Christine Voiron-Canicio

Emmanuel Doveri
email: emmanuel.doveri@wanadoo.fr.

Johanna Fusco
email: johannafusco@hotmail.fr.

SPATIAL ARCHAEOLOGY USING IMAGE ANALYSIS AND MATHEMATICAL MORPHOLOGY

Keywords: Spatial Analysis, Image Analysis, Mathematical Morphology, Archaeology, Building alignment

For the last ten years, geographers, archaeologists and historians work together around the analysis of space transformation processes at various time scales, in particular long-term ones. However, the approaches which were developed until then to study the arrangement of archaeological remains and the links they share with their environment, are on one hand most related to statistical and counting methods, and on the other hand often take into account only their functional or symbolical dimension, overshadowing their purely physical aspect. Image analysis by Mathematical Morphology permits to overtake this limit. Two applications of this method are proposed: the first consists in detecting spatial configurations in the arrangement of funerary structures in a Bronze Age necropolis of central Mongolia, in order to point out some particular organisations. The purpose of the second study is to set up an image processing protocol applied to urban maps from different dates, permitting to detect the various architectonic orientations.

INTRODUCTION

Geography and Archaeology have in parallel experienced a huge upheaval in terms of reflexion and methods during the twentieth century, and particularly in the nineties. Archaeologists became progressively aware of the necessity to consider a true spatial dimension in their researches [START_REF] Djindjian | Méthodes pour l'archéologie[END_REF], and geographers shown the wiliness to associate time to space in a more rigorous and explicit way [START_REF] Lepetit | Remarques sur la contribution de l'espace à l'analyse historique[END_REF]. The conjunction of these new perspectives gave birth to a discipline, whose originality is first given by its interdisciplinarity, and named "spatial archaeology" (Van der Leeuw, [START_REF] Fiches | De la structuration et de l'analyse spatiale aux processus historiques[END_REF]. However, as this discipline is really recent, only a few works can be really considered as such. But all of them claim to adhere to Archaeomedes program [START_REF] References Favory | ARCHAEOMEDES, Des oppida aux métropoles, Archéologues et géographes en vallée du Rhône[END_REF], which certainly constituted the impulse of spatial archaeology. The ambition of this study is to position itself in the field of this discipline: the objective here is to detect invisible structures from visible shapes, and to link these shapes to the processes which gave them birth [START_REF] Voiron | Analyse spatiale et analyse d'images[END_REF], on the basis of archaeological data, to detect structures, underlying spatial organisations of remains arrangement. Image analysis by Mathematical Morphology is for that purpose an original and interesting method, which permits to introduce a supplementary dimension in spatial archaeology: the remains' shape one. Indeed, archaeological remains are not taken here for what they do represent, or classified, categorized, conceptualized according to their nature, but taken in a raw way, without stripping them of their proper originality, of their individual shape and variations. Moreover, Mathematical Morphology presents the advantage to be an extremely flexible method, where each image transformation is a step towards a precise aim, defined by the researcher: the results of the study are directly linked to the reasoning built by the operator, and of its interpretations. Moreover, the retrospective geographic study by Mathematical Morphology analysis can be helpful to the archaeologists to better foresee and anticipate the potentially interesting zones to conduct excavations. This method will be here tackled through two works, which take into account two spatial and temporal scales: the first study consists in detecting spatial configurations in the arrangement of funerary structures in a Bronze Age necropolis of central Mongolia. The purpose of the second study is to set up an image processing protocol applied to urban maps from different dates, permitting to detect the various architectonic orientations.

FINDING SPATIAL STRUCTURES IN THE ARRANGEMENT OF FUNERARY MONUMENTS ON A BRONZE AGE NECROPOLIS (CENTRAL MONGOLIA) WITH MATHEMATICAL MORPHOLOGY

The purpose of this study is to question the relevance of image analysis to resolve an issue belonging to archaeology and spatial analysis at the same time. Its aim is to detect spatial structures of the arrangement of funerary monuments building up the Bronze Age necropolis of Tsatsiin Ereg (figure 1A) in Central Mongolia, by the use of Mathematical Morphology. Indeed, the sacred and ceremonial dimensions of funerary complexes discovered by archaeologists can lead us to think that they had been arranged in a very particular way, following a certain number of rules which it would be interesting to reveal, by detecting visible or invisible spatial structures, and the spatial links which unite them. Three binary images have been built in order to realize the treatments under Micromorph software, on the basis of an initial satellite image (figure 1B) showing the localization and the shape of the tombs on a little part of the necropolis. The first image built shows the distribution of the tombs (figure 1C) on Tsatsiin Ereg slope. Two other images were built: two imaginary lines, the first one materialising the summit of Tsatsiin Ereg slope, and the second one separating it on two equal parts from the top to the bottom. This study's frame is constituted with four main objectives: the first is to determine if the tombs size and position on the slope is causally linked. Then, the purpose is to see if the tombs are distributed uniformly on the slope, or if some zones contain more tombs than others. The third question was to make appear the distance separating the tombs, according to their localization on the slope, and finally, to determine if these monuments are oriented in a particular way. Some preliminary treatments have first been realized: their aim was to build first images, which could be taken as a basis of some further manipulations. Even if their interpretation power is limited when you consider them independently, they will reveal all their subtlety simultaneously -this will be the final step of the analysis. The first step is to realise a granulometry test on the tombs: the purpose of this sieve is to sort out monuments according to their size, by taking them out a certain number of pixels -erosion-and then adding them the same number of pixels -dilation-on the resulting image. The variation of this pitch allowed us to build three classes: the big, medium and small tombs (figure 2A). A second granulometry test is then realised, in order to highlight the voids between the tombs, in other words the distances which separate them, to materialise their arrangement according to their proximity. The process is the opposite of what we did before: here, we first dilated and then eroded the tombs image of a determined pitch (figure 2B). We now attempted to reveal the zone where the tombs are the most numerous, relatively to the summit and the centre of the slope (figure 3), by cutting virtually this one in 20 meters large stripes. The summit image undergoes successive dilations of 7 pixels (equivalent to 20 meters), and each obtained image is superposed with the tombs image. The number of monuments contained on each dilated surface is counted and referenced. The same manipulations are now realised taking into account the image of the centre of the slope instead of the summit one. After these processes, we finally crossed the two obtained images, in order to localize the 400 meters square zone of the slope where the tombs are the most numerous. As a result of all these preliminary manipulations, we can observe that the biggest tombs are actually situated at the extremities of the slope, or even in the plain, and seem to delimit a "sacred space" in opposition with the "living space" of the actual nomad pastoralists. Their size increases gradually from the periphery towards the centre of the slope: this phenomenon could show the wiliness to highlight the most important monuments in terms of size and social hierarchy. It is furthermore possible to detect that the slope zone containing the greatest density of tombs (at 80 meters from the summit) is planer than its superior part -it is directly situated under the granitic tor-but stonier than the plain. This observation can be explained by some obvious practical reasons (less effort to build the tomb and access to first material) but also by the strength of the symbolic dimension of stone in Mongolian beliefs [START_REF] Ruhlmann | L'enterrement chez les Mongols contemporains : le cercueil, la tombe et la yourte miniature du mort[END_REF].

The second step of the analysis is to cross the different images obtained, in order to realise more interesting interpretations. Now, if we superpose the "skeleton" of the tombs (figure 4A), representing their general structure, and the image distinguishing the different size of the tombs, we notice that the arrangement of the tombs on the slope seems to form a pyramid (figure 4B), whose summit would be composed with the biggest A B Big Medium Small tombs, and its base with the smallest and medium monuments, maybe materializing the social hierarchy of the deceased. We also notice than the median of this triangle is materialised by the one of the branches of the delta formed by the tombs' skeleton, which confirms the observation of a central position of the biggest tombs. In addition, the little branches of the skeleton are perfectly parallel to the west part of the triangle, and to the bed of an ancient river. The sacred dimension of rivers in Mongolia [START_REF] Lavrillier | S'orienter avec les rivières chez les Evenks du Sud-est sibérien : Un système d'orientation spatial, identitaire et rituel[END_REF] could justify this structure, and allows us to emit the hypothesis that this element could have played an important role in the arrangement of the tombs.

Then, if we cross the image representing the 400 meters square zone containing the greatest tombs' density, and the "skeleton" formed by the smallest tombs (figure 5A), we notice that these two images are perfectly superposing: it means that the tombs contained in this part of the slope are the smallest. This peripheral space could be eventually seen as a zone of less importance: the presence of a great number of tombs could be only the effect of their small size and of the convenience of the field, and as the consequence of the possibility to build many of them in a restricted space, for the benefit of most important zones.

In order to interpret the distance between the tombs, we successively superposed the images comporting the three tomb size classes, with the one showing the sieve of their spacing (figures 5B 1 , B 2 , B 3 ). It is actually possible to notice that the space between the tombs seems not directly linked with their size, -each size class presents some tombs at the inside and at the outside of the closest spaces zonebut rather with their position on the slope: the tombs are more spaced in the lain. This observation could be possibility linked with the fact that the monuments we find in the plain are surrounded by small satellite structures -which we could not take into account in this study-whose building asks much available space, inexistent on the slope.

If we consider the orientation of the tombs, which Micromorph software is able to detect, it seems that all these structures are oriented towards the Tamir river -South-West. This observation reinforces the hypothesis about the importance of liquid element, and of the possibility of its role on the necropolis' building.

A B 1 A B B 2 B 3

ANCIENT URBAN PLANNING RECONSTRUCTION

The second study aims to identify ancient urban planning thanks to recent archaeological diggings and to high definition charts, showing constructions in detail: we think that some of the oldest buildings of that Fortress kept into their walls orientations of past urban planning.

The Fort Royal is located on the highest relief of the Sainte-Marguerite Island, the largest island of the Lérins archipelago (figure 6). Facing Cannes' shores, this ancient State Fortress has been built over the remains of an antique oppidum, of which evidence of existence have been found in 1972 during some emergency excavations (figure 7B). The well preserved traces of dwellings, streets or even sewers presented evidence of planned urbanisation (figure 7B) and archaeologists recognised the grid plan of hippodamian style [START_REF] Phoca | Architecture and city planning[END_REF]. Today, thanks to a very good experience of field, we wonder if nowadays buildings into the Fort Royal don't keep into their very orientation, their walls and situation into the limits of the Fort, the trace of previous urban or palatial planning. Common scientific knowledge incline towards the following chronology for the evolution of this site: an indigenous oppidum until the roman period with a change of the orientation of the buildings, opening on a 10 centuries period of darkness until a map drawn under Charles IX, showing a fortress recently up-dated to gun warfare and siege (figure 7A). Today's Fort is the result of Spanish and French constructions very well documented, starting in 1630's during the 30 years long war, but taking advantage of previous constructions obviously never chartered before.

The aim of this study is twofold: i) Showing that they are still some buildings and walls following ancient plans, pre-roman and roman in order to indicate to archaeologists the possible extension of the ancient sites, ii) Providing an image processing protocol which enables archaeologists to depict automatically the morphology of these urban plans.

In order to reconstitute the cities plans, archaeologists have used to extend the bits of the ways brought to light by the excavations or detected by remote sensing, by using a ruler and a square. Our method differs by the fact that on one hand, it is based on spatial analysis by image analysis and that on the other hand, it is carried out with various documents: a diggings plan and maps giving the current arrangement of the buildings and more precisely, the walls located within. In figure 8, all the buildings of the Fort have been depicted. Some buildings have been left apart from this study because they are historically well documented: we know who and how they've been built. They have been drawn with light grey color in figures 8 and 11.

We can observe two North-South directions corresponding to the alignment of the ruins of the oppidum and the ruins of roman buildings. Directions more or less perpendicular can be observed in different buildings (figure 8). We isolate 4 directions, more or less at right angle, with the North-South directions.

Figure 8:

The relevant building orientations N1, N2, E3, E4, E5, E6 and a focus on the archaeological diggings done at the North West corner of the Fort.

