = R and Y θ1 = S, + ∞, else, K(Y) :=        1 2 θ1 θ0 | Ẏθ | 2 + |∇g(θ, Y θ )| 2 η(θ) dθ, if Y ∈ H 1 ([θ 1 , θ 1 ]; R p ) Y θ0 = R and Y θ1 = S, + ∞, else.
Then

(K ε ) ε>0 Γ-converges to K for the topology on uniform convergence of C 0 ([θ 0 , θ 1 ]; R p ). Lemma 9.
With the notations of Theorem 5, let us call

θ 0 := log t 0 /2, θ 1 := log t 1 /2, p = dN and for θ ∈ [θ 0 , θ 1 ], ε > 0 and Y ∈ (R d ) N : g ε (θ, Y ) := f ε (exp(2θ), exp(θ)Y ) exp(2θ) and g(θ, Y ) := f (exp(θ)Y ) exp(2θ) . (13)
Then (g ε ) ε>0 pointwise converges to g, and they satisfy (H1), (H2), (H3), (H4) and (H5) from Assumptions 6.

In the next subsections, we will prove these three lemmas one by one. The most involved one is undoubtedly Lemma 8, which can be seen as the main step in the proof of Theorem 5. Let us start by proving Theorem 5 using Lemmas 7, 8 and 9.

Proof of Theorem 5. In this proof, the notation X = X t will stand for a generic curve from

[t 0 , t 1 ] to (R d ) N . Associated with X , we define Y = Y θ the curve from [θ 0 , θ 1 ] to (R d ) N , where θ 0 := log t 0 /2, θ 1 := log t 1 /2, and for all θ ∈ [θ 0 , θ 1 ], Y θ := X exp(2θ) / exp(θ). Note that X is H 1 if and only if Y is H 1 . If (X ε ) ε>0 is a family of curves from [t 0 , t 1 ] to (R d ) N ,
we define in the same way the family of corresponding curves

(Y ε ) ε>0 from [θ 0 , θ 1 ] to (R d ) N . A quick computation shows that for all X ∈ H 1 ([θ 0 , θ 1 ]; (R d ) N ), considering η(θ) := β(exp(2(θ)
) and (g ε ) ε>0 , g as defined in Lemma 9, we have:

L ε (X ) = t1 t0 Ẋt - X t -∇f ε (t, X t ) 2t 2 β(t) dt = 1 2 θ1 θ0 Ẏθ + ∇g ε (θ, Y θ ) 2 η(θ) dθ (14) = 1 2 θ1 θ0 | Ẏθ | 2 + |∇g ε (θ, Y θ )| 2 η(θ) dθ + θ1 θ0 Ẏθ • ∇g ε (θ, Y θ )η(θ) dθ. (15)
and:

L(X ) = t1 t0 Ẋt - X t -∇f (X t ) 2t 2 β(t) dt = 1 2 θ1 θ0 Ẏθ + ∇g(θ, Y θ ) 2 η(θ) dθ = 1 2 θ1 θ0 | Ẏθ | 2 + |∇g(θ, Y θ )| 2 η(θ) dθ + θ1 θ0 Ẏθ • ∇g(θ, Y θ )η(θ) dθ. (16)
(Note that due to Lemma 9, g is convex with respect to the space variable, and so ∇g is well defined.) Proof of the Γ -lim inf. Let X ε → ε→0 X for the topology of uniform convergence. Of course, we also have

Y ε → ε→0 Y.
Without loss of generality, we can suppose

sup ε>0 L ε (X ε ) < +∞. (17)
Indeed, if the lim inf of this quantity is infinite, there is nothing to prove, and if the lim inf is finite, up to an extraction, we can reduce ourselves to the case where the sup is finite.

As ∇g ε (θ, Y ) is bounded uniformly in ε, θ, Y (this is (H4)), we easily deduce with ( 14) that this assumption implies

sup ε>0 θ1 θ0 | Ẏ ε θ | 2 dθ < +∞.
In particular, by lower semi-continuity of this H 1 seminorm with respect to uniform convergence, all the curves Y ε , ε > 0 as well as

Y are in H 1 ([θ 0 , θ 1 ]; (R d ) N ).
In particular, applying Lemma 7 thanks to Lemma 9, we have:

θ1 θ0 Ẏ ε θ • ∇g ε (θ, Y ε θ )η(θ) dθ -→ ε→0 θ1 θ0 Ẏθ • ∇g(θ, Y θ )η(θ) dθ. ( 18
)
On the other hand, it is clear that under (17), for ε > 0 sufficiently small, the endpoints of X ε are stationary, that is X ε t0 = P σ0 and X ε t1 = Q σ1 with σ 0 , σ 1 independent of ε. So for such ε, Y ε satisfies the endpoint constraint for K ε with R := P σ0 / √ t 0 and S := Q σ1 / √ t 1 . Hence, applying Lemma 8 thanks to Lemma 9, we have:

1 2 θ1 θ0 | Ẏθ | 2 +|∇g(θ, Y θ )| 2 η(θ) dθ = K(Y) ≤ lim inf ε→0 K ε (Y ε ) = lim inf ε→0 1 2 θ1 θ0 | Ẏ ε θ | 2 + |∇g ε (θ, Y ε θ )| 2 η(θ) dθ. (19)
The result follows easily by gathering ( 15), ( 18), ( 19) and [START_REF] Ya | Gravitational instability: An Approximate theory for large density perturbations[END_REF]. Proof of the Γ -lim sup. Let X ∈ C 0 ([t 0 , t 1 ]; (R d ) N ). Without loss of generality, we can suppose that

X ∈ H 1 ([t 0 , t 1 ]; (R d ) N
) and that it satisfies the endpoint constraint for L. In particular, Y belongs to

H 1 ([θ 0 , θ 1 ]; (R d ) N
) and satisfies the endpoint constraint for K with R := X t0 / √ t 0 and S := X t1 / √ t 1 . Lemmas 8 and 9 let us find a family (Y ε ) ε>0 converging to the corresponding Y such that:

lim sup ε→0 K ε (Y ε ) ≤ K(Y). (20)
In particular Y ε is in H 1 for sufficiently small ε, and by Lemmas 7 and 9,

θ1 θ0 Ẏ ε θ • ∇g ε (θ, Y ε θ )η(θ) dθ -→ ε→0 θ1 θ0 Ẏθ • ∇g(θ, Y θ )η(θ) dθ. (21)
The result follows easily from ( 15), ( 20), ( 21) and ( 16), by noticing that because of (20), Y ε satisfies the endpoint constraint for K ε . Hence for such ε, X ε satisfies the endpoint constraint for L ε .

Proof of Lemma 7

The proof of Lemma 7 just consists in integrating by parts and using the convergence properties of (g ε ) ε>0 .

