L Meziou

Aymeric Histace
email: aymeric.histace@u-cergy.fr

F Precioso

O Romain

X Dray

B Granado

B J Matuszewski

Computer-Assisted Segmentation of Videocapsule Images Using Alpha-Divergence-Based Active Contour In The Framework of Intestinal Pathologies Detection

Keywords: Videocapsule, Active Contour, Alpha-Divergence, Gastrointestinal Pathologies

Visualization of the entire length of the gastrointestinal tract through natural orifices is a challenge for endoscopists. Videoendoscopy is currently the "gold standard" technique for diagnosis of different pathologies of the intestinal tract. Wireless Capsule Endoscopy (WCE) has been developed in the 1990's as an alternative to videoendoscopy to allow direct examination of the gastrointestinal tract without any need for sedation. Nevertheless, the systematic post-examination by the specialist of the 50,000 (for the small bowel) to 150,000 images (for the colon) of a complete acquisition using WCE remains time-consuming and challenging due to the poor quality of WCE images. In this article, a semiautomatic segmentation for analysis of WCE images is proposed. Based on active contour segmentation, the proposed method introduces alpha-divergences, a flexible statistical similarity measure that gives a real flexibility to different types of gastrointestinal pathologies. Results of segmentation using the proposed approach are shown on different types of real-case examinations, from (multi-) polyp(s) segmentation, to radiation enteritis delineation.

Introduction

Visualization of the entire length of gastrointestinal tract through natural orifices is a challenge for endoscopists. Moreover, radiologic techniques are relatively insensitive for diminutive, flat, vascular, or inflammatory lesions of the small bowel. Currently, videoendoscopy is the "gold standard" technique for the diagnosis of various gastrointestinal diseases. Noticeably, using a videoendoscope, gastroenterologists can perform and record a complete examination of the colon, aiming to detect, to characterize, to sample, and to remove suspicious tissular structures like adenomas (pre-neoplastic lesions from which colorectal cancer can emerge). In most cases, videocolonoscopy is performed under general anaesthesia.

Alternatively, mini-invasive techniques such as computed-tomography-based colonography and Wireless Capsule Endoscopy (WCE) have been proposed when sedation is contraindicated or refused. In this article, we focus our attention on WCE devices.

Since 1994 [Moglia 09], WCEs have been developed to allow direct visualization of the small bowel (a hardly accessible part of the gastrointestinal tract) without any need for sedation.

Small bowel WCE are best indicated for obscure gastrointestinal bleeding, for evaluation of Crohn's disease and coeliac disease, and when small bowel polyposis (Peutz-Jeghers syndrome) or tumors are suspected. The Pillcam © video capsule designed by Given Imaging Company is the most popular of them (see Fig. 1 for illustration).

PillCam © Endocapsule © Mirocam © Omom ©

Fig. 1 Illustration of the existing WCE devices

This autonomous device images the gastrointestinal tract during at least 8 hours. However, acquired images are of low resolution and imaging conditions are sometimes very poor, as the movements, the speed of the WCE, and the illumination, are not entirely controlled (see Fig. 2 for illustration). Moreover a complete acquisition is made of more than 50,000 images for the small bowel (and up to 150,000 for colon WCE), and the analysis of such an amount of data is therefore demanding and tedious for physicians. Image segmentation and analysis would be of great interest to decrease the physician's workload. However, segmentation of WCE images is a challenging problem with two main issues:

1. A substantial reducing of the initial 50,000+ images to analyse is necessary since a lot of them do not show meaningful information from a clinical standpoint, and, consequently do not need to be analysed.

2. The development of an (semi-) automatic segmentation approach to robustly and precisely delineate detected structure for further characterization (size, shape, texture for instance) is a real need from a Computer-assisted diagnostic perspective.

Regarding the first issue, in [Silva,14], we proposed an automatic boosting-based detection approach for abnormal Region of Interest (ROI) identification, with a particular focus on polyp detection. A database of 500 polyps and 1200 non-polyps was used to learn by boosting the optimal classifier in terms of false positive (FP) and true positive (TP) rates. In this study, texture parameters were used (computed from the co-occurrence matrix) and a global detection rate of 94% was reached with a FP rate of 4%. This work, achieved on classic videoendoscopic images, is currently being extended to other type of structure than polyps and to WCE images, for which the image database is far more difficult to enrich because of limited access to raw images data. Fig. 3 displays three examples of obtained results. In each case, non-bold squares show candidate regions before the classifying step, whereas the bold square shows the region eventually identified as a polyp using the boosted classifier. It is important to highlight here that a resolution of only 5 bits (32 grey-levels) was used to compute the co-occurrence matrices from which the Haralick's parameters are then computed.

It shows that even for low-resolution WCE images (with respect to classic resolution of videocolonoscopy images), the method remains adapted. To illustrate this, Fig. 4 shows a detection/recognition obtained on one of the WCE images of Fig. 2 without any learning step dedicated to WCE images 1 .

Fig. 4 Detection and recognition of a polyp area using the approach proposed in [Silva 14] on a WCE images. It can be noticed, that the only 5-bits resolution of the co-occurrence matrices used for the learning step of the algorithm, avoid to be too dependant of the "quality" of the image.

Starting from these results, a fine segmentation is required to address the second issue mentioned above. In this article, a method for WCE-image-segmentation based on statistical active contours is proposed and a prospective study on various types of gastrointestinal structures is presented.

The remainder of this article is divided into 4 sections: Section 2 introduced the data considered and their specificity in terms of illustrated pathology. Section 3 is focused on the theoretical aspect of the segmentation method proposed; Mainly a recall on histogram-based active contour is first presented and the details on the developed method follow. Segmentation 1 Currently, most of the WCE companies use a patented codec for the encryption of the videos which makes quite hard the gathering of a sufficient amount of images to create a database with a significant amount of WCE images.

experiments on synthetic images and WCE images are shown in Section 4; the objective of this section is, firstly, to evaluate the method on controlled set-ups (noise and texture) and secondly to show comparative performance on the particular targeted clinical application.

Finally Conclusion and Discussion are proposed in Section 5.

Materials

Images used for the study purpose were acquired using the PillCam © device and were taken from the freely available database "Gastro Vidéo Web" (http://dr.dm.gastro.free.fr/).

Resolution of images is 320x320 pixels.

To show the flexibility of the proposed active contour approach, different segmentation tasks were considered including some demanding ones. Nine conditions selected for the study (Fig. 9): gastric polyps (a, b), colon sigmoid polyp (c), lymphangiectasia (d), small-bowel metastasis (e), radiation enteritis (f), angiodysplasia (g), lipoma (h), and hypertrophied anal papilla (i).

In each case, the segmentation task consisted in delineating, with precision and reproducibility, the structure of interest in order to estimate clinical parameters.

Method

Main principle and notations

Among the existing proposed segmentation methods in medical imaging, active contour models have attracted extensive research in the past two decades. Originally proposed in 1988

[ Kass 88], the basic idea of the active contour is to iteratively evolve an initial curve towards the boundaries of target organ driven by the combination of internal forces determined by the geometry of the evolving curve and the external forces induced from the image (see Fig. 5 for illustration). Image segmentation method using active contour is usually based on minimizing a functional, which is so defined that for curves close to the target boundaries it has small values. The minimization process is iteratively performed until convergence is reached i.e. the value of the computed energy does not meaningfully change from an iteration to the following one. In the following, we will refer to the active contour and the related evolution equation using the notation of Eq. (1) and Eq. (2). (2)

∂Γ(p,τ ) ∂t = v(p, t) Γ(p,0) = Γ0(p) # $ % % & % % [ ] [ ] ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = Γ → ℜ → × ) , ( ) , ( ) x ) , ( ) , (

Alpha-divergence

In To overcome the limitations of KL-divergence, a particular statistical distance is proposed:

the alpha (α)-divergence. This similarity measure is a generalization of the KL-divergence, which becomes an asymptotic case (α tends to one) of the alpha-divergence. Main advantage of proposed approach is the flexibility of the alpha-divergence which metric can be adapted to very difficult PDF separation tasks [Hero06].

In the following, D α . ||.

( ) will denote the alpha-divergence defined by:

D α p in || p out ( ) = ϕ α p in , p out , λ ( ) dλ ℜ m ∫ (3)
with ϕ α the corresponding alpha-metric such as: In Eq (3), λ is the grey-level resolution of the image (i.e. in our case λ ∈ 0, 255 [ ] ).

ϕ α p in , p out , λ ( ) = α p in + 1-α ( ) p out -p in α p out 1-α α 1-α ( ) , α ∈ ℜ \ 0,1 { } p out log p out p in $ % & ' ( ) + p in -p out , α = 0 p in log p in p out $ % & ' ( ) + p out -p in , α = 1 * + , , , , - , , , , (4)
In Eq. (3) and Eq. ( 4), it can be noticed that for particular values of α , some usual distances of the literature can be related to alpha-divergences. For instance,

D 2 p in || p out ( ) = 1 2 D χ 2 p in || p out ( ) D 0.5 p in || p out ( ) = 2D Hellinger p in || p out ( ) D KL p in || p out ( ) = lim α→1 D α p in || p out ( ) " # $ $ $ % $ $ $ (5)
where D KL . ||.

( ) is the KL--divergence.

As stated before, this makes alpha-divergence a generic divergence estimation, with multiple tuning possibilities via alpha parameter and, as a consequence, a very flexible measure in the context of non-Gaussian PDF characterizing inner and outer regions delimited by the active curve all along the segmentation process.

PDF modeling

Practically speaking, the use of alpha-divergence in the framework of histogram-based active contour requires the estimation of the inner and outer PDF. More precisely, the fact that Eq.

(4) will be optimized using an Euler-derivative approach imposes a continuous and derivable formulation of the PDF. Classically, two kinds of approaches are considered: a parametric estimation or a non-parametric one. In the first case, the PDF can be analytically expressed as part of a general family like the exponential one. Nevertheless, it can be seen as a limitation since in practical medical case, PDF can be of very complex shapes (see Fig. 5 for instance) that cannot be easily expressed in a particular statistical family.

The second solution proposes to estimate the PDF using a Gaussian-mixture model based on

Parzen-window estimator [Parzen 62]. The PDF is then given by:

p i I(x, y), Ω i ( ) = 1 Ω i g σ I(x, y) -λ ( ) dλ Ω I ∫ (6) with i ∈ in, out [ ] ,
Ω i representing all the pixels of region i and g σ a Gaussian kernel function of standard deviation σ.

