C Dominguez
email: christina.dominguez@irsn.fr

Effect of pre-oxide and hydrogen on creep of Zircaloy-4 at 1123 K

Highlights (85 caractères max espaces compris)

 Creep behaviour of pre-oxidized and pre-hydrided Zircaloy-4 was studied at 1123 K  Pre-oxidation decreases creep rate depending on pre-oxide thickness and location  Increasing the hydrogen concentration results in increasing the creep rate

Abstract

The creep behaviour of pre-oxidized and pre-hydrided Zircaloy-4 was investigated at 1123 K.

Outer only and outer + inner pre-oxidation were considered in the study. The cladding creep rate decreases when the thickness of the pre-oxide layer increases and stabilizes for thicknesses higher than about 20 µm. The effect of pre-oxide is stronger in double-sided than in single-sided pre-oxidized samples and depends on the applied stress. Hydrogen increases Zircaloy-4 creep rate on both bare and pre-oxidized samples. The presence of high amounts of hydrogen, which induces the formation of significant quantities of  phase at high temperature, modifies the creep mechanism. During preoxidation, a network of regularly spaced radial cracks is formed on thick pre-oxide layers. At the beginning of the creep test, these cracks spread out throughout the oxide layer up to the (O) layer, where they stop. Finally, strain develops in the oxide-free zones.

Introduction

The purpose of the PERFROI project is to study the cooling of a nuclear reactor core during a Loss-Of-Coolant-Accident (LOCA), considering both the thermo-mechanical and thermohydraulic phenomena within the fuel assemblies. During this accident, the fuel assemblies of a nuclear reactor core can be partially or completely dried out. Under these conditions, the increase in fuel temperature and the pressure drop in the reactor core can lead to significant cladding strain, or even to fracture of the fuel rod cladding. The consequences of these geometry modifications on the core cooling efficiency during the injection of water by the safety systems are an important safety issue. The first axis of the PERFROI project concerns the thermo-mechanical study of the deformation and fracture of fuel rods under LOCA conditions. It includes thermo-mechanical creep and sample rupture tests (ELFE tests). The main objective of these tests is to accurately characterise the thermo-mechanical properties of the materials used in the DRACCAR data (IRSN software for the simulation of LOCA type accidents).

The deformation of the fuel rods during a LOCA transient is determined by the creep behaviour of the cladding. However, we mainly know the behaviour of bare (non-hydrided, non-oxidized) Zircaloy-4 (Zy-4) in the  and domains. In the  domain, creep behaviour of bare Zy-4 is very complex and experimental results are highly scattered. The mechanisms involved in the creep of bare Zy-4 depend on the domain considered, and on the applied stresses [START_REF] Rosinger | Steady-state creep of Zircaloy-4 fuel cladding from 940 to 1873K[END_REF][START_REF] Kaddour | Experimental determination of creep properties of Zirconium alloys together with phase transformation[END_REF]:  In the  domain at low stress (<15 MPa), creep is governed by the diffusion at the grain boundaries with an exponent n of the Norton law = K( n ) close to 1.



In the  domain at higher stress (> 15 MPa), the creep mechanism is sliding and dislocation climb. The values of the exponent n are in the vicinity of 5.



In the  domain, it is the dislocation induced creep that governs the strain, with values of n in the vicinity of 4.



In the  +  domain, the experimental data are dispersed due to the strong influence of the microstructure and the composition of the material on the mechanical behaviour. When Zircaloy is mainly in the  phase, a superplasticity mechanism by grain boundary sliding at the  - interface accommodated by diffusion is proposed at around 1123 K for low stresses. At this temperature, the small amount of  phase (much more ductile than the  phase) forms a continuous film around the  grains, which promotes the interface sliding. When the temperature increases, the amount of  phase increases and forms grains that break the continuity around the  grains. Thus, the  - interface sliding becomes less and less significant with a minimum ductility at around 1223 K. For high stresses, dislocation induced creep becomes dominant. The deformation laws proposed in the  +  domain relate only to low stresses with values of n close to 1.

The mechanical behaviour of zirconium alloys is highly dependent on the oxygen and hydrogen contents. Oxygen and hydrogen have opposite effects on the behaviour of Zircaloy [START_REF] Brachet | Influence of hydrogen content on the α/β phase transformation temperatures and on the thermal-mechanical behavior of Zy-4, M4 (ZrSnFeV), and M5™ (ZrNbO) alloys during the first phase of LOCA transient[END_REF][START_REF] Chosson | Creep behavior at high temperature of the oxygen stabilized zirconum alpha phase of fuel cladding tubes oxidized in LOCA conditions[END_REF][START_REF] Chosson | Creep flow and fracture behavior of the oxygen-enriched alpha phase in zirconium alloys[END_REF]: the phase transformation temperatures and creep resistance increase with the amount of oxygen while they decrease with the increase in the hydrogen concentration. There are no data in the literature regarding the combined effect of both elements on the mechanical behaviour of Zy-4 in the  +  domain. This paper presents the effect of pre-oxide and hydrogen on the creep of Zircaloy-4 at 1123 K. The study is limited to one temperature but addresses the influence of other parameters as thermal and loading history, applied stress, pre-oxide thickness and location (single and double sided pre-oxide) and phase distribution on creep behaviour. Several thermal tests were also performed to characterize the changes in Zy-4 microstructure during the temperature plateau at 1123 K.

Materials and methods

Samples

The samples were machined from stress-relieved annealed, low-tin Zircaloy-4 cladding tubes provided by FRAMATOME (see figure 1). The outer diameter and thickness of the tubes are respectively 9.5 mm and 570 µm. The composition of the alloy is (supplier data): 1.3 wt% Sn, 0.21 wt% Fe, 0.10 wt% Cr, 0.13 wt% O, 10 wppm H.

Various states of the fuel cladding were tested:  Bare Zircaloy;  Pre-hydrided Zircaloy (hydrogen content 130 wppm);  Pre-oxidized Zircaloy. In service, the thickness of the inner pre-oxide formed inside the cladding in contact with the fuel can reach 10 µm, while the outer pre-oxide, formed by oxidation by the pressurized water coolant can reach or even exceed 100 µm [START_REF] Nakano | Mitsubishi PWR Fuel Experience and Reliability, Water Reactor Fuel Performance Meeting[END_REF][START_REF] Bouineau | Analysis of the Waterside Corrosion Kinetics of Zircaloy-4 Fuel Cladding in French PWRs[END_REF]. In order to limit the duration of pre-oxidation, the maximum thickness of the waterside corrosion pre-oxide has been limited to 70 µm.

Two different pre-oxide configurations were studied: single-sided (SS, on external surface) or doublesided (DS, on external and internal surfaces, having similar or different thicknesses);  Pre-hydrided (up to 1000 wppm) and pre-oxidized Zircaloy (up to 70 µm of pre-oxide thickness, both single-sided and double-sided).

Various hydriding and pre-oxidation methods were used:

 Gaseous hydriding at 693 K in a vacuum furnace;  Pre-oxidation in dry oxygen at two different temperatures: 698 K (low temperature preoxidation) or 743 K (high temperature pre-oxidation). Under these conditions, the oxidation rates are respectively about 0.1 and 0.6 µm/day;  Hydriding + pre-oxidation: gaseous hydriding followed by pre-oxidation in dry oxygen at a high temperature;

 Simultaneous pre-oxidation and hydriding: Pre-oxidation in a mixture of O 2 -2% steam. This method allows hydrogen to be introduced into the Zircaloy during the pre-oxidation.

The materials are called Hx-Oy/z where x is the hydrogen content in ppm by mass and y and z are the thicknesses (in µm) of the outer and inner zirconia layers respectively. Thus, H400-O20/10 is a double-sided pre-oxidized Zircaloy with 20 µm (outer) and 10 µm (inner) thick pre-oxides and 400 wppm of hydrogen. x,y and z are rounded off to the nearest 10 (values lower than 150) or 100 (values higher than 150). The omission of Hx indicates a pre-oxidation in dry oxygen, with negligible hydrogen pick-up. However, few quantities of hydrogen are absorbed during pre-oxidation. The omission of z indicates a single-side pre-oxidized material. The different materials tested are presented in table 1.

The standard deviations presented in the table are those of the set of samples concerned. The standard deviations of the measurements on one sample are comprised between 1 and 4 µm (for preoxide thicknesses) or between 2 and 26 wppm (hydrogen contents), the higher values, the higher standard deviations.

The samples are machined prior to pre-oxidation, in order to avoid damage to the oxide layer during machining. Hydriding is performed either before machining or during pre-oxidation. The preoxidized samples have pre-oxide layers on all surfaces (internal, external and lateral). To obtain samples with pre-oxide on the outer surface only (single-sided pre-oxide), the internal pre-oxide layer is removed by sandblasting over the reduced section. For samples with double-sided pre-oxide having different thicknesses at inner and outer surfaces (called e int and e ext ), pre-oxidation was performed in two steps. The samples were first pre-oxidized up to obtain a first pre-oxide layer e ext -e int thick. Then, the internal pre-oxide layer was removed by sandblasting. Finally, the samples were pre-oxidized again up to obtain an additional pre-oxide layer e int thick.

Thermal test protocol

Thermal tests are carried out in a vertical resistance furnace (Model STF 15/610 from Carbolite). Segments of bare cladding and pre-oxidized cladding (10 and 50 µm of double-sided preoxide) are encapsulated in vacuum in quartz phials to prevent oxidation. The segments are subjected to 2, 10 and 20 minutes thermal plateaus at 1123 K, then quenched in water. The temperature is controlled by one Type S thermocouple placed next to the furnace resistors. A second Type K thermocouple is used to measure the temperature next to the phial. At the end of the test, the phial drops into a water container placed under the furnace and is immediately fractured to allow quenching of the samples. For measuring and regulating the temperature during the creep tests, a K type thermocouple is welded onto the inner surface of the sample at mid-height of the reduced section. The temperature expanded uncertainty was estimated to be 6.0 K at 933 K and 2.9 K at 1233 K. This uncertainty is greater than the thermal gradient undergone by the sample region where the strain is measured: the axial gradient estimated to be lower than 1 K at 973 K and 1.5 K at 1273 K over 4 mm.

Marks perpendicular to the tensile axis are affixed to the external surface of the sample before the creep test. These marks, made of ceramic powder, are used for measuring the deformation during the test. The sample strain is measured without contact using a laser extensometer (Fiedler Optoelektronic brand P-50 model). The laser beam of the extensometer is focalised and reflected by marks. The system detects, over time, the position along the scanner axis of the upper and lower limits of the marks. This device enables the strain to be measured with an uncertainty lower than 0.05%.

The sample is heated at a rate of 2 K/s up to the test temperature. Once the set temperature is reached, the load is applied at a rate of 30 N/s. With the aim of saving time and material, each sample is loaded with different constant loads during one test. At the end of the test, the heating is switched off and the sample is unloaded.