Those 6 directions vectors actually, have been set on gridded images, in order to make directional dilations using mathematical morphology algorithms (figure 9).

vector N1 dilation of N1 vector E3 dilation of E3 The result of intersections of two images gives a parallelogram the shape of which is analysed using a qualitative matching (figure 10). From that six vectors (two North/South, and four East/West), by Mathematical Morphology processing, we will try to get parallelograms showing the squareness of their angle, through dilation and intersection. The most square is the parallelogram, the most likely the directions match with an hippodamian grid urban plan, the indigenous one (Oppidum) with N1 and the roman one with N2.

Intersection between: dir. N1and E4 dir. N2and E5 dir. N1 and E5 The origins of the direction (the vector from the chart) on the map indicate the buildings and structures having this direction, enabling the spatial analysis of the Fort evolution and establishing a chronology of the successive constructions, or even their evolution and fate (figure 11). The processing of images shows two groups of positive results: N1 with E3 and E4, and N2 with E5, of which final parallelograms are almost perfectly square. According to the original directions found on the map, we determine the building possessing those directions. Apart the barracks of the XVII th century, three groups of constructions appear in figure 11: the most southern group of buildings, which seem to belong to the most ancient plan of the oppidum, the North West group of buildings, perfectly square, belonging to the roman period (palatial?), and a vey interesting group of constructions, called "the castle", where we find all the directions from every period. This castle seems to have been built, rebuilt and extended many times during 2000 years, keeping the directions of past urban plans.

It is important to notice that all these groups of constructions are situated on the margins of the Fort, close to the walls of the Fortress. This would confirm that the Spanish troops destroyed drastically the centre of the Fortress to flatten the ground in order to built barracks that finally the French did.

CONCLUSIONS AND DISCUSSION

The two presented studies belong to spatial archaeology. This field, which still remains few widespread, aims to distinguish, on the basis of the spatial and spatio-temporal marks given by archaeological data, networks of chronological relations but also spatial and dynamic structures, and the way these relations vary through space. The detected spatial organisation has to contribute to retrace the functioning of past anthroposystems.

Our studies position themselves into this research frame, but are distinct from the previous works of spatial archaeology for three reasons. They are conducted at the micro scales of an archaeological site and of a little Mediterranean city, while the works realised until then were realised at a small scale, the regions' one. Our spatial analysis approach favours the morphological aspect of the objects considered. The objective is indeed to detect invisible spatial structures from visible shapes, to embody the detected structures and to link them with the processes which gave them birth. Finally, image analysis, with the help of Mathematical Morphology Algorithms, was not used to realise a remote-sensing analysis but to find spatio-morphological structures [START_REF] Voiron | Analyse spatiale et analyse d'images[END_REF].

The two proposed studies take into account types of relations existing between the objects through space. Each one presents particular scientific interests.

With the help of Mathematical Morphology, we could detect some underlying spatial structures of the monuments' disposition on the Bronze Age necropolis. Some hints about the organisation of the tombs, their position on the slope, and their link with local topography have been revealed for the first time at a similar scale, on a funerary site, and by the use of Mathematical Morphology. Further perspectives can be here envisaged: realising similar treatments on the totality of the necropolis, and on other necropoleis, could be interesting in order to compare the organisation obtained. This would permit to create an exhaustive organisation model of Mongolian Bronze Age necropoleis, with a real explicative dimension, as the lack of comparison of the present study gives it only a descriptive one. This model would be one of the elements built to understand space perceptions of Mongolian nomadic groups, and its evolutions through time and space.

The study about the Sainte-Marguerite fortress presents two methodological originalities: i) the research was conducted on the basis of current architectural data, and not on archaeological remains, ii) by considering the directions revealed by a little zone of excavations, the approach permits to determine the building groups belonging to the hippodamian plan and to estimate the extent of the oppidum. From the hypothesis of this plan, the obtained results show the membership of some building groups to really ancient plans, from the oldest period of the site. They open interesting historical research perspectives about the existence of an antique city on the Sainte-Marguerite Island.

Figure 1 :

 1 Figure 1: From Tsatsiin Ereg tombs picture (A) and satellite image (B) to build a binary image (C)

Figure 2 :

 2 Figure 2: Size classes of the tombs (A) and space between them (B)

Figure 3 :

 3 Figure 3: 400 m² zone of higher density of tombs, and their link with topography

Figure 4 :

 4 Figure 4: Skeleton of the tombs (A) and pyramidal structure of the necropolis (B)

Figure 5 :

 5 Figure 5: A: small tombs and zone of higher density of monuments. B: size of tombs and space between them, B 1: big tombs, B 2 medium tombs, B 3 small tombs

Figure 6 :

 6 Figure 6: The fortress, where have been found remains of an ancient oppidum.

Figure 7A :

 7A Figure 7A: Attempt to reconstruct the Charles IX fortress plan according to spanish planning; Figure 7B: Actual findings and one of the supposed directions: vector N1. After G.Vindry's mapping, 1980's

Figure 9 :

 9 Figure 9: Examples of directional dilations

Figure 10 :

 10 Figure 10: Qualitative matching of the parallelograms: analysis of the squareness of the angle

Figure 11 :

 11 Figure 11: The chronology of the successive constructions


Ilaria Rivolta

Alice Panariti

Maddalena Collini

Barbara Lettiero

Laura D'alfonso

Laura Sironi

Giuseppe Miserocchi

Giuseppe Chirico
email: giuseppe.chirico@mib.infn.it

A biophysical model of intracellular distribution and perinuclear accumulation of particulate matter

We have measured in human alveolar cells the cytoplasmic distribution of the fluorophore coumarin-6 carried by Solid Lipid Nanoparticles (SLNs) and observed a perinuclear accumulation of the fluorescence that can be described by a single exponential growth along an ideal line joining the plasma membrane to the nuclear border and by a sigmoidal relationship as a function of time. Intracellular distribution was affected by lowering the temperature from 37 to 4°C and by the disruption of cytoskeleton by cytochalasin D, but it was minimally perturbed by the inhibition of ATP dependent molecular motors. A biophysical model was developed for an accumulation of loaded particles against a diffusion gradient based on a mean field interaction energy, whose origin we ascribe to the actin structure of the cytoskeleton. The estimated value for the load diffusion coefficient was four and two orders of magnitude less than that of free coumarin-6 and of SLNs in aqueous solutions, respectively, suggesting that the load moves within the cell cytoplasm in a form still reminiscent of the nanocarrier structure.

A C C E P T E D M A N U S C R I P T

Introduction

The uptake of extracellular material across the plasma membrane of eukaryotic cells is well described and characterized by a variety of mechanisms going from the passive diffusion to the active transport [START_REF] Sugano | Coexistence of passive and carrier-mediated processes in drug transport[END_REF]. The former, defined as a concentration gradient-driven mass transport of a compound, is considered the dominant route [START_REF] Sugano | Coexistence of passive and carrier-mediated processes in drug transport[END_REF]. The latter, including the endocytosis mediated processes, is in charge of taking up larger molecules and molecular complexes.

Once entered the cells, the intracellular traffic and distribution of substances that do not follow the endocytotic pathway are related to aspecific movements or microtubule-based active mechanisms [START_REF] Lai | Characterization of the intracellular dynamics of a non-degradative pathway accessed by polymer nanoparticles[END_REF][START_REF] Suh | Hanes Efficient active transport of gene nanocarriers to the cell nucleus[END_REF].

Furthermore, exogenous material of various origin tends to accumulate in the perinuclear region after cellular uptake [START_REF] Lai | Characterization of the intracellular dynamics of a non-degradative pathway accessed by polymer nanoparticles[END_REF][START_REF] Chavanpatil | Surfactant-polymer nanoparticles overcome Pglycoprotein-mediated drug efflux[END_REF][START_REF] Chakraborty | Cholera toxin B conjugated quantum dots for live cell labeling[END_REF][START_REF] Lai | Privileged delivery of polymer nanoparticles to the perinuclear region of live cells via non-clathrin, non-degradative pathway[END_REF][START_REF] Gemeinhart | Cellular fate of a modular DNA delivery system mediated by silica nanoparticles[END_REF][START_REF] Schulze | Compaction of Cell Shape Occurs Before Decrease of Elasticity in CHO-K1 Cells Treated With Actin Cytoskeleton Disrupting Drug Cytochalasin D[END_REF][START_REF] Wei | Preparation of Hierarchical Hollow CaCO3 Particles and the Application as Anticancer Drug Carrier[END_REF]. In a recent study involving human alveolar epithelial cell line, we confirmed this observation also for Solid Lipid Nanoparticles, SLNs, loaded with the fluorophore coumarin-6 (c-SLNs) and we hypothesized a pathway for their uptake [START_REF] Rivolta | Cellular uptake of coumarin-6 as a model drug loaded in Solid Lipid Nanoparticles (SLN)[END_REF].

Here we want to analyse, from a biophysical point of view, the cytoplasmic distribution of the fluorophore carried by the SLNs, and the way it is affected by the incubation temperature, by the state of the cytoskeleton and the efficiency of the molecular motors. We focused our efforts on the perinuclear accumulation of coumarin-6, taken here as a model hydrophobic compound and develop a biophysical model of the fluorophore distribution in the cell that assumes an interaction energy between the fluorophore carried by the SLNs and cellular components. This interaction energy counterbalances the free thermal diffusion of the molecules within the cytoplasm and could be specifically related to the cytoskeleton or to active transport mechanisms such as those related to the molecular motors. The modelling of the perinuclear accumulation of nanoparticulate that we outline here, may be of potential interest for the biomedical use of nanoparticulate structures, in particular for the delivery of drugs or genetic material in the nuclear region.

Materials and methods

Nanoparticles. SLNs were produced by NANOVECTOR, by choosing tripalmitin as lipidic matrix. SLNs have been loaded with coumarin-6 (c-SLNs) to allow their visualization by means of fluorescence microscopy (λ exc =450 nm, λ em =505 nm). The molar ratio coumarin-6:tripalmitin was about 80 (NANOVECTOR, personal communication). The average diameter of heat sterilised c-SLNs dispersion,

A C C E P T E D M A N U S C R I P T

determined by Dynamic Light Scattering (Brookhaven 90Plus system, Brookhaven Instruments Corp.), was 116±15 nm, and remained essentially stable (≤130 nm) either when the particles were dispersed in buffer or in cell culture medium supplemented with 1% FBS. The Z potential of the SLNs was -24±2 mV (Brookhaven 90Plus system, Brookhaven Instruments Corp).

In experiments where coumarin-6 was used as free drug, a stock solution of 20 µg/ml in DMSO was prepared and diluted at the desired working concentration.

Cell culture. A30 cells [START_REF] Botto | Hypoxia-induced modifications in plasma membranes and lipid microdomains in A549 cells and primary human alveolar cells[END_REF] were grown on Petri dishes in DMEM medium supplied with 10% FBS, 1% of L-Glutamine and 1% of Penicillin/Streptomycin and incubated in a controlled environment at 37°C with 5% CO 2 . During the experiments the cells were incubated with medium supplemented with 1% FBS to prevent NPs aggregation.

Effect of temperature on NPs intracellular distribution. We estimated the cytoplasmic concentration of coumarin-6 by monitoring the fluorescence distribution in cells exposed to c-SLNs (0.01 mg/ml) for 45 min either at 37°C or at 4°C. In the latter case the cells were pre-cooled for 30 min at 4°C. At the end of the incubation period, cells were immediately fixed in paraformaldehyde 4% in PBS.

Image acquisition. The experimental set up consisted of a wide field NIKON Eclipse FN1 equipped with a 63X objective, numerical aperture 1.2, motorized table (Prior Scientific, Rockland, MA, USA), Photometric coolSNAP ES Digital Camera and Metamorph (Molecular Device, Downington, PA, USA) software for image acquisition and analysis. We used fields of view that encompassed at least 5 cells for analysis. The fluorescence signal form the c-SLNs was discriminated by a band pass filter (λ em =505 nm) and the images were averaged over an exposure time of 400 milliseconds.