Such approach does not limit the type of PDF to be estimated and compel with the requirement of class C 1 . Such properties definitely fit with our context of applications and in the following Parzen Window technique is used.

For fast implementation under Matlab © 2012, the AOS scheme formerly introduced in [Weickert 88] was used.

Parameters setting

As it can be noticed in Eq. ( 9), several parameters, namely β , ξ , σ and α influence the optimisation process.

Considering and , we arbitrarily set ξ to one and then empirically tuned β to have a sufficient "amount of energy" for the segmentation process to start and evolve. Main interest of the proposed method is to automatically balance the confidence in the divergence term of Eq. ( 9) using alpha-parameter and not being too dependent of β value.

σ value is empirically tuned to 0.1. A lower value does not bring much to the PDF estimation and a higher value results in a too important smoothing effect on the shape of the PDF (loss of information).

Finally, about alpha-parameter that can be seen as the key point of the proposed approach, we propose an automatic setting based on a joint optimization process with Eq. ( 9). More precisely, optimal value of alpha-parameter, α opti , is obtained using a classic gradient descent such as to solve:

α opti = argmax α D α ( p in || p out ) ( ) , ( 10
)
Eq. ( 9) and Eq. (10) being optimized in turn. α opti value is initialized to 1 in order to start from the classic KL-divergence that corresponds to Shannon context for which no prior information on the shape of the PDF to separate are taken into account. Main idea is to let the statistical discriminative properties arise from the iterative optimisation process of alpha parameter.

The corresponding proposed algorithm is the following one:

β ξ

Experiments 4.3.1 Synthetic images

First of all, to illustrate the properties of the proposed method, we show the results obtained on a synthetic image (Fig. 7). The "peanut" binary shape is corrupted using a strong Gaussian noise such as the PSNR is only of 10dB. The segmentation process is initialized using a multiple circles strategy as illustrated Fig. 6a. Fig. 6b shows the segmentation result obtained with the KL-divergence and Fig. 6c with the proposed approach coupling the maximizations of the alpha-divergence and of alpha value.

ξ β

As it can be noticed, if KL-divergence fails because of the too strong amount of noise, even for Gaussian PDF, proposed method and strategy of optimization for alpha-value permit the obtaining of a satisfying segmentation. Fig. 7(b) shows that with KL divergence, the segmentation process finally stops in a local maximum that does not correspond to the desired segmentation. Optimizing the metric in parallel with the evolution of the active curve makes possible an adaptation to the statistic properties of the PDF all along the process. At convergence, final alpha-value is 0.5 in this particular case. In that particular case, at convergence optimal alpha value is 0.4. Again, KL divergence fails to segment the peanut shape, being stuck into a local maximum whereas proposed approach leads to the right segmentation.

In terms of robustness, a first statistical evaluation of the final segmentation error (percentage of misclassified pixels) on the peanut shape of Fig. 7 was carried out. Tab. 1 summarizes obtained results. Statistics of Tab. 1 were computed on 100 hundred independent experiments with a different realisation of the Gaussian noise in each case. Proposed method for each experiment outperforms the classic KL divergence and remains stable in terms of final segmentation results.

WCE image segmentation

In the following, we now consider application of the proposed approach to the segmentation of different type of structures in WCE images. Considering the initialization issue, we hypothesis that a pre-detection step permitted to automatically identify the barycentre of particular ROI identified as structure of interest by the learning-based approach of Silva et al [Silva 14] (see Fig. 3 for illustration). The initializing curve is a 20-pixels-diameter circle, which centre is the barycentre of the square in which the structure is included after the detection step.

For all the experiments, same values of hyper parameters were used ( =1, β=10, σ=0.1) except of course for alpha, which was optimized in accordance with the proposed strategy. As it can be noticed, in every case shown, the final segmentation is satisfying and makes possible the extraction of features including for instance, shape and texture ones. More precisely, as it can be noticed in Fig. 9, the maximisation of alpha-divergence is a flexible approach that could fit to different kind of segmentation structures acquired in challenging conditions. For instance, even in the case of polyp segmentation, which luminance characteristics are quite similar to the ones of the associated local background (Fig. 9(a-c)), the proposed segmentation approach makes delineation of the boundaries possible whereas usual approach like the Chan and Vese one, necessarily fail. In each case, it can be noticed that the proposed method outperforms classic KL-based approach.

Finally, in Fig. 11, semi-automatic segmentation results are compared with the corresponding manual delineation performed by the clinicians using the "RatSnake" dedicated free software [Iakovidis 14].

We provide here visual results showing that, in most cases, the semi-automatic segmentation is very close from the clinician's one. In addition, in Tab. 2, for each case, the segmentation error is proposed (computed as the number of misclassified pixels compare to the manualsegmentation mask).

These quantitative results must be considered with care because some values that may be seen as "bad" ones (for instance with the Gastric polyp (b)) does not highlight the fact that the segmentation obtained with the semi-automatic proposed method is definitely of better interest than the one obtained with KL divergence. One can also notice that for particular cases like the "enteritis" image or the "metastasis" one, the clinician does not take into account the possible complex topology of the ROI to segment. Because, we use level-set algorithms, the obtained results can highlight particular topological properties that can be of real interest for further characterizations.

Conclusion and Discussion

In this article, a flexible statistical region-based active contour criterion has been presented for WCE image segmentation. Based on alpha-divergence similarity, this criterion, embedded into a maximisation scheme of the corresponding segmentation energy, shows a real flexibility regarding the particular type of data considered which are acquired in non usual conditions. The obtained results on a set of WCE images showed that the proposed method overcome the usual drawback of histogram-based method using KL divergence for similarity measure.

From a clinical point of view, we now need to improve the performance (robustness and reproducibility mainly) on a larger set of data. For this purpose, an acquisition campaign is currently on the run with the clinical partner of the project (APHP, Hôpital Lariboisière, Department of Gastroenterology, Paris Sorbonne Paris 7 University).

Prospective work in terms of image processing will focus on three main issues:

(i) To take into account colour features of WCE images that can be related to textural information;

Fig. 2

 2 Fig. 2 Three images extracted from three different video made with a Pillcam © WCE (taken from the freely available database "Gastro Vidéo Web" (http://dr.dm.gastro.free.fr/).

Fig. 3

 3 Fig. 3 Three examples of the proposed boosting-based polyp detection approach in [Silva 14]

Fig. 5

 5 Fig. 5 Illustration of active contour segmentation: Γ = Γ(p, τ) denotes the coordinate of the point p of the curve at iteration τ of the segmentation process.

Fig. 6

 6 Fig. 6 Illustration of the non-Gaussianity of the PDF (luminance) extracted from a WCE image presenting with a polyp. Left: delineation of the polyp, middle: inner PDF of the luminance (polyp), right: outer PDF (background).

  the particular framework of histogram-based active contour segmentation [Aubert03, J-Besson 03, Herbulot 07, Lecellier 10, Lecellier 14], external forces involve optimization of distance (or divergence) between the luminance PDF computed from the inner (p in ) and the outer (p out ) regions delimited by the boundaries of the active curve. If usually the Kullback-Leibler (KL) divergence is used, as shown in [Hero06], when inner and outer PDF have too strong overlapping, KL-divergence may fail in separating the two PDF.

Fig. 7

 7 Fig. 7 Segmentation results: Comparison between KL-divergence based approach and our method. (a) Original noisy image (Gaussian noise) and initialization of the segmentation process, (b) Segmentation result with KL divergence, (c) Segmentation result with proposed approach ( =1, =1, σ=0.1).

Fig. 8 Fig. 8

 88 Fig.8illustrates performance of the proposed approach on a texture image computed from the Brodatz database. Again, a comparison between KL-divergence and our approach is shown.

Tab. 1 :

 1 Average segmentation error in percentage of misclassified pixels function of the strategy used for the setting of α parameter (KL or optimized).

Fig. 9 :

 9 Fig. 9: WCE image segmentation using the proposed histogram-based active contour approach. (a) and (b) gastric polyps in Peutz-Jeghers syndrome (c) Colon sigmoid polyp, (d) Lymphangiectasia, (e) Small-bowel metastasis, (f) radiation enteritis (g) Angiodysplasia, (h) Lipoma, (i) Hypertrophied anal papilla. Indicated values of alpha correspond to the optimal value reach at convergence of the joint optimisation process.

Fig. 10 :

 10 Fig. 10: WCE image segmentation using histogram-based active contour approach with KL divergence. (a) Colon sigmoid polyp, (b) Gastric polyps in Peutz-Jeghers syndrome, (c) Hypertrophied anal papilla. Same initialisation and hyper parameters than those of previous experiments were chosen.

Fig. 11 WCETab. 2 :

 112 Fig. 11 WCE image segmentation of Fig. 8 plus, in yellow colour, the manual segmentation performed by the clinician.

Acknowledgement

Part of this work was achieved in the framework of TERAFS project (EPSRC Grant EP/F013698/1).

Alpha-divergence-based active contour

As explained above, the segmentation task is formulated as an optimization problem. More precisely, a maximisation of Eq. (3) using the classic "shape gradients" framework formerly proposed in [Auber03] leads to the following evolution equation, as shown in [Meziou 12],

for non parametric estimation of the PDF using Parzen window technique:

with

where i, j

are the derivatives of ϕ α with respect to the first (p in ) and the second (p out ) variable, and * denotes the convolution operator.

Results

Implementation

In order to be able to segment images presenting multiple target objects, we propose to embed the alpha-divergence maximization within the classic level-set framework [ Osher 88].

Moreover, because a regularization of the obtained curve is needed not to take into account to small regions in the final segmentation result, a minimisation of the final length of the curve is added to Eq. ( 7). Considering the classic level-set embedding function Φ : ℜ 2 × ℜ → ℜ , the final evolution equation is then given by:

with β and ξ two weighting parameters, δ(.) the Dirac function and ∇ the usual gradient operator. In Eq. ( 9), first term corresponds to the regularization term (minimization of the final length) and second term to the maximisation of the alpha-divergence between p in and p out .

(ii) To extract meaningful clinical parameters from the segmented area in order to better differentiate benign from malignant lesions;

(iii) To embed within the capsule the proposed approach, in order to make it compatible with real-time in situ processing.