The sample temperature, the applied load and the position of the boundaries of the marks placed on the sample are measured, during the test, as a function of time. If p 1 and p 2 are the positions of edges 1 and 2 respectively, the true strain between edges 1 and 2 is calculated by the formula:

(

where p 10 and p 20 are the positions of the edges at the beginning of the test. The true strain uncertainty was estimated to be lower than 0.03% for the typical distance between marks (7 mm). The true stress ( ) is calculated assuming that the volume (V) of the deformed portion of the sample remains unchanged:

(2)

F is the applied force, L = is the distance between the two marks, S is the crosssectional area of the reduced section and subscript 0 stands for initial value. For the pre-oxidized samples, the cross-sectional areas were calculated without the pre-oxide layer. Thus the true stress is mean stress on the metal part of the sample, by removing the area occupied by the pre-oxide (considering that the pre-oxide does not support any load). The stress expanded uncertainty (k=2) was estimated to be between 1,4% at 10 MPa and 0.95 % at 50 MPa.

The strain rate ( or d/dt) is calculated by means of the slope of the regression line strain-time (-t). The associated uncertainty varies from 60% (for strains rates of about 110 -5 s -1

) to less than 6%

(strain rates close to 110 -3 s -1

). To compare the results obtained at different stresses, the strain rates For tests which lead to rupture of the samples, the distance between marks is measured before and after the test with a KEYENCE VHX-5000 microscope, in order to calculate the elongation after rupture, by the formula:

(

and are the distances between the marks delimitating the rupture zone (generally the central area of the sample) before and after the test. The uncertainty of is 0,4 %.

Sample characterization

Two segments are taken from each sample after pre-oxidation and/or pre-hydriding, as well as after testing: one for the hydrogen content measurement and the other for the metallographic examination. Hydrogen contents were determined by catharometry using a Brücker ONH mat 286 analyser. The first segment taken from the sample is cut into pieces of about 25 mg. Each piece is melted in a graphite crucible and the released hydrogen gas is then driven with the aid of a carrier gas to the measuring cell. The calibration of the device is periodically checked with certified standards. The second segment taken from the sample is embedded in resin for polishing and metallographic observation. The metallurgical observation of the samples is carried out using an optical microscope (LEICA DMR), a digital microscope (KEYENCE VHX-5000) and an electron microprobe (CAMECA SX 100) equipped with five WDS (Wavelength Dispersive Spectrometers).

The metallographic examination reveals the formation, during the creep test, of an O) phase layer on the surface of the bare samples and under the oxide layer of the pre-oxidized samples. For the bare samples, it is an indication that the very small residual quantities of oxygen in the furnace are sufficient to induce some Zircaloy oxidation during the thermal transients. The thicknesses of the zirconia layers and the (O) phase were measured on the inner face and outer face at 4 different positions.

Chemical mapping of the samples was performed with the electron microprobe to characterize the Fe distribution in the Zircaloy microstructure. This mapping was performed on 100x100 μm² surfaces with 0.25 µm steps and acquisition time of 300 ms/dot. Considering the high contrast in Fe partitioning between  and  phases, the Fe maps were transformed into binary images with the aim of measuring the  phase fraction for each sample after the tests. Nevertheless, the  phase fraction could only be estimated in samples with noticeable quantities of  phase and clearly delimited phaseboundaries, which corresponds to samples with high H-contents and thermal treatment samples subjected to 20 minutes plateaus. Oxygen concentration profiles in thermal test samples were measured by microprobe over the entire thickness of the cladding at 1 to 2 µm intervals.

Results

Considering the overall creep test results, the strain rates can be affected by several parameters: the oxygen and hydrogen contents, the quantity and location of the pre-oxide (single or double-sided), as well as by the thermal and mechanical history undergone by the material. All of these factors will be analysed in the following sections.

Effect of the pre-oxidation method

In order to determine the influence of the pre-oxidation process on the creep rates, Zy-4 samples were pre-oxidized by different methods. However, no differences have been observed in the microstructure or in the creep rates for similar materials pre-oxidized with the different methods.

Circumferential cracks and corrugation of the pre-oxide-metal interface are observed in all types of pre-oxide (see fig 2). Conversely, the appearance of the outer surface of the samples varies depending on the thickness of the pre-oxide layer (see figure 3). The 10 µm pre-oxide surface, which is smooth and greyish, becomes light brown on pre-oxides of thickness greater than about 20 µm and crazed on pre-oxides thicker than about 28 µm.

Effect of thermal and mechanical history

Variations in the creep rate were observed as a function of the duration of the thermal plateau and the loading history. With the aim of saving time and substance, a single sample was loaded with different constant loads during the same test. In most tests, the sample was subjected to increasing and decreasing loads to check whether or not there was any loading history effect. It was observed that, in tests conducted at 1123 K and under certain conditions (detailed below), the creep rate increases when the sample has previously been submitted to high stresses. This was observed in the short-duration tests carried out on pre-oxidized Zy-4 (with low hydrogen contents), when the sample is subjected to stresses  1 15 MPa after plateaus at  2 17MPa. Conversely, in the absence of preoxide, Zy-4 does not seem to be affected by an earlier loading.

Two tests were performed at 1023 K on pre-oxidized Zy-4 (e ZrO2 =12µm) with the purpose of checking whether the effect of the loading history can also be observed in the  domain. However, the values of the creep rates measured in the increasing and decreasing load plateaus were similar.

Consequently, the Zy-4 creep in the  domain does not seem to be sensitive to the sample loading history.

The tests performed on bare Zy-4 at 1123 K showed that, in low-stress tests (which are slow), the creep rate diminishes with the duration of the test, which is an effect that is not observed at lower temperatures, or with high stresses (quick tests). In order to check whether this is still valid in the presence of pre-oxide, a test was performed at 1123 K on Zy-4 with 10 µm of single-sided zirconia. In this test, the increasing and decreasing load plateaus were applied after 30 and 60 minutes of stabilization time at 1123 K. The results of the test are presented in figure 4 and show that the creep rate diminishes with the stabilization time. The values obtained after 1 hour at 1123 K are of the order of 100 times lower than those measured without hold time.

Various creep rates are measured during each ELFE test. Thus, except for creep rates measured during the first stress plateau of each test, some time elapses between the start of the thermal plateau and the beginning of each creep rate measurement. Figure 5 shows the changes in the normalized creep rate for bare and pre-oxidized (10 µm, single-sided) Zy-4 with the time elapsed at 1123 K before the start of the measurement. As illustrated in the graph on the right, the decrease in the creep rate is negligible for plateaus lasting less than 200 s. However, as from 1800 s, it is significant (see the graph on the left in figure 5). The creep rate values affected by the thermal and mechanical history were systematically disregarded.

Effect of pre-oxide

In order to study the effect of the pre-oxide on the creep rate, tests were performed on samples with two pre-oxide configurations: single-sided (SS, with pre-oxide only on the outer surface) and double-sided (DS, with pre-oxide on the inner and outer surfaces). Figure 6 shows the creep rates obtained with both configurations and compare them with those of bare Zy-4. The creep rate decreases in the presence of pre-oxide layers and depends on the applied stress and the location of the pre-oxide. At 10 MPa, the creep rates of both kinds of pre-oxide are close and disperse. They can be up to 10 times lower than those of bare Zy-4. At 20 MPa, the decrease in the strain rate is very significant for cladding with double-sided pre-oxide, while the creep rates measured on samples with single-sided pre-oxide layers are only slightly lower than those measured for bare cladding. Beyond 30

MPa, the creep rates of double-sided pre-oxidized Zy-4 approach those of single-sided pre-oxidized and bare Zy-4. The normalised strain rate values are shown in figure 7 as a function of the pre-oxide thickness for low hydrogen samples ([H]<130 wppm). As SS and DS pre-oxidized samples are not differentiated, the values are highly dispersed. As can be seen in the figure, the presence of a preoxide layer decreases the cladding creep rate, mainly for small pre-oxide thicknesses. As the thickness increases, this effect is mitigated and it becomes independent from the pre-oxide layer thickness for thicknesses higher than about 20 µm.

No crack was observed at the surface of non-pre-oxidized samples. However, when the samples are pre-oxidized, the pre-oxide layer is always cracked after having undergone a creep test (see figure 8). The cracks are always perpendicular to the direction of the mechanical load; however their shape depends on the initial thickness of the pre-oxide layer: they are straight on 10 µm pre-oxides, but crooked on very thick pre-oxides (higher than about 28 µm). No difference was observed in the cracks formed on the oxide of the pre-hydrided Zy-4 relative to those formed on the non-hydrided material.

Effect of Hydrogen

The effect of low concentrations of hydrogen (130 wppm) on the strain rate was studied for both non-pre-oxidized and single-sided pre-oxidized Zircaloy-4, while higher hydrogen contents (400-1000 wppm) were investigated for pre-oxidized samples either single-sided or double-sided. The creep rate measurements at 1123 K are illustrated in figure 9 and compared to those of non hydrided Zy-4.

The presence of around 130 wppm of hydrogen can slightly increase the Zy-4 creep rate with a stress of 10 MPa (triangles in upper graph). Conversely, small quantities of hydrogen do not seem to have any noticeable effect on the cladding strain at 20 MPa. No differences have been observed

between creep rates of both SS pre-oxidized + pre-hydrided materials (H90-O10 and H700-O60, white diamonds in upper graph). Thus, at 1123 K, hydrogen content and thickness of the external pre-oxide layer seem to have little effect on strain rate for this type of material.

The effect of hydrogen on DS pre-oxidized samples is presented in the lower graph of figure 9.

Despite the scattering of creep rates of DS-pre-oxidized + pre-hydrided samples (squares), it can be noticed that pre-hydriding increases creep rates for DS-pre-oxidized Zy-4, particularly at high stresses.

This apparent inconsistency between results for SS and DS pre-oxidized samples can be explained by the behaviour of hydrogen in the samples during the tests. The hydrogen contents of H700-O60 samples (single-sided pre-oxidized samples), that are around 680 wppm before the tests, halve down to around 300 wppm at the end of the tests. On the contrary, the hydrogen concentration remains rather constant in double-sided pre-oxidized samples: hydrogen contents measured after the tests are comprised between 400 wppm (in H400-O20/10 samples) and 770 wppm (in H1000-O60/10 samples). Figure 10 presents the microstructure of both materials after the creep tests. The figure

shows the absence of hydrides in a zone at the vicinity of the internal surface of H700-O60 sample (the surface that is not covered by a pre-oxide layer). On the contrary, hydrides are observed under all the interfaces metal-pre-oxide near the outer surface, even where oxide spalling has occurred. The normalised strain rate values as a function of hydrogen content at different pre-oxide thicknesses are presented on figure 11. In spite of the high dispersion of the values, it can be seen that hydrogen increases the Zy-4 creep rate on both bare and pre-oxidized samples (SS and DS). It has to be noticed that for high stresses and high hydrogen contents, the effect of the creep rate due to hydriding compensates and even overcomes the decrease of the strain rate caused by pre-oxidation. This way, H1000-O60/10 samples present faster deformations than bare Zy-4 at 30 MPa (see lower graph of figure 9).