Starting from the top of the cell, 40 Z planes were acquired along the Z axis at 0.5 µm inter-distance.

We typically considered the overall fluorescence acquired in the wide field. In specific cases, we used the fluorescence estimated on a Z plane corresponding to the mid cell height, obtained through a deconvolution process (AutoDeblur, Media Cybernetics, Bethesda, MA, USA). 2-deoxy-D-glucose (DDG) [START_REF] Tsuruta | Microfilament-dependent movement of the β3 integrin subunit within focal contacts of endothelial cells[END_REF]. After this pre-treatment, the cells were incubated also with c-SLN (0.01 mg/ml) for additional 45 minute at 37°C. At the end of the incubation time the intracellular fluorescence distribution was monitored and the images were acquired and analysed as described. Nuclei were labelled with DAPI 1 µM.

Immunocytochemistry. Cells plated on glass coverslips were fixed with paraformaldehyde 4% in PBS at room temperature for 20 min, washed three times in PBS, LS (Low Salt PBS) and in HS (High Salt PBS)

respectively and then permeabilized with digitonin 0.01% in GDB for 30 min. Fixed and permeabilized cells were incubated with phalloidin conjugated with Texas RED (dil 1:200. Molecular Probes) for actin filaments staining or primary antibody antimyosin (dil 1:500, Covance) diluted in GDB at RT for 2 hours. After washing with HS for three times, cells labeled for myosin were incubated with Alexa conjugated secondary antibody diluted in GDB (1:100). Cells were then washed with HS AND LS three times respectively. DAPI was used to stain cell nuclei at a concentration of 1µM in PBS for 5 min. Coverslips were mounted with glycerol.

Statistical analysis: Statistical analysis were carried out by t-test and the significance level was set at p<0.001 (indicated by symbol * or # in the figures).

Results

The intracellular fluorescence distribution of the fluorophores loaded in the c-SLNs was measured on monolayers of human lung epithelial A30 cells, very similar to the more studied A549, [START_REF] Botto | Hypoxia-induced modifications in plasma membranes and lipid microdomains in A549 cells and primary human alveolar cells[END_REF] The cells were exposed to an estimated concentration of 1.8 x 10 10 c-SLNs/ml, corresponding to a tripalmitin concentration ≅ 0.01 mg/ml, one order of magnitude lower than the toxic concentration (0.2 mg/ml) estimated by LDH release assay and MTT test. [START_REF] Rivolta | Cellular uptake of coumarin-6 as a model drug loaded in Solid Lipid Nanoparticles (SLN)[END_REF] Cytoplasmic distribution of c-SLNs.

We followed the kinetics of c-SLNs intracellular distribution by monitoring the ROI's average fluorescence signal along the EL line at 3, 9, 15 and 30 minutes after the addition of NPs to the cell culture (T=32°C during the images acquisition, Fig. 2). A minimum of seven cells was used to average the data. The intracellular fluorescence increased as a function of the distance from the plasma membrane, reaching the maximum value in the proximity of the nuclear membrane (Fig. 2A). The fluorescence spatial distribution, F(x), observed during the uptake process and at equilibrium (Fig. 2A), can be described by an exponential growth as a function of the distance, x, from the cell membrane along the EL according to:

( ) [ ] L x A F x F / 100 exp ) ( 0 - + = (1)
It is important to notice that the cell membrane value of the fluorescence is

[ ] L A F F / 100 exp ) 0 ( 0 - + ≅ .
The best fit parameters of the solid lines reported in Fig. 2A are: A(15 min)=135±40 [arbitrary units, a.u.], L(15min)=26±5% and F 0 (15min)=12±5 [a.u.], and A(30min)=330±50 [a.u.], L(30min)=29±3% and

F 0 (30min)=10±2 [a.u.].
The only relevant difference in the time evolution of the fluorescence distribution concerns the inhomogeneity ratio A/F 0 , that offers a quantification of the perinuclear accumulation of the fluorophore. This ratio varies from 12±7 to 25±7 when passing from 15min to 30min of c-SLNs exposure time, indicating a progressive perinuclear accumulation. The fluorescence signal followed in time at x=50% and 75% along the EL (Fig. 2B) increases at first gradually and seems to reach a saturation value that depends on the position along the EL after approximately 30 min (Fig. 2B). Such bimodal behaviour, can be fit to the following sigmoidal function:

A C C E P T E D M A N U S C R I P T ( ) ( ) ( ) δ / exp 1 ) ( 2 / 1 0 1 0 t t f f f t f - + - + = (2)
The mid-point time, 2 / 1 t , is relatively insensitive to the position along the EL line, 2 / 1 t ≅18min (Fig. 2B). The transition is instead sharper (δ(50%)≅3.5min compared to δ(75%)≅7min) as we move closer to the cell membrane through which the c-SLNs carried fluorophore is internalized. This observation is an indication that the kinetics of the fluorophore distribution in the cytoplasm is affected also by the internalization process, as discussed later.

Effect of temperature

In order to evaluate the effect of the temperature on the distribution of the fluorophore, the cells were exposed to c-SLNs for 45 minutes either at 4°C or 37°C, fixed as described in the Materials and Methods, and the distribution of the fluorophore was then measured (Fig. 3). The absolute value of the intracellular fluorescence is lower for cells incubated with the c-SLNs at the lowest temperature: we find F(75%)=155±10 [a.u.] at T=37 o C and F(75%)=80±12 [a.u.] at T=4 o C. Changes in the cell thickness due to temperature do not affect substantially this conclusion since the emission F(75%) computed on the deconvoluted images (focal plane at 50% of the cell thickness) confirms a marked difference between the two temperatures investigated (350±80 [a.u.] compared to 262±6 [a.u.] for 37°C and 4°C respectively, p<0.001, 22 cells). Moreover, the total amount of fluorophore loaded by the cell at T=37 o C, as measured by the integral under the lines reported in Fig. 3, is approximately twice (8800 [a.u.]) that measured for T=4 o C (4800 [a.u.]).

The intensity normalized to its maximum value increases with the distance from the cell membrane independent of the experimental temperature, and it can still be described by Eq. 1. The best fit decay length values are L=31±1% and L=43±2% for T=37 o C and T=4 o C, respectively.

Role of molecular motors in determining the intracellular fluorescence distribution.

The reduction of the cytoplasmic fluorescence intensity observed at 4°C might reflect the temperature dependent inhibition of active ATP-dependent mechanisms, the so called "molecular motors" that are known to promote cell intracellular transport [START_REF] Smith | Model of motor assisted transport of intracellular particles[END_REF]. In order to assess the role of the molecular

A C C E P T E D

M A N U S C R I P T motors, we depleted cells from ATP, applying NaN 3 or DDG before and during the incubation with the c-SLN. Despite the inhibition of energy production by the metabolic poisons, as Fig. 4 shows, we still found the characteristic intracellular distribution and perinuclear accumulation of the fluorophore.

Role of cytoskeleton on the intracellular distribution

In order to probe the role of cytoskeleton on the steady state fluorescence distribution, we incubated the cells with cytochalasin D (cytoD) at a concentration (2.0 µM for 1 hour) [START_REF] Schulze | Compaction of Cell Shape Occurs Before Decrease of Elasticity in CHO-K1 Cells Treated With Actin Cytoskeleton Disrupting Drug Cytochalasin D[END_REF] suitable to disintegrate actin bundles but low enough to avoid cytotoxic damage, as measured by a lactate dehydrogenase releasing assay (data not shown). We found that the treatment with cytoD led to a more homogeneous pattern of the fluorescence signal inside the cells, suggesting that the fluorophore undergoes a redistribution in the cytoplasm (Fig. 5A,A'). While in the physiological condition the perinuclear/cytoplasmic fluorescence ratio increased 5 fold, in the CytoD treated cells, this ratio reaches now only a plateu value ≅1.7 (Fig. 5E). A similar ratio, ≅1.5, was obtained when CytoD was applied before and maintained during the incubation of cells with c-SLN (Fig. 5E). It is worth noting that, upon cytoD treatment, not only the actin filaments (Fig. 5B,B') but also the myosin fibres were distrupted in the cells (Fig. 5C,C'), while the microtubules remained almost intact (Fig. 5D,D').

Discussion

Our data indicate that when A30 epithelial cells are exposed to c-SLNs, the intracellular fluorescence signal progressively increases (20-80 fold) along the EL line with a monotonic accumulation in the perinuclear region. As can be observed from Figs. 2 and3, at least two processes are active in determining this inhomogeneous distribution, with a cross-over at x≅30% along the EL that depends on the incubation temperature (see Fig. 3). The experiments with metabolic poisons (NaN 3 and DDG) do not substantially modify the cytoplasmic fluorescence distribution (Fig. 4) In fact, as from the plot in Fig. 4, the effect of NaN 3

and DDG correspond to maximum decreases, with respect to the control cell line, of 14 ± 18 % and 22 ± 26 %, respectively. Though these values indicate a slight systematic decrease of the perinuclear accumulation effect when treating the cells with metabolic poisons, this effect is still largely within the experimental uncertainties. The increase in the fluorescence signal when approaching the nuclear membrane is still relevant, being ≅ 7, 6 and 5 for the control line and the lines treated with NaN 3 and DDG, respectively.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 9
Therefore, the data reported in Fig. 4 do not allow us to assume a clear direct effect of the molecular motors on the perinuclear accumulation of the fluorophore at least in the present case in which SLNs have been used as extracellular carriers. However, it is worth noting that a minor direct role of the molecular motors in the generation of the perinuclear accumulation cannot be completely ruled out. [START_REF] Tabb | Transport of ER vesicles on actin filaments in neurons by myosin V[END_REF] On the other hand the treatment with cytoD does affect the fluorescence cytoplasma distribution indicating a major role of the cytoskeleton. The observed effect of cytoD on myosin (Fig. 5C,C') seems to imply an indirect effect of cytoD on the fluorescence distribution mediated by myosin related to its role as a component of the cellular stress fibers. [START_REF] Luo | A multi-modular tensegrity model of an actin stress fiber[END_REF] A direct effect of myosin acting as a molecular motor seems to be excluded due to the very limited effect of metabolic poisons on the fluorescence distribution (Fig. 4D).

It appears therefore reasonable to develop a physical model for the perinuclear accumulation based on the role of the cytoskeleton, whose integrity is crucial to favour the fluorophore accumulation close to the nuclear membrane, thus against a diffusion gradient. With our choice of normalized distance, x, along the EL line, the solution of the diffusion equation for the stationary case can be written as (see Supplementary

Material, "Derivation of Eq.3"):

( )       - ∆ = 100 100 exp ) ( x T K U c x c B N eq (3)
The comparison of Eq.1 to Eq.3 indicates that the best fit decay length, L, gives the strength of the average interaction energy gain, <∆U> = <U cell > -<U nucleus >, that the fluorophore suffers when moving from the cell membrane to the nuclear membrane ( 0 ≥ ∆U

): L T K U B 100 = ∆ (4)
The interaction energy gain ∆U is equivalent to the action of an effective force field acting on the fluorophore over the thickness of the cytoplasm. This force field may be related to a difference in pressure between cytoplasm areas close to the cell and to the nuclear membrane. However, since from the data

A C C E P T E D M A N U S C R I P T

ACCEPTED MANUSCRIPT 10 analysis (see Eqs.3, 4) we have a direct access only to the overall interaction energy gain that corresponds to this force field, we have preferred to evaluate the degree of perinuclear accumulation (i.e. the exponent in Eq. 3) in terms of the energy gain ∆U.

The L values reported in Table I correspond to a decrease of 0.9±0.14 T K B units of U ∆ , when decreasing the temperature from T=37 o C to T=4 o C. This may reflect a change in the cytoskeleton structure and/or dynamics that affects the fluorophore distribution.