Proof of Lemma 7. Integration by parts. First, notice that as soon as Y ∈ H 1 ([θ 0 , θ 1 ]; R p ) and ε > 0, then θ → g ε (θ, Y θ ) and θ → g(θ, Y θ ) are also in H 1 , with for almost every θ:

d dθ g ε (θ, Y θ ) = ∂ θ g ε (θ, Y θ ) + ∇g ε (θ, Y θ ) • Ẏθ and d dθ g(θ, Y θ ) = ∂ θ g(θ, Y θ ) + ∇g(θ, Y θ ) • Ẏθ .
It is clear in the case of g ε because g ε is smooth, and it is the assumption (H3) in the case of g. As a consequence, by an integration by parts, it suffices to prove that whenever (Y ε ) ε>0 converges to Y as ε → 0 for the topology of uniform convergence,

g ε (θ 1 , Y ε θ1 )η(θ 1 ) -g ε (θ 0 , Y ε θ0 )η(θ 0 ) - θ1 θ0 g ε (θ, Y ε θ )η (θ) dθ - θ1 θ0 ∂ θ g ε (θ, Y ε θ )η(θ) dθ -→ ε→0 g(θ 1 , Y θ1 )η(θ 1 ) -g(θ 0 , Y θ0 )η(θ 0 ) - θ1 θ0 g(θ, Y θ )η (θ) dθ - θ1 θ0 ∂ θ g(θ, Y θ )η(θ) dθ.
Convergence term by term. The convergence

g ε (θ 1 , Y ε θ1 )η(θ 1 ) -g ε (θ 0 , Y ε θ0 )η(θ 0 ) -→ ε→0 g(θ 1 , Y θ1 )η(θ 1 ) -g(θ 0 , Y θ0 )η(θ 0 )
is an easy consequence of the pointwise convergence and of the uniform Lipschitz bound (H4).

For the same reason, we have for all θ

∈ [θ 0 , θ 1 ], g ε (θ, Y ε θ ) -→ ε→0 g(θ, Y θ ).
But on the other hand, because of (H1) and (H4), g ε is locally bounded, uniformly in ε. Hence,

θ1 θ0 g ε (θ, Y ε θ )η (θ) dθ -→ ε→0 θ1 θ0 g(θ, Y θ )η (θ) dθ
is a consequence of the dominated convergence theorem. Because of (H1) and (H5), for all θ, (∂ θ g ε (θ, •)) ε>0 is compact for the topology of local uniform convergence. But its only possible limit point is the distributional derivative ∂ θ g. As a consequence, (∂ θ g ε ) ε>0 converges pointwise to ∂ θ g, and because of the uniform bound (H5), for all θ, ∂ θ g

ε (θ, Y ε θ ) -→ ε→0 ∂ θ g(θ, Y θ ).
Because of (H1) and (H5), ∂ θ g ε is locally bounded, uniformly in ε, and so

θ1 θ0 ∂ θ g ε (θ, Y ε θ )η(θ) dθ -→ ε→0 θ1 θ0 ∂ θ g(θ, Y θ )η(θ) dθ
is also a consequence of the dominated convergence theorem.

Proof of Lemma 8

Before entering the proof of Lemma 8, we need to state a few standard results concerning the extended gradient ∇ as defined in Definition 4, and its links with the so-called resolvent map. These tools could even be set in the infinite dimensional setting, that is in Hilbert spaces [START_REF] Strömberg | The operation of infimal convolution[END_REF], or in metric spaces [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], and we refer to these works for the proofs. Consider h : R p → R a convex function. It is easily shown that for all

X ∈ R p , |∇h(X)| = sup Y =X (h(X) -h(Y )) + |X -Y | .
The following proposition, that we state without a proof, is an easy consequence of this formula and of the elementary fact that in finite dimension, pointwise convergence of convex functions to a finite valued convex functions implies Γ-convergence.

Proposition 10. Let (h ε ) ε>0 be a family of convex functions on R p pointwise converging to h, and let (X ε ) ε>0 be a family of points in R p converging to X. Then

|∇h(X)| ≤ lim inf ε→0 |∇h ε (X ε )|.
For τ > 0 and X ∈ R p , define the resolvent operator by:

J τ,h (X) := argmin Y ∈R p h(Y ) + |Y -X| 2 2τ .
Once again, the following proposition is standard, and we state it here without a proof.

Proposition 11.

1. We have for all X ∈ R p and τ > 0:

|∇h(J τ,h (X))| ≤ X -J τ,h (X) τ ≤ |∇h(X)|. (22)
2. If h is everywhere differentiable and X ∈ R p , then the following first order condition holds:

X -J τ,h (X) τ = ∇h(J τ,h (X)).
3. If (h ε ) ε>0 is a family of convex functions on R p pointwise converging to h, then for all τ > 0 and

X ∈ R p , J τ,hε (X) -→ ε→0 J τ,h (X). (23)
We are now ready for the proof of Lemma 8.

Proof of Lemma 8. Proof of the Γ -lim inf. It is straightforward using Fatou's lemma, Proposition 10 and the lower semi-continuity of Y → θ1 θ0 | Ẏθ | 2 dθ with respect to the topology of uniform convergence. Proof of the Γ -lim sup. Let us consider a curve Y ∈ H 1 ([θ 0 , θ 1 ]; R p ) with Y θ0 = R and Y θ1 = S (else there is nothing to prove). For all ε > 0 and τ > 0, we define:

Y τ,ε : θ → J τ,gε(θ,•) (Y θ ),
and correspondingly:

Y τ : θ → J τ,g(θ,•) (Y θ ).
First, we prove:

lim sup τ →0 lim sup ε→0 1 2 θ1 θ0 | Ẏ τ,ε θ | 2 + |∇g ε (θ, Y τ,ε θ )| 2 η(θ) dθ ≤ 1 2 θ1 θ0 | Ẏθ | 2 + |∇g(θ, Y θ )| 2 η(θ) dθ. (24)
We will then choose τ as a function of ε and show how to fix the endpoints. Proof of (24). By the second point of Proposition 11, for all ε, τ, θ, we have:

Y θ = Y τ,ε θ + τ ∇g ε (θ, Y τ,ε θ ).
Using the smoothness and convexity of g ε , and Y ∈ H 1 , we easily deduce that Y τ,ε is in H 1 and that for almost all θ,

Ẏθ = I + τ D 2 g ε (θ, Y τ,ε θ ) • Ẏ τ,ε θ + τ ∂ θ ∇g ε (θ, Y τ,ε θ )
. By convexity of g ε , we have I ≤ I + τ D 2 g ε in the sense of symmetric matrices, and hence:

| Ẏ τ,ε θ | ≤ | Ẏθ -τ ∂ θ ∇g ε (θ, Y τ,ε θ )| ≤ | Ẏθ | + τ M. (25)
Recall that M was defined in the uniform integrability assumption (12) on ∂ θ ∇g ε . (In the case when ∂ θ ∇g ε = 0, we recover the known fact that for h independent of time, J τ,h is contractive.) Then, we deduce:

lim sup ε→0 1 2 θ1 θ0 | Ẏ τ,ε θ | 2 + |∇g ε (θ, Y τ,ε θ )| 2 η(θ) dθ (22)(25) ≤ lim sup ε→0 1 2 θ1 θ0 | Ẏθ | + τ M 2 + Y θ -Y τ,ε θ τ 2 η(θ) dθ (23) ≤ 1 2 θ1 θ0 | Ẏθ | + τ M 2 + Y θ -Y τ θ τ 2 η(θ) dθ (22) ≤ 1 2 θ1 θ0 | Ẏθ | + τ M 2 + ∇g(θ, Y θ ) 2 η(θ) dθ.
Formula (24) follows.