The Fe-maps of the DS-pre-oxidized + pre-hydrided samples after the creep tests are presented in figure 12 while the  phase fractions measured are presented in table 2. It can be clearly observed that the quantity of  phase measured after the tests increases with the hydrogen content in the samples: it rises from 20 % to more than 30 % when the initial hydrogen content increases from 400 to 1000 wppm.

Only few isothermal tests were conducted until rupture. Nevertheless, these tests allow observing the effect of pre-oxidation and pre-hydriding on rupture strain (see figure 13). This figure

shows that rupture strain  R , decreases when the pre-oxide thickness increases:  R , is close to 200% in bare samples and decreases to about 50 % in samples with outer pre-oxide thicknesses about 60 µm. Hydrogen do not seem to affect the rupture strain under isothermal conditions. The small number of tests performed did not enable to observe if the location of pre-oxide (DS or SS) influences the rupture strain.

3.5 Evolution of the sample microstructure as a function of the high-temperature exposure time

In their initial state, before creep tests, the samples are made of Zircaloy in the  phase (with some hydrogen) either covered or not with a layer of zirconia. During high temperature exposure, oxygen and hydrogen exchanges may take place between the sample and the atmosphere of the furnace causing variations in the content of these two elements in the samples. In parallel, in the preoxidized samples, the oxygen contained in the oxide layer migrates toward the metal. Thus, a -O layer is formed at the interface between the oxide and the metal and at the same time Zircaloy becomes enriched with oxygen. At the same time, once the + transformation temperature has been exceeded, a certain fraction of  phase is transformed progressively into  phase, and this proportion varies as a function of temperature and hydrogen and oxygen contents.

The measurements of the oxidation layer thickness and phase fraction contents are carried out after the tests and may be influenced by the atmosphere of the furnace and the cooling time. To overcome these problems and in order to measure the changes in the microstructure and the distribution of the  and  phases over time at 1123 K, some specific thermal tests were performed:

segments of bare cladding and DS pre-oxidized + pre-hydrided cladding (O10/10 and H200-O50/50) were encapsulated in vacuum in quartz phials and subjected to thermal plateaus lasting 2, 10 and 20 minutes at 1123 K, and then quenched in water. No noticeable change in the zirconia thickness has been measured after these tests. Conversely, the thickness of the (O) layer increases with the duration of the plateau, regardless of the thickness of the initial pre-oxide (see figure 14). On bare samples, the mass uptake is null and no oxidation layer is formed, which shows that the samples are not subjected to external oxygen sources during these specific thermal tests.

Changes in the microstructure of bare and pre-oxidized + pre-hydrided samples observed after the thermal tests as a function of the oxide thickness and of the duration of the thermal plateau are presented in figure 15. To make the figure more understandable, the microstructures of the thin preoxide sample (O10/10) and those obtained after 10 min are omitted. At the beginning of the thermal plateau (2 min), small  phase grains (black dots on the BSE images, white dots on the maps) precipitate homogeneously in the  phase of the bare sample. On pre-oxidized samples, the  phase also forms in the  grain boundaries (see the cavities in the BSE images): locally in the material with a fine layer of pre-oxide (O10/10) and in a more generalised way in the H200-O50/50 material. As the duration of the thermal plateau increases, the  phase migrates from the inside of the  grain towards boundaries. After 20 minutes at 1123 K, the  phase is mainly located around  grains. However, differences in the grain boundary distribution can be found: in the bare and the O10/10 samples, the grain boundaries are partially surrounded with  phase; in H200-O50/50 samples, the  phase entirely surrounds the  grains. The  phase fractions for samples exposed during 20 min at 1123 K are presented in table 2. It can be observed that the  phase content increases with pre-oxidation + prehydriding treatment. As previously explained, the small size and dispersion of the  phase observed in the samples subjected to shorter thermal plateaus avoid the measurement of reliable  phase fractions.

Figure 16 shows an example of an oxygen profile measured over the cladding section in the radial direction. The dotted line indicates the exact location of the profile measurement. Three different layers can be identified inside the cladding using the oxygen profile: the pre-oxide layer at the outer surface of the cladding, with a high oxygen concentration; the metallic layer at the inner part of the cladding, with oxygen content close to the nominal content of the bare alloy (0.13 wt%); and between the two, the (O) layer with an oxygen concentration ranging between 7 wt% at the ZrO 2 /(O) boundary and the nominal content of the metal at the (O)/metal boundary. The oxygen profiles measured did not show any variation in the oxygen concentration of the metal with the duration of the thermal plateau, only the growth of the (O) layer was evidenced.

Discussion

On pre-oxide layers with thicknesses greater than around 30 µm, a network of regularly spaced radial cracks has been observed by various authors [START_REF] Guilbert | Effect of pre-oxide on Zircaloy-4 high temperature steam oxidation and post-quench mechanical properties[END_REF][START_REF] Busser | Modelling of corrosion induced stresses during Zircaloy-4 oxidation in air[END_REF]. This network of cracks gives the outer surface of the sample a snake-skin appearance, like that observed on the ELFE samples with 40 and 70 µm of pre-oxide (see figure 3). The cracks are caused by the creep of the metal subjected to tensile stresses generated by the oxide growth [START_REF] Busser | Modelling of corrosion induced stresses during Zircaloy-4 oxidation in air[END_REF]. In the absence of an external mechanical load, cracks have a length shorter than the pre-oxide thickness and do not reach the metal. Thus, in pre-oxides at least 30 µm thick, some of the cracks formed during the pre-oxidation develop and propagate, during the creep test, through the pre-existing network of cracks, with the crooked appearance seen in figure 8. In 10 µm thick pre-oxides, where no crack has formed during pre-oxidation, cracks developed during the creep test are perfectly straight. As illustrated in figure 9, at high stresses, creep rate of DS pre-oxidized samples is more sensitive to stress when they are pre-hydrided. For H1000-O60/10 samples (orange squares), the exponent n of the Norton law is close to 10, while for H400-O20/10 materials (white squares), n ranges between 5 (DS-pre-oxide, blue circles) and 10 (H1000-O60/10). Values of n higher than 8 have been observed in dispersed phase alloys [START_REF] Poirier | Plasticité à haute température des solides cristallins[END_REF]. At 1123 K, Zircaloy-4 is in the  +  domain. At this temperature, the amount of  phase strongly increases with the hydrogen content while it is minimally affected by the presence of up to 2 wt% of oxygen [START_REF] Guilbert-Banti | Influence of hydrogen on the oxygen solubility in Zircaloy-4[END_REF]. Under equilibrium conditions, the amount of  phase formed in Zy-4 at 1123 K increases with hydrogen content from  20 % in bare materials [START_REF] Forgeron | Experiment and modeling of advanced fuel rod cladding under LOCA conditions : alpha-beta phase transformation kinetics and EDGAR methodology[END_REF] to  90 % in samples with 1000 wppm of hydrogen [START_REF] Massih | Transformation kinetics of zirconium alloys under non-isothermal conditions[END_REF]. It has to be taken into account that, during the tests, the samples remains less than 200 s at 1123 K and that  phase does not form instantaneously but progressively. According to Koo [START_REF] Yoo | Effect of (+) heat treatment on the mechanical properties of Zircaloy-4[END_REF], the quantity of  phase formed at the end of the tests should be less than 50% of that expected at equilibrium. Consequently, the quantity and distribution of  phase evolves all along the test.

Hydrogen contents in single-sided oxidized samples (H700-O60) halve during the test while the hydrogen loss is significantly lower on pre-hydrided materials protected with double-sided pre-oxide.

This change in hydrogen concentration probably explains the small effect of hydrogen on the strain rate of single-sided pre-oxidized samples compared to double-sided ones (see figure 9). ELFE tests are performed under vacuum and hydrogen can be released from the sample through the internal surface of the cladding which is not covered by a pre-oxide layer. Under vacuum, hydrogen can leave the sample in various ways:

 Mainly through the non-pre-oxidized surfaces (as the inner surface in SS pre-oxidized samples);

 To a lesser extent through the pre-oxide. The pre-oxide layer has a protective effect and the hydrogen loss diminishes in DS pre-oxidized samples;

 Small losses are also expected through the oxide-less zones after cracks propagation. In ELFE samples, hydrogen concentration increases from the edges to the center of the reduced section (which concentrates the strain of the samples). Hydrogen migrates towards the zones submitted to tensile stresses [START_REF] Kammenzind | The long range migration of hydrogen through zircaloy in response to tensile and comprensive stress gradients[END_REF], as the most strained zones (under the cracks). Finally, hydrogen can leave the sample by the oxide-free surfaces on the crack tip.

Under steam, hydrogen should penetrate into the sample by similar mechanisms but in the opposite direction. However, hydrogen intake will be limited by the oxidation of oxide-free surfaces.

During the tests, variations in the creep rate were observed at constant temperature and stress, as a function of the duration of the thermal plateau and the loading history. There is no significant variation in the creep rate of non hydrided materials with the duration of the plateau at 1123 K for periods shorter than 200 s. However, it is significant beyond 1800 s. The decrease in the creep rate over time at high temperatures for bare Zircaloy had already been observed by Garde, who explains this behaviour by the changes in the microstructure with the duration of the thermal plateau [START_REF] Garde | Micrograin superplasticity in Zircaloy at 850[END_REF]. At 1123 K, the equilibrium structure of Zircaloy-4 consists in a large volume fraction of  phase and a small volume fraction of  phase, more ductile than for the  phase. Initially,  phase forms both as a continuous thin film at the  grain boundaries and within the  grains. As the hold time at high temperature increases,  phase tends to coalesce at the triple points of the  grains. When the quantity of  phase increases, continuity of the  phase film diminishes, as well as plasticity. In ELFE tests, the effect of the thermal history was observed for both bare and pre-oxidized samples.

Nevertheless, oxygen and hydrogen can affect the changes in the microstructure with the duration of the thermal plateau, modifying both the phase fraction and the phase distribution. The thermal tests were performed to observe changes in the microstructure as a function of the thermal plateau duration at 1123 K. At the start of the thermal plateau, small  phase grains are precipitated uniformly in the  phase of the bare sample, while on the pre-hydrided samples small quantities of  phase form also at the  grain boundaries. As the duration of the thermal plateau increases, the quantity of  phase increases at the same time as the  phase, previously formed inside the  grains, migrates towards the grain boundaries. The presence of pre-oxide, on the contrary, does not seem to affect the distribution of the phases in the metal since no increase in the oxygen concentration was measured on the samples exposed to thermal tests, only the growth of the (O) layer is observed.

For most of the tests, the pre-oxidized samples underwent increasing and decreasing loading plateaus. In some cases, the creep rate measured for decreasing plateaus can be higher than that measured on increasing plateaus for a given stress. This phenomenon is observed when the following conditions are met:



For stresses lower than around 15 MPa. Most of the tests were performed on samples with 10 µm of SS pre-oxide, for which the boundary between the two creep regimes (diffusional creep and dislocation induced creep) is located between 10 and 20 MPa. We can thus assume that the loading history affects only the creep mechanism dominating at low stresses (grain boundary sliding + diffusion)



For pre-oxidized samples (whether they have been pre-hydrided or not). Thus, oxygen plays a role in the creep rate increase caused by the loading history: either by the modification of the phase distribution (which can facilitate grain boundary sliding), or by the increase in the diffusion rate.