By analyzing Eq.1 we gain also information on the relative perinuclear concentration of loaded dye at the equilibrium. In fact, if the perinuclear and the cell concentrations of the fluorophore are c N and c C , respectively, the excess perinuclear concentration, ∆C = (c N -c C )/c C , can be related to the area, A, under the fluorescence distribution curves reported in Fig. 3 (normalized to the background level F 0 ) and the decay length, as (see Supplementary Material, section "Derivation of Eq.5"):

( ) ( ) L A c c c C C N C 100 1 + = - = ∆ (5)
The value of ∆ C estimated on the plots of the fluorescence distribution (see Table I), indicates that the perinuclear excess of concentration increases by approximately 55% when increasing the temperature from

T=4 o C to T=37 o C.
The limited changes in the fluorophore internalization observed when decreasing the cell temperature, suggest that endocytotic pathways, that are sensibly dependent on the temperature [START_REF] Weigel | Temperature dependence of endocytosis mediated by the asialoglycoprotein receptor in isolated rat hepatocytes. Evidence for two potentially rate-limiting steps[END_REF][START_REF] Wolkers | Temperature dependence of fluid phase endocytosis coincides with membrane properties of pig platelets[END_REF] in the range 4-37 o C, are only partially effective in the upload process of the c-SLNs and that the internalization is also determined by a direct interaction between the membrane and the c-SLNs phospholipids. This picture agrees with previous analyses [START_REF] Rivolta | Cellular uptake of coumarin-6 as a model drug loaded in Solid Lipid Nanoparticles (SLN)[END_REF] that showed that the internalization of the c-SLNs carried fluorophore occurs with the modification of the lipid (SLNs) physical and chemical thermodynamic parameters, probably due to their interaction with phospho-lipids (membrane). Indeed, calorimetric studies indicate [START_REF] Rivolta | Cellular uptake of coumarin-6 as a model drug loaded in Solid Lipid Nanoparticles (SLN)[END_REF] that the c-SLNs interacting with the cell membrane, only partially preserve their native structure. A fraction of the intracellular fluorescence could then result from coumarin-6 released from the SLNs upon their interaction with the cell membrane.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 11

The treatment with CytoD markedly reduced the inhomogeneity of the intracellular distribution (Fig. 5 A'): L was as large as 80%, almost three times larger than the value measured in the absence of CytoD. This finding corresponds to a decreased energy gain (see Eq.4, <∆U>≅K B T compared to <∆U>≅3 K B T prior to the CytoD treatment, Table I). It must be noted that the analysis proposed here, is not substantially affected by the assumption of a one dimensional diffusion process (see "The distribution of particles in a polar coordinate system" in the Supplementary Materials).

Upload kinetics.

Concerning the origin of the hypothesized NPs-cytoskeleton interaction, we suggest the possibility of local overpressures generated by the cytoskeleton dynamics. The cytoplasm of mammalian cells possesses actin and myosin structures arranged as stress fibers and organized to generate contractile structures responsible for cellular tension [START_REF] Luo | A multi-modular tensegrity model of an actin stress fiber[END_REF][START_REF] Goeckeler | Nonmuscle myosin II is responsible for maintaining endothelial cell basal tone and stress fiber integrity[END_REF][START_REF] Cai | Force propagation across cells: mechanical coherence of dynamic cytoskeletons[END_REF]. These fibers in intact living cells, may shorten to generate tension as well as elongate, thereby determining also cellular elasticity. One possible hypothesis about the generation of the fluorescence perinuclear accumulation in the cells may rely on the presence of these forces [START_REF] Schulze | Compaction of Cell Shape Occurs Before Decrease of Elasticity in CHO-K1 Cells Treated With Actin Cytoskeleton Disrupting Drug Cytochalasin D[END_REF] that should be relaxed when treating the cells with cytoskeleton destabilising drugs, such as CytoD, as indeed is found here (Fig. 5A,A').

We develop and discuss a simple model based on the diffusion of the fluorophore in the cytoplasm under the action of a mean field interaction force. If we assume the following initial condition,

) ( ) 0 , ( x c x c ext δ =
, that describes the case in which the extracellular fluorophore concentration is constant, c ext at time t=0, the intracellular fluorophore concentration follows a simple exponential law (Eq.B8 in the Supplementary Material):

[ ] 0 100 , 0 exp ) ( ) ( ) , ( Γ - + = = t x b x a t x c x (6)
where a(x) and b(x) are parameters that depend on the initial fluorophore concentration at the membrane, c ext , and on the position along the EL line. The relaxation rate Γ 0 is related to the effective energy gain, <∆U>, the fluorophore diffusion coefficient, D, and the average cell thickness, ∆ (from the cell to the nuclear membrane) as:

A C C E P T E D M A N U S C R I P T 2 2 0 4         ∆ ∆ = Γ T K U D B (7)
A more realistic model takes into account that the SLNs enter the cell membrane at a characteristic rate, γ, which depends on the details of the nanoparticle interaction with the membrane. Moreover, the local extracellular SLNs concentration, c ext , might decay slowly (with a second relaxation rate Γ) with time due to the continuous internalization of the SLNs by all the cells in the culture. Within these two hypotheses the initial conditions for the intracellular fluorophore concentration close to the cell membrane, is actually

described by: [ ] ( ) [ ] Γ - - - = t t c t c ext exp exp 1 ) , 0 ( γ .
The first exponential growth (the term within the round parentheses) accounts for the increase in the concentration of the fluorophore close to the cell membrane, due to the internalization process. The second exponential decay, with relaxation rate Γ, accounts instead for the local depletion of the SLNs in the cell culture again at the cell membrane. Within such model, Eq.6 changes to a sum of exponential decays (see Supplementary Material) that, with the assumption that Γ << γ, can be written as:

[ ] [ ] ( ) [ ] ( ) γ γ t x c t t B t x c - - + Γ - - - ≅ ∞ exp 1 ) ( exp exp ) , ( 0 (8)
From Eq.8, we can obtain the fluorescence signal along the EL as a function of time: 2B) is determined by the relaxation rate γ, and the slope of the fluorescence signal at larger times is related to the rate Γ 0 . The marked sigmoidal shape of the kinetics derives then from the difference between the two relaxation rates, Γ 0 and γ, that account for the intracellular diffusion and the internalization of the fluorophore, respectively.

[ ] [ ] ( ) [ ] ( ) γ γ t F t t b x c t x c F t x F - - + Γ - - - = ≅ ∞ ∞ ∞ exp 1 exp exp ) ( / ) , ( )

A C C E P T E D

M A N U S C R I P T

ACCEPTED MANUSCRIPT 13

From the diffusion relaxation rate, Γ 0 , we can estimate a value of the diffusion coefficient of the fluorophore in the cytoplasm. By taking an average value of the energy loss, ∆U/K B T≅2.8 (Table 1, without CytoD) and by assuming an average size of 5 µm for the cell cytoplasm (∆≅5 µm), we estimate D≅0.02 µm 2 /s. This value is more than four orders of magnitude lower than the coumarin-6 diffusion in aqueous solution at room temperature (D coum ≅ 280 µm 2 /s), and about two orders of magnitude lower than the diffusion coefficient of the SLNs in the same conditions. Literature data on the diffusion coefficient of simple fluorophore in the cell cytoplasm and on the cell membrane [START_REF] Schwille | Molecular dynamics in living cells observed by fluorescence correlation spectroscopy with one-and two-photon excitation[END_REF] indicate that the fluorophore diffusion decreases by almost two orders of magnitudes in the cytoplasm and about four orders of magnitude on the cell membrane. In the view of these data, our estimate of the diffusion coefficient associated with the coumarin-6 fluorescence signal suggests that the fluorophore moves within the cell cytoplasm in a form that is still reminiscent of the nanocarrier structure and not in the form of free dye. This hypothesis is also supported by the results of calorimetric studies [START_REF] Rivolta | Cellular uptake of coumarin-6 as a model drug loaded in Solid Lipid Nanoparticles (SLN)[END_REF] from which it seems that the c-SLNs preserve partially their native structure after entering the cell.

In conclusion, the analysis of the fluorescence microscopy data indicates that the combined action of the diffusion of the dye in the cytoplasm and of a mean force field acting on the dye induces its accumulation around the nucleus. Data collected in conditions in which the cytoskeleton is perturbed suggest that these forces, that correspond to an energy gain of the order of few K B T from the cell to the nuclear membrane, originate from the cytoskeleton structure and dynamics but are not directly related to the action of molecular motors. The biophysical model that we propose and discuss analytically in the Supplementary Materials, allows us to quantify the interaction energy gain of the fluorophore from the cell to the nucleus membrane.

This study also sets a warning about non-specific effects in the intracellular distribution of drugs due to the cytoskeleton structure and state. The signal is shown in arbitrary units.

A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T

  Drugs treatment: The role of cytoskeleton in affecting fluorescence cytoplasmic distribution was addressed by treating cells, previously exposed to c-SLNs ( 0.01 mg/ml for 45 minute at 37°C) and analysing the perinuclear fluorescence that these cells displayed when incubated with cytochalasin D (CytoD) at a concentration of 2 µM for 1 hour at 37°C. In other experiments, before the exposure to the NPs, cellular A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 5 energy stores were depleted by a pre-treatment for 1h at 37°C with 0.2 mM sodium azide (NaN 3 ) or 50 mM

  an important component of the blood-air barrier and a sensitive site for particulate-body interaction. The analysis of the coumarin-6 fluorescence distribution was performed by selecting, on individual cells, a line (emission line, EL) running from the plasma to the nuclear membrane (Fig 1, image) and four cytoplasmic regions of interest (10 pixel square ROIs) positioned along such EL at 25%, 50% and 75% of the total length. The location of the plasma membrane was obtained by saturating the colour levels to half the maximum colour depth in the images. We verified that this allowed us to locate the cell membrane by comparing with images in which the actin filaments were specifically labelled. Since the width of the cell cytoplasm varies among the cells we needed to define a normalized distance along the EL in order to compare and average the A C C E P T E D M A N U S C R I P T fluorophore distributions measured on different cells. Therefore in the following we will refer to the distance along the EL as x = 100 (r-r N )/(r C -r N ), where r N , r C and r are the positions of the nuclear and the cell membrane and the selected ROI along the EL (x=100% indicates the nuclear membrane).

, ( 0 ..

 0 We have globally fitted the two data sets in Fig.2Bto Eq.8 by assuming common values of 0 Γ and γ , whose best fit values are: The time delay observed in the rise of the fluorophore concentration (Fig.

Figure 1 .

 1 Figure 1.Image: an Emission Line (EL) is drawn on a representative cell image acquired after 30 min of c-SLNs incubation (λ exc =488 nm, integration time 400 ms). The EL runs from the cell to the perinuclear region and four different ROIs (at 0, 1, 2.4, 3.6 µm along the EL) on which the fluorescence analysis was carried out, are marked on the image. Plot: we report the representative traces of the fluorescent signal collected on a single cell along the EL and recorded at increasing incubation time from bottom to top (0, 9, 15 and 30 min).

Figure 2 .t

 2 Figure 2. Panel A: fluorescence intensity as a function of the distance (estimated in normalized units ( ) ( ) N C N r r r r x --= / 100 ) from the plasma membrane along the EL for 15 min (open squares) and 30 min

Figure 3 .

 3 Figure 3. Main panel: effect of the c-SLN incubation temperature decrease on the intracellular fluorescence: T=37°C (closed circles) and T=4°C (open circles). The data result from the average over seven cells and the error bars represent the standard deviations. The solid lines are the best fit of Eq. 1 to the data. The inset reports the residuals of the data to the best fit lines (solid line for T=4 o C and dashed line for T=37 o C).

A

  C C E P T E D M A N U S C R I P T

Figure 4 .

 4 Figure 4. Panel A: intracellular fluorescence distribution after exposure of A30 to c-SLN for 45 min at 37°C in control condition. Panel B and C: Intracellular fluorescence distribution after treatment with the metabolic poisons NaN 3 and DDG, respectively. Panel D: quantification of the fluorescence intensity as a function of the distance from the plasma membrane along the EL in the three conditions presented above: black squares (control cell line), red circles (NaN 3 treated cell line) and green triangles (DDG treated cell line). The solid lines (same color code as the symbols) are the best fit to Eq. 1.