Choice of τ = τ (ε). Because of (24), and because:

∀ε > 0, Y τ,ε θ0 -→ τ →0 R and Y τ,ε θ1 -→ τ →0 S,
it is possible to find a non-increasing function τ = τ (ε) converging sufficiently slowly to 0 so that:

lim sup ε→0 1 2 θ1 θ0 | Ẏ τ (ε),ε θ | 2 + |∇g ε (θ, Y τ (ε),ε θ )| 2 η(θ) dθ ≤ 1 2 θ1 θ0 | Ẏθ | 2 + |∇g(θ, Y θ )| 2 η(θ) dθ, (26)
Y τ (ε),ε θ0 -→ ε→0 R and Y τ (ε),ε θ1 -→ τ →0 S. (27)
Fixing the endpoints. For fixed ε and small δ > 0, we will define Z δ,ε as a slight modification of the curve Y ε,τ (ε) in such a way that Z δ,ε joins R to S. For this, we just set for θ ∈ [θ 0 , θ 1 ]:

Z δ,ε θ =            R + θ -θ 0 δ Y τ (ε),ε δ0+δ -R , if θ ∈ [θ 0 , θ 0 + δ], Y τ (ε),ε θ , if θ ∈ [θ 0 + δ, θ 1 -δ], S + θ 1 -θ δ Y τ (ε),ε δ1-δ -S , if θ ∈ [θ 1 -δ, θ 1 ],
A quick computation shows:

1 2 θ1 θ0 | Żδ,ε θ | 2 + |∇g ε (θ, Z δ,ε θ )| 2 η(θ) dθ ≤ 1 2 θ1 θ0 | Ẏ τ (ε),ε θ | 2 + |∇g ε (θ, Y τ (ε),ε θ )| 2 η(θ) dθ + η ∞ |Y τ (ε),ε θ0+δ -R| 2 2δ + |Y τ (ε),ε θ1-δ -S| 2 2δ + δL 2 , ( 28
)
where L is defined in the uniform Lipschitz assumption (11) for g ε .

Let us estimate |Y τ (ε),ε θ0+δ -R| 2 /2δ. We have:

|Y τ (ε),ε θ0+δ -R| 2 2δ ≤ |Y τ (ε),ε θ0 -R| 2 δ + |Y τ (ε),ε θ0+δ -Y τ (ε),ε θ0 | 2 δ ≤ |Y τ (ε),ε θ0 -R| 2 δ + θ0+δ θ0 | Ẏ τ (ε),ε θ | 2 dθ.
Because of (25), [START_REF] Frisch | A reconstruction of the initial conditions of the universe by optimal mass transportation[END_REF] and Y ∈ H 1 , the integral

θ0+δ θ0 | Ẏ τ (ε),ε θ
| 2 dθ → 0 as δ → 0, uniformly in ε: we bound it by a function v i = v i (δ) tending to 0 as δ → 0. In the same way,

|Y τ (ε),ε θ1-δ -S| 2 2δ ≤ |Y τ (ε),ε θ1 -S| 2 δ + v f (δ),
where v f (δ) → 0 as δ → 0.

Plugging these bounds into (28), we get:

1 2 θ1 θ0 | Żδ,ε θ | 2 + |∇g ε (θ, Z δ,ε θ )| 2 η(θ) dθ ≤ 1 2 θ1 θ0 | Ẏ τ (ε),ε θ | 2 + |∇g ε (θ, Y τ (ε),ε θ )| 2 η(θ) dθ + η ∞ u(ε) δ + v(δ) ,
where u(ε 27), and v(δ) := v i (δ) + v f (δ) + δL 2 → 0 as δ → 0. Hence, choosing δ(ε) := u(ε), we find with the help of (26) that Z δ(ε),ε is a recovery sequence for the Γ -lim sup of K ε towards K.

) := |Y τ (ε),ε θ0 -R| 2 + |Y τ (ε),ε θ1-δ -S| 2 → 0 as ε → 0 by (

Proof of Lemma 9

The proof is straightforward, and relies on explicit computations.

Proof of Lemma 9. Let us define for

X ∈ (R d ) N : h(X) := log 1 N ! σ∈S N exp(X • A σ ) . (29)
For ε > 0, θ ∈ [θ 0 , θ 1 ] and Y ∈ (R d ) N , we have by definition of f ε and g ε (formulas ( 8) and ( 13) respectively):

g ε (θ, Y ) = εh Y ε exp(θ) . ( 30
)
Proof of (H1). It is obvious.

Proof of (H2). By (30), it suffices to check that h is convex. Differentiating twice (29), we get for all X ∈ (R d ) N :

D 2 h(X) = A σ ⊗ A σ X -A σ X ⊗ A σ X = A σ -A σ X X ⊗ A σ -A σ X X , (31)
where if a is a function of σ, a(σ) X stands for:

a(σ) X := σ∈S N a(σ) exp(X • A σ ) σ∈S N exp(X • A σ )
.

It follows that D 2 h(X) is a nonnegative symmetric matrix. Proof of (H3). By the definitions (9) of f and (13) of g, we have for all θ ∈ [θ 0 , θ 1 ] and Y ∈ (R d ) N :

g(θ, Y ) = f (Y ) exp(θ) .
The convexity is obvious, let us check [START_REF] Dembo | Large deviations techniques and applications[END_REF]. Let us consider Y ∈ H 1 ([θ 0 , θ 1 ]; (R d ) N ). The function g is clearly locally Lipschitz in both θ and Y . As a consequence, the map G : θ → g(θ, Y θ ) is also H 1 . Let us take θ ∈ (θ 0 , θ 1 ) a point where both Y and G are differentiable (this happens for almost every θ). We have:

G (θ) = lim δ↓0 1 δ f (Y θ+δ ) exp(θ + δ) - f (Y θ ) exp(θ) = - f (Y θ ) exp(θ) + 1 exp(θ) lim δ↓0 f (Y θ+δ ) -f (Y θ ) δ ≥ - f (Y θ ) exp(θ) + 1 exp(θ) lim sup δ↓0 ∇f (Y θ ) • Y θ+δ -Y θ δ = ∂ θ g(θ, Y θ ) + ∇f (Y θ ) exp(θ) • Ẏθ = ∂ θ g(θ, Y θ ) + ∇g(θ, Y θ ) • Ẏθ , where we used f (Y θ+δ ) ≥ f (Y θ ) + ∇f (Y θ ) • (Y θ+δ -Y θ )
to get the second line. In the same way, we have:

G (θ) = lim δ↓0 1 δ f (Y θ ) exp(θ) - f (Y θ-δ ) exp(θ -δ) ≤ ∂ θ g(θ, Y θ ) + ∇g(θ, Y θ ) • Ẏθ .
The result follows from gathering these two inequalities. Proof of (H4). In view of (30) and as θ 0 > -∞, it suffices to check that ∇h is bounded. Differentiating (29) at X ∈ (R d ) N leads to:

∇h(X) = A σ X ,
which is clearly bounded by |A|. Proof of (H5). Using (30), we get for all ε > 0, θ ∈ [θ 0 , θ 1 ] and Y ∈ (R d ) N :

∂ θ ∇g ε (θ, Y ) = - 1 exp(θ) ∇h Y ε exp(θ) + D 2 h Y ε exp(θ) • Y ε exp(θ)
.