After stresses greater than around 17 MPa, which are the typical stresses for the dislocation induced creep regime. Thus, the introduction of defects produced by this type of creep can promote diffusional creep. Another explanation could be that when the pre-strain is carried out at lower stresses, the material deforms slowly and the thermal aging can cancel out the effect of the loading history.



In the + domain (at 1123 K) while the Zy-4 creep in the  domain (1023 K) does not seem to be sensitive to the sample loading history. Under these conditions, the effect is linked to the presence of the  phase.



In short tests; when hold time at 1123 K is less than 200 s. Conversely, when pre-strain is applied after a thermal plateau of at least 1800 s, this effect is not observed and the creep rate decreases instead of increasing. Thus, the decrease in the creep rate due to hold time at high temperature dominates over the increase due to pre-strain. However, Garde observed that the effect of thermal history disappears on bare Zy-4 when pre-strain is applied before the thermal plateau (0.3%

pre-strain at 973 K). He explains this fact by the introduction of defects and dislocations, which allow the precipitation of  under fine edges resulting in the limitation of the  phase coalescence and ductility maintenance after 120 minutes.

In summary, the increase in the creep rate for pre-strained cladding observed in the two-phase domain in the presence of oxygen could be explained by the introduction of defects and dislocations that promote diffusion and/or grain boundary sliding.

Conclusions

In order to simulate the in service corrosion of the cladding, samples have been pre-oxidized and pre-hydrided. Various pre-oxidation and pre-hydriding methods have been tested, without however revealing first order differences in the microstructure of the materials or in the creep rates of the samples obtained with different methods. Thus, the most simple and quickest method was chosen for preparing the samples: pre-oxidation at 743 K in a mixture of O 2 and steam.

The phenomenology governing the creep behaviour of Zy-4 in the two-phase domain is complex and depends on many parameters that affect the phase distribution: temperature, duration of the heat plateau, thermal and loading history, applied stress, presence of oxygen and how it is supplied (preoxide, test atmosphere, pre-oxide location), hydrogen content... However, the decrease in the creep rate observed in some cases appears to be related to the coalescence of the  phase at the periphery of the  grains.

The presence of a pre-oxide layer decreases the cladding creep rate. This effect is already important for small pre-oxide thicknesses, and increases for oxide thicknesses up to about 20 µm where it stabilizes. Creep rate decrease depends on the applied stress and on the location of the preoxide: it is stronger in double-sided than in single-sided pre-oxidized samples, in particular at stress levels close to 20 MPa.

The short duration of the tests limits the effect of pre-oxide and hydrogen on the cladding microstructure. This way, the diffusion of oxygen from the pre-oxide to the metal results only in the growth of a small (O) layer at the metal/pre-oxide interface while no additional supply of oxygen is detected in the metal. In parallel, the  phase that develops and redistributes during the creep test,

2. 3

 3 Tensile test protocolCreep tests are carried out in the Figaro facility. They are performed in primary vacuum (2) in a double-walled and water-cooled sealed furnace. The furnace resistors and the insulating basket are made of molybdenum. The load is applied with an electromechanical actuator with a nominal capacity of 2.5 kN. The stroke of the cylinder is 500 mm and the displacement resolution is 0.1 µm with speeds ranging between 0.001 and 500 mm/min. The applied load measurement is performed with a load cell calibrated up to 1kN in class 0.5. The expanded uncertainty (k = 2) of the load varies between 0.7 N (at 10 N) and 0.8 N (at 175 N).

(

  were normalised in relation to the stress ( in MPa) by A( )/ n , where A (MPa n s) is a constant and n depends on the stress and on the state of the material. n is equal to 1.3 for bare and single-sided pre-oxidized Zy-4 samples up to stresses close to 10 MPa and up to about 26 MPa in double-sided pre-oxidized samples (diffusion creep area in fig. 6). Otherwise, n is equal to 5 (dislocation creep area in the same figure). These stress exponents are proposed by Kaddour [2] for bare Zircaloy in the  domain. The values of constant A depend on the stress exponent and have been fixed to approach the normalized creep rates obtained with the two values of n: A=1 when n=1.3 and A=16000 when n=5.

Figure 17

 17 Figure 17 shows various cracks observed after a creep test on a sample with 40 µm of preoxide arranged from the finest opening to the largest, giving an insight into the process of formation and growth of cracks. Cracks formed on the outer surface of the pre-oxide during pre-oxidation, advance up to the (O) layer where they stop. Then, the cracks open out and the edges of the cracks move parallel to the cladding surface. At the same time, the (O) layer at the end of the crack is thinning. Finally, once the (O) layer has practically disappeared at the front of the crack, straincontinues to a level leading to rupture in zones stripped of oxide and (O) layer (see the picture at the bottom of figure17). Consequently, the strain is concentrated in the oxide-free zones. A similar mechanism has been previously observed by different authors on bare samples under steam at temperatures higher than 1273 K[START_REF] Chung | High temperature deformation and rupture behavior of internally pressurized Zircaloy-4 cladding in vaccum and steam environments[END_REF][START_REF] Sagat | Deformation and failure of Zircaloy fuel sheaths under LOCA conditions[END_REF]. Nevertheless, it is necessary to keep in mind the fact that ELFE tests are performed in vacuum and that in the presence of an oxidizing atmosphere (steam during a LOCA) the naked crack tip should oxidize, which is likely to change the strain process (particularly during low strain rate creep).

  remains in quantities far below the values expected at equilibrium for Zy-4 with high hydrogen contents.Two flow regimes can be distinguished in Zy-4 at 1123 K: diffusional creep at low stresses and dislocation induced creep at high stresses. Creep laws of pre-oxidized Zy-4 (both single-sided and double-sided) at 1123 K seem to follow the stress exponents proposed by Kaddour[START_REF] Kaddour | Experimental determination of creep properties of Zirconium alloys together with phase transformation[END_REF] for bare Zircaloy: n=1.3 at low stresses and n=5 at high stresses. The boundary between the two creep regimes, lower than 20 MPa in SS pre-oxidized cladding and bare Zircaloy-4, increases to more than

Acknowledgements

The author is very grateful to S. Charbaut, D. Drouan, L. Druel, P. Lacote and G. Villevieille for their collaboration in tests conduct and post-test examinations and to J. Desquines, P. March and T. Haste for the careful review of this paper.

Funding: This work was supported by the French National Agency of Research (program of

Investments for the Future, reference n° ANR-11-RSNR-0017). Particular acknowledgments are also given to EDF and US-NRC, for their financial support.

20 MPa in DS pre-oxidized samples. This fact can explain the large gap observed between the strain rates of DS and SS pre-oxidized cladding at 20 MPa.

Hydrogen increases Zy-4 creep rate on both bare and pre-oxidized samples at 1123 K. In prehydrided + DS pre-oxidized samples, the exponent of the Norton law at high stresses and high hydrogen contents increases up to values close to 10. The presence of high amounts of hydrogen, which induces the formation of significant quantities of  phase at high temperature, modifies the creep mechanism. This way, hydriding can compensate and overcome the decrease of the strain rate caused by pre-oxidation. Thus, H1000-O60/10 samples present faster deformations than bare Zy-4 at high stresses.

ELFE tests were performed under vacuum. In SS pre-oxidized samples, hydrogen can easily desorb from the sample through the internal surface of the cladding which is not covered by a preoxide layer. As a result, the apparent non-effect of hydrogen content on the strain rate of SS preoxidized Zy-4 is probably an artefact due to the loss of hydrogen during the creep test.

Variations in the creep rate were also observed at constant temperature and stress, as a function of the duration of the heat plateau and loading history. The creep rate shows no significant variations with the duration of the plateau at 1123 K for periods shorter than 200 s. However, these variations are significant beyond 1800 s. Consequently, it is necessary to pay attention to thermal and mechanical scenario of the tests when using the data of the bibliography.

During pre-oxidation, a network of regularly spaced radial cracks is formed on pre-oxide layers thicker than about 30 µm. During the creep tests, these cracks, which are initially located in the outer part of the oxide layer only, spread out throughout the oxide up to the (O) layer, where they stop.

Then, the (O) layer at the crack tip becomes deformed. Finally, strain develops in the oxide-free zones. This mechanism is similar to that previously observed by different authors on bare cladding under steam at temperatures higher than 1273 K.


D Bresch

M Gisclon

I Lacroix-Violet

A Vasseur

ON THE EXPONENTIAL DECAY FOR COMPRESSIBLE NAVIER-STOKES-KORTEWEG EQUATIONS WITH A DRAG TERM

Keywords: AMS Classification. 35B40, 35B45, 35K35, 76Y05 Navier-Stokes-Korteweg model, long-time behaviour, exponential decay, quantum models, drag term

In this paper, we consider global weak solutions to compressible Navier-Stokes-Korteweg equations with density dependent viscosities, in a periodic domain Ω = T 3 , with a linear drag term with respect to the velocity. The main result concerns the exponential decay to equilibrium of such solutions using log-sobolev type inequalities. In order to show such a result, the starting point is a global weak-entropy solutions definition, introduced in D. Bresch, A. Vasseur and C. Yu [12]. Assuming extra assumptions on the shear viscosity when the density is close to vacuum and when the density tends to infinity, we conclude the exponential decay to equilibrium. Note that our result covers the quantum Navier-Stokes system with a drag term.

Introduction

The goal of this paper is to study the long-time behaviour of solutions of Navier-Stokes-Korteweg models with a drag term of type r 2 ρ u (for r 2 > 0 a constant) in a periodic domain Ω = T 3 using a κ relative entropy generalizing the one by [START_REF] Bresch | On Navier-Stokes-Korteweg and Euler-Korteweg Systems: application to quantum fluids models[END_REF]. Without loss of generality we fix |Ω| = 1. More precisely the system under consideration reads

∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) -2 div(µ(ρ)D(u)) -∇(λ(ρ)div u) -2 ε ρ ∇ K(ρ)∆ ρ 0 K(s) ds + r 2 ρ u = 0
where ε > 0 is the Planck constant and D(u) = ∇u + t ∇u /2 stands for the symmetric part of the velocity gradient. The pressure state law reads p(ρ) = a ρ γ where a > 0 and γ ≥ 1 are constants. The shear and bulk viscosities are assumed to satisfy the BD relation [START_REF] Alazard | Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics[END_REF] λ(ρ) = 2(µ ′ (ρ)ρ -µ(ρ)).