Figure 5 .

 5 Figure 5. Typical images acquired on A30 cells in control condition (CTRL) and after treatment with CytoD 2 µM for 1 h (+CytoD). Fluorescence distribution in A30 cells after 30 min of incubation with c-SLNs before (A) and after CytoD treatment, (A'). The treatment with CytoD breaks the major part of actin (B vs B') and myosin (C vs C') filaments in the cells, while microtubules integrity was preserved (D vs D'). The horizontal white bar corresponds to 10 µm. Actin and myosin filaments are labelled in red, microtubules are labelled in green, the cell nuclei in blue. Panel E: the plot represents the trend of the fluorescence as a function of the distance from the plasma membrane for the control (closed circles), the samples treated with CytoD after incubation with c-SLN (open circles), the samples treated with CytoD before incubation with c-SLN (open squares). All data were normalized to the fluorescence signal measured at EL=25%. The data result from the average over ten cells and the error bars represent the standard deviations. Statistical analysis was carried out by t-test and the significance level relative to control was set at p<0.001 (indicated by symbol # * in the plot).

Acknowledgements

The research was partially funded by the European Community Contract STREP N° LSHB-CT-2006-037639-BONSAI (Bio-imaging with Smart Functional Nanoparticles) to G.M. and by the European Community contract No: 201842, ENCITE (European Network for Cell Imaging and Tracking Expertise) to G.C. The authors thank NANOVECTOR for having kindly provided SLNs.

Research highlights

We have measured the cytoplasmic distribution of the coumarin-6 carried by Solid Lipid Nanoparticles in human alveolar cells and observed a perinuclear accumulation of the fluorescence that we model with the a mean field interaction of the load with the cytoskeleton. Experiments as a function of temperature, the cytoskeleton state and metabolic poisons validate this analysis.

A C C E P T E D M A N U S C R I P T

Online supplementary material:

1. Derivation of the Intracellular Equilibrium Load Distribution, Eq. 3.

We derive the equilibrium distribution of the load assuming a Cartesian coordinate system and a constant average force field between the cell and the nuclear membrane.

2. Derivation of the Total Intracellular Load, Eq. 5.

We derive a possible measurement of the total intracellular load from the computation of the integral of the fluorescence signal along the emission line.

Time Evolution of the Fluorescence

During the Uptake: Eq.8.

We derive a simplified expression of the time dependence of the intracellular load concentration observed during the cellular uptake.

The Distribution of Particles in a Polar Coordinate System.

We discuss the differences of the equilibrium Intracellular distribution of the load when assuming a polar and a Cartesian coordinate system.

Footnote

Abbreviations list

Arbitrary units (a.


Elissar Nasreddine
email: elissar.nasreddine@math.univ-toulouse.fr

A model of individual clustering with vanishing diffusion

We consider a model of individual clustering with two specific reproduction rates and small diffusion parameter in one space dimension. It consists of a drift-diffusion equation for the population density coupled to an elliptic equation for the velocity of individuals. We prove the convergence (in suitable topologies) of the solution of the problem to the unique solution of the limit transport problem, as the diffusion coefficient tends to zero.

Introduction

In [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF], a model for the dispersal of individuals with an additional aggregation mechanism is proposed. More precisely, the population density u(t, x) at location x ∈ Ω where Ω is an open bounded domain of R N , 1 ≤ N ≤ 3, and time t > 0 solves the convection-diffusion equation

∂ t u = δ ∆u -∇ • (u ω) + r u E(u), (1)
where δ > 0, r ≥ 0 and E is the net rate reproduction per individual . This equation is coupled to an elliptic equation for the velocity ω which is assumed to be in the direction of increasing E(u), say, of the form λ∇E(u) with λ > 0. The evolution of the velocity ω is described by

-ε ∆ω + ω = λ ∇E(u), (2)
where ε > 0 and ε ∆ω is simply to smooth out any sharp local variation in ∇E(u) so that ω represents a local average of the velocity λ ∇E(u).

We supplement (1) and ( 2) with no-flux boundary conditions

n • ∇u = n • ω = 0, x ∈ ∂Ω, t ≥ 0, (3)
as suggested in [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF] where n is the outward normal of ∂Ω. In addition, in dimension 2 or 3, we impose the following additional condition given in [START_REF] Dias | A simplified variational model for the bidimensional coupled evolution equations of a nematic liquid crystal[END_REF][START_REF] Dias | Un problème aux limites pour un système d'équations non linéaires tridimensionnel[END_REF][START_REF] Schoenauer | Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées[END_REF] to guarantee the well-posedness of the elliptic system (2)

∂ n ω × n = 0, x ∈ ∂Ω, t ≥ 0. ( 4
)
As usual, v × ω is the number v 1 ω 1 + v 2 ω 2 if N = 2 and the vector field (v 2 ω 3 -

v 3 ω 2 , -v 1 ω 3 + v 3 ω 1 , v 1 ω 2 -v 2 ω 1 ) if N = 3.
We are interested here in the case where the aggregation mechanism is dominant, that is, the diffusivity δ is small. For biological models, this can change dramatically the dynamical behaviour of the solutions, and might generate finite time blow-up such as for the Keller-Segel system, see [START_REF] Rascle | Finite time blow-up in some models of chemotaxis[END_REF] for instance. Nevertheless, there are situation for which the small diffusivity limit is somehow "stable", including some models from semiconductor physics, see Markowich and Szmolyan [START_REF] Markowhich | A system of convection-diffusion equations with small diffusion coefficient arising in semiconductor physics[END_REF], and for the Keller-Segel system with volume-filling effect, see [START_REF] Dolak | The Keller-Segel model with logistic sensitivity function and small diffusivity[END_REF][START_REF] Perthame | Existence of solutions of the hyperbolic Keller-Segel model[END_REF]. There, the authors prove the convergence, in the small diffusivity limit, of the solutions of the parabolic systems to weak entropy solutions of the corresponding hyperbolic systems. A related field of research which is currently very active is the analysis of the so-called aggregation equation ∂ t u + div(K(u)) = 0 where K is a nonlocal linear operator, see [START_REF] Laurent | Local and global existence for an aggregation equation[END_REF][START_REF] Carrillo | Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations[END_REF], and the references therein. Taking the case of small diffusivity as a motivation, we will study the system (1), ( 2), ( 3) and (4), in the limit of vanishing diffusivity δ. More precisely, given a sufficiently smooth function E, parameters δ ∈ (0, 1), ε > 0 and r ≥ 0, our aim in this paper is to investigate the limit δ → 0 of the following one dimensional system

       ∂ t u δ = δ ∂ 2 x u δ -∂ x (u δ ϕ δ ) + r u δ E(u δ ), x ∈ (-1, 1), t > 0 -ε ∂ 2 x ϕ δ + ϕ δ = ∂ x E(u δ ), x ∈ (-1, 1), t > 0 ∂ x u δ (t, ±1) = ϕ δ (t, ±1) = 0, t > 0 u δ (0, x) = u 0 (x),
x ∈ (-1, 1).

(

) 5
where E has two specific forms suggested in [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF], namely

E(u) = 1 -u (6)
or

E(u) = (1 -u)(u -a)
, for some a ∈ (0, 1).

Given u 0 ∈ W 1,2 (-1, 1), the existence and uniqueness of a global solution of (5) have been shown in [START_REF] Nasreddine | Well-posedness for a model of individual clustering[END_REF], and the purpose of this paper is to prove that (u δ , ϕ δ ) converges to a solution of the nonlocal transport problem

       ∂ t u = -∂ x (u ϕ) + r u E(u), x ∈ (-1, 1), t > 0, -ε ∂ 2 x ϕ + ϕ = ∂ x E(u), x ∈ (-1, 1), t > 0, ϕ(t, ±1) = 0, t > 0, u(0, x) = u 0 (x), x ∈ (-1, 1), (8)
as the diffusion coefficient δ approaches zero. This leads, in a natural way, to the existence of a smooth solution of (8), the uniqueness being established in Proposition 4.2.

Our paper is organized as follows. In Section 2, we state the main results and focus on the two specific forms of E suggested in [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF]: the "bistable case" [START_REF] Laurent | Local and global existence for an aggregation equation[END_REF], see Theorem 2.1, and the "monostable case" [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF], see Theorem 2.2. In Section 3, we recall some results of existence and uniqueness of a global solution of (5) obtained in [START_REF] Nasreddine | Well-posedness for a model of individual clustering[END_REF]. Section 4 is devoted to the uniqueness issue of smooth solutions of the transport problem [START_REF] Markowhich | A system of convection-diffusion equations with small diffusion coefficient arising in semiconductor physics[END_REF]. In Section 5, we focus on the bistable case [START_REF] Laurent | Local and global existence for an aggregation equation[END_REF]. We derive a priori estimates on (u δ , ϕ δ ), which are uniformly valid in δ, and particularly we derive a lower bound for ∂ x ϕ δ and an L ∞ (W 1,1 ) estimate on u δ which leads to an L ∞ (W 2,2 ) bound on u δ . these estimates imply, by a compactness argument, the existence of accumulation points of any sequence (u δ , ϕ δ ) δ . Thanks to Section 3, we conclude that the limit of (u δ , ϕ δ ) δ is unique, and the whole family (u δ , ϕ δ ) converges to the unique solution of ( 8) with E(u) = (1 -u)(u -a). In Section 6, we analyse the limit δ → 0 of (5) in the monostable case [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF]. This analysis is quite similar to that of the previous case, except for the first estimate.

Main results

Throughout this paper, and unless otherwise stated, we assume that δ ∈ (0, 1), ε > 0, r ≥ 0.

In [START_REF] Nasreddine | Well-posedness for a model of individual clustering[END_REF], the global existence and uniqueness of smooth solution of (5) are shown when E(u) has the structure ( 6) or [START_REF] Laurent | Local and global existence for an aggregation equation[END_REF]. Our first result gives the limit δ → 0 in [START_REF] Dolak | The Keller-Segel model with logistic sensitivity function and small diffusivity[END_REF] in the bistable case, that is when E(u) = (1 -u)(u -a) for some a ∈ (0, 1).

Theorem 2.1. Assume that u 0 is a nonnegative function in W 1,2 (-1, 1) and E(u) = (1 -u)(u -a)
for some a ∈ (0, 1). For δ > 0, let u δ be the global nonnegative solution to [START_REF] Dolak | The Keller-Segel model with logistic sensitivity function and small diffusivity[END_REF] given by Theorem 3.2 below. Then, for all T > 0 lim δ→0 ||u δ (t) -u(t)|| C = 0 for all t ∈ (0, T ),

where u ∈ C [0, T ]; L 2 (-1, 1) ∩ L ∞ (0, T ); W 1,2 (-1, 1)
is the unique smooth solution of the corresponding transport system

∂ t u = -∂ x (u ϕ) + r u (1 -u)(u -a), x ∈ (-1, 1), t > 0, (9)
-ε ∂ 2 x ϕ + ϕ = (-2 u + a + 1) ∂ x u, x ∈ (-1, 1), t > 0 ( 10
)
with boundary and initial conditions ϕ(t, ±1) = 0, for all t > 0, and u(0, x) = u 0 (x), x ∈ (-1, 1).

As a consequence of [START_REF] Rascle | Finite time blow-up in some models of chemotaxis[END_REF] no boundary conditions for u are needed. The proof of the previous theorem is performed by deriving estimates which are uniformly valid for 0 < δ < 1. This proof starts with the suitable cancellation of the coupling terms in two equations which gives an estimate for u δ in L ∞ (L 2 ) and for ϕ δ in L 2 (W 1,2 ). Then we derive a lower bound for ∂ x ϕ δ and an L ∞ (W 1,1 ) bound on u δ which leads to an L ∞ (W 1,2 ) bound on u δ . We will, by a compactness argument, show the convergence of u δ to the smooth solution of the transport system (9), [START_REF] Perthame | Existence of solutions of the hyperbolic Keller-Segel model[END_REF] and [START_REF] Rascle | Finite time blow-up in some models of chemotaxis[END_REF].