Conflict of interest

The authors have not declared any conflicts of interest.


Théophane Nicolas

Ronan Gaugne

Cédric Tavernier

Valérie Gouranton

Bruno Arnaldi

Preservative Approach to Study Encased Archaeological Artefacts

Keywords: Tomography, 3D modelling, 3D printing, Digital heritage

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

Introduction

Preserving archaeological material while analysing it, is a major concern for archaeologists. This problematic is often difficult to address because studied artefacts may be encased in corroded materials or a block of block of cremated bones, or integrated with, and inseparable from, larger assemblies, such as manufactured objects composed of several pieces. Once a material is destroyed, hypotheses on the original structure become unverifiable.

In this paper, we aim to propose a workflow for analysing archaeological artefacts by combining computed tomography with 3D printing, which is non-destructive, and accurate enough to perform further analysis. We can apply this methodology on three different contexts:

-Analysis of the internal iron-made axis of an Iron Age weight; -Evaluation of the interest to restore a completely corroded piece; -Study of objects sealed in a block of block of cremated bones. Computed tomography (CT) [START_REF] Herman | Fundamentals of computerized tomography: Image reconstruction from projection[END_REF] is an imaging technology widely used in medicine, CT scanners use computer-processed x-rays to produce tomographic images (virtual «slices») of specific areas of the scanned object.

The use of CT technologies for the analysis of historical artefacts is not new but remains limited, due to the difficulties of carrying this type of analysis on a large scale with appropriate access to equipment and trained personnel, and for exploiting the obtained digital data. Furthermore, the existing examples of such images for cultural heritage are mostly related to analysis of human or animal remains (e.g. mummified materials) but there are few cases of other materials being examined, such as non-organic archaeological material [START_REF]X-rays for Archaeology[END_REF], [START_REF] Hughes | CT Scanning in Archaeology[END_REF]. All of previous projects exploit the acquired images in their usual format, and do not combine them with other 3D technologies. The work presented in [START_REF] Laycock | Combining X-Ray Micro-CT Technology and 3D Printing for the Digital Preservation and Study of a 19th Century Cantonese Chess Piece with Intricate Internal Structure[END_REF] proposes to use of 3D printers to reproduce a chess piece for study purposes. This work illustrates one possible bridge between medical images and 3D technologies.

3D technologies are widely used in archaeology and cultural heritage domains [START_REF] Arnold | 3D-COFORM: Tools and Expertise for 3D Collection Formation[END_REF], [START_REF] Havemann | The Presentation of Cultural Heritage Models in Epoch, Open Digital Cultural Heritage Systems[END_REF], but mainly associated to 3D modelling, photogrammetry or lasergrammetry.

Our paper proposes to combine these two approaches to provide new tools for the preservation and analysis of archaeological artefacts.

Archaeological Context of the Work

The excavation of the site la Claraiserie, in Ossé (Brittany, France), revealed part of an Iron Age farm and field systems (Excavation J. Gall, Inrap). The site was occupied from the second century BCE. The remains of several earth and timber buildings were uncovered, together with a large number of iron objects contained within a pit. These were identified as agricultural tools, hunting weapons, horsebits as well as elements from a set of scales, probably used for weighing the farm produce. These included several spherical weights made out of granite. The weight studied here has a diameter of 8 cm for a weight of 620 grams. It is perforated axially, with an iron made axis attached to a ring inserted in the perforation, in order to allow the suspension of the weight (Fig. 1a). The excavation of the site of Domaine de la Bizaie in Guipry (Brittany, France) uncovered a trapezoidal shaped funerary enclosure the central area housing ten cremation burials containing pottery vessels dating to the Iron Age (Excavation L. Aubry, Inrap). The exceptional state of preservation of some of these cremations prompted us to use tomography to analyse their contents. In some case, a number of metal objects were highlighted as in the F42A cremation (Fig. 1b) containing a fibula and a knife blade.

The excavation of the site la Salmondière in Cesson-Sévigné (Brittany, France) revealed a long occupation dating from the Bronze Age to medieval times (Excavation J.-C. Durand, Inrap). From about 250 BC to the end of the Iron Age (80 BCE), the occupation of the site is characterized by the establishment of a large a b c agricultural area. Many finds, in particular metallic objects, have been recovered from the site. A complete radiographic survey for iron objects was performed, combined with tomography for the more complex objects. This survey allowed us to identify corroded objects that would not otherwise have been possible. This is the case of the helical small piece, 4.5 cm long and 1.3 cm in diameter with 7 turns, studied in this work. This piece was completely covered by a thick layer of rust (Fig. 1c).

Presentation of the Approach

While the contexts of the three use cases studied in this paper are completely different, there exists a common characteristic: the parts of the artefacts we wanted to study were unreachable. In the case of the Iron Age weight, the metal axis was encased in the stone part, and covered by rust. Concerning the cremation, the different objects were sealed in the block of cremated bones. In the case of the helical piece, the archaeologists wanted to identify and evaluate the interest the piece completely covered by a corrosion layer before launching a long and costly conservation and restoration process.

Before any destructive analysis, all the archaeological material was digitized with computed tomography. The aim of the work was to provide 3D views and tangible material to archaeologists in order to ease the study of unreachable parts of the original material. In order to fulfil this goal, we combined 3D images technologies and 3D printing.

The same approach was applied to the three object we wanted to study: 1. A computed tomography of the archaeological material was performed generating a database of X-Ray images. 2. Volume 3D views are generated for interesting parts of the material 3. Surface 3D models are generated, processed and printed

The volume 3D views and 3D printed objects are specifically dedicated to the archaeologist work for study purposes. Their generation required engineers with specific skills, with the participation of an archaeologist to validate the interest of the resulting data.

Work Description

We will now present the technical workflow applied to the different artefacts studied in this work.

Technical Environment

The technical environment is composed of three parts, tomography scanner, 3D images processing, and 3D printing.

Tomography. The CT scanner used in this study is a Siemens SOMATOM sensation 16 (Fig. 2). It is owned by the French company Image ET1 who delivers services and expertise on X-Ray images and associated tools. The three-dimensional explorations were performed through two passes (acquisition) and 2D/3D images post-processing. The two acquisitions were a topogram (or radio mode) for positioning the slices to be realized and a helix scan.

The global methodology is similar for all the scans processed in this work. The only differences are in kV and mAs values, and in the resolution of the cutting map (XY) which depends of the field of view, the matrix of the 2D slice images being constant (512x512 pixels) on the scanner. The scan generates a dataset under DICOM format [START_REF] Mustra | Overview of the DICOM Standard[END_REF].

Image processing. The data generated during the CT scan was processed with the software Osirix2, an image processing application for Mac dedicated to DICOM images, under free software license. Two post-processing functionalities of Osirix were used in the current work, the 3D volume rendering and the 3D surface rendering. The 3D surface rendering generates 3D meshes that were processed with the 3D modelling tools Blender3, and Meshlab4 to be ready-to-print.

3D printing

We used two different 3D printers (Fig. 3) for the production of the copies. For the weight, we used a MarkerBot Replicator 2x. For the fibula and the helical piece, we used a Stratasys MOJO. The MakerBot Replicator 2X 3D Printer uses Fused Filament Fabrication to print objects from ABS. Using 2 extruder head(s), it prints objects in layers as thin as 100 Microns, with a maximum dimension of 15.2cm wide, 15.5cm high and 24.6cm long. The Stratasys Mojo 3D Printer uses Plastic Jet Printing to print objects from ABS. Using 2 extruder head(s), it prints objects in layers as thin as 178 Microns, with a maximum dimension of 12.7cm wide, 12.7cm high and 12.7cm long.

We chose to use the Mojo printer for smaller pieces because the resulting pieces were more accurate. The Replicator printer was used for larger pieces and better coloured rendering. The kV and mAs values represent the parameters of the X rays intensity, influencing the penetration of the X rays (kV) and the clearness of the image (mAs). The FoV, for Field of View, depends on the size of the zone we want to focus on. The resulting image is projected on an invariant 512x512 pixels matrix, inducing the resolution.

The scan of the weight allowed us to focus on the iron axis inserted in granite and covered by rust. The highlight of the different components of the material is based on density information calculated during the CT scan, called the radiodensity, and expressed relatively to the Hounsfield scale [START_REF] Molteni | From CT numbers to Hounsfield Units in cone beam volumetric Imaging: the effect off artifacts[END_REF]. The 3D volume reference image presented in the left of Fig. 4 was generated with four colours, modifying the opacity of each colour with respect to the level of radiodensity. The display rules are presented in Fig. 5, the cyan layer around the weight generated by the beam hardening artifacts (corresponding to the air / material interface) is used to strengthen the visualisation. The image on the right of Fig. 4 was generated with only two colours, with different parameters in order to focus on the iron axis. The 3D surface renderer of Osirix is used to produce the mesh destinated to the 3D printer. The mesh is constructed as the surface of the point cloud correspondig to an interval of radiodensity values. For the granite part, the values considered are in the interval [300;2000]. For the metal part, the interval considered is greater than 3000.

The mesh obtained with Osirix was exported in obj format and processed with Blender and Meshlab to clean the generated mesh in order to obtain a ready-to-print 3D model. The mesh for the granite part was cut in two parts along the axis hole in order to access to the iron axis and study the positioning of the axis inside the granite sphere. The 3D models were printed with the Replicator printer, in two colours to have a clear distinction between the different parts of the weight (Fig. 7) Fig. 7: 3D printed copy of the weight

Helical Piece Study

The technical data for the scan of the weight is the following: kV mAs FoV Resolution 140 kV 350 mAs 50mm x 50mm 97 µ

In order to highlight the helical piece, the 3D volume rendering was generated with two colours according to the values of Fig. 8. The resulting rendering is presented in Fig. 9. These images provide a very good quality view of the helical piece inside rust. The 3D surface rendering was generated for radiodensity values greater the 6500 (Fig10). The mesh was exported in obj format and processed with Meshlab to be cleaned and exported in stl format which is the fomat acceptable for the Mojo printer. The lateral and front views of the resulting 3D printing, compared to the real object are presented in Fig. 11.