1

The shear viscosity and the capillarity coefficient are linked through the relation K(ρ) = (µ ′ (ρ)) 2 /ρ. Readers interested by Korteweg type systems are referred to the following articles and books: [START_REF] Korteweg | Sur la forme que prennent les équations du mouvement si l'on tient compte de forces capillaires causées par les variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité[END_REF][START_REF] Van Der Waals | Thermodynamische theorie der capillariteit in de onderstelling van continue dichteisverandering[END_REF][START_REF] Cahn | Free energy of a non uniform system I, interfacial free energy[END_REF][START_REF] Dunn | On the thermomechanics of interstitial working[END_REF][START_REF] Rowlinson | Molecular theory of capillarity[END_REF][START_REF] Nigmatulin | Methods of mechanics of a continuous medium for the description of multiphase mixtures[END_REF][START_REF] Heida | On compressible Korteweg fluid-like materials[END_REF][START_REF] Antonelli | On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics[END_REF][START_REF] Antonelli | The Quantum Hydrodynamics System in Two Space Dimensions[END_REF][START_REF] Antonelli | Global existence of weak solutions to the Navier-Stokes-Korteweg equations[END_REF] and references cited therein. Note that the Korteweg type models contain the class of quantum models which corresponds to K(ρ) = 1/ρ which gives µ(ρ) = ρ and, with the BD relation [START_REF] Alazard | Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics[END_REF], λ(ρ) = 0.

Quantum fluid models have attracted a lot of attention in the last decades due to the variety of applications. Indeed, such models can be used to describe superfluids [START_REF] Loffredo | On the creation of quantum vortex lines in rotating He II[END_REF], quantum semiconductors [START_REF] Ferry | Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling[END_REF], weakly interacting Bose gases [START_REF] Grant | Pressure and stress tensor expressions in the fluid mechanical formulation of the Bose condensate equations[END_REF] and quantum trajectories of Bohmian mechanics [START_REF] Wyatt | Quantum Dynamics with Trajectories[END_REF]. Recently some dissipative quantum fluid models have been derived. In particular, under some assumptions and using a Chapman-Enskog expansion in Wigner equation, the authors have obtained in [START_REF] Brull | Derivation of viscous correction terms for the isothermal quantum Euler model[END_REF] the so-called quantum Navier-Stokes model. Roughly speaking, it corresponds to the classical Navier-Stokes equations with a quantum correction term. The main difficulties of such models lie in the highly nonlinear structure of the third order quantum term and the proof of positivity (or non-negativity) of the particle density.

In [START_REF] Vasseur | Global weak solutions to compressible quantum Navier-Stokes equations with damping[END_REF], A. Vasseur and C. Yu have proven the global-in-time existence of weak solutions of the quantum Navier-Stokes equations with two drag terms of type r 0 u and r 1 ρ|u| 2 u. Their result is still valuable in the case r 1 = 0 and their proof is based on a Faedo-Galerkin approximation (following the ideas of [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF]) and the BD entropy (see [START_REF] Bresch | Existence of global weak solutions for 2D viscous shallow water equations and convergence to the quasi-geostrophic model[END_REF][START_REF] Bresch | On some compressible fluid models: Korteweg, lubrication and shallow water systems[END_REF]). Note that the result is also still valuable with the add of a third drag force term of type r 2 ρ u because the term does not perturb the uniform estimates and stable in limit process. In the same time, M. Gisclon and I. Lacroix-Violet [START_REF] Gisclon | About the barotropic compressible quantum Navier-Stokes equations[END_REF] also obtained the existence of global-in-time weak solutions for the same model using a cold pressure instead of drag terms with the same method of Faedo-Galerkin approximation. In [START_REF] Vasseur | Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations[END_REF], the authors use the result obtained in [START_REF] Vasseur | Global weak solutions to compressible quantum Navier-Stokes equations with damping[END_REF] and pass to the limits ε, r 0 , r 1 tend to zero to prove the existence of globalin-time weak solutions to degenerate compressible Navier-Stokes Equations with µ(ρ) = µ ρ and λ(ρ) = 0. Such existence of global existence of weak solutions has also been obtained at the same time and independently in [START_REF] Li | Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities[END_REF]. Note that to prove the result in [START_REF] Vasseur | Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations[END_REF], they need uniform (with respect to r 0 , r 1 ) estimates to pass to the limit r 0 , r 1 tend to 0. To this end they have to firstly pass to the limit ε tends to 0.

Recently in [START_REF] Lacroix-Violet | Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit[END_REF], global existence of weak solutions for the quantum Navier-Stokes Equations has been proved without drag terms. The method is based on the construction of weak solutions that are renormalized in the velocity variable for the system with drag terms r 0 u and r 1 ρ|u| 2 u and uses the stability part of the result to pass to the limit r 0 , r 1 tend to zero to obtain weak renormalized solutions of the system without drag terms. All weak renormalized solution being a weak solution, the existence of global weak solutions for the quantum Navier-Stokes system without drag terms is shown.

Adding an extra drag term r 2 ρ u to the system does not change the result: it does not perturb the estimates and is stable in limit passage. Note that the construction being uniform with respect to the Planck constant ε, the authors also perform the semi-classical limit ε → 0 to the associated compressible Navier-Stokes equations. It is important to remark that a global weak solution of the quantum Navier-Stokes equations in the sense of [START_REF] Lacroix-Violet | Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit[END_REF] is also weak solution of the corresponding augmented system as introduced in [START_REF] Bresch | Two-velocity hydrodynamics in fluid mechanics : Part II Existence of global κ-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities[END_REF]. Note also the paper [START_REF] Antonelli | Global existence of finite energy weak solutions of the quantum Navier-Stokes equations[END_REF] concerning the global existence for the quantum Navier-Stokes system using the approximate procedure introduced by J. Li and Z.P. Xin (see [START_REF] Li | Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities[END_REF]).

We also mention the recent paper [START_REF] Antonelli | Global existence of weak solutions to the Navier-Stokes-Korteweg equations[END_REF] where they consider the global existence of weak solutions to Navier-Stokes-Korteweg model extending the method of truncation, regularization and renormalization introduced by A. Vasseur and C. Yu in [START_REF] Vasseur | Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations[END_REF] and generalized by I. Lacroix-Violet and A. Vasseur in [START_REF] Lacroix-Violet | Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit[END_REF].

Finally, let us cite [START_REF] Bresch | Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities[END_REF] where the existence of global weak solutions for some Navier-Stokes-Korteweg system in Ω = T 3 is obtained. The proof is performed around four main ideas. The first one is to introduce the equation satisfied by µ(ρ) in the definition of weak solutions assuming µ(ρ 0 ) in L 1 (Ω) to relax the constraints on the pressure exponent. The second one consists in extending the two-velocities framework, introduced by D. Bresch, B. Desjardins and E. Zatorska in [START_REF] Bresch | Two-velocity hydrodynamics in fluid mechanics : Part II Existence of global κ-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities[END_REF], with capillarity and more general drag terms using a generalization of the quantum Böhm identity developed in [START_REF] Bresch | A generalization of the quantum Bohm identity: Hyperbolic CFL condition for Euler-Korteweg equations[END_REF]. The third one is the proof of a generalized high-order log-Sobolev dissipation inequality similar to the one used in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF] and established by A. Jüngel and D. Matthes in [START_REF] Jüngel | The Derrida-Lebowitz-Speer-Spohn equations: Existence, uniqueness and decay rates of the solutions[END_REF]. Finally the last one is the extend of the renormalized framework, introduced by I. Lacroix-Violet and A. Vasseur in [START_REF] Lacroix-Violet | Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit[END_REF] for the quantum case, to viscosities satisfying the BD relation. Note that in this paper we use their definition of weak solution that we recall, for reader convenience, in the next section (see Definition 2).

Denoting by r the mean value of ρ on Ω i.e. r = Ω ρ the goal of the paper is to show that a global weak solution (ρ, u) in the sense of [START_REF] Bresch | Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities[END_REF] is exponentially decaying in time to (r, 0). To obtain such a result we use the augmented formulation of the system firstly introduced in [START_REF] Bresch | Two-velocity hydrodynamics in fluid mechanics : Part II Existence of global κ-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities[END_REF] and we use a relative entropy version of the κ entropy similar to the one introduced in [START_REF] Bresch | On Navier-Stokes-Korteweg and Euler-Korteweg Systems: application to quantum fluids models[END_REF].

The paper is organized as follows. In Section 2 we give the definition of weak solution used in the paper, all the assumptions and we state the main result of exponential decay (see Theorem 5). Section 3 is devoted to the proof of this exponential time decay. Finally we give in Appendix the proof of a technical lemma (Lemma 13) used in Section 3. Note that in all the paper the notation C is used for constants independent of time t but which can be different from line to line.

Assumptions and main result

In this section, we first recall the definition of weak solution used in [START_REF] Bresch | Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities[END_REF]. We complete this definition with some assumptions on the shear and bulk viscosities and finally we state the main result of the paper.

Definition of weak solutions.

Let us first recall the definition of weak solution used in [START_REF] Bresch | Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities[END_REF]. The authors consider shear and bulk viscosities such that:

• denoting by

R + = [0, ∞) and R * + = (0, ∞). (2) µ ∈ C 0 (R + ; R + ) ∩ C 2 (R * + , R), • there exists two positive numbers α 1 , α 2 satifying 2/3 < α 1 < α 2 < 4 such that for any ρ > 0 (3) 0 < 1 α 2 ρ µ ′ (ρ) ≤ µ(ρ) ≤ 1 α 1 ρ µ ′ (ρ),
• there exists a constant C > 0 such that

(4) ρ µ ′′ (ρ) µ ′ (ρ) ≤ C < +∞.
Note that, thanks to (3) and ( 4), there exists ν > 0 such that

(5) λ(ρ) + 2 3 µ(ρ) ≥ νµ(ρ)
and µ(0) = λ(0) = 0. The method, used in [START_REF] Bresch | Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities[END_REF], to prove global existence of weak solution to the Navier-Stokes-Korteweg is linked to the two-hydrodynamic system introduced in [START_REF] Bresch | Two-velocity hydrodynamics in fluid mechanics : Part II Existence of global κ-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities[END_REF], the extension of the Böhm identity proved in [START_REF] Bresch | A generalization of the quantum Bohm identity: Hyperbolic CFL condition for Euler-Korteweg equations[END_REF], the generalization of the dissipation inequality used in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF] for the quantum Navier-Stokes system and established in [START_REF] Jüngel | The Derrida-Lebowitz-Speer-Spohn equations: Existence, uniqueness and decay rates of the solutions[END_REF] and the renormalized solutions introduced in [START_REF] Lacroix-Violet | Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit[END_REF].

Here, the addition of a drag term r 2 ρ u does not change the result because the κ-entropy remains uniform and the term ρ u is already in the time derivative for compactness.