As we already saw in (H4) that ∇h is bounded, it suffices to prove that X → D 2 h(X) • X is bounded. Let us expand everything in (31) and apply X to the right. We get:

D 2 h(X) • X = σ,η∈S N X • (A σ -A η )A σ exp X • (A σ + A η ) σ,η∈S N exp X • (A σ + A η )
.

As a consequence, it suffices to show that for each σ, η ∈ S N ,

T (σ, η, X) := X • (A σ -A η ) exp X • (A σ + A η ) σ ,η ∈S N exp X • (A σ + A η )
is bounded, uniformly in X. First, if η = σ, then T (σ, σ, X) = 0. Else, let us use the bound:

σ ,η ∈S N exp X • (A σ + A η ) ≤ exp 2X • A σ + exp 2X • A η ,
obtained by only keeping the terms corresponding to σ = η = σ and σ = η = η in the sum. This leads to:

|T (σ, η, X)| ≤ |X • (A σ -A η )| exp X • (A σ + A η ) exp 2X • A σ + exp 2X • A η = |X • (A σ -A η )| exp -|X • (A σ -A η )| + exp |X • (A σ -A η )|
, which is clearly bounded uniformly in X. The result follows.

The case of dimension 1: sticky collisions

In this section, we will study the global minimizers of the functional Λ obtained in Subsection 2.5, in dimension d = 1. If we call t the time variable and if we replace θ 0 and θ 1 by 0 and T respectively, due to the invariance of the functional through translation in time, Λ reads:

Λ (Z) =        T 0 | Żt | 2 + |Z t -∇f (Z t )| 2 dt, if Z ∈ H 1 ([0, T ]; R N ), Z 0 ∈ {P σ } and Z T ∈ {Q σ }, + ∞, else, (32)
where:

f (X) = max σ∈S N X • A σ , X ∈ R N . (33)
Here, we chose a strictly ordered A = (a 1 , . . . , a N ), that is such that a

1 < • • • < a N , P, Q ∈ R N and T > 0.
Once again, when X = (x 1 , . . . , x N ) ∈ R N and σ ∈ S N , X σ := (x σ(1) , . . . , x σ(N ) ), and {P σ } and {Q σ } refer to {P σ , σ ∈ S N } and {Q σ , σ ∈ S N } respectively. Of course P = (p 1 , . . . , p N ) and Q = (q 1 , . . . , q N ) can be supposed to be ordered, that is

p 1 ≤ • • • ≤ p N and q 1 ≤ • • • ≤ q N .
We recall that we defined the extended gradient ∇f in Definition 4. As already noticed in Subsection 2.4, the existence of global minimizers for Λ follows from the direct method of calculus of variations. Uniqueness does not hold in general, even up to permutations. The purpose of the section is twofold. On the one hand, we will show that the model has nice regularity properties: any global minimizer of Λ is smooth except on a finite number of "sticking" or separation" times3 . On the other hand, we will justify as claimed in Section 2 that Λ describes a model with sticky collisions in the sense that a minimizer Z = (z 1 (t), . . . , z N (t)) of Λ will typically exhibit some sticking effects as z i (t) = z j (t) for i = j on non-trivial intervals.

To describe the sticking effect, it is convenient to introduce the following definition:

Definition 12 (Partition of 1, N ). Let X ∈ R N . We say that X is divided according to π(X) when π(X) is the partition of 1, N induced by the relation:

∀(i, j) ∈ 1, N 2 , i ∼ j ⇔ x i = x j .
We call C(X, i) the class of i ∈ 1, N in π(X).

The main result of the section is the following result:

Theorem 13 (Regularity of the optimal trajectories). For given A, P, Q ∈ R N and T > 0 as before, let Z be a global minimizer of Λ defined in (32). Then Z is continuous and there exist:

0 = t 0 < t 1 < • • • < t p = T
a family of times such that for each i = 1, . . . , p, Z is smooth on [t i-1 , t i ], and π(Z) is constant on (t i-1 , t i ).

It will be quite clear from the proof that sticking effects do occur. This exactly means that there exist trajectories Z for which with the notations of Subsection 2.6, Λ (Z) < Λ + (Z). For such trajectories, Z t is located on the set where f is not differentiable for a set of times of positive Lebesgue measure. But in dimension 1, this set is exactly the set where at least two particles are located at the same place. Otherwise stated, the set of times when π(Z) = {{1}, . . . , {N }} is typically of positive Lebesgue measure. As a consequence of Theorem 13, it is even a finite union of intervals.

Still it might be convenient to illustrate the sticking effects included in the model by the following easy proposition. It asserts that the set of times when all the particles are stuck is an interval: if all the particles are stuck at two different times, the cheapest behaviour between these two times is to remain stuck. It also shows that this phenomenon occurs: if all the particles are sufficiently close at the initial and final time, then they necessarily stick together during a non-trivial interval along the evolution.

Proposition 14 (Intervals of full degeneration).

1. For given A, P, Q ∈ R N and T > 0 as before, let Z = (z 1 (t), . . . , z N (t)) be a global minimizer of Λ . Suppose there exist two times 0 ≤ t 1 < t 2 ≤ T such that:

z 1 (t 1 ) = • • • = z N (t 1 ) and z 1 (t 1 ) = • • • = z N (t 2 ).
Then for all t

∈ [t 1 , t 2 ], z 1 (t) = • • • = z N (t).
2. For given A ∈ R N and T > 0 as before, the set U of endpoints P, Q ∈ R N with the property that for all minimizer Z = (z 1 (t), . . . , z N (t)) of Λ , the set of times:

t ∈ [0, T ] z 1 (t) = • • • = z N (t) is a non-trivial interval, is a neighbourhood of {P, Q ∈ R N | p 1 = • • • = p N and q 1 = • • • = q N }.
The proof of Proposition 14 uses almost nothing and is given in Subsection 4.2. Except for that, the whole section is dedicated to the proof of Theorem 13. For this we take once for all A, P, Q ∈ R N and T > 0, A being strictly ordered and P, Q being ordered.

Even if all the arguments are elementary, we will need a certain number of steps, including the explicit computation of the potential |X -∇f (X)| 2 (Subsection 4.1 and 4.4) and the justification of a priori knowledge on the optimal trajectories: they can be supposed to be ordered at all time (Subsection 4.3), and the conservation of energy and momentum holds during shocks4 (Subsection 4.5). The main ingredient in the proof of Theorem 13 is an estimate given in Subsection 4.6: during a non-pathological shock (pathological shocks are excluded a posteriori ), at least one particle has a below-bounded jump in its velocity (Proposition 23). We finally provide the proof of Theorem 13 in Subsection 4.7.

Throughout the section, we will work with several type of finite sets: the partitions of type π(X) and the class of particles of type C(X, i). Some of the arguments or computations will deal with their cardinal. Thus, if F is a finite set, we will denote by #F its cardinal.