Fibula Study

The technical data for the scan of the weight is the following: kV mAs FoV resolution 120 kV 350 mAs 320mm x 320mm 625 µ

The aim of the volume rendering was to highlight objects with different density than ashes. We worked in an extended Hounsfield scale (from -10.000 to +40.000), to get a finer view of the metallic objects and focused on two density intervals, one between 1600 and 2300 corresponding to bones represented in red, and one between 4500 and 10950 corresponding to metallic objects, represented in blue. In order to keep a view of the ash block, which has a heterogeneous density, we took a large interval with a variation of the opacity to get a uniform representation, in grey, of the urn shape. The association of the radiodensity values is presented in Fig 12, and the volume 3D rendering, in Fig. 13. The resulting 3D mesh is presented in Fig. 14.

Discussion

New technology in imaging, that has until now been exclusively used by museums and applied to the study of remarkable objects, has opened up new perspectives in Preventive Archaeology. This hitherto unused method has revealed to be an amazing tool for the scientific study of archaeological artefacts providing precious information as to the preservation and the protection of the most fragile of objects. It has many advantages. It is a non destructive method that can be used for the examination and study of fragile or non-visible objects in need of immediate consolidation (conglomerate of objects, oxidized object) or for a preliminary examination of objects that will undergo laboratory excavation (for example a cremation burial in an urn), as it provides a 3D visualisation or can be used to create a facsimile using 3D printing. This considerably reduces the manipulation of the object itself. Also, as excavation and the cleaning of artefacts are non-reversible, the imagery and 3D printing become a permanent virtual record of the object's original state thus allowing ulterior observations and study. This process saves time as it facilitates the identification and analysis and so is perfectly adapted to preventive archaeology. It is also possible to determine material density providing valuable information as to the preservation of the object and its fragility, helping to make the right choice concerning the methods required for cleaning and conservation.

Analysis of the weight with tomography highlights elements that are invisible to the eye such as the ring which is encased in the gangue corrosion. It also allows to access some parts without having to break the object such as the iron axis inserted in the granite sphere. The 3D printing of the sphere and axis allows the manipulation of these elements, and identifying and/or discussing technical choices of the implementation (insertion of the metal stem in the sphere, drawing of two perforated plates placed on either side of the latter), but also clues to the manufacture of the axis (facets hammering...) hardly noticeable with tomography.

If in the case of the balance weight, the object is identifiable, this is not the case with the helical piece. First, tomography characterized an object that cannot be identified by the naked eye and hardly recognizable with simple radiography. The acquisition of a 3D model of this item helped with identification and description; associated to a 3D printing, that process appears to be a quick and inexpensive alternative to study this kind of object. It also gives clear indications to the archaeologists as to later restorations (object category and preservation state) and to the restorer (location of the object in the matrix, differential density of materials, composite assemblies identification).

In the case of the cremations, tomography allows spatial localisation of all artifacts and their identification. If modelling indicates how the objects were organized in the burial (in particular the cremated bone), it mainly serves to rapidly determine if the deposits contain sensitive objects such as metal. This information allows us to anticipate optimally fine analysis of clusters and to implement protective measures for artefacts. 3D acquisition associated to 3D printing offers the possibility to duplicate an artefact amongst others, leading to an immediate typological identification of the artefact, as well as morphological, typo-metric or technological observations, regardless of contingencies relative to conservation and restoration operations often required by such objects.

Nevertheless, there are still some efforts to do in order to adapt these technologies for the specificities of archaeological and cultural heritage. In particular, the software related to DICOM image processing is very well suited for medical applications but would require some adaptations for archaeological objects (focus on the volume and surface 3D rendering, better handling of extended Hounsfield scale more adapted to dense material such as metal).

Associating tomography and 3D printing offers many opportunities for the study of archaeological objects especially composite or oxidized metal; if this process is a precious tool in the identification of corroded objects, it also helps to identify and characterize complex objects, or technical characteristics. It also contributes to the development of specific protective measures for artifacts, as well as operational support during the restoration phase

The copies made of objects can be used for exhibition in particular when the state of preservation does not allow restoration. It is also a great educational tool highlighting the hidden side of some composite objects or even the functionality of certain mechanisms.

It appears that if, today, these tools are mainly used in the cases of exceptional discoveries, it is clear that their fields and conditions of application are fully in tune with those of preventive archaeology.

Fig. 1 -

 1 Fig. 1-a: granite weight, b: F42A cremation from Guipry, in situ, c: External view of the helical piece

Fig. 2 :

 2 Fig. 2: The CT scanner in Image ET

Fig. 3 :

 3 Fig. 3: Left : MakerBot Replicator 2X Right : Stratasys Mojo

Fig. 4 :Fig. 5 :

 45 Fig. 4: 3D views of the weight

Fig. 8 :

 8 Fig. 8: 3D volume rendering values Fig. 9: 3D views of the helical piece

Fig. 12

 12 Fig. 12: rendering values Fig. 13: 3D volume rendering for the urn

http://www.image-et.fr/

http://www.osirix-viewer.com/

http://www.blender.org/

http://meshlab.sourceforge.net/

Acknowledgments. The authors wish to thank Guillermo Andrade (Inria) and Yann Macé (ENS de Rennes) for their work on 3D printing, and Rebecca Peake (Inrap) for her kind support.


M Gisclon

I Lacroix-Violet

ABOUT THE BAROTROPIC COMPRESSIBLE QUANTUM NAVIER-STOKES EQUATIONS

Keywords: AMS Classification. 35K35, 65N12, 76Y05 Quantum Navier-Stokes equations, global weak solutions, asymptotic analysis

In this paper we consider the barotropic compressible quantum Navier-Stokes equations with a linear density dependent viscosity and its limit when the scaled Planck constant vanish. Following recent works on degenerate compressible Navier-Stokes equations, we prove the global existence of weak solutions by the use of a singular pressure close to vacuum. With such singular pressure, we can use the standard definition of global weak solutions which also allows to justify the limit when the scaled Planck constant denoted by ε tends to 0.

Introduction-Motivations

In this paper, the model of interest belongs to quantum fluid models. 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 [START_REF] Gualdani | Analysis of the viscous quantum hydrodynamic equations for semiconductors[END_REF] the authors derived viscous quantum Euler models using a moment method in Wigner-Fokker-Planck equation. In [START_REF] Brull | Derivation of viscous correction terms for the isothermal quantum Euler model[END_REF], under some assumptions, using a Chapman-Enskog expansion in Wigner equation, the quantum Navier-Stokes equations are obtained.

In this paper, we are interesting in the barotropic quantum Navier-Stokes equations which read as, for x ∈ Ω and t > 0 (1)

         ∂ t n + div(n u) = 0, ∂ t (n u) + div(n u ⊗ u) + ∇ x (p(n)) -2 ε 2 n∇ ∆ √ n √ n = 2 ν div(n D(u)), n| t=0 = n 0 , (n u)| t=0 = n 0 u 0 ,
where the unknowns are the particle density n and the particle velocity u. Here, Ω = T d is the torus in dimension d (in this article 1 ≤ d ≤ 3), and u ⊗ u is the matrix with components u i u j . The function p(n) = n γ with γ > 1 is the pressure function and D(u) stands for the symetric part of the velocity gradient, namely D(u) = (∇u + t ∇u)/2. Finally, the physical parameters are the Planck constant ε > 0 and the viscosity constant ν > 0.

The quantum correction (∆ √ n)/ √ n can be interpreted as a quantum potential, the socalled Bohm potential, which is well known in quantum mechanics. This Bohm potential arises from the fluid dynamical formulation of the single-state Schrödinger equation. The non-locality of quantum mechanics is approximated by the fact that the equations of state do not only depend on the particle density but also on its gradient. These equations were employed to model field emissions from metals and steady-state tunneling in metalinsulatormetal structures and to simulate ultra-small semiconductor devices .

Formally, setting ε = 0 in (1) leads to the compressible Navier-Stokes equation with densitydependent viscosity. Our goal is to perform the limit ε tends to zero. To this end we need some global existence result on [START_REF] Bresch | On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF].

Existence results for the stationary isothermal model in one space dimension were shown in [START_REF] Jüngel | Quantum navier-Stokes equations[END_REF]. The main mathematical difficulty, besides of the highly nonlinear structure of the thirdorder quantum terms, is the proof of positivity (or non-negativity) of the particle density.

In [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF], A. Jüngel, proves the global existence of this system when the scaled Planck constant is bigger than the viscosity constant (ε > ν). In [START_REF] Dong | A note on barotropic compressible quantum navier-Stokes equations[END_REF], J. Dong extends this result where the viscosity constant is equal to the scaled Planck constant (ε = ν), and, in [START_REF] Jiang | A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations[END_REF], F. Jiang shows that the result still holds when the viscosity constant is bigger than the scaled Planck constant (ν > ε). Therefore, thanks to [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF], [START_REF] Dong | A note on barotropic compressible quantum navier-Stokes equations[END_REF] and [START_REF] Jiang | A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations[END_REF], we have the global existence for all physically interesting cases of the scaled Planck and viscosity constants. Note that the definition of global weak solutions used in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF], [START_REF] Dong | A note on barotropic compressible quantum navier-Stokes equations[END_REF] and [START_REF] Jiang | A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations[END_REF] follows the idea introduced in [START_REF] Bresch | On some compressible fluid models: Korteweg, lubrication and shallow water systems[END_REF] by testing the momentum equation by n φ with φ a test function. Here the problem of such formulation is that it requires γ > 3 for d = 3 which is not a suitable assumption for physical case, and the estimates on the solution are badly dependent on ε due to extra terms coming from the diffusive term tested against nφ.