Next, we turn to the monostable case, that is when E(u) = 1 -u, and we study the limit δ → 0. Theorem 2.2. Assume that u 0 is a nonnegative function in W 1,2 (-1, 1) and E(u) = (1 -u). For δ > 0 let u δ be the global nonnegative solution of (5) given by Theorem 3.2 below. Then, for all T > 0 lim δ→0 ||u δ (t) -u(t)|| C = 0 for all t ∈ (0, T ),

where u ∈ C [0, T ]; L 2 (-1, 1) ∩ L ∞ (0, T ); W 1,2 (-1, 1)
is the unique smooth solution of the following transport system,

∂ t u = -∂ x (u ϕ) + r u (1 -u), x ∈ (-1, 1), t > 0, ( 12
) -ε ∂ 2 x ϕ + ϕ = -∂ x u, x ∈ (-1, 1), t > 0, ( 13
)
with boundary and initial conditions ϕ(t, ±1) = 0, for all t > 0, and u(0, x) = u 0 (x), x ∈ (-1, 1). ( 14)

The proof of Theorem 2.2 follows the same lines as that of Theorem 2.1. As in the bistable case, there is a cancellation between the two equations but it only gives an L ∞ (L log L) bound on u δ and an L 2 (W 1,2 ) bound on ϕ δ .

3 Well-Posedness of [START_REF] Dolak | The Keller-Segel model with logistic sensitivity function and small diffusivity[END_REF] We first recall the notion of solution to [START_REF] Dolak | The Keller-Segel model with logistic sensitivity function and small diffusivity[END_REF] to be used in this paper. Definition 3.1. Let T > 0, E ∈ C 2 (R), and an initial condition u 0 ∈ W 1,2 (-1, 1) . For 0 < δ < 1, a strong solution to (5) on [0, T ) is a function

u δ ∈ C [0, T ), W 1,2 (-1, 1) ∩ C (0, T ), W 2,2 (-1, 1) , such that    ∂ t u δ = δ ∂ 2 x u δ -∂ x (u δ ϕ δ ) + r u δ E(u δ ), a.e. in [0, T ) × (-1, 1) u δ (0, x) = u 0 (x), a.e. in (-1, 1) ∂ x u δ (t, ±1) = 0, a.e. on [0, T ),
where, for all t ∈ [0, T ), ϕ δ (t) is the unique solution in W 2,2 (-1, 1) of

-ε∂ 2 x ϕ δ (t) + ϕ δ (t) = ∂ x E(u δ (t)) a.e. in (-1, 1) ϕ δ (t, ±1) = 0
We now recall the global existence theorem which is proved in [START_REF] Nasreddine | Well-posedness for a model of individual clustering[END_REF], where E(u) is given by ( 6) or [START_REF] Laurent | Local and global existence for an aggregation equation[END_REF].

Theorem 3.2. Assume that u 0 is a nonnegative function in W 1,2 (-1, 1), and E(u) = (1 -u)(u -a) for some a ∈ (0, 1) or E(u) = 1 -u.

Uniqueness

In this section we prove the uniqueness of the solution of [START_REF] Markowhich | A system of convection-diffusion equations with small diffusion coefficient arising in semiconductor physics[END_REF]. Let us first give the definition of the strong solution of [START_REF] Markowhich | A system of convection-diffusion equations with small diffusion coefficient arising in semiconductor physics[END_REF].

Definition 4.1. Let T > 0, E ∈ C 2 (R), and an initial condition u 0 ∈ W 1,2 (-1, 1) . A strong solution on [0, T ) to the transport system (8) is a function u ∈ C [0, T ), L 2 (-1, 1) ∩ C (0, T ), W 1,2 (-1, 1) , such that ∂ t u = -∂ x (u ϕ) + r u E(u), a.e. in [0, T ) × (-1, 1) u(0, x) = u 0 (x), a.e. in (-1, 1),
where, for all t

∈ [0, T ), ϕ(t) is the unique solution in W 2,2 (-1, 1) of -ε∂ 2 x ϕ(t) + ϕ(t) = ∂ x E(u(t)) a.e. in (-1, 1) ϕ(t, ±1) = 0 .
The main result is contained in

Proposition 4.2. Assume that u 0 is a nonnegative function in W 1,2 (-1, 1) and E ∈ C 2 (R).
Then for all T > 0, there exists at most one solution u of (8) in the sense of Definition 4.1, such that

u ∈ L ∞ (0, T ), W 1,1 (-1, 1) , and ϕ ∈ L ∞ (0, T ); W 1,∞ (-1, 1) . ( 15
)
Proof. Let us assume that there exist two different solutions u 1 and u 2 to (8) corresponding to the same initial conditions, and fix T > 0. We put

(u, ϕ) = (u 1 -u 2 , ϕ 1 -ϕ 2 ), in [0, T ] × (-1, 1). Then (u, ϕ) satisfies        ∂ t u = -∂ x (u ϕ 1 ) -∂ x (u 2 ϕ) + r u 1 E(u 1 ) -r u 2 E(u 2 ), in (0, T ) × (-1, 1) -ε∂ 2 x ϕ + ϕ = E ′ (u 1 ) ∂ x u 1 -E ′ (u 2 ) ∂ x u 2 , in (0, T ) × (-1, 1) ϕ(t, ±1) = 0 on (0, T ) u(0, x) = 0, in (-1, 1
). ( 16) We multiply the first equation in ( 16) by sign(u), and integrate it by parts over (-1, 1) to obtain

d dt ||u|| 1 = - 1 -1 ϕ 1 ∂ x |u| dx - 1 -1 ∂ x ϕ 1 |u| dx - 1 -1 sign(u) ∂ x ϕ u 2 dx - 1 -1 ϕ ∂ x u 2 sign(u) dx + r 1 -1 (u 1 E(u 1 ) -u 2 E(u 2 ))) sign(u) dx ≤ ||∂ x ϕ|| 1 ||u 2 || ∞ + ||∂ x u 2 || 1 ||ϕ|| ∞ + r ||u 1 E(u 1 ) -u 2 E(u 2 )|| 1 , (17)
since the first line in the right-hand side vanishes. Using the fact that u 1 and u 2 are bounded by ( 15) and the embedding of

W 1,1 (-1, 1) in L ∞ (-1, 1) we estimate ||u 1 E(u 1 ) -u 2 E(u 2 )|| 1 ≤ C ||u|| 1 . (18)
Using ( 15), (18), and the continuous embedding of W 1,1 (-1, 1) in L ∞ (-1, 1) , (17) becomes

d dt ||u|| 1 ≤ C ||∂ x ϕ|| 1 + C ||ϕ|| ∞ + C ||u|| 1 . (19)
To complete the proof of Proposition 4.2, it remains to estimate ||∂ x ϕ|| 1 and ||ϕ|| ∞ .

For x, y ∈ (-1, 1), we integrate the second equation in ( 16) to obtain

-ε x y ∂ 2 x ϕ(z) dz + x y ϕ(z) dz = x y (∂ x E(u 1 ) -∂ x E(u 2 )) dz -ε (∂ x ϕ(x) -∂ x ϕ(y)) + x y ϕ(z) dz = [E(u 1 (x)) -E(u 2 (x)) -E(u 1 (y)) + E(u 2 (y))].
Next we integrate the above equality with respect to y over (-1, 1) to obtain

∂ x ϕ(x) = 1 2 ε 1 -1 x y ϕ(z) dzdy- 1 ε E(u 1 (x))+ 1 ε E(u 2 (x))+ 1 2 ε 1 -1 (E(u 1 (y))-E(u 2 (y))) dy.
This gives

|∂ x ϕ(x)| ≤ ||ϕ|| 1 ε + 1 ε |E(u 1 (x)) -E(u 2 (x))| + 1 2 ε ||E(u 1 ) -E(u 2 )|| 1 .
Therefore

||∂ x ϕ|| 1 ≤ 2 ||ϕ|| 1 ε + 1 ε ||E(u 1 ) -E(u 2 )|| 1 + 1 ε ||E(u 1 ) -E(u 2 )|| 1 .
Since u 1 and u 2 are bounded and E ∈ C 2 (R) we obtain

||∂ x ϕ|| 1 ≤ 2 ||ϕ|| 1 ε + C 2 ε ||u|| 1 . (20)
It remains to prove an L 1 estimate to ϕ. For that purpose, we define, for i = 1, 2, the function

ψ i ∈ L ∞ ((0, T ), W 2,∞ (-1, 1)) solution of -∂ 2 x ψ i (t, x) = ϕ i (t, x), in (-1, 1) ψ i (t, ±1) = 0. ( 21
)
We multiply the second equation in ( 16) by ψ = ψ 1 -ψ 2 and integrate it over (-1, 1) to obtain

||ϕ|| 2 2 + ||∂ x ψ|| 2 2 = 1 -1 ∂ x (E(u 1 ) -E(u 2 )) ψ dx ≤ ||E(u 1 ) -E(u 2 )|| 1 ||∂ x ψ|| ∞ ≤ C ||u|| 1 ||∂ x ψ|| ∞ .
By the continuous embedding of W 1,2 (-1, 1) in L ∞ (-1, 1) and by ( 21), the previous inequality reads

||∂ x ψ|| W 1,2 ≤ C ||u|| 1 and ||ϕ|| 1 ≤ C ||ϕ|| 2 = C ||∂ 2 x ψ|| 2 ≤ C(||∂ x ψ|| 2 + ||∂ 2 x ψ|| 2 ) ≤ C ||u|| 1 . (22)
Substituting ( 22) into (20), and by the continuous embedding of

W 1,1 (-1, 1) in L ∞ (-1, 1) we obtain ||ϕ|| ∞ ≤ C ||∂ x ϕ|| 1 ≤ C ||u|| 1 . (23)
Finally, we substitute (23) into (19) we obtain

d dt ||u|| 1 ≤ C ||u|| 1 + r C ||u|| 1 . (24)
Gronwall's inequality applied to inequality (24) implies that the two solutions are identical, which proves Proposition 4.2.