Properties of the extended gradient

The extended gradient of f can be computed explicitly In Lemma 16, we gather easy properties of ∇f that will be needed in the following. Before doing so, let us introduce some notations. Definition 15. Let π be a partition of 1, N . We call E π the linear subspace of R N of all X s.t. π is a refinement of π(X), that is:

E π := C∈π i,j∈C X = (x 1 , . . . , x N ) ∈ R N x i = x j .
Here is the lemma: Lemma 16 (Properties of ∇f ).

1. The extended gradient ∇f has the following symmetry:

∀X ∈ R N , ∀σ ∈ S N , ∇f (X σ ) = ∇f (X) σ . (34)
2. The function X → |X -∇f (X)| is symmetric:

∀X ∈ R N , ∀σ ∈ S N , |X σ -∇f (X σ )| 2 = |X -∇f (X)| 2 . ( 35
)
3. If X is ordered, then ∇f (X) is the orthogonal projection of A on E π(X) .

4. If X is ordered and i ∈ {1, . . . , N }:

∀i = 1, . . . , N, ∇f (X) i = 1 #C(X, i) j∈C(X,i) a j . ( 36
) (Recall that C(X, i) is defined in Definition 12.)
Remark 17. The extended gradient ∇f is completely characterized by points 1. and 3. (or 4.) of Lemma 16.

Proof. Point 1. Let σ ∈ S N . By the definition (33) of f , for all X ∈ R N , f (X σ ) = f (X). Calling I σ : X → X σ , we easily deduce that at the level of subdifferentials:

∂ -f (X σ ) = I σ ∂ -f (X)
. We conclude by the fact that I σ is orthogonal. Point 2. It is a direct consequence of Point 1. Point 3. Let X = (x 1 , . . . x N ) ∈ R N be an ordered vector. Considering the definition (33) of f and noticing that the maximum is achieved exactly for those σ such that X σ = X, it appears that ∇f (X) belongs to the convex hull:

Conv A σ σ ∈ S N such that X σ = X .
For a given i ∈ {1, . . . , N }, we call V i ∈ R N the vector whose j-th coordinate is 1 if j ∈ C(X, i) and 0 otherwise. On the one hand, we have E π(X) = Span{V i | i = 1, . . . , N }, and on the other hand, for all i, the scalar product V i • Y is constant on the above-mentioned convex hull. So we deduce:

A -∇f (X) ∈ E π(X) ⊥ .
Hence, we just have to prove that ∇f (X) ∈ E π(X) . If i, j ∈ {1, . . . , N } are such that x i = x j , let us apply formula (34) to the permutation σ := (i, j):

∇f (X) i = ∇f (X) σ j = ∇f (X σ ) j = ∇f (X) j .
The result follows. Point 4. Let X be ordered and i ∈ {1, . . . , N }. As ∇f (X) ∈ E π(X) , with the notations of the proof of Point 3:

∇f (X) i = 1 #C(X, i) j∈C(X,i) ∇f (X) j = 1 #C(X, i) ∇f (X) • V i = 1 #C(X, i) A • V i = 1 #C(X, i) j∈C(X,i) a j ,
where we used A -∇f (X) ⊥ V i to get the first identity in the second line.

The three next subsections will be dedicated to consequences of this lemma:

• A proof of Proposition 14;

• When proving Theorem 13, it is enough to consider ordered trajectories (Proposition 18);

• For ordered trajectories, the potential in Λ can be decomposed as sum of a smooth "external" potential and an "internal" energy only depending on π(X) (Proposition 19).

Proof of Proposition 14

With the help of Lemma 16, we are ready to prove Proposition 14.

Proof of Proposition 14. Point 1. Without loss of generality, we can suppose t 1 = 0 and t 2 = T , that is P = (p 1 , . . . , p N ) and Q = (q 1 , . . . , q N ) are such that

p 1 = • • • = p N and q 1 = • • • = q N . Call Ψ the orthogonal projection on the line E 1,N := {X = (x , . . . , x N ) ∈ R N | x 1 = • • • = x N }.
It suffices to prove that when Z is a continuous trajectory joining P to Q, then Λ (Ψ(Z)) ≤ Λ (Z), and with equality if and only if Z = Ψ(Z). As Ψ is 1-Lipschitz, it reduces the kinetic part of Λ . For the potential part, we remark that for all

X ∈ R N , E π(Ψ(X)) = E 1,N ⊂ E π(X)
. As a consequence, by Point 3. of Lemma 16, we have as soon as X is ordered ∇f (Ψ(X)) = Ψ(∇f (X)). Hence:

|Ψ(X) -∇f (Ψ(X))| 2 = Ψ X -∇f (X) 2 ≤ |X -∇f (X)| 2 ,
with equality if and only if X ∈ E 1,N , i.e. if and only if Ψ(X) = X. This property is extended to non-ordered X using (35), and the result follows. Point 2. The function Λ = Λ (P, Q) defined for all P, Q ∈ R N as the minimal value of Λ is continuous. Indeed, if P, P , Q, Q ∈ R N are chosen so that |P -P | + |Q -Q| 1 and if Z is a trajectory joining P to Q, we can find a trajectory Z joining P to Q with:5

Λ ( Z) ≤ Λ (Z) + o (P ,Q )→(P,Q) (1). (37)
To do so, it suffices to choose τ ∼ |P -P | + |Q -Q|, and to define Z as the trajectory joining P to P in straight line between times 0 and τ , joining P to Q between times τ and T -τ by following Z with a proper affine change of time, and finally joining Q to Q in straight line between times T -τ and T . This shows that Λ is lower semi-continuous, but the continuity is obtained by noticing that the o in (37) is locally uniform on P, Q ∈ R N . The argument is easily adapted to show that Λ = Λ (P, Q) defined for P, Q ∈ R N by:

Λ (P, Q) := inf Λ (Z) Z whose set of t s.t. Z t ∈ E 1,N is negligible
is also continuous. Besides, the set U defined in the statement clearly satisfies:

V := P, Q ∈ R N Λ (P, Q) < Λ (P, Q) ⊂ U.
By continuity of Λ and Λ , V is an open set. Hence it remains to prove that:

P, Q ∈ R N p 1 = • • • = p N and q 1 = • • • = q N = E 1,N × E 1,N ⊂ V.
To do so, we take

P, Q ∈ E 1,N , Z a curve joining P to Q such that {t | Z t ∈ E 1,N
} is negligible, we still call Ψ the orthogonal projection on E 1,N , and we prove that

Λ (Z) ≥ Λ (Ψ(Z)) + a,
where a > 0 does not depend on Z. Let us call Φ := Id -Ψ the orthogonal projection on the orthogonal of E 1,N . As in the proof of the first point, ∇f • Ψ = Ψ • ∇f . As a consequence:

Λ (Z) = T 0 |Ψ( Żt )| 2 + |Ψ(Z t ) -Ψ(∇f (Z t ))| 2 dt + T 0 |Φ( Żt )| 2 + |Φ(Z t ) -Φ(∇f (Z t ))| 2 dt = Λ (Ψ(Z)) + T 0 | Ż⊥ t | 2 + |Z ⊥ t -Φ(∇f (Z t ))| 2 dt,
where Z ⊥ = Z ⊥ t := Φ(Z t ) is a curve joining 0 to 0. But for almost all t, Z t / ∈ E 1,N , so as we saw in the proof of the first point, ∇f (Z t ) / ∈ E 1,N . As ∇f only takes a finite number of values (see Lemma 16), for almost all t, Φ(∇f (Z t )) belongs to some finite set, say G, which does not contain 0. Hence,

T 0 | Ż⊥ t | 2 + |Z ⊥ t -Φ(∇f (Z t ))| 2 dt ≥ T 0 | Ż⊥ t | 2 + dist(Z ⊥ t , G) 2 dt,
where dist(Z, G) denotes the distance from Z to G. Because Z ⊥ joins 0 to 0 and G does not contain 0, this last integral is easily seen to be below bounded away from 0 independently of Z, and the result follows.