Then, the goal of this paper is to prove existence of global weak solutions without the assumption γ > 3 if d = 3 and with uniform estimates allowing to perform the limit of (1) when ε tends to 0. To this end we will use an another formulation. In more details, our proof of existence will be split in two steps. In a first time we construct some approximate solutions and in a second time, using some a priori estimates, we prove the stability of solutions. The construction of approximate solutions will follow exactly the ideas introduced by A. Jüengel in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF]. We will recall this approximate system which gives an other way to construct approximate solutions than the one introduced by [START_REF] Mucha | Chemically reacting mixtures in terms of degenerated parabolic setting[END_REF] after some hints given in [START_REF] Bresch | On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF]. The new lines in our paper, compared to [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF], will be the asymptotic limit with respect to the regularized parameter. Indeed we replace the concept of global weak solutions obtained by multiplying the momentum equation by nφ by a more standard formulation which required the addition of an extra cold pressure as introduced in [START_REF] Bresch | On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF] and fully developed for chemical reactive flows in [START_REF] Mucha | Chemically reacting mixtures in terms of degenerated parabolic setting[END_REF] and [START_REF] Zatorska | Fundamental problems to equations of compressible chemicaly reacting flows[END_REF]. The idea is to prove global existence of weak solutions of the following system, for x ∈ Ω and t > 0:

(2)

         ∂ t n + div(n u) = 0, ∂ t (n u) + div(n u ⊗ u) + ∇ x (p(n) + p c (n)) -2 ε 2 n∇ ∆ √ n √ n = 2 ν div(n D(u)), n| t=0 = n 0 , (n u)| t=0 = n 0 u 0
where p c is a singular continuous function namely a suitable increasing function satisfying

lim n→0 p c (n) = +∞
and called cold pressure. More precisely, we assume

(3) p ′ c (n) = c n -4k-1 , for n ≤ 1, k > 1, n γ-1 , for n > 1, γ > 1
for some constant c > 0.

Remark 1. As mentioned in [START_REF] Bresch | On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF], the physical relevance of the compressible Navier-Stokes equations is very questionnable in regions where the density are close to vacuum: the medium is not only unlikely to be in a liquid or gas state (elasticity and plasticity have to be considered for such solid materials for which by the way low densities may lead to negative pressures), but also the rarefied regime of vanishing densities violates the assumptions on the mean free path of particles suitable for fluid models. The singular pressure is a way to modelize such plastification in fluid models. Of course, it could be interesting from a mathematical view-point to understand if such singular pressure is necessary.

Remark 2. Such degeneracy will help (at the level of stability) to conclude about the strong convergence of term of type √ n u and thus pass to the limit in the nonlinearity nu ⊗ u.

We write the quantum term in a different form to pass to the limit:

< n∇ ∆ √ n √ n , φ > = < ∇ √ n∆ √ n , φ > -< 1 √ n ∇n∆ √ n, φ > = -< √ n∆ √ n, divφ > -2 < ∇ √ n∆ √ n, φ > = -< ∇ √ n∇ √ n , divφ > + < |∇ √ n| 2 , divφ > -2 < div(∇ √ n ⊗ ∇ √ n), φ > +2 < ∇ √ n • ∇ ∇ √ n, φ > = < √ n∇ √ n, ∇divφ > + < |∇ √ n| 2 , divφ > +2 < ∇ √ n ⊗ ∇ √ n, ∇φ > + < ∇|∇ √ n| 2 , φ > = < √ n∇ √ n, ∇divφ > +2 < ∇ √ n ⊗ ∇ √ n, ∇φ > .
Associated to (2), we can define now the following weak formulation of the momentum equation:

Ω n 0 u 0 • φ(•, 0) dx + T 0 Ω (nu • ∂ t φ + n(u ⊗ u) : ∇φ) dx dt + T 0 Ω p(n) + p c (n) divφ dx dt = T 0 Ω 2 ε 2 √ n ∇ √ n • ∇divφ + 4ε 2 ∇ √ n ⊗ ∇ √ n : ∇φ + T 0 Ω nD(u) : ∇φ dx dt.
In this paper, we will first prove that, for a fixed ε, there exists a global weak solution (n ε , u ε ) of [START_REF] Bresch | On some compressible fluid models: Korteweg, lubrication and shallow water systems[END_REF]. Secondly, we will prove that when ε tends to zero, (n ε , u ε ) tends to (n 0 , u 0 ) which is a global weak solution of:

(4)      ∂ t n 0 + div(n 0 u 0 ) = 0, ∂ t (n 0 u 0 ) + div(n 0 u 0 ⊗ u 0 ) + ∇(p(n 0 ) + p c (n 0 )) = 2 ν div(n 0 D(u 0 )), n 0 | t=0 = n 0 , (n 0 u 0 )| t=0 = n 0 u 0 .
Our existence result relies on a careful used of what has been done to prove global existence of weak solutions for the degenerate compressible Navier-Stokes equations by [START_REF] Bresch | On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF] and more recently in the very interesting complete studies [START_REF] Zatorska | Fundamental problems to equations of compressible chemicaly reacting flows[END_REF] (published in [START_REF] Mucha | Chemically reacting mixtures in terms of degenerated parabolic setting[END_REF]). We will then mix these ingredients with some that may be found in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF] and [START_REF] Jiang | A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations[END_REF] for the construction of approximate solutions. For the asymptotic limit we will strongly use the fact that ε may vanish letting ν fixed: the key point being the identity div(n∇

2 log n) = 2n∇(∆ √ n/ √ n) ( * )
and the presence of a singular pressure close to vacuum allowing to pass to the limit in the nonlinear term nu ⊗ u. In conclusion that means that the construction of approximate solutions build by A. Jüengel is consistent with the stability procedure with singular pressure initiated by the works of D. Bresch, B. Desjardins.

The paper is organized as follow. In Section 2 we state the two main results about existence of solutions and the low Planck constant limit. In Section 3 we prove some a priori estimates and we precise which of them are independent of ε. Section 4 is devoted to the proof of the existence result and will be split in two parts: one for the construction of approximate solution and the second one for the stability of solutions. Finally Section 5 is devoted to the proof of the low Planck constant limit.

Main results

In this section we present our two main results. The first one gives the existence of a global weak solution to (2) (in the sense of Definition 3) without any assumption on γ even if the dimension is equal to three. The second one is devoted to the low Planck constant limit and shows that global weak solutions (as defined in Definition 3) of ( 2) tends to a global weak solution of (4) (in the sense of Definition 4) when ε tends to zero.

Let us first of all give the definitions we will use of weak solution for ( 2) and ( 4).

Definition 3. We say that (n, u) is a weak solution of (2) if the continuity equation

(5) ∂ t n + div ( √ n √ nu) = 0, n(0, x) = n 0 (x)
is satisfied in the sense of distributions and the weak formulation of the momentum equation ( 6)

Ω n 0 u 0 • φ(•, 0) dx + T 0 Ω (nu • ∂ t φ + n(u ⊗ u) : ∇φ) dx dt + T 0 Ω p(n) + p c (n) divφ dx dt = T 0 Ω 2 ε 2 √ n ∇ √ n • ∇divφ + 4ε 2 ∇ √ n ⊗ ∇ √ n : ∇φ + 2 ν nD(u) : ∇φ dx dt.
holds for any smooth, compactly supported test function φ such that φ(T, .) = 0.

Definition 4. We say that (n 0 , u 0 ) is a weak solution of (4) if the continuity equation

(7) ∂ t n 0 + div ( √ n 0 √ n 0 u 0 ) = 0, n 0 (0, x) = n 0 (x)
is satisfied in the sense of distributions and the weak formulation of the momentum equation

(8) Ω n 0 u 0 • φ(•, 0) dx + T 0 Ω (n 0 u 0 • ∂ t φ + n 0 (u 0 ⊗ u 0 ) : ∇φ) dx dt + T 0 Ω p(n 0 ) + p c (n 0 ) divφ dx dt = 2 ν T 0 Ω n 0 D(u 0 ) : ∇φ dx dt.
holds for any smooth, compactly supported test function φ such that φ(T, .) = 0. Now we can state the main results of this paper. First of all, let us introduce the energy of the system which is given by the sum of the kinetic, internal and quantum energies:

(9) E ε (n, u) = n 2 |u| 2 + H(n) + H c (n) + 2 ε 2 |∇ √ n| 2 ,
where H and H c are given by:

H ′′ (n) = p ′ (n) n and H ′′ c (n) = p ′ c (n) n .
Remark 5. Here, with p(n) = n γ and γ > 1, we have:

H(n) = n γ γ -1 .
The first main result of this paper is devoted to the existence of global weak solution in the sense of Definition 3.

Theorem 6. Let ν > 0, ε > 0, 1 ≤ d ≤ 3, T > 0, γ ≥ 1. Let (n 0 , u 0 ) such that n 0 ≥ 0 and E ε (n 0 , u 0 ) < ∞.
Then there exists a weak solution (n, u) to System (2) in the sense of definition 3 such that

n ≥ 0 in T 3 , √ n ∈ L ∞ (0, T ; H 1 ) ∩ L 2 (0, T ; H 2 ), n ∈ L ∞ (0, T ; L γ ), n γ ∈ L 5/3 (0, T ; L 5/3 ), √ nu ∈ L ∞ (0, T, L 2 ), n|∇u| ∈ L 2 (0, T ; L 2 ), √ n|∇u| ∈ L 2 (0, T ; L 2 ), ∇ 1 √ n ∈ L 2 (0, T ; L 2 ).
Let us now state the second main result of this paper which gives the convergence of a sequence of global weak solutions (n ε , u ε ) of ( 2) (in the sense of Definition 3) to (n 0 , u 0 ) a global weak solution to (4) in the sense of Definition 4.

Theorem 7. Let 1 ≤ d ≤ 3, T > 0, 0 < ε < ν, γ ≥ 1. Let (n 0 , u 0 ) such that n 0 ≥ 0 and E ε (n 0 , u 0 ) < ∞. Then for (n ε , u ε ) solution of (2)
we have, when ε tends to 0:

(n ε ) ε → n 0 strongly in L 2 (0, T ; L ∞ (Ω)), ( √ n ε ) ε ⇀ √ n 0 weakly in L 2 (0, T ; H 2 (Ω)), ( √ n ε ) ε → √ n 0 strongly in L 2 (0, T ; H 1 (Ω)), (1/ √ n ε ) ε → 1 √ n 0 almost everywhere, ( √ n ε u ε ) ε → √ n 0 u 0 strongly in L 2 (0, T ; L 2 (Ω)), (u ε ) ε ⇀ u 0 , weakly in L p (0, T ; L q ⋆ (Ω)),
with p = 8k/(4k + 1), q ⋆ = 24k/(12k + 1) and (n 0 , u 0 ) solution of (4).

Remark 8. Note that, even if the results are valuable for 1 ≤ d ≤ 3, in the proofs we will only focus on the 3d-case. Indeed, the most interesting case in term of difficulties is the 3d-one and the hypothesis γ > 3 in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF] was only necessary in this case.

A priori estimates

In this section we establish all the a priori estimates we need in order to prove existence and convergence results. We pay a particular attention to the dependence or independence of each one with regards to ε.