5 The bistable case:

E(u) = (1 -u)(u -a)
Let T > 0, the system (5) now reads

       ∂ t u δ = δ ∂ 2 x u δ -∂ x (u δ ϕ δ ) + r u δ (u δ -a)(1 -u δ ), in (0, T ) × (-1, 1) -ε ∂ 2 x ϕ δ + ϕ δ = (-2u δ + (a + 1)) ∂ x u δ , in (0, T ) × (-1, 1) ∂ x u δ (t, ±1) = ϕ δ (t, ±1) = 0, on (0, T ), u δ (0, x) = u 0 (x), in (-1, 1), ( 25
) for some a ∈ (0, 1). Thanks to Theorem 3.2, (25) has a unique global nonnegative solution in the sense of the Definition 3.1. Integrating (25) over (0, T ) × (-1, 1) and using the nonnegativity of u δ , we first observe that

||u δ (t)|| 1 ≤ ||u 0 || 1 + 2 r (1 -a) T, for all t ∈ [0, T ]. ( 26
)

Estimates

Lemma 5.1. There is C 1 (T ) > 0 independent of δ such that

T 0 ε 2 ||∂ x ϕ δ || 2 2 + ||ϕ δ || 2 2 + 2 δ ||∂ x u δ || 2 2 dt ≤ C 1 (T ), for all t ∈ [0, T ] (27) ||u δ (t)|| 2 ≤ C 1 (T ), for all t ∈ [0, T ], (28)
and

T 0 ||ϕ δ || 2 ∞ dt ≤ C 1 (T ) for all t ∈ [0, T ]. ( 29
)
Proof. Multiplying the first equation in (25) by 2 u δ and integrating it over (-1, 1), we obtain

d dt 1 -1 |u δ | 2 dx = -2 δ 1 -1 |∂ x u δ | 2 dx + 2 1 -1 u δ ϕ δ ∂ x u δ dx + 2 r 1 -1 u 2 δ E(u δ ) dx. ( 30
)
Multiplying now the second equation in (25) by ϕ δ and integrating it over (-1, 1) we obtain

ε 1 -1 |∂ x ϕ δ | 2 dx + 1 -1 |ϕ δ | 2 dx = -2 1 -1 u δ ϕ δ ∂ x u δ dx + (a + 1) 1 -1 ∂ x u δ ϕ δ dx. (31)
At this point we notice that the cubic terms on the right hand side of (30) and (31) cancel one with the other, and summing (31) and ( 30) we obtain

d dt ||u δ || 2 2 + ε ||∂ x ϕ δ || 2 2 + ||ϕ δ || 2 2 + 2 δ ||∂ x u δ || 2 2 = 2 r 1 -1 u 2 δ E(u δ ) dx + (a+ 1) 1 -1 ∂ x u δ ϕ δ dx.
(32) We integrate by parts and use Cauchy-Schwarz inequality to obtain (a + 1)

1 -1 ∂ x u δ ϕ δ dx = -(a + 1) 1 -1 u δ ∂ x ϕ δ dx ≤ (a + 1) 2 2 ε ||u δ || 2 2 + ε 2 ||∂ x ϕ δ || 2 2 .
On the other hand,

u 2 δ E(u δ ) ≤ 0 if u δ / ∈ (a, 1) so that 1 -1 u 2 δ E(u δ ) dx ≤ 2 (1 -a)
The previous inequalities give that

d dt ||u δ || 2 2 + ε 2 ||∂ x ϕ δ || 2 2 + ||ϕ δ || 2 2 + 2 δ ||∂ x u δ || 2 2 ≤ (a + 1) 2 2 ε ||u δ || 2 2 + 4 r (1 -a).
Therefore, by a time integration, there exists C 1 (T ) such that ( 27) and (28) hold. By the continuous embedding of W 1,2 (-1, 1) in L ∞ (-1, 1) we obtain (29).

Lemma 5.2. There is C 2 (T ) > 0 independent of δ such that

||ϕ δ (t)|| 2 ≤ C 2 (T ), for all t ∈ [0, T ]. ( 33
)
Proof. We define the function ψ ∈ L 2 0, T ; W 2,2 (-1, 1) solution of

-∂ 2 x ψ δ = ϕ δ in (-1, 1) ψ δ (t, ±1) = 0, in [0, T ). ( 34
)
Multiplying the second equation in (25) by ψ δ and integrating it over (-1, 1) we obtain

-ε 1 -1 ϕ δ ∂ 2 x ψ δ dx - 1 -1 ∂ 2 x ψ δ ψ δ dx = - 1 -1 E(u δ ) ∂ x ψ δ dx ε ||∂ 2 x ψ δ || 2 2 + ||∂ x ψ δ || 2 2 ≤ ||E(u δ )|| 1 ||∂ x ψ δ || ∞ .
Using the embedding of W 1,2 (-1, 1) in L ∞ (-1, 1) and the specific form ( 7) of E we obtain

||∂ x ψ δ || W 1,2 ≤ C ||E(u δ )|| 1 ≤ C (1 + ||u δ || 2 2 ). ( 35
)
Using ( 34) and (28) the above inequation becomes

||ϕ δ || 2 = ||∂ 2 x ψ δ || 2 ≤ ||∂ x ψ δ || W 1,2 ≤ C. ( 36
)
Lemma 5.3. For 0 < δ < 1, there exists C(T ) > 0 independent of δ such that

∂ x ϕ δ (x) ≥ -4 ||ϕ δ || ∞ - (a + 1) 2 4 ε -C(T ). (37)
Proof. For x, y ∈ (-1, 1), we integrate the second equation in (25) to obtain

-ε x y ∂ 2 x ϕ δ (z) dz + x y ϕ δ (z) dz = x y (-2u δ + a + 1) ∂ x u δ dz -ε (∂ x ϕ δ (x) -∂ x ϕ δ (y)) + x y ϕ δ (z) dz = -[u 2 δ (x) -u 2 δ (y)] + (a + 1) (u δ (x) -u δ (y)).
Next we integrate the above equality with respect to y over (-1, 1) to obtain

-2 ε ∂ x ϕ δ (x) = - 1 -1 x y ϕ δ (z) dzdy -2 u 2 δ (x) + 2 (a + 1) u δ (x) + ||u δ || 2 2 -(a + 1) ||u δ || 1 . Since 1 ε u 2 δ -(a+1) ε u δ ≥ -(a+1) 2 4 ε , and
1 -1 x y ϕ δ (z) dzdy ≥ - 1 -1 1 -1 ||ϕ δ || ∞ dzdy ≥ -4 ||ϕ δ || ∞ ,
it follows from (28) that

∂ x ϕ δ (x) = 1 2 ε 1 -1 x y ϕ δ (z) dzdy + 1 ε u 2 δ (x) - a + 1 ε u δ (x) - 1 2 ε ||u δ || 2 2 + a + 1 2 ε ||u δ || 1 ≥ - 2 ε ||ϕ δ || ∞ - (a + 1) 2 4 ε -C(T ).
We continue with estimates for the derivatives of u δ .

Lemma 5.4. There is C 3 (T ) > 0 independent of δ such that

||∂ x u δ (t)|| 1 ≤ C 3 (T ) for all t ∈ [0, T ]. (38)
Proof. We set g δ = ∂ x u δ to simplify the notation, and differentiate the first equation in (25) with respect to x. This yields

∂ t g δ + ∂ 2 x (u δ ϕ δ ) -r ∂ x (u δ (1 -u δ ) (u δ -a)) = δ ∂ 2 x g δ . (39)
We define an approximation of the sign function by σ γ (z) = σ( z γ ), 0 < γ ≪ 1, with σ smooth and increasing, σ(0) = 0, and σ(z) = sign z for |z| > 1. Then, with abs γ (z) = z 0 σ γ (ξ) dξ, the convergence of abs γ (z) to |z| as γ → 0 is uniform in z ∈ R. Multiplying (39) by σ γ (g δ ) and integrating with respect to x yields

1 -1 σ γ (g δ ) ∂ t g δ dx + 1 -1 σ γ (g δ ) [∂ x (g δ ϕ δ ) + g δ ∂ x ϕ δ + u δ ∂ 2 x ϕ δ ] dx -r 1 -1 σ γ (g δ ) (-3 u 2 δ + 2 (a + 1) u δ -a) g δ dx = δ 1 -1 σ γ (g δ ) ∂ 2 x g δ dx. ( 40
) Since 1 -1 σ γ (g δ ) ∂ x (g δ ϕ δ ) dx = - 1 -1 σ ′ γ (g δ ) ∂ x g δ g δ ϕ δ dx, and
1 -1 σ γ (g δ ) ∂ 2 x g δ dx = - 1 -1 (∂ x g δ ) 2 σ ′ γ (g δ ) dx, we obtain d dt 1 -1 abs γ (g δ ) dx - 1 -1 σ ′ γ (g δ ) ∂ x g δ g δ ϕ δ dx + 1 -1 σ γ (g δ ) g δ ∂ x ϕ δ dx + 1 -1 σ γ (g δ ) u δ ∂ 2 x ϕ δ dx -r 1 -1 σ γ (g δ ) (-3 u 2 δ + 2 (a + 1) u δ -a) g δ dx = -δ 1 -1 (∂ x g δ ) 2 σ ′ γ (g δ ) dx ≤ 0. ( 41
)
The function

f γ (z) = σ γ (z) z -abs γ (z) satisfies f ′ γ (z) = σ ′ γ (z)
z and converges to 0 uniformly in z ∈ R. We integrate the second term in (41) by parts and we use the second equation in ( 25) in the fourth one to obtain

d dt 1 -1 abs γ (g δ ) dx + 1 -1 f γ (g δ ) ∂ x ϕ δ dx + 1 -1 σ γ (g δ ) g δ ∂ x ϕ δ dx + 1 ε 1 -1 σ γ (g δ ) u δ (2u δ -a -1) g δ dx + 1 ε 1 -1 σ γ (g δ ) u δ ϕ δ dx ≤ r 1 -1 σ γ (g δ ) (-3 u 2 δ + 2 (a + 1) u δ -a) g δ dx. (42) Now, |g δ | ≥ σ γ (g δ ) g δ ≥ 0 and (2u δ -a -1) ≥ 0 if u δ ≥ a+1 2 , so that 1 -1 σ γ (g δ ) u δ (2u δ -a -1) g δ dx = u δ ≥ a+1 2 σ γ (g δ ) u δ (2u δ -a -1) g δ dx + 2 0<u δ ≤ a+1 2 u 2 δ σ γ (g δ ) g δ dx -(a + 1) 0<u δ ≤ a+1 2 σ γ (g δ ) g δ (u δ ) dx ≥ - (a + 1) 2 2 0<u δ ≤ a+1 2 |g δ |dx
and thus

1 -1 σ γ (g δ ) u δ (2u δ -a -1) g δ dx ≥ - (a + 1) 2 2 1 -1 |g δ | dx. ( 43
)
Also, since -3

u 2 δ + 2 (a + 1) u δ -a ≤ (a+1) 2 2 and |g δ | ≥ σ γ (g δ ) g δ ≥ 0, r 1 -1 σ γ (g δ ) (-3 u 2 δ + 2 (a + 1) u δ -a) g δ dx ≤ r (a + 1) 2 2 1 -1 |g δ | dx. ( 44
)
Passing to the limit γ → 0 in (42) the first term on the right-hand side vanishes. And it follows from Lemma 5.3, ( 43), ( 44) and ( 26) that

d dt 1 -1 |g δ | dx - 2 ε ||ϕ δ || ∞ + (a + 1) 2 4 ε + C 2 (T ) 1 -1 |g δ | dx - (a + 1) 2 2 ε 1 -1 |g δ | dx ≤ ||u δ || 1 ε ||ϕ δ || ∞ + r (a + 1) 2 2 1 -1 |g δ | dx ≤ C(T ) ε ||ϕ δ || ∞ + r (a + 1) 2 2 1 -1 |g δ | dx. ( 45
)
Integrating (45) in time, and using (29) yield that there exists C 3 (T ) such that (38) holds.

Lemma 5.5. There is C 4 (T ) > 0 independent of δ such that

||u δ (t)|| ∞ ≤ C 4 (T ) for all t ∈ [0, T ], ( 46
)
and

||∂ x ϕ δ (t)|| ∞ ≤ C 4 (T ) for all t ∈ [0, T ]. ( 47
)
Proof. For all T > 0, (26) and Lemma 5.4 guarantee that

||u δ (t)|| W 1,1 ≤ C(T ) for all t ∈ [0, T ]. ( 48
)
Therefore, the continuous embedding of W 1,1 (-1, 1) in L ∞ (-1, 1) implies that (46) holds.

On the other hand, Lemma 5.4, Lemma 5.2, (46) and the second equation in (25) ensure that ∂ x ϕ δ (t) is bounded in W 1,1 (-1, 1) and by the continuous embedding of W 1,1 (-1, 1) in L ∞ (-1, 1) we get (47).