Ordering of the particles

The purpose of this subsection is to show that when proving Theorem 13, we can restrict ourselves to study trajectories that remain ordered (see Figure 1). This is due to the following proposition.

Proposition 18. Let Z = Z t be a global minimizer of Λ . We call Z = Z t the trajectory obtained by reordering the coordinates of Z in increasing order. Then Z is also a global minimizer of Λ . Moreover, Z has the regularity stated in Theorem 13 if and only if Z does.

In particular, Λ always admits an ordered minimizer, and it it is enough to prove Theorem 13 for such minimizers.

Thanks to this proposition, from now on, we only work with ordered minimizers of Λ . These minimizers Z = Z t satisfy in particular Z 0 = P and Z T = Q (as we chose them to be ordered in the first place).

0 T p 1 p 2 p 3 q 1 q 2 q 3 0 T p 1 p 2 p 3 q 1 q 2 q 3
Figure 1: These two trajectories share their initial and final position up to ordering and their actions. But to the right, the order is preserved while to the left, this is not the case.

Proof. Let Z and Z be as in the statement of the proposition. Point 2 of Lemma 16 implies:

T 0 | Z t -∇f ( Z t )| 2 dt = T 0 |Z t -∇f (Z t )| 2 dt.
We call Ψ : R N → R N the operator that reorders the coordinates of a vector in increasing order, so that in particular for all t, Z t = Ψ(Z t ). A simple application of the rearrangement inequality shows that Ψ is 1-Lipschitz. In particular, it reduces the action of curves:

T 0 | ˙ Z t | 2 dt ≤ T 0 | Żt | 2 dt.
By adding the two last formulas, and by noticing that the endpoint constraint is fulfilled, we get Λ ( Z) ≤ Λ (Z). As Z is a minimizer, this inequality is in fact an equality, and Z is also a minimizer.

Remark that both Z and Z are continuous because they have finite action. Hence, the second claim of the proposition is a consequence of the two following facts:

• For all t ∈ [0, T ], #π( Z t ) = #π(Z t ).

• For any continuous trajectory t ∈ I → X t ∈ R N where I is an interval, t → π(X t ) is constant if and only if t → #π(X t ) is constant.

Indeed in that case, t → π(Z t ) and t → π( Z t ) are constant on the same intervals, and the result follows. The first point and the "only if" part of the second point are trivial.

For the "if" part of the second one, we reason by contraposition. Suppose s → π(X s ) has a discontinuity at time t and we prove that s → #π(X s ) also does. If s → π(X s ) has a discontinuity at time t, we can find two distinct accumulation points π 1 and π 2 of s → π(X s ) at time t. As for all π, the set E π is closed, X t belongs to E π1 ∩ E π2 . But this set is noting but E π where π is the finest partition of which π 1 and π 2 are refinements, that is the partition corresponding to the relation:

i ∼ j ⇔ ∃C ∈ π 1 ∪ π 2 s.t. {i, j} ⊂ C.
In particular, π(X t ) is a refinement of π and as π 1 = π 2 , we easily get:

#π(X t ) ≤ #π < max #π 1 , #π 2 .
So s → #π(X s ) has a discontinuity at time t, and the result follows.

Decomposition of the potential

Here, we compute explicitly the values of the potential X → |X -∇f (X)| 2 on ordered vectors X ∈ R N . Notice that for such vectors X, π(X) has an additional structure: if C ∈ π(X), then C is an interval of integers. We say that such partitions are ordered. We prove the following:

Proposition 19. For all ordered X ∈ R N : |X -∇f (X)| 2 = |X -A| 2 + h(π(X)) -|A| 2 , ( 38
)
where h is defined on a partition π of 1, N by:

h(π) := C∈π 1 #C j∈C a j 2 . ( 39
)
In particular, h has the following monotonicity property: if π and π are two ordered partitions and if π is a strict refinement of π, then h(π) < h(π ).

The more particle are stuck together, the lower h. This is the reason for which Λ favours the sticking of particles. The function -h can be understood as the internal energy of the system.

Dropping the constant term |A| 2 /2 in (38) and defining Λ on a trajectory Z by:

Λ (Z) =        T 0 | Żt | 2 + |Z t -A| 2 + h(π(Z t )) dt, if Z ∈ H 1 ([0, T ]; R N ), Z 0 = P and Z T = Q, + ∞, else, (40)
it is clear that Λ and Λ have the same minimizers in the class of ordered trajectories. Hence, as a consequence of Proposition 18, it suffices to prove the conclusion of Theorem 13 for the minimizers of Λ in the class of ordered trajectories.

Proof of Proposition 19. Let X ∈ R N be an ordered vector. By Point 3. of Lemma 16, we have A-∇f (X) ∈ E π(X) ⊥ and both X and ∇f (X) ∈ E π(X) . So using twice the Pythagorean theorem, we get:

|X -∇f (X)| 2 = |X -A| 2 -|A -∇f (X)| 2 = |X -A| 2 + |∇f (X)| 2 -|A| 2 .
The identities (38) and (39) are obtained by computing |∇f (X)| 2 using (36). If we recap, h(π) is the squared norm of the orthogonal projection of A on E π . But if π is a refinement of π, E π ⊂ E π , and hence h(π) ≤ h(π ). The strict inequality is obtained by noticing with the help of (36) and using the strict ordering of A that if in addition π and π are ordered and π = π, then the projection of A on E π does not belong to E π .

Conserved quantities

In this subsection, we discuss two simple and yet structural properties of the dynamic prescribed by the functionals Λ , Λ : the Hamiltonian of the system is conserved (Proposition 20), and its center of mass is smooth (Proposition 21). In particular, the momentum of the system is conserved during shocks.

Proposition 20. Let Z be an ordered minimizer of Λ . Then:

E = E(t) := | Żt | 2 -|Z t -A| 2 -h(π(Z t )) (41)
is constant in the sense of distributions.

Proof. The proof is completely standard and consists in comparing the value of Λ on Z and t → Z t+εϕ(t) for small ε and functions ϕ that are smooth and compactly supported in (0, T ).

Proposition 23. Let Z = (z 1 (t), . . . , z N (t)) be an ordered minimizer of Λ (or equivalently a minimizer of Λ in the class of ordered trajectories), and let t ∈ [0, T ].

1. If particle i is not involved in a shock at time t, then for s in the neighbourhood of t, C := C(Z s , i) is constant and z i is a smooth solution of:

zi (s) = z i (s) - 1 #C j∈C a j . (42)
In particular, if i is involved in an isolated shock at time t, then z i admits left and right derivatives at time t, denoted by żi (t-) and żi (t+) respectively.