Using a formal computation, we easily obtain

(10) d dt Ω E ε (n, u) dx + ν Ω n|D(u)| 2 dx = 0.
Directly from [START_REF] Jüngel | Quantum navier-Stokes equations[END_REF] we deduce the following estimates:

Lemma 9. Under the hypothesis of Theorem 6, there exists a constant C independent of ε,

such that √ nu L ∞ (0,T ;L 2 (Ω)) ≤ C, (11)
n γ L ∞ (0,T ;L 1 (Ω)) ≤ C, (12) √ nD(u) L 2 (0,T ;L 2 (Ω)) ≤ C. ( 13
)
The following entropy inequality can be shown following the lines of the Bresch-Desjardins entropy and the Bohm potential identity ( * ).

Proposition 10. Under the hypothesis of Theorem 6, we have:

d dt Ω n 2 |u + ν∇ log n| 2 + H(n) + H c (n) + (2ε 2 + 4ν 2 )|∇ √ n| 2 dx (14) +ν Ω H ′′ (n)|∇n| 2 + H ′′ c (n)|∇n| 2 + ε 2 n|∇ 2 log n| 2 + 2n |∇u| 2 dx = 0.
Proof : We have:

∂ t (n∇logn) + div(n t ∇u) + div(n u ⊗ ∇logn) = 0.
We multiply this new equation by ν, and we add it to:

∂ t (n u) + div(n u ⊗ u) + ∇(p(n) + p c (n)) -2 ε 2 n∇ ∆ √ n √ n = 2 ν div(n D(u)),
to obtain

∂ t (n (u + ν∇ log n)) + div(n u ⊗ (u + ν∇ log n)) + ∇(p(n) + p c (n)) -2 ε 2 n∇ ∆ √ n √ n = 2 ν div(n ∇u).
Multiplying by u + ν∇logn and integrating over Ω we have:

Ω ∂ t (n (u + ν∇logn)) . (u + ν∇logn) dx + Ω div (nu ⊗ (u + ν∇logn)) . (u + ν∇logn) dx + Ω ∇p(n). (u + ν∇logn) dx -2ε 2 Ω n∇ ∆ √ n √ n . (u + ν∇logn) dx = 2ν Ω div (n∇u) . (u + ν∇logn) dx. (15)
Moreover,

• Ω ∂ t (n (u + ν∇logn)) . (u + ν∇logn) dx = d dt Ω n 2 |u + ν∇logn| 2 dx - 1 2 Ω div(nu)|u + ν∇logn| 2 dx • Ω ∇(p(n) + p c (n)) • (u + 2ν∇logn)dx = Ω ∂ t (H(n) + H c (n))dx +ν Ω H ′′ (n)|∇n| 2 dx + ν Ω H ′′ c (n)|∇n| 2 dx • -2 ε 2 Ω n∇ ∆ √ n √ n .(u + 2ν∇logn) = 2ε 2 Ω ∂ t ((∇ √ n) 2 ) + νε 2 Ω n|∇ 2 logn| 2 • Ω div (nu ⊗ (u + ν∇logn)) . (u + ν∇logn) dx = 1 2 Ω div(nu)|u + ν∇logn| 2 dx.
Finally, using integration by parts,

• 2ν Ω div (n∇u) . (u + ν∇logn) dx = -2ν Ω n |∇u| 2 dx + 2ν 2 Ω u.div n∇ 2 logn dx = -2ν Ω n |∇u| 2 dx + 4ν 2 Ω nu∇ ∆ √ n √ n dx = -2ν Ω n |∇u| 2 dx -4ν 2 Ω div(nu) ∆ √ n √ n dx = -2ν Ω n |∇u| 2 dx + 4ν 2 Ω ∂ t n ∆ √ n √ n dx = -2ν Ω n |∇u| 2 dx -4ν 2 Ω ∂ t |∇ √ n| 2 dx.
Using all the above inequalities in [START_REF] Wyatt | Quantum Dynamics with Trajectories[END_REF] we obtain proposition 10.

Directly from Proposition 10 we deduce the following estimates:

Lemma 11. Under the hypothesis of Theorem 6, there exists a constant C independent of ε,

such that √ n(u + ν∇ log n) L ∞ (0,T ;L 2 (Ω)) ≤ C, (16)
∇ √ n L ∞ (0,T ;L 2 (Ω)) ≤ C, (17)
∇n γ/2 L 2 (0,T ;L 2 (Ω)) ≤ C, (18) √ n∇u L 2 (0,T ;L 2 (Ω)) ≤ C, (19) ε √ n∇ 2 log n L 2 (0,T ;L 2 (Ω)) ≤ C, (20) ε ∇ 2 √ n L 2 (0,T ;L 2 (Ω)) ≤ C. ( 21
)
Remark 12. Contrary to [START_REF] Jüngel | Quantum navier-Stokes equations[END_REF], the entropy equality (14) allows us to obtain (17) with C independent of ε thanks to the coefficient (2ε 2 + 4ν 2 ) which can be minor by 4ν 2 .

Proof : Estimates ( 16), ( 17), ( 19) and (20) are just direct consequence from [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]. Estimate (18) derived from

H ′′ (n)|∇n| 2 = p ′ (n)|∇n| 2 n = γn γ-2 |∇n| 2 and ∇n γ/2 = γ 2 n γ/2-1 ∇n.
The last one is obtain using the following identity (which may be found for instance in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF])

T 0 Ω n|∇ 2 logn| 2 dx dt ≥ T 0 Ω |∇ 2 √ n| 2 dx dt
and (20).

Let us now show some estimates concerning the two different pressures.

Lemma 13. Under the hypothesis of Theorem 6, assuming that d = 3, there exists a constant C independent of ε, such that

n γ L 5/3 (0,T ;L 5/3 (Ω)) ≤ C, (22) p c (n) L 5/3 (0,T ;L 5/3 (Ω)) ≤ C. ( 23
)
Proof : Using ( 12) and (18) and Sobolev embedding for d = 3, we obtain

n γ/2 L 2 (0,T ;L 6 (Ω))
≤ C, which can also be written n γ L 1 (0,T ;L 3 (Ω)) ≤ C. Using interpolation, last inequality and [START_REF] Mellet | On the barotropic compressible Navier-Stokes equations[END_REF] give

n γ L 5/3 (0,T ;L 5/3 (Ω)) ≤ C. Let us recall that p c (n) = c -4k n -4k , n ≤ 1, k > 1.
Let ζ be a smooth function such that ζ(y) = y for y ≤ 1/2 and ζ(y) = 0 for y > 1.

Following the idea developed in [START_REF] Bresch | On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF] (page 71) and more recently in [START_REF] Zatorska | Fundamental problems to equations of compressible chemicaly reacting flows[END_REF] (page 62), using [START_REF] Jüngel | Quantum navier-Stokes equations[END_REF] and ( 14) we obtain (through

H c ) for a constant C independent of ε T 0 Ω |∇ζ(n) -2k | 2 dx dt ≤ C.
That means that ∇ζ(n) -2k ∈ L 2 (0, T ; L 2 (Ω)) and close to vacuum, (24) sup

t Ω n -4k dx ≤ C, with C independent of ε. This gives ζ(n) -2k ∈ L ∞ (0, T ; L 2 (Ω))
, and therefore we have

ζ(n) -2k ∈ L 2 (0, T ; H 1 (Ω)). Using Sobolev embedding in dimension d = 3, (25) ζ(n) -2k
L 2 (0,T ;L 6 (Ω)) ≤ C, with still C independent of ε. Now using as previously interpolation and (24), (25) we obtain (23). This ends the proof of Lemma 13.

Let us now prove some estimates that will be used to pass to the limit in the nonlinear term nu ⊗ u. That means to get the strong convergence of √ nu in L 2 (0, T ; L 2 (Ω)). Such convergence is required for the stability process to build global weak solution but also for the asymptotic analysis ε tends to 0. This follows the lines that may be found in [START_REF] Bresch | On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF] and [START_REF] Zatorska | Fundamental problems to equations of compressible chemicaly reacting flows[END_REF] but taking care of the new quantum term. We will only recall the main steps and refer the interested reader to very nice PhD Thesis [START_REF] Zatorska | Fundamental problems to equations of compressible chemicaly reacting flows[END_REF] for all details.

The idea is to prove that √ nu is uniformly bounded in L p ′ (0, T, L q ′ (Ω)) for p ′ , q ′ > 2.

This will be used with almost pointwise convergence to deduce the strong convergence in L 2 (0, T ; L 2 (Ω)). The main steps are the following: look at the estimates on ∇u (using the estimates on √ n∇u and the estimates on some negative power of n) and therefore on u by Sobolev embeddings. Mix it with the bounds on √ nu and n to increase the uniform integrability in space on √ nu uniformly with respect to ε.

Lemma 14. Under the hypothesis of Theorem 6, assuming that d = 3, there exists a constant C independent of ε, such that (26) ∇u L p (0,T ;L q (Ω)) ≤ C, with p = 8k/(4k + 1) and q = 24k/(12k + 1).

Proof : Let us write

∇u = 1 √ n √ n∇u.
Using (19) we have an estimate in L 2 (0, T ; L 2 (Ω)) for √ n∇u. Then it remains to obtain an estimate for 1/ √ n in the appropriate space. Using (25) and the remark that close to vacuum,

ζ(n) -2k 2 L 2 (0,T ;L 6 (Ω)) = T 0 Ω 1 √ n 24k dx 8k/24k dt, we obtain (27) 1/ √ n L 8k (0,T ;L 24k (Ω)) ≤ C, with C independent of ε.
As previously said, since ∇u = (1/ √ n) √ n∇u, using ( 19) and ( 27), we have the result.

Using Sobolev embedding, a direct consequence of Lemma 14 is the following one.

Lemma 15. Under the hypothesis of Theorem 6, assuming that d = 3, there exists a constant C independent of ε, such that (28) u L p (0,T ;L q ⋆ (Ω)) ≤ C, with p = 8k/(4k + 1) and q ⋆ = 24k/(12k + 1).