Remark 1. Note the recent paper [START_REF] Alazard | Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics[END_REF] where the authors have proved a Sobolev inequality which could help to enlarge the range of viscosities namely

µ(ρ) = ρ α , λ(ρ) = 2(α -1)ρ α with 2 3 < α < +∞.
Let us now recall the definition of the global weak solutions for the Navier-Stokes-Korteweg system similar to what has been developped in [START_REF] Bresch | Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities[END_REF] Definition 2. We say that (ρ, u) is a global weak solution to the compressible Navier-Stokes-Korteweg equations as constructed in [START_REF] Bresch | Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities[END_REF] if

• The density satisfies ρ ∈ C 0 ([0, +∞); L γ weak (Ω)) with ρ ≥ 0 in (0, +∞)× Ω, ρ| t=0 = ρ 0 a.e. in Ω with the viscosity µ(ρ) ∈ C 0 ([0, +∞); L 3/2 weak (Ω)). • The momentum satisfies ρ u ∈ C([0, +∞); L 2γ/(γ+1) weak (Ω)) with ρu| t=0 = m 0 .
• The following energy estimate holds a.e t ∈ [0, +∞)

Ω ρ |w| 2 2 + [(1 -κ)κ + ε] |v| 2 2 + H(ρ) + 2 r 2 κ h(ρ) +ν t 0 Ω |T µ | 2 + r 2 t 0 Ω ρ|u| 2 (6) +ε C α 1 , ν t 0 Ω |∇ 2 Z(ρ)| 2 + |∇Z 1 (ρ)| 4 + |T ε | 2 +2κ t 0 Ω µ ′ (ρ)p ′ (ρ) ρ |∇ρ| 2 ≤ Ω ρ 0 |w 0 | 2 2 + [(1 -κ)κ + ε] |v 0 | 2 2 +H(ρ 0 ) + 2 r 2 κ h(ρ 0 )
for some C α 1 , ν > 0 depending on α 1 and ν and some κ ∈ (0, 1) with

w = u + κ v and v = 2∇s(ρ) where s ′ (ρ) = µ ′ (ρ) ρ (7) H(ρ) = ρ ρ 1 p(s) s 2 and h(ρ) = ρ ρ 1 µ(s) s 2 (8) Z(ρ) = ρ 1 µ(s)µ ′ (s) s and Z 1 (ρ) = ρ 1 µ ′ (s) (µ(s)) 1/4 s 1/2 (9)
and where T µ is defined through

µ(ρ)T µ = ∇ √ ρu µ(ρ) √ ρ - √ ρu ⊗ √ ρ∇s(ρ) (10) λ(ρ) 2µ(ρ) Tr( µ(ρ)T µ )Id (11) = div λ(ρ) µ(ρ) √ ρu µ(ρ) √ ρ - √ ρu • √ ρ∇s(ρ) ρµ ′′ (ρ) µ ′ (ρ) Id.
The same definitions and compatibility condition are satisfied for T ε , replacing u by v = 2∇s(ρ) respectively in [START_REF] Bresch | Two-velocity hydrodynamics in fluid mechanics : Part II Existence of global κ-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities[END_REF] and [START_REF] Bresch | On Navier-Stokes-Korteweg and Euler-Korteweg Systems: application to quantum fluids models[END_REF]. • The following extra estimates hold

µ(ρ) L ∞ (0,+∞;W 1,1 (Ω)) + µ(ρ)u L ∞ (0,∞;L 3/2 (Ω))∩L 2 (0,+∞;W 1,1 (Ω)) < +∞ ∂ t µ(ρ) L ∞ (0,+∞;W -1,1 (Ω)) + Z(ρ) L 1+ (0,+∞;L 1 (Ω)) < +∞.
• The density ρ satisfies the mass equation in D ′ ((0, +∞) × Ω): [START_REF] Bresch | Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities[END_REF] ∂ t ρ + div(ρu) = 0.

• The velocity satisfies the momentum in D ′ ((0, +∞) × Ω) :

∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) + r 2 ρ u -2div( µ(ρ)S µ + λ(ρ) 2µ(ρ) Tr( µ(ρ)S µ )Id) (13) -2εdiv( µ(ρ)S ε + λ(ρ) 2µ(ρ) Tr( µ(ρ)S ε )Id) = 0 with S µ = (T µ + T t µ ) 2 , S ε = (T ε + T t ε ) 2 .
• The viscosity satisfies in D ′ ((0, +∞) × Ω):

∂ t µ(ρ) + div(µ(ρ)u) + λ(ρ) 2µ(ρ) Tr( µ(ρ)T µ ) = 0. (14)
• The global weak solution is obtained through regularized solutions

(ρ ζ , w ζ , v ζ ) (letting ζ tend to zero) satisfying dE ζ dt (t) + εC ν,α 1 Ω |∇ 2 Z(ρ ζ )| 2 +κ Ω µ ′ (ρ ζ )H ′′ (ρ ζ )|∇ρ ζ | 2 + Ω r 2 ρ ζ |u ζ | 2 ≤ 0 (15)
where

(16) E ζ (t) = Ω ρ ζ (|w ζ | 2 + (κ(1 -κ) + ε)|v ζ | 2 ) + H(ρ ζ |r) + 2 r 2 κ h(ρ ζ |r)
where H(ρ ζ |r) and h(ρ ζ |r) are defined by:

H(ρ ζ |r) = H(ρ ζ ) -H(r) -H ′ (r)(ρ ζ -r), h(ρ ζ |r) = h(ρ ζ ) -h(r) -h ′ (r)(ρ ζ -r).
Remark 3. The presence of the drag term r 2 ρ u with r 2 > 0 implies (using the κ-entropy estimate) the control of sup t∈(0,+∞)

Ω h(ρ)(t, x) if Ω h(ρ 0 ) initially. This reads sup t∈(0,+∞) Ω ρ ρ 1 µ(s) s 2 ≤ C < +∞.
Using Hypothesis (3), this implies that

µ(ρ) ∈ L ∞ (0, +∞; L 1 (Ω)).
This is an important information compared to previous papers such as [START_REF] Li | Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities[END_REF] or [START_REF] Bresch | Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities[END_REF]. Indeed, to control µ(ρ) in L ∞ (0, +∞; L 1 (Ω)) without using drag terms information, we would need as in [START_REF] Li | Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities[END_REF] to consider a pressure law as

p(ρ) = a ρ γ with γ ≥ max(2α 2 -1, 1)
but this condition is restrictive. Note that ρ γ ∈ L ∞ (0, +∞;

L 1 (Ω)) implies µ(ρ) ∈ L ∞ (0, +∞; L 1 (Ω)) because √ ρ ∈ L ∞ (0, +∞; L 2 (Ω)
) deduced from the mass conservation. Note that the hypothesis γ ≥ 2α 2 -1 is not present in [START_REF] Bresch | Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities[END_REF]. It suffices to assume µ(ρ 0 ) ∈ L 1 (Ω) and use the equation satisfied by µ(ρ) to get the result µ(ρ) ∈ L ∞ (0, T ; L 1 (Ω)) for all T < +∞ but this result is not valid for T = +∞.

2.2. Main result: exponential decay. In order to be able to get exponential decay, we add the following assumptions on the shear viscosity:

• Behaviour of µ(ρ) for ρ small enough:

(17) lim ρ→0 ρµ ′ (ρ) µ(ρ) = α, with 2 3 < α ≤ 4, assuming -If α < 1 or if there exists ρ 0 > 0 such that µ(ρ) ≥ Cρ for ρ < ρ 0 that the pressure exponent satisfies γ ≥ 1, -If α ≥ 1 that the pressure exponent satisfies γ < α.
• Behaviour of µ(ρ) for ρ big enough: [START_REF] Grant | Pressure and stress tensor expressions in the fluid mechanical formulation of the Bose condensate equations[END_REF] lim

ρ→∞ ρµ ′ (ρ) µ(ρ) = β, with 1 < β < 4
or letting M > 0 a given constant

(19) µ(ρ) = ρ for any ρ ≥ M.
Remark 4. The quantum case satisfies (17)-( 19) without any restriction on the pressure law namely p(ρ) = aρ γ with γ ≥ 1 is elligible.

As mention before, denoting by r the mean value of ρ on Ω, the goal is to show that a global weak solution (ρ, u) in the sense of Definition 2 with p(ρ) = a ρ γ with γ ≥ 1 and the shear and bulk viscosities satisfying (1)-( 2) and ( 17)-( 19) is exponentially decaying in time to (r, 0). More precisely, the goal is to prove the following result: Theorem 5. Let (ρ, u) be a global weak solution of the compressible Navier-Stokes-Korteweg System in the sense of Definition 2 with p(ρ) = a ρ γ with γ ≥ 1 and the shear and bulk viscosities satisfying (1)-( 2) and ( 17)- [START_REF] Heida | On compressible Korteweg fluid-like materials[END_REF]. We define the relative entropy

E(ρ, u, v|r, 0, 0) = E(t) = Ω [ρ(|w| 2 +(κ(1-κ)+ε)|v| 2 )+H(ρ|r)+2 κ r 2 h(ρ|r)],
where v, w are given by [START_REF] Bresch | Quelques modèles diffusifs capillaires de type Korteweg[END_REF] and H(ρ|r) and h(ρ|r) are defined by

H(ρ|r) = H(ρ) -H(r) -H ′ (r)(ρ -r), h(ρ|r) = h(ρ) -h(r) -h ′ (r)(ρ -r),
with h, H given by [START_REF] Bresch | Existence of global weak solutions for 2D viscous shallow water equations and convergence to the quasi-geostrophic model[END_REF].

Then, for some constant C > 0, E(t) ≤ E(0) exp(-Ct).

Proof of Theorem 5

The goal of this section is to prove Theorem 5. As global weak solutions are constructed using regular enough solutions, the formal calculations we will do may be justified using these approximate solutions and their initial conditions. We will then write first the exponential decay for E ζ and then pass to the limit with respect to ζ to obtain Theorem 5 assuming strong convergence initially. The goal of this section is then to prove the following Theorem. Theorem 6. Let the hypothesis on the viscosities and on the pressure law be satisfied namely p(s) = a s γ with γ ≥ 1 and the shear and bulk viscosities satisfying (2)-( 1) and ( 17)- [START_REF] Heida | On compressible Korteweg fluid-like materials[END_REF]. Then there exists a constant C > 0, uniform in ζ, such that for any t > 0, the quantity E ζ (t) defined in ( 16) verifies

E ζ (t) ≤ E ζ (0) exp(-Ct).
Note that thanks to (15) of Definition 2, the regular approximation satisfies the following inequality

dE ζ dt (t) ≤ -ε C ν,α 1 Ω |∇ 2 Z(ρ ζ )| 2 -κ Ω µ ′ (ρ ζ )H ′′ (ρ ζ )|∇ρ ζ | 2 - Ω r 2 ρ ζ |u ζ | 2 (20)
It remains then to prove the following proposition to obtain theorem 6 for the regular approximation.

Proposition 7. There exists a constant C > 0 such that for every t > 0

-ε C ν,α 1 Ω |∇ 2 Z(ρ ζ )| 2 -κ Ω µ ′ (ρ ζ )H ′′ (ρ ζ )|∇ρ ζ | 2 - Ω r 2 ρ ζ |u ζ | 2 ≤ -CE ζ (t).
The proof of this proposition is divided in three steps corresponding to the three following subsections. To simplify the presentation, we omit the subscript ζ but all the calculations have to be thinking for the approximate solutions.