2. There is α = α(N, A) > 0 such that for any isolated shock (t, q, C), calling i := min C:

żi (t-) -żi (t+) ≥ α. (43)
Proof. Point 1. If particle i is not involved in a shock at time t, by definition of a shock, it means that C := C(Z t , i) ∈ π(Z s ) for all s in a neighbourhood of t. In particular, for all j ∈ C and s sufficiently close to t, by (36):

∇f (Z s ) j = 1 #C k∈C a k .
On the other hand, it is easy to find a neighbourhood U of (t, z i (t)) in [0, T ]×R such that for all j ∈ {1, . . . , N } and all s ∈ [0, T ], (s, z j (s)) ∈ U implies j ∈ C. As a consequence, if ξ : [0, T ] → R is smooth and compactly supported in a sufficiently small neighbourhood of t, and if ε is sufficiently small, by defining Z = ( z 1 (s), . . . , z N (s)) for any j ∈ {1, . . . , N } and s ∈ [0, T ] by:

z j (s) := z j (s) + εξ(s) if j ∈ C, z j (s) else,
then π(Z) and π( Z) (and hence ∇f (Z) and ∇f ( Z)) coincide at all time. The ODE follows from comparing the values of Λ on Z and trajectories of type Z.

In particular, by boundedness of Z, if particle i is not involved in a shock at time t, |z i | is bounded by a constant not depending on t. The existence of żi (t-) and żi (t+) at the times of isolated shocks follows easily. Point 2. This is the heart of our study of the dynamical system, and maybe the less standard part of Section 4. But still the idea is very easy: With the notations of the statement, if żi (t-) -żi (t+) is too small, then it is cheaper to stick particle i with other particles, as shown in Figure 3. The proof goes as follows.

Step 1: Definition of a competitor.

Let us consider (t, q, C) an isolated shock. Because it is isolated, we can find τ > 0 such that the particles of C are not involved in an other shock between times t -τ and t + τ . By definition of a shock, we cannot have C ∈ π(Z s ) for all s ∈ (t -τ, t + τ ), so either for all s ∈ (t -τ, t), C / ∈ π(Z s ) or for all s ∈ (t, t + τ ), C / ∈ π(Z s ). Without loss of generality, we suppose that the second one holds: the particles of C are not all stuck right after the shock. Moreover, by our choice of τ , for all C ⊂ C, the assertion C ∈ π(Z s ) is either true of false independently on s ∈ (t, t+τ ). Then, for s ∈ (t, t+τ ), the following definitions of C 1 , C 2 ∈ π(Z s ) do not depend on s:

C 1 := C(Z s , i) for i = min C and C 2 := C(Z s , i) for i = min C\C 1 .
(The classes C 1 and C 2 are the two leftmost packs of particles of C right after the shock.) Let us define for j = 1, 2:

k j := #C j , v j := żi (t+) for i ∈ C j , and p := k 1 v 1 + k 2 v 2 k 1 + k 2 . ( 44
) t q t + λσ t + σ t q t + λσ t + τ Figure 3:
To the left, a piece of the trajectory Z, and to the right, the competitor Z σ,λ that we describe in the proof.

For 0 ≤ σ < τ and λ ∈ [0, 1), we define a competitor Z σ,λ = (z σ,λ 1 (s), . . . , z σ,λ N (s)) by setting for all i = {1, . . . , N } and s ∈ [0, T ]:

z σ,λ i (s) =          z i (s) if i / ∈ C 1 ∪ C 2 or s / ∈ (t, t + σ), q + (s -t)p if i ∈ C 1 ∪ C 2 and s ∈ (t, t + λσ), t + σ -s (1 -λ)σ q + λσp + s -(t + λσ) (1 -λ)σ z i (t + σ) if i ∈ C 1 ∪ C 2 and s ∈ (t + λσ, t + σ).
(See Figure 3 for an illustration of this competitor.) We will get a below bound on v 2 -v 1 by comparing the value of Λ on Z and Z σ,λ , and by differentiating the corresponding inequality first with respect to σ at σ = 0 (we zoom so that the particles of Z only travel along straight lines), and then with respect to λ at λ = 0 (we compute the first variation of the action when we let the particles stick together).

Step 2: A below bound on v 2 -v 1 .

The partitions π(Z σ,λ s ) and π(Z s ) coincide at all time except between time t and t + λσ, when π(Z s ) is a strict refinement of π(Z σ,λ s ). Hence, calling:

δ = δ(N, A) := min h(π) -h(π ) | (π, π ) ordered partition of 1, N , π strict refinement of π > 0,
we have for all s ∈ (t, t + λσ):

h(π(Z λ,σ s )) + δ ≤ h(π(Z s )). (45)
As Z σ coincide with Z for times outside (t, t+σ) and for coordinates that are not in C 1 ∪C 2 , by definition (40) of Λ , we have:

Λ (Z σ,λ ) -Λ (Z) = i∈C1∪C2 t+σ t | żσ,λ i (s)| 2 + |z σ,λ i (s) -a i | 2 -| żi (s)| 2 -|z i (s) -a i | 2 ds + t+λσ t {h(π(Z σ,λ s )) -h(π(Z s ))} ds ≤ i∈C1∪C2 t+σ t | żτ,σ i (s)| 2 -| żi (s)| 2 ds -δλσ + o σ→0 (σ), (46)
where to obtain the second line, we used (45) and the fact that between times t and t + σ, both z i and z σ,λ i remain at a distance of order σ of q.

Let us consider i ∈ C j for j = 1, 2. One the one hand, as z i admits v j as a right derivative at time t, we have:

t+σ t | żi (s)| 2 ds = v 2 j σ + o σ→0 (σ). (47)
On the other hand, we can compute explicitly:

By plugging (47) and ( 48) in (46) and by using the definition (44) of k 1 , k 2 and p, we get:

Λ (Z σ,λ ) -Λ (Z) ≤ (k 1 + k 2 )λp 2 + k 1 (v 1 -λp) 2 + k 2 (v 2 -λp) 2 1 -λ -k 1 v 2 1 -k 2 v 2 2 -δλ σ + o σ→0 (σ) = (k 1 + k 2 )p 2 + k 1 v 2 1 + k 2 v 2 2 -2p k 1 v 1 + k 2 v 2 -δ(1 -λ) λ 1 -λ σ + o σ→0 (σ) = k 1 v 2 1 + k 2 v 2 2 - (k 1 v 1 + k 2 v 2 ) 2 k 1 + k 2 -δ(1 -λ) λ 1 -λ σ + o σ→0 (σ) = k 1 k 2 k 1 + k 2 (v 2 -v 1 ) 2 -δ(1 -λ) λ 1 -λ σ + o σ→0 (σ).
By minimality of Λ (Z), this quantity must be nonnegative. If we divide it by λσ, and if we let σ and then λ go to zero, we end-up with:

k 1 k 2 k 1 + k 2 (v 2 -v 1 ) 2 ≥ δ. (49)
Step 3: Conservation of momentum during an isolated shock and conclusion. Because (t, q, C) is isolated, it is easy to justify that we can replace V by the vector V C whose j-th coordinate is 1 if j ∈ C and 0 otherwise in the proof of Proposition 21. Doing so, we obtain the "local" conservation of momentum: by ordering of the particles, we have for i = min C:

żi (t-) ≥ P(t) = 1 #C i∈C żi (t+) ≥ k 1 #C v 1 + #C -k 1 #C v 2 .
(Indeed, j ∈ C → żj (t-) and j ∈ C → żj (t+) are clearly non-increasing and non-decreasing respectively.) By recalling that v 1 = żi (t+) and using (49), we get:

żi (t-) -żi (t+) ≥ #C -k 1 #C (v 2 -v 1 ) ≥ #C -k 1 #C k 1 + k 2 k 1 k 2 δ.
The minimal right hand side's value is δ/(#C 2 -#C), obtained for k 1 = #C -1 and k 2 = 1. Hence, we get the result by choosing α = δ/(N 2 -N ).