In the following lemma, we have an estimate on the gradient of negative power of n:

Lemma 16. Under the hypothesis of Theorem 6, there exists a constant C independent of ε, such that

(29) ∇(1/ √ n) L 2 (0,T ;L 2 (Ω)) ≤ C. Proof : We know from (17) that ∇ √ n ∈ L 2 (Ω). But ∇ 1 √ n = - 1 2 ∇n n 3/2 = - ∇ √ n n . Then, if n > 1, 1 n < 1 and ∇(1/ √ n) ∈ L 2 (Ω).
Now look at the case n ≤ 1. From Proposition 10, we have

T 0 Ω H ′′ c (n)|∇n| 2 dx < +∞ where H ′′ c (n) = p ′ c (n) n . For n ≤ 1, p ′ c (n) = cn -4k-1 then H ′′ c (n) = cn -4k-2 , H ′′ (c)|∇n| 2 = c ∇n n 2k+1 2 = c 4k 2 ∇ 1 n 2k 2 .
Make the connection between ∇ 1 n 2k and ∇ 1 √ n :

∇ 1 √ n = ∇ 1 n 2k n 2k-1/2 = n 2k-1/2 ∇ 1 n 2k + 1 n 2k ∇ n 2k-1/2 = n 2k-1/2 ∇ 1 n 2k + (2k -1/2)n -2k ∇(n)n 2k-3/2 = n 2k-1/2 ∇ 1 n 2k + (2k -1/2)∇(n)n -3/2 = n 2k-1/2 ∇ 1 n 2k -2(2k -1/2)∇ 1 √ n ,
we have

(1 + 4k -1)∇ 1 √ n = n 2k-1/2 ∇ 1 n 2k , and
∇ 1 √ n 2 = 1 16k 2 n 4k-1 ∇ 1 n 2k 2 = 1 16k 2 n 4k-1 4k 2 1 c H ′′ c (n)|∇n| 2 = 1 4c n 4k-1 H ′′ c (n)|∇n| 2 .
Such as n ≤ 1 and

T 0 Ω H ′′ c (n)|∇n| 2 dx < +∞, we have ∇ 1 √ n ∈ L 2 (0, T ; L 2 (Ω)).
Using previous lemma, we are now able to establish the following proposition.

Proposition 17. Under the hypothesis of Theorem 6, assuming that d = 3, there exists a constant C independent of ε, such that

(30) √ nu L p ′ (0,T ;L q ′ (Ω)) ≤ C, with p ′ , q ′ > 2.
Proof : Let r > 0 to be chosen later on. We write

√ nu = √ nu 2r u 1-2r n 1/2-r .
Using ( 12) 11), ( 28) and (29), choosing

n 1/2-r L ∞ (0,T ;L γ/(1/2-r) (Ω)) ≤ C, with C a constant independent of ε. Using (
1 p ′ = 1 -2r p and 1 q ′ = 2r 2 + 1 -2r q ⋆ + 1/2 -r γ ,
with r = 2/3, we get the conclusion. Indeed, with such definition of p ′ and q ′ , the condition q ′ > 2 is equivalent to r > 1/2 and the condition p ′ > 2 is equivalent to r > 1/(8k + 2) with 1/(8k + 2) < 1/10 since k > 1.

Remark 18. Estimate (30) is the one proved in [START_REF] Zatorska | Fundamental problems to equations of compressible chemicaly reacting flows[END_REF]. Note that it is also possible to choose drag terms of form r 0 u + r 1 n|u|u instead of cold pressure. Such a choice provides a bound on n u 2+δ in L 1 (0, T ; L 1 (Ω)) uniformly with δ > 0 directly from the energy estimate (cf the trick from Mellet and Vasseur [START_REF] Mellet | On the barotropic compressible Navier-Stokes equations[END_REF]).

Proof of global existence of solutions

In this section we prove existence of a global weak solution to (2) i.e. we prove Theorem 6. The proof can be split into two parts: in the first one we show how to construct an approximate solution and in the second one we show the stability of solution. This is the classical way to prove existence of solution for Navier-Stokes equations. For clarity of presentation we will detail each part in one subsection.

Construction of an approximate solution.

In this subsection we present the construction of an approximate solution which follows the lines given in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF]. In few words, the approximate solution is build through a fixed point argument at the level of the Galerkin approximate system, and, thanks to appropriate uniform estimates with respect to the Galerkin parameters, we can pass to the limit and obtain existence of global weak solution.

Let us now describe precisely the procedure. As in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF], let us first transform (2) by the use of the so-called effective velocity w = u + ν∇ log n. Then a simple computation shows that ( 2) is equivalent to: for x ∈ Ω and t > 0:

(31)          ∂ t n + div(n w) = ν∆n, ∂ t (n w) + div(n w ⊗ w) + ∇ x (p(n) + p c (n)) -2 ε 0 n∇ ∆ √ n √ n = ν ∆(nw), n| t=0 = n 0 , (n w)| t=0 = n 0 w 0 ,
with w 0 = u 0 + ν∇ log n 0 and ε 0 = ε 2 -ν 2 . We associate to (31) the following definition of weak solution.

Definition 19. We say that (n, w) is a weak solution of (31) if the equation

(32) ∂ t n + div (nw) = ν∆n, n(0, x) = n 0 (x)
is satisfied in the sense of distributions and the weak formulation

(33) Ω n 0 w 0 • φ(•, 0) dx + T 0 Ω (nw • ∂ t φ + n(w ⊗ w) : ∇φ) dx dt + T 0 Ω p(n) + p c (n) divφ dx dt = T 0 Ω 2 ε 2 0 ∆ √ n √ n div(n φ) + ν ∇(nw) : ∇φ dx dt.
holds for any smooth, compactly supported test function φ such that φ(T, .) = 0. Now the goal is to prove existence of solution to (31) in the sense of Definition 19. To this end we use the same technic as the one described Sections 3, 4, 5 and 6 in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF].

Global existence of solutions.

Let T > 0 and (e p ) an orthonormal basis of L 2 (Ω). Note that (e p ) is also an orthogonal basis of H 1 (Ω). We introduce the finite space 2 some initial data such that n 0 (x) ≥ δ > 0 for all x ∈ Ω and for some δ > 0. Let v ∈ C 0 ([0, T ]; X N ) a given velocity, v can be written for x ∈ Ω and t ∈

X N = span{e 1 , • • • , e N }, for N ∈ N * . Let (n 0 , w 0 ) ∈ C ∞ (Ω)
[0, T ] v(x, t) = N i=1 λ i (t)e i (x),
for some functions λ i . The norm of v in C 0 ([0, T ]; X N ) can be formulated as

v C 0 ([0,T ];X N ) = max t∈[0,T ] N i=1 |λ i (t)|,
which has for consequence that v is bounded in C 0 ([0, T ]; C n (Ω)) for any n ∈ N, and there exists a constant C depending on n such that

(34) v C 0 ([0,T ];C n (Ω)) ≤ C v C 0 ([0,T ];L 2 (Ω)) .
As in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF], the approximate system is defined as follows. Let n ∈ C 1 ([0, T ]; C 3 (Ω)) be the classical solution to (35)

∂ t n + div (nv) = ν∆n, x ∈ Ω, t ∈ [0, T ] n(0, x) = n 0 (x), x ∈ Ω.
Using the maximum principle, which provides lower and upper bounds, the assumption n 0 ≥ δ > 0 and (34), n is strictly positive and for all (x, t)

∈ Ω × [0, T ], 0 < n(c) ≤ n(x, t) ≤ n(c).
Moreover, for a given n N solution of (35), we are looking for a function

w N ∈ C 0 ([0, T ]; X N ) such that (36)
-

Ω n 0 w 0 • φ(•, 0) dx = T 0 Ω n N w N • ∂ t φ + n N (v ⊗ w N ) : ∇φ + (p(n N ) + p c (n N ))divφ -2 ε 0 ∆ √ n N √ n N div(n N φ) -ν ∇(n N w N ) : ∇φ -δ(∇w N : ∇φ + w N • φ) dx dt.
As detailed in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF], using a Banach fixed point theorem, there exists a unique local-in-time solution (n N , w N ) to ( 35) and (36) with n N ∈ C 1 ([0, T ′ ]; C 3 (Ω)) and w N ∈ C 1 ([0, T ′ ]; X N ), for T ′ ≤ T . In order to prove the global nature of the solution (n N , w N ) constructed above we use the following energy estimate.

Lemma 20. Let T ′ ≤ T , and let

n N ∈ C 1 ([0, T ′ ]; C 3 (Ω)), w N ∈ C 1 ([0, T ′ ]; X N ) be a local-in- time solution to (35) and (36) with n = n N and v = w N . Then (37) dE ε 0 dt (n N , w N ) + ν Ω n N |∇w N | 2 + H ′′ (n N )|∇n N | 2 + H ′′ c (n N )|∇n N | 2 dx + ε 0 ν Ω n N |∇ 2 log n N | 2 dx + δ Ω (|∇w N | 2 + |w N | 2 )dx = 0, where E ε 0 (n N , w N ) = Ω n N 2 |w N | 2 + H(n N ) + H c (n N ) + 2ε 2 0 |∇ √ n N | 2 dx.
We refer the interested reader to [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF] for the proof of such lemma. Then the both limits N → ∞ and δ → 0 are considered separately.

Remark 21. Note that as in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF], equations for u and w being equivalent, using (37), the estimates obtained previously on u are still true for w N and w δ with constants C independent of N and δ.

4.1.2. Limits N → ∞ and δ → 0. First we perform the limit N → ∞, δ > 0 being fixed. This is achieved by the use of regularities of the solution and Aubin-Simon's lemma. Here, the use of a cold pressure term avoid the hypothesis γ > 3 for d = 3 which was required in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF].

We can prove the following proposition.

Proposition 22. Under the hypothesis of Theorem 6, for a fixed ε, up to a subsequence, the following convergences hold when N tends to ∞.

√ n N → √ n δ , strongly in L 2 (0, T ; H 1 (Ω)),

p(n N ) → p(n δ ), strongly in L 1 (0, T ; L 1 (Ω)), p c (n N ) → p c (n δ ), strongly in L 1 (0, T ; L 1 (Ω)), 1/ √ n N → 1/ √ n δ , almost everywhere , √ n N w N → √ n δ w δ , strongly in L 2 (0, T ; L 2 (Ω)),
∇w N ⇀ ∇w δ , weakly in L p (0, T ; L q (Ω)),

w N ⇀ w δ , weakly in L p (0, T ; L q ⋆ (Ω)), with p, q and q ⋆ given in Lemmas 14 and 15.