First step of proof of proposition 7. First of all, let us prove this following lemma.

Lemma 8. There exists a constant C > 0 such that for every t > 0

- Ω |∇ 2 Z(ρ)| 2 - Ω µ ′ (ρ)H ′′ (ρ)|∇ρ| 2 - Ω ρ |u| 2 (21) ≤ -C Ω ρ 2 |w| 2 + (κ(1 -κ) + ε)|v| 2
Proof. From Poincaré's inequality, we find

Ω |∇ 2 Z(ρ)| 2 ≥ C Ω |∇Z(ρ)| 2 ≥ C Ω µ(ρ) ρ ρ|v| 2 .
Moreover,

µ ′ (ρ)H ′′ (ρ)|∇ρ| 2 = 1 4 ρ µ ′ (ρ) H ′′ (ρ)ρ|v| 2 ,
hence there exists C > 0 such that

- Ω |∇ 2 Z(ρ)| 2 - Ω µ ′ (ρ)H ′′ (ρ)|∇ρ| 2 ≤ -C Ω µ(ρ) ρ + ρH ′′ (ρ) µ ′ (ρ) ρ|v| 2 .
If now we prove that there exists a constant C > 0 such that for any ρ ≥ 0

(22) µ(ρ) ρ + ρH ′′ (ρ) µ ′ (ρ) ≥ C, then (23)
-

Ω |∇ 2 Z(ρ)| 2 - Ω µ ′ (ρ)H ′′ (ρ)|∇ρ| 2 ≤ -C Ω ρ|v| 2 .
Proof of [START_REF] Korteweg | Sur la forme que prennent les équations du mouvement si l'on tient compte de forces capillaires causées par les variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité[END_REF]. To do so let us consider large and small density values, the inequality being true for intermediate values. Note that for large density values or small density values with α < 1, we only use the term µ(ρ)/ρ to obtain the inequality [START_REF] Korteweg | Sur la forme que prennent les équations du mouvement si l'on tient compte de forces capillaires causées par les variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité[END_REF]. But for small density values with α ≥ 1, we have to use the term ρH ′′ (ρ)/µ ′ (ρ) to obtain [START_REF] Korteweg | Sur la forme que prennent les équations du mouvement si l'on tient compte de forces capillaires causées par les variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité[END_REF]. First part (large density values). From ( 18) or ( 19), for M > 0 big enough and ρ > M , we have ρ µ ′ (ρ) ≥ µ(ρ). Integrating the ODE gives µ(ρ) ≥ Cρ for ρ > M .

Second part (small density values). We have to consider two cases to control the small values of ρ. If α < 1 in [START_REF] Gisclon | About the barotropic compressible quantum Navier-Stokes equations[END_REF], then, for ρ 0 > 0 small enough and ρ < ρ 0 , we have ρ µ ′ (ρ) ≥ µ(ρ). Integrating the ODE gives µ(ρ) ≥ Cρ for all values of ρ ≥ 0. Now, we consider the case α ≥ 1 in [START_REF] Gisclon | About the barotropic compressible quantum Navier-Stokes equations[END_REF]. For any η > 0 there exists ρ η > 0 such that for any ρ < ρ η we have

ρµ ′ (ρ) ≥ (α -η)µ(ρ).
Integrating backward in ρ the ODE gives that for ρ < ρ η :

µ(ρ) ≤ Cρ α-η .
Using [START_REF] Gisclon | About the barotropic compressible quantum Navier-Stokes equations[END_REF] again gives that for ρ < ρ η :

ρH ′′ (ρ) µ ′ (ρ) ≥ Cρ γ-α+η .
If γ < α, taking η > 0 small enough we have:

ρH ′′ (ρ) µ ′ (ρ) ≥ C, for ρ < ρ η .
This gives estimate [START_REF] Korteweg | Sur la forme que prennent les équations du mouvement si l'on tient compte de forces capillaires causées par les variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité[END_REF] and then [START_REF] Lacroix-Violet | Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit[END_REF].

Note now that we have

|w| 2 = |u + κv| 2 ≤ 2(|u| 2 + κ 2 |v| 2 ). So -|u| 2 ≤ κ 2 |v| 2 - |w| 2 2 .
Hence, using [START_REF] Lacroix-Violet | Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit[END_REF],

- Ω |∇ 2 Z(ρ)| 2 - Ω µ ′ (ρ)H ′′ (ρ)|∇ρ| 2 - Ω ρ|u| 2 ≤ -C ( Ω ρ|v| 2 + Ω ρ|u| 2 ) ≤ -C Ω ρ |w| 2 + (κ(1 -κ) + ε)|v| 2 ,
which concludes the proof of Lemma 8.

Remark 9. Note that we can obtain the result of Lemma 8 without the capillarity term and then the exponential time decay but we then need conditions on the exponent γ of the pressure. In other terms, the use of the capillarity term allow to obtain the exponential decay for all exponent γ of the pressure law.

Second step of proof of proposition 7.

The goal now is to prove the following lemma.

Lemma 10. There exists a constant C > 0 such that

(24) - Ω |∇ 2 Z(ρ)| 2 - Ω H ′′ (ρ)µ ′ (ρ)|∇ρ| 2 ≤ -C Ω h(ρ|r) + Ω H(ρ|r) .
To prove this lemma, we use two results: lemmas 11 and 12. Note that lemma 11 will be used to prove lemma 12 which gives lemma 10.

Lemma 11. For any constant η > 0 small enough, there exist 0 < ρ 0 < r < M and C > 0 such that

Ω |∇ 2 Z| 2 ≥ C Ω |ρ -r| 2 1 {ρ 0 ≤ρ≤M } (25) +C Ω 1 {ρ<ρ 0 } + C Ω ρ 3(β-η)-2 1 {ρ≥M } . Moreover, if β ≥ 1 then (26) Ω H ′′ (ρ)µ ′ (ρ)|∇ρ| 2 ≥ C Ω H(ρ)1 {ρ≥M } .
Proof. The proof is divided into three parts: The first deals with the large values of ρ, the second one with the small values of ρ, and finally the third with the intermediate values of ρ.

First part (large density values).

In this part we consider the large values of ρ. We first focus on the proof of (25) and then, using a generalization of the classical log-Sobolev inequality we prove [START_REF] Nigmatulin | Methods of mechanics of a continuous medium for the description of multiphase mixtures[END_REF].

Proof of (25): Since the function ρ → Z(ρ) is an increasing function, we have |{Z(ρ) ≥ Z(2r)}| = |{ρ ≥ 2r}|. By the Tchebychev's inequality, we get ( 27)

|{Z(ρ) ≥ Z(2r)}| = |{ρ ≥ 2r}| ≤ 1 2r Ω ρ = 1 2 .
Since |Ω| = 1, using [START_REF] Rowlinson | Molecular theory of capillarity[END_REF] we have

|{Z(ρ) < Z(2r)}| ≥ 1 2 .
Using Lemma 13 given in Appendix on (Z(ρ) -Z(2r)) + we get ( 28)

Ω (Z(ρ) -Z(2r)) 2 + ≤ C Ω |∇Z| 2 .
Thanks to assumptions [START_REF] Grant | Pressure and stress tensor expressions in the fluid mechanical formulation of the Bose condensate equations[END_REF] or [START_REF] Heida | On compressible Korteweg fluid-like materials[END_REF], the function ρ → Z(ρ) converges to infinity when ρ goes to infinity. So there exists M > 0 such that for any

ρ ≥ M (29) Z(ρ) 2 ≤ 2(Z(ρ) -Z(2r)) 2 + .
Moreover for any η > 0, there exists a possibly bigger M > 0 such that we have for any ρ > M :

Z(ρ) 2 ≥ Cρ 3(β-η)-2 .
Therefore with ( 28) and ( 29), we have:

Ω ρ 3(β-η)-2 1 {ρ≥M } ≤ C Ω |Z(ρ)| 2 1 {ρ≥M } ≤ C Ω |∇Z| 2 .
Proof of (26): If γ > 1, working with the function J(ρ) instead of Z(ρ) where J(ρ) is defined such that

J ′ (ρ) = H ′′ (ρ)µ ′ (ρ),
with similar arguments, we obtain [START_REF] Nigmatulin | Methods of mechanics of a continuous medium for the description of multiphase mixtures[END_REF]. If γ = 1, then H(ρ) = ρ ln ρ, and we need a generalization of the classical log-Sobolev inequality. Let us denote f = √ ρ. As in [START_REF] Rowlinson | Molecular theory of capillarity[END_REF], we have

(30) |{f ≤ √ 2r}| ≥ |{ρ ≤ 2r}| ≥ 1 2 .
So, from Sobolev's injections and Lemma 13, we obtain

1 + (f - √ 2r) 2 + L 3 ≤ 1 + (f - √ 2r) 2 + L 3 ≤ 1 + (f - √ 2r) + 2 L 6 ≤ 1 + C ∇(f - √ 2r) + 2 L 2 + (f - √ 2r) + 2 L 2 ≤ 1 + C ∇(f - √ 2r) + 2 L 2 .
Denoting by

m = Ω 1 + (f - √ 2r) 2 + ds = 1 + (f - √ 2r) 2 + m ,
(note that s is a positive measure of mass 1), the logarithmic function being increasing and concave, using Jensen's inequality, we have

C ∇(f - √ 2r) + 2 L 2 ≥ ln(1 + C ∇(f - √ 2r) + 2 L 2 ) ≥ 1 3 ln Ω 1 + (f - √ 2r) 2 + 3 dx = 1 3 ln Ω m 1 + (f - √ 2r) 2 + 2 ds ≥ Ω 1 3 ln m 1 + (f - √ 2r) 2 + 2 ds (31) = 2 3 Ω ln √ m(1 + (f - √ 2r) 2 + ) 1 + (f - √ 2r) 2 + m dx ≥ 2 3(1 + r) Ω ln (1 + (f - √ 2r) 2 + ) (1 + (f - √ 2r) 2 + ) dx,
where we have used that 1 ≤ m ≤ 1 + r.

Defining M = sup(8r, 16), for f = √ ρ ≥ √ M , we have both 1 + (f - √ 2r) 2 + ≥ ρ 4 and ln(1 + (f - √ 2r) 2 + ) ≥ 1 2 ln ρ.
Therefore, [START_REF] Wyatt | Quantum Dynamics with Trajectories[END_REF] gives the following control

C Ω H(ρ)1 {ρ>M } = C Ω ρ ln ρ1 {ρ>M } ≤ C ∇(f - √ 2r) + 2 L 2 .
From Properties [START_REF] Grant | Pressure and stress tensor expressions in the fluid mechanical formulation of the Bose condensate equations[END_REF] and ( 19)

sup ρ≥2r 1/ρ H ′′ (ρ)µ ′ (ρ) ≤ C,
and so

C Ω H(ρ)1 {ρ>M } ≤ C ∇(f - √ 2r) + 2 L 2 ≤ C ∇ √ ρ1 {ρ≥2r} 2 L 2 ≤ C Ω H ′′ (ρ)µ ′ (ρ)|∇ρ| 2
which gives [START_REF] Nigmatulin | Methods of mechanics of a continuous medium for the description of multiphase mixtures[END_REF].