Conclusion: proof of Theorem 13

We are now ready to give the proof of Theorem 13. We give ourselves Z a global minimizer of Λ . Thanks to Proposition 18, we can suppose that Z is ordered, and thanks to Proposition 19, we can consider Λ instead of Λ . Because of Proposition 23, it suffices to prove that there is a finite number of shocks. Indeed, in that case one can take for 0 = t 0 < t 1 • • • < t p = T the moments of these shocks (and the endpoints of [0, T ]).

The smoothness of Z on each [t i-1 , t i ], i = 1, . . . , p follows directly from the Proposition 23. Then π(Z) is constant on each (t i-1 , t i ), i = 1, . . . , p because by Definition 22 of a shock, at each time of discontinuity of π(Z), there is at least one shock.

The set: (t , q ) ∃C ⊂ 1, N s.t. (t , q , C ) is a shock is easily seen to be compact. So if it is not finite, it admits at least one accumulation point. Otherwise stated, if there is an infinite number of shocks, then there is at least one shock which is not isolated. Let us consider such a shock (t, q, C) with minimal number of particles involved, i.e. with minimal #C. The rest of the proof consists in showing that the existence of (t, q, C) leads to a contradiction.

Step 1: The velocities are bounded.

As Z is continuous on [0, T ], it is bounded. On the other hand, by definition, h ≤ |A| 2 . Now if i ∈ {1, . . . , N } and t ∈ [0, T ] is such that Z is differentiable at t (which is true for almost any t), recalling the definition (41) of E:

żi (t) 2 ≤ | Żt | 2 ≤ E + |Z t -A| 2 + h(π(Z t )),
which is bounded uniformly in t.

Step 2: All the shocks in the neighbourhood of (t, q) are isolated.

Let U be a neighbourhood of (t, q) in [0, T ]×R such that for all s ∈ [0, T ] and i ∈ {1, . . . , N }, (s, z i (s)) ∈ U implies i ∈ C. This is possible since Z is continuous and for all j / ∈ C, z i (t) = q by Definition 22 of a shock. Let us consider (t , q , C ) a shock with (t , q ) ∈ U . If #C < #C, then (t , q , C ) is isolated by minimality of #C. If #C = #C, then C = C by definition of U . But then it is easy to adapt the proof of Point 1. of Proposition 14 to prove that C ∈ π(Z s ), for all s between t and t and so there is no shock in U between t and t . Hence there exists at most one such shock in U : either one before t or one after t, but not both because else (t, q, C) would contradict the third point of the definition of a shock. Up to reducing U , we can then exclude (t , q , C ).

Step 3: Conclusion using Proposition 23.

As (t, q, C) is not isolated, there is an infinite number of (isolated) shocks in U . Without loss of generality, we can assume that there is an infinite number of shocks in U after time t. Call i ∈ C the smallest index such that particle i is involved in an infinite number of shocks in U after time t. When i = min C, up to reducing U and by minimality of i, we can assume that no particle j ∈ C with j < i is involved in a shock in U after time t.

As the shocks in U involving i after time t are isolated (Step 2.), we can enumerate their times in decreasing order (t p ) p∈N . The boundedness of Z as well as (42) allows us to take M an upper bound for zi between the times of shocks. For all p ∈ N and s ∈ (t p+1 , t p ), taking α as in (43), we have:

| 2 = λp 2 σ + 1 ( 1 2 =

 2112 żσ,λ i (s)| 2 ds = λp 2 σ + (1 -λ)σ z i (t + σ) -(q + λpσ) (1 -λ)τ -λ)σ q + v j σ + o σ→0 (σ) -q -λpσ λp 2 σ + (v j -λp)

  ) =: P C (t).

  żi (s) = żi (t 0 -) + p k=1 żi (t k -) -żi (t k +) -t k t k-1 zi (τ ) dτ -s tp zi (τ ) dτ ≥ żi (t 0 -) + pα -M (t 0 -t),which contradicts Step 1. as soon as p is sufficiently large.

Due to the lack of uniqueness in the discrete optimal transport problem, this system is not always well defined a priori but we don't want to enter into the details here.

Once again up to a suitable change of time, see Subsection 2.5.

Notice that Λ is invariant under time inversion, so that if particles are allowed to stick, they are also allowed to separate.

We say that Z presents a shock at time t if t is a discontinuity point of π(Z), see Definition 22.

With a slight abuse of notation, we do not refer explicitly to the dependence of Λ on P, Q.

q 2 q 3 0 T Figure 2: A shock with three particles which does not seem to be isolated. We will see later on that this kind of shock cannot occur in our model.

Proposition 21. Let Z = (z 1 (t), . . . , z N (t)) be an ordered minimizer of Λ . Call a := (a

and for t ∈ [0, T ]:

(M is well define for all t, and P for almost all t.) Then M, P solve distributionally:

In particular, M is smooth and P coincide almost surely with a smooth function.

Proof. Here the proof consists in comparing the value of Λ on Z and t → Z t + εϕ(t)V for small ε, smooth and compactly supported ϕ, and where we call V = (1, . . . , 1). The only somehow unusual thing to remark is that π and hence h • π are invariant under translations in the direction of V .

Shock, isolated shocks and minimal deviation

This subsection contains the main estimate that allows to prove Theorem 13. Roughly speaking, if at time t some of the particles stick or separate, there is a below bound on the change of the velocity of at least one particle. The proof of Theorem 13 will then consist in showing that this cannot happen an infinite number of time. Let us first define as "shocks" these sticking and separating behaviours:

Definition 22 (Shocks). Let X = X t = (x 1 (t), . . . , x N (t)) be a continuous trajectory on R N .

1. We call a shock of X a triplet (t, q, C) with t ∈ [0, T ], q ∈ R and C ⊂ 1, N such that:

• for all i ∈ C, x i (t) = q,

• for all τ > 0, there exists s ∈ (t -τ, t + τ ) such that C / ∈ π(X s ).

2. If (t, q, C) is a shock of X , we say that it is isolated if (t, q) is isolated in:

(t , q ) ∃C ⊂ 1, N s.t. (t , q , C ) is a shock , i.e. if there is no other shock than (t, q, C) in the neighbourhood of (t, q) ∈ [0, T ] × R.

We provide in Figure 2 a picture of a shock which does not seem to be isolated. The following result is the main step in the proof of Theorem 13.