Proof : First of all, let say that using (37) and the same technics as used in Section 3, we can prove estimates ( 11), ( 17)-( 28) with n = n N and u = w N and with constants C which are all independent of N and δ but can depend on ε. Using ( 11), ( 17), ( 19), ( 21), ( 27) and rewriting equation (32) as

∂ t ( √ n N ) + 1 2 √ n N div(n N w N ) = ν ∆ √ n N + |∇ √ n N | 2 √ n N , we can show that (38) ∂ t ( √ n N ) L 2 (0,T ;H -1 (Ω)) ≤ C,
with C a constant independent of ε. Then using ( 21) and (38) we can apply the Aubin-Simon's Lemma (see [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]) to obtain the strong convergence of √ n N to √ n in L 2 (0, T ; H 1 (Ω)).

Note that (21) being badly dependent on ε, this strong convergence is only true for a fixed ε. Using ( 22), (23) and the almost everywhere convergence of n N (which is a direct consequence of the previous strong convergence for √ n N ), we obtain the strong convergence of the two pressures in L 1 (0, T ; L 1 (Ω)).

Rewriting equation (32) as

∂ t 1 √ n N + ∇ w N √ n N + 3 2 √ n N div(w N ) = -ν ∆ √ n N n N + |∇ √ n N | 2 n N 3/2 ,
and using ( 11), ( 17), ( 19), ( 21), ( 24) and ( 27), we have

(39) ||∂ t 1 √ n N || L ∞ (0,T ;W -1,1 (Ω)) ≤ C,
with C a constant independent of N and δ but which can depend on ε. Then (39) and ( 27) allow us to apply the Aubin-Simon's Lemma and to obtain the almost everywhere convergence of 1/ √ n N .

Moreover, we have

∇(n N w N ) = n N ∇w N + w N ∇n N = √ n N √ n N ∇w N + 2 √ n N w N ∇( √ n N ) ∈ L 2 (0, T ; L 1 (Ω)).
Therefore n N w N ∈ L 2 (0, T ; W 1,1 (Ω)). Using the second equation in (31), we can get information on ∂ t (n N w N ) and therefore through Aubin-Lions-Simon's Lemma convergence almost everywhere of n N w N . With (30) and since √ n N w N converges almost everywhere then ( √ n N w N ) 2 converges strongly in L 1 (0, T ; L 1 (Ω)) and therefore √ n N w N converges strongly in L 2 (0, T ; L 2 (Ω)).

Finally the last two weak convergences directly come from estimates (26) and (28).

Using Proposition 22, passing to the limit N tends to ∞, we obtain the existence of a solution (n δ , w δ ) satisfying (37). Note that since estimates [START_REF] Loffredo | On the creation of quantum vortex lines in rotating He II[END_REF], ( 17)-( 28) for n = n N and u = w N were true for constants C independent of N and δ, they are still true for n = n δ and u = w δ and with constants C independent of δ but which can depend on ε. In a same way that previously this allows us to obtain the following convergences.

Proposition 23. Under the hypothesis of Theorem 6, for a fixed ε, up to a subsequence, the following convergences hold when δ tends to 0.

√ n δ → √ n, strongly in L 2 (0, T ; H 1 (Ω)), p(n δ ) → p(n), strongly in L 1 (0, T ; L 1 (Ω)), p c (n δ ) → p c (n), strongly in L 1 (0, T ; L 1 (Ω)), 1/ √ n δ → 1/ √ n, almost everywhere , √ n δ w δ → √ n w, strongly in L 2 (0, T ; L 2 (Ω)).
∇w δ ⇀ ∇w, weakly in L p (0, T ; L q (Ω)), w δ ⇀ w, weakly in L p (0, T ; L q ⋆ (Ω)), with p, q and q ⋆ given in Lemmas 14 and 15.

The proof of this proposition being really similar to the one for Proposition 22, we skip it here. Thanks to Proposition 23 we obtain the existence of solution to (31) in the sense of Definition 19 without the hypothesis γ > 3 even if we have used the same procedure as the one presented in [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF]. This achieves the construction of an approximate solution using the relation between u and w.

Stability.

In this subsection we look at the stability. Then let us assume that there exists a sequence (n τ , u τ ) of global weak solutions to (2) satisfying uniformly the energy and entropy inequalities. The goal is to prove that there exists a subsequence which converges to a global weak solution of (2) satisfying also the energy and entropy inequalities. Then it remains to prove that we can pass to the limit in

n τ u τ , n τ u τ ⊗ u τ , n τ D(u τ ), p(n τ ), p c (n τ ), ∇ √ n τ ⊗ ∇ √ n τ and in the quantum term n τ ∇( 1 √ n τ ∆ √ n τ ) namely in the terms with √ n τ or ∇ √ n τ .
To this end we need some strong convergences which can be obtained by the use of Aubin-Simon's Lemma. More precisely, we have the following proposition.

Proposition 24. Under the hypothesis of Theoerem 6, we have for a fixed ε

√ n τ → √ n, strongly in L 2 (0, T ; H 1 (Ω)), p(n τ ) → p(n), strongly in L 1 (0, T ; L 1 (Ω)), p c (n τ ) → p c (n), strongly in L 1 (0, T ; L 1 (Ω)), 1/ √ n τ → 1/ √ n, almost everywhere, √ n τ u τ → √ n u, strongly in L 2 (0, T ; L 2 (Ω)).
Proof : Using ( 11), ( 19) and the mass equation, we can show that (40)

∂ t ( √ n τ ) L 2 (0,T ;H -1 (Ω)) ≤ C,
with C a constant independent of ε. Then using ( 21) and (40) we can apply the Aubin-Simon's Lemma (see [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]) to obtain the strong convergence of

√ n τ to √ n in L 2 (0, T ; H 1 (Ω)).
Note that (21) being badly dependent on ε, this strong convergence is only true for a fixed ε. Using ( 22), ( 23) and the almost everywhere convergence of n τ (which is a direct consequence of the previous strong convergence for √ n τ ), we obtain the strong convergence of the two pressures in L 1 (0, T ; L 1 (Ω)).

Again using the mass equation and ( 11), ( 19) and (24), we have

(41) ∂ t (1/ √ n τ ) L ∞ (0,T ;W -1,1 (Ω)) ≤ C,
with C a constant independent of ε. Then (41) and ( 27) allow us to apply the Aubin-Simon's Lemma and to obtain the almost everywhere convergence of 1/ √ n τ .

Finally, we have

∇(n τ u τ ) = n τ ∇u τ + u τ ∇n τ = √ n τ √ n τ ∇u τ + 2 √ n τ u τ ∇ √ n τ ∈ L 2 (0, T ; L 1 (Ω)).
Therefore n τ u τ ∈ L 2 (0, T ; W 1,1 (Ω)). Using the momentum equation, we can get information on ∂ t (n τ u τ ) and therefore through Aubin-Lions-Simon's Lemma convergence almost everywhere of n τ u τ . With (30) and since √ n τ u τ converges almost everywhere then ( √ n τ u τ ) 2 converges strongly in L 1 (0, T ; L 1 (Ω)) and therefore √ n τ u τ converges strongly in L 2 (0, T ; L 2 (Ω)).

We can pass to the limit δ → 0 thanks to the strongly convergence on √ n τ u τ and the weakly convergence on ∇ √ n τ because we write

n τ u τ = √ n τ √ n τ u τ , n τ u τ ⊗ u τ = √ n τ u τ ⊗ √ n τ u τ , and
< n τ D(u τ ), ∇φ >=< D(n τ u τ ), ∇φ > -< (u τ ) j ∂ i n τ , ∂ i φ j > -< (u τ ) i ∂ j n τ , ∂ i φ j > = -< √ n τ √ n τ u τ , D∇φ > -< √ n τ (u n ) j ∂ i √ n τ , ∂ i φ j > -< √ n τ (u τ ) i ∂ j √ n τ , ∂ i φ j > .

Low Planck limit

Let us consider (n ε , u ε ) ε a sequence of solutions of (2) in the sense of Definition 3. The goal of this section is to prove Theorem 7 i.e. that, up to a subsequence, (n ε , u ε ) ε tends to (n 0 , u 0 ) solution of (4) in the sense of Definition 4 when ε tends to 0.

Using a priori estimates showed in Section 3 we can prove the same convergence as in the stability Section 4.2. The unique difference concerns the strong convergence of √ n ε which is here only in L 2 (0, T ; L 2 (Ω)) instead of L 2 (0, T ; H 1 (Ω)). Indeed this last one was obtain using an estimate which badly depend on ε.

Proposition 25. Under the hypothesis of Theorem 7, we have, when ε tends to 0, √ n ε → √ n, strongly in L 2 (0, T ; L 2 (Ω)), p(n ε ) → p(n), strongly in L 1 (0, T ; L 1 (Ω)), p c (n ε ) → p c (n), strongly in L 1 (0, T ; L 1 (Ω)), 1/ √ n ε → 1/ √ n, almost everywhere, √ n ε u ε → √ n u, strongly in L 2 (0, T ; L 2 (Ω)).

Proof : To prove the first convergence we use the estimate on ∇ √ n in L 2 (0, T ; L 2 (Ω)), the estimate on ∂ t √ n in L 2 (0, T ; H -1 (Ω)) (obtained in Section 4.2) and the Aubin-Simon's Lemma with H 1 (Ω) ⊂⊂ L 2 (Ω) ⊂ H -1 (Ω). We refer the reader to Section 4.2 for the establishment of the others convergences since they are completely similar.

Convergences of Proposition 25 allow us to pass to the limit ε tends to 0 in the second and third integrals of the left hand side and last one of the right hand-side of the weak formulation [START_REF] Gualdani | Analysis of the viscous quantum hydrodynamic equations for semiconductors[END_REF]. Using the fact that √ n lies in L 2 (0, T ; L 2 (Ω)) and ∇ √ n in L ∞ (0, T ; L 2 (Ω))

(due to (17)) we can show that:

2 ε 2 T 0 Ω √ n ∇ √ n • ∇divφ + 4ε 2 T 0 Ω ∇ √ n ⊗ ∇ √ n : ∇φ ≤ Cε 2 ,
with C a constant independent of ε. Then these two integrals go to 0 when ε tends to 0, and we obtain the result.

Acknowledgements

The authors would like to thank Prof. Didier Bresch. All discussions with him were very helpful to achieve this paper.