Lemma 5.6. There is C 5 (T ) > 0 independent of δ such that

||∂ x u δ (t)|| 2 ≤ C 5 (T ) for all t ∈ [0, T ]. ( 49
)
Proof. Coming back to (39) , g δ = ∂ x u δ satisfies

∂ t g δ = -∂ x g δ ϕ δ -2 g δ ∂ x ϕ δ -u δ ∂ 2 x ϕ δ + r ∂ x (u δ (1 -u δ ) (u δ -a)) + δ ∂ 2 x g δ . ( 50
)
We multiply (50) by g δ and integrate it over (-1, 1) to obtain

1 2 d dt ||g δ || 2 2 = - 1 -1 ∂ x g δ ϕ δ g δ dx -2 1 -1 |g δ | 2 ∂ x ϕ δ dx - 1 -1 u δ ∂ 2 x ϕ δ g δ dx + r 1 -1 ∂ x (u δ (1 -u δ ) (u δ -a)) g δ dx + δ 1 -1 ∂ 2 x g δ g δ dx.
We integrate by parts the last term of the right-hand side, and use the second equation in (25) to obtain

1 2 d dt ||g δ || 2 2 = - 1 -1 ∂ x ( |g δ | 2 2 ) ϕ δ dx -2 1 -1 |g δ | 2 ∂ x ϕ δ dx + 1 ε 1 -1 u δ (-ϕ δ + E ′ (u δ ) g δ ) g δ dx + r 1 -1 -3 u 2 δ + 2(a + 1)u δ -a |g δ | 2 dx -δ 1 -1 |∂ x g δ | 2 dx.
We integrate the first term in the right-hand side by parts and use Hölder inequality to obtain

1 2 d dt ||g δ || 2 2 = 1 -1 |g δ | 2 2 ∂ x ϕ δ dx -2 1 -1 |g δ | 2 ∂ x ϕ δ dx - 1 ε 1 -1 u δ ϕ δ g δ dx + 1 ε 1 -1 u δ (-2u δ + a + 1) |g δ | 2 dx + r 1 -1 -3 u 2 δ + 2(a + 1)u δ -a |g δ | 2 dx ≤ 3 2 ||g δ || 2 2 ||∂ x ϕ δ || ∞ + 1 ε ||u δ || ∞ ||ϕ δ || 2 ||g δ || 2 + 1 ε ||u δ (-2u δ + a + 1)|| ∞ ||g δ || 2 2 + r || -3 u 2 δ + 2(a + 1)u δ -a|| ∞ ||g δ || 2 2 .
Using (46), (47) and Young inequality we obtain

d dt ||g δ || 2 2 ≤ C ||g δ || 2 2 + C ||ϕ δ || 2 2 , (51)
it follows from (27) after integration that (49) holds.

Lemma 5.7. There is C 6 (T ) > 0 independent of δ such that

||∂ t u δ (t)|| 2 (W 1,2 ) ′ ≤ C 6 (T ) for all t ∈ [0, T ], (52)
where (W 1,2 ) ′ denotes the dual space of W 1,2 .

Proof. Consider ψ ∈ W 1,2 (-1, 1) and t ∈ (0, T ). We have by the first equation in (25)

1 -1 ∂ t u δ ψ dx = 1 -1 [∂ x (δ ∂ x u δ -u δ ϕ δ ) + r u δ E(u δ )] ψ dx = 1 -1 (-δ ∂ x u δ ∂ x ψ + u δ ϕ δ ∂ x ψ + r u δ (1 -u δ ) (u δ -a) ψ) dx ≤ δ ||∂ x ψ|| 2 ||∂ x u δ || 2 + ||∂ x ψ|| 2 ||u δ || ∞ ||ϕ δ || 2 + r ||u δ (1 -u δ )(u δ -a)|| 2 ||ψ|| 2 .
Using (49), (46) and Lemma 5.2 we end up with

1 -1 ∂ t u δ ψ dx ≤ (δ + 1 + r) C(T ) ||ψ|| W 1,2 ≤ C(T ) ||ψ|| W 1,2
since 0 < δ < 1. A duality argument gives

||∂ t u δ (t)|| (W 1,2 ) ′ ≤ C 6 (T ), t ∈ [0, T ]
and the proof of Lemma 5.7 is complete.

Convergence

In this section we discuss the limit of (u δ , ϕ δ ) as δ → 0. For that purpose, we study the compactness properties of (u δ , ϕ δ ).

Proof of Theorem 2.1. Thanks to Lemma 5.6 and ( 27), (u

δ ) δ is bounded in L ∞ (0, T ); W 1,2 (-1, 1) while (∂ t u δ ) δ is bounded in L ∞ (0, T ); (W 1,2 ) ′ (-1, 1) by Lemma 5.7. Since W 1,2 (-1, 1) is compactly embedded in C[-1, 1] and C[-1, 1] is continuously embed- ded in (W 1,2 ) ′ (-1, 1), it follows from [13, Corollary 4] that (u δ ) δ is relatively compact in C ([0, T ] × [-1, 1]
). Therefore, there are a sequence (δ j ) of positive real numbers, δ j → 0, and u ∈ L ∞ (0, T ); W 1,2 (-1, 1) such that

u δ j ⇀ u in L 2 (0, T ); W 1,2 (-1, 1) , (53)
and

u δ j -→ u in C ([0, T ] × [-1, 1]) . (54)
Owing to Lemma 5.1 and Lemma 5.6, we may also assume that

ϕ δ j ⇀ ϕ in L 2 (0, T ); W 1,2 (-1, 1) as δ j → 0, (55)
and

δ j ∂ x u δ j -→ 0 in L 2 ((0, T ) × (-1, 1)) as δ j → 0. ( 56
)
Owing to (53)-(56), it is straightforward to deduce from (25) that (u, ϕ) satisfies

T 0 ∂ t u, ψ dt = T 0 1 -1 (u ϕ ∂ x ψ + r u (1 -u)(u -a) ψ) dx dt, (57)
and

ε T 0 1 -1 ∂ x ϕ ∂ x ψ dxdt + T 0 1 -1 ϕ ψ dxdt = T 0 1 -1 (-2 u + a + 1) ∂ x u ψ dxdt. ( 58
)
for all test functions ψ ∈ C 2 ([0, T ] × (-1, 1)). Since (u, ϕ) satisfies (57) and (58), then (u, ϕ) is a weak solution of ( 9), [START_REF] Perthame | Existence of solutions of the hyperbolic Keller-Segel model[END_REF] and [START_REF] Rascle | Finite time blow-up in some models of chemotaxis[END_REF]. Recalling that u ∈ L ∞ (0, T ); W 1,2 (-1, 1) and ϕ ∈ L ∞ (0, T ); W 1,2 (-1, 1) by Lemma 5.5 and Lemma 5.6, we deduce from (57) that ∂ t u ∈ L 2 ((0, T ) × (-1, 1)) and from (58) that ϕ ∈ L ∞ (0, T ); W 2,2 (-1, 1) , so that u solves ( 9) and [START_REF] Rascle | Finite time blow-up in some models of chemotaxis[END_REF] in the sense of Definition 4.1 with the regularity (15). By Proposition 4.2, such a solution is unique so that u is the only possible cluster point of (u δ ) δ in C ([0, T ] × [-1, 1]). Therefore, the whole family (u δ ) δ converges to u in C ([0, T ] × [-1, 1]) as δ -→ 0.

6 The monostable case, E(u) = 1u

Let T > 0, system (5) now reads

       ∂ t u δ = δ ∂ 2 x u δ -∂ x (u δ ϕ δ ) + r u δ (1 -u δ ) x ∈ (-1, 1), t > 0 -ε ∂ 2 x ϕ δ + ϕ δ = -∂ x u δ , x ∈ (-1, 1), t > 0 ∂ x u δ (t, ±1) = ϕ δ (t, ±1) = 0 t > 0, u δ (0, x) = u 0 (x)
x ∈ (-1, 1), (59) Thanks to Theorem 3.2, system (59) has a unique global nonnegative solution in the sense of Definition 3.1. Unlike the previous case, it does not seem to be possible to begin the proof with an L ∞ (L 2 ) estimate on u δ . Nevertheless, there is still a cancellation between the two equations which actually gives an L ∞ (L log L) bound on u δ and a L 2 bound on ∂ x √ u δ as we shall see below. Integrating (59) over [0, T ] × (-1, 1) and using the nonnegativity of u δ , we first observe that,

||u δ (t)|| 1 ≤ ||u 0 || 1 + 2 r t, for all t ∈ [0, T ]. ( 60
)
6.1 Estimates Lemma 6.1. There is C 7 (T ) > 0 independent of δ such that

T 0 ε ||∂ x ϕ δ || 2 2 + ||ϕ δ || 2 2 + 4 δ ||∂ x √ u δ || 2 2 dt ≤ C 7 (T ), for all t ∈ [0, T ], (61)
T 0 ||ϕ δ || 2 ∞ dt ≤ C 7 (T ), for all t ∈ [0, T ]. ( 62
)
Proof. The proof goes as follows. On the one hand, we multiply the first equation in (59) by (log u δ +1) and integrate it over (-1, 1). Since u δ (1-u δ ) log u δ ≤ 0 and u δ (1-u δ ) ≤ 1,

d dt 1 -1 u δ log u δ dx = - 1 -1 (δ ∂ x u δ -u δ ϕ δ ) ( 1 u δ ∂ x u δ ) dx + r 1 -1 u δ (1 -u δ ) (log u δ + 1) dx ≤ - 1 -1 δ u δ (∂ x u δ ) 2 dx + 1 -1 ϕ δ ∂ x u δ dx + 2 r. (63)
On the other hand, we multiply the second equation in (59) by ϕ δ and integrate it over (-1, 1) to obtain

ε 1 -1 |∂ x ϕ δ | 2 dx + 1 -1 |ϕ δ | 2 dx = - 1 -1 ∂ x u δ ϕ δ dx. (64)
Adding ( 63) and (64) yields

d dt 1 -1 u δ log u δ dx + ε ||∂ x ϕ δ || 2 2 + ||ϕ δ || 2 2 ≤ -4 δ 1 -1 |∂ x √ u δ | 2 dx + 2 r. (65)
Then, (61) is obtained by a time integration of (65). Finally, by the continuous embedding of W 1,2 (-1, 1) in L ∞ (-1, 1) we obtain (62). Lemma 6.2. For 0 < δ < 1, there exists C 8 (T ) > 0 independent of δ such that

∂ x ϕ δ (x) ≥ -4 ||ϕ δ || ∞ -C 8 (T ) for all t ∈ [0, T ]. (66)
Proof. For x, y ∈ (-1, 1), we integrate the second equation in (59) to obtain

-ε x y ∂ 2 x ϕ δ (z) dz + x y ϕ δ (z) dz = - x y ∂ x u δ dz -ε (∂ x ϕ δ (x) -∂ x ϕ δ (y)) + x y ϕ δ (z) dz = -[u δ (x) -u δ (y)].
Next we integrate the above equality with respect to y over (-1, 1) to obtain

-2 ε ∂ x ϕ δ (x) = - 1 -1 x y ϕ δ (z) dzdy -2 u δ (x) + ||u δ || 1 . Since 1 -1 x y ϕ δ (z) dzdy ≥ -4 ||ϕ δ || ∞ ,
and u δ ≥ 0, it follows from (60) that

∂ x ϕ δ (x) = 1 2 ε 1 -1 x y ϕ δ (z) dzdy + 1 ε u δ (x) - 1 2 ε ||u δ || 1 ≥ - 2 ε ||ϕ δ || ∞ -C 8 (T ). Lemma 6.3. There is C 9 (T ) > 0 independent of δ such that ||ϕ δ (t)|| 2 ≤ C 9 (T ), for all t ∈ [0, T ]. ( 67
)
Proof. The proof is similar to that of Lemma 5.2. Now, we continue with estimates for the derivatives of u δ . Lemma 6.4. There is C 10 (T ) > 0 independent of δ such that

||∂ x u δ (t)|| 1 ≤ C 10 (T ) for all t ∈ [0, T ]. (68)
Proof. Differentiating the first equation in (59) with respect to x and setting

g δ = ∂ x u δ yield ∂ t (g δ ) + ∂ 2 x (u δ ϕ δ ) -r ∂ x (u δ (1 -u δ )) = δ ∂ 2 x g δ . (69)
We define as in the bistable case an approximation of the sign function by σ γ (z) = σ( z γ ), 0 < γ ≪ 1, with σ smooth and increasing, σ(0) = 0, and σ(z) = sign z for |z| > 1. Then, with abs γ (z) = z 0 σ γ (ξ) dξ, the convergence of abs γ (z) to |z| as γ → 0 is uniform in z ∈ R. Multiplying (69) by σ