Second part (small density values). In this part we consider the small values of ρ. From ( 17), there exists σ > 0 such that

Z ′ (ρ) ≥ Cρ σ-1 ,
and so (32)

Ω |∇ρ σ | 2 ≤ C Ω |∇Z| 2 .
Let us fix ρ 0 = r 4 and δ = inf

r 16 , 1 2 . Note that, if C Ω |∇ 2 Z| 2 ≥ δ, Ω 1 {ρ≤ρ 0 } ≤ 1 ≤ C δ Ω |∇ 2 Z| 2 ,
which immediately gives [START_REF] Loffredo | On the creation of quantum vortex lines in rotating He II[END_REF] for ρ < ρ 0 . Then, without loss of generality, we can assume that

C Ω |∇ 2 Z| 2 ≤ δ,
such that the result of the first part of the proof implies

Ω 1 {ρ≥M } (ρ -r) + ≤ M -3(β-η)+3 Ω ρ 3(β-η)-2 1 {ρ≥M } ≤ C Ω |∇ 2 Z| 2 ≤ δ.
We claim that

(33) |{ρ > r/2}| ≥ δ M + 1 .
To prove this claim let us show that if it is not the case we obtain a contradiction. Assuming

|{ρ > r/2}| < δ M + 1 , we have M |{ρ ≥ r}| ≤ M |{ρ > r/2}| ≤ δ, and so Ω (ρ -r) + ≤ Ω 1 {ρ≥M } (ρ -r) + + M |{ρ ≥ r}| ≤ 2δ. Since r = Ω ρ, we have Ω (ρ -r) + = Ω (r -ρ) + . Using Tchebychev's inequality, we get |{ρ ≤ r/2}| = |{(r -ρ) + ≥ r/2}| ≤ 2 r Ω (r -ρ) + ≤ 4δ r ≤ 1 4 . Therefore |{ρ > r/2}| ≥ 1 -1/4 = 3 4 , which is a contradiction, since δ M + 1 ≤ δ ≤ 1 2
. Therefore the claim (33) is proved.

Let us now rewrite the claim (33) (since σ > 0)

|{(r/2) σ -ρ σ < 0}| ≥ δ M + 1 .
Since, |{((r/2) σ -ρ σ ) + = 0}| = |{(r/2) σ -ρ σ < 0}| = 0, using Lemma 13 with the function ((r/2) σ -ρ σ ) + , (32) and Poincaré's inequality, we get

Ω ((r/2) σ -ρ σ ) 2 + ≤ C Ω |∇((r/2) σ -ρ σ ) + | 2 ≤ C Ω |∇ρ σ | 2 ≤ C Ω |∇Z| 2 ≤ C Ω |∇ 2 Z| 2 .
Using the Tchebychev's inequality and the previous one we get

|{ρ ≤ r/4}| ≤ C (2 σ -1) 2 4 r 2σ Ω |∇ 2 Z| 2 .
This gives [START_REF] Loffredo | On the creation of quantum vortex lines in rotating He II[END_REF] for ρ < ρ 0 .

Third part (Intermediate density values).

In this part we deal with the intermediate values of ρ i.e. ρ 0 ≤ ρ ≤ M . Let us introduce ρ T (t, x) = sup(ρ 0 , inf(M, ρ(t, x))).

Note that |Z

′ (ρ)| is bounded by below on [ρ 0 , M ], hence |∇ρ T | = |∇ρ|1 {ρ 0 ≤ρ≤M } = ∇Z(ρ) Z ′ (ρ) 1 {ρ 0 ≤ρ≤M } ≤ C|∇Z|.
Therefore, using Poincaré's inequality, since |Ω| = 1 and ρ 0 = r/4,

Ω |ρ -r| 2 1 {ρ 0 ≤ρ≤M } ≤ Ω |ρ T -r| 2 ≤ 2 Ω ρ T - Ω ρ T 2 + Ω Ω ρ T -r 2 ≤ C Ω |∇Z| 2 + Ω ρ T - Ω ρ 2 = C   Ω |∇Z| 2 + {ρ>M } (M -ρ) + {ρ<ρ 0 } (ρ 0 -ρ) 2   ≤ C Ω |∇Z| 2 + r {ρ>M } ρ + r 4 {ρ<ρ 0 } ρ 0 ≤ C Ω |∇Z| 2 + r M -3(β-η)+3 Ω 1 {ρ>M } ρ 3(β-η)-2 + ρ 0 r 4 Ω 1 {ρ<ρ 0 } .
We get the desired result using the first and second parts of the proof.

As mention before to complete the proof of lemma 10, we need also this following lemma:

Lemma 12. Assume that G is a function of ρ verifying G is C 2 on [ρ 0 , M ],
and that there exists a constant C > 0 such that

G(ρ) ≤ C sup(ρ 3(β-η)-2 , H(ρ)), for ρ ≥ M, (34) G(ρ) ≤ C, for ρ ≤ ρ 0 . (35)
Then, there exists an other constant C > 0 such that

Ω G(ρ|r) ≤ C Ω |∇ 2 Z| 2 + C Ω H ′′ (ρ)µ ′ (ρ)|∇ρ| 2 ,
where G(ρ|r) is given by

G(ρ|r) = G(ρ) -G(r) -G ′ (r)(ρ -r). Proof. We have Ω G(ρ|r) = Ω G(ρ|r)1 {ρ 0 ≤ρ≤M } + Ω G(ρ|r)1 {ρ<ρ 0 } + Ω G(ρ|r)1 {ρ>M } ≤ C sup [ρ 0 ,M ] |G ′′ | Ω |ρ -r| 2 1 {ρ 0 ≤ρ≤M } +C Ω 1 {ρ<ρ 0 } + C Ω sup(ρ 3(β-η)-2 , H(ρ))1 {ρ>M } .
Lemma 11 finishes the proof.

To finish the proof of lemma 10, the idea is now to apply Lemma 12 with G equals h and H. Note that (35) is valid for both quantities in all configurations. We want to show that (34) is also true for both quantities in all configurations. To this end, we consider two cases.

Case 1 γ > 1. Since β > 1, for η small enough, (36) 1 < β + η < 3(β -η) -2.
Solving the ODE (18) for ρ > r, we find that there exists a constant C > 0 such that for any ρ > r: µ(ρ) ≤ Cρ β+η , and using again [START_REF] Grant | Pressure and stress tensor expressions in the fluid mechanical formulation of the Bose condensate equations[END_REF] to get a bound by above on µ ′ from the bound on µ, we get µ ′ (ρ) ≤ Cρ β+η-1 .

Since h ′′ (ρ) = µ ′ (ρ) ρ , using the bound above on µ ′ and integrating in ρ twice, we find that there exists a constant C > 0 such that for every ρ > M , we have h(ρ) ≤ Cρ β+η , which, thanks to (36), leads to

h(ρ) ≤ C sup(ρ 3(β-η)-2 , ρ γ ).
This proves (34) for h.

Since γ > 1, then obviously, for ρ big enough, we have

H(ρ) = ρ γ γ -1 ≤ Cρ γ . Case 2 γ = 1. Then we have h(ρ) = H(ρ) = ρ ln ρ. Since 3(β -η) -2 > 1,
for ρ big enough h(ρ) = H(ρ) = ρ ln ρ ≤ ρ 3(β-η)-2 and H satisfies (34).

Then in all cases, we can apply Lemma 12 to find

dE ζ dt (t) ≤ -ε C ν,α 1 Ω |∇ 2 Z(ρ ζ )| 2 -κ Ω µ ′ (ρ ζ )H ′′ (ρ ζ )|∇ρ ζ | 2 - Ω r 2 ρ ζ |u ζ | 2 .
Applying Lemmas 8 and 10, we obtain: which directly gives the result of Theorem 6. Then, since E ζ is lower semicontinuous and since we assume the strong convergence initially, passing to the limit ζ tends to zero leads to E(t) ≤ E(0) exp (-Ct) , which proves Theorem 5.

dE ζ dt (

Appendix

Let us prove the following classical technical Lemma.

Ω 2 κ 3 . 3 .

 233 r 2 h(ρ|r)+ Ω H(ρ|r) ≤ C Ω |∇ 2 Z(ρ)| 2 + Ω H ′′ (ρ)µ ′ (ρ)|∇ρ| 2 ,which ends the proof of lemma 10. Ends of proof of proposition 7 and theorem 5. We can now state and prove the proposition which is the goal of this section since it allows us to obtain the exponential time decay given in Theorem 5 letting ζ tends to zero if we obtain the exponential decay for E ζ with constant independent on ζ. Here we adopt again the indice ζ for reader's convenience.Let us recall inequality (15) of Definition 2:

Lemma 13 . 1 0Ω Ω 1 0Ω

 1311 For any function f ≥ 0 such that |{f = 0}| ≥ δ > 0, there exists a constant C depending on δ and Ω, such thatΩ f 2 dx ≤ C Ω |∇f | 2 dx.Proof. From the fundamental theorem of calculus, for any x, y ∈ Ωf (x) -f (y) = ∇f (tx + (1 -t)y) • (x -y) dt. Therefore Ω |f (x)| 2 dx ≤ 1 δ {f =0} Ω |f (x)| 2 dx dy = 1 δ {f =0} Ω |f (x) -f (y)| 2 dx dy ≤ 1 δ Ω Ω |f (x) -f (y)| 2 dxdy ≤ 1 δ |∇f (tx + (1 -t)y)| 2 |x -y| 2 dx dy dt (tx + (1 -t)y)| 2 dy dt dx t)Ω |∇f (tx + z)| 2 dz (1 -t) 3 dt dx |∇f (z)| 2 dz.

  |w ζ | 2 + (κ(1 -κ) + ε)|v ζ | 2 + H(ρ ζ |r) + 2κr 2 h(ρ ζ |r) ,with C a constant which only depends on ε, ν, α 1 , α 2 , κ and r 2 , but which is independent of ζ. Now using the definition of E ζ given in[START_REF] Ferry | Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling[END_REF] we finally obtain:

	(37)	dE ζ dt	(t) ≤ -CE ζ (t).
	Then using the Gronwall Lemma, we obtain
	(38)	E ζ (t) ≤ E ζ (0) exp (-Ct) ,

t) ≤ -C Ω ρ ζ

Acknowledgements

The third author acknowledges support from the team INRIA/RAPSODI and the Labex CEMPI (ANR-11-LABX-0007-01). The first author acknowledges the project TELLUS INSU-INSMI "Approche croisée pour fluides visco-élasto-plastiques: vers une meilleure compréhension des zones solides/fluides". The first and the second authors acknowledge the ANR Project FRAISE managed by C. Ruyer-Quil. The fourth author was partially supported by the NSF, DMS 1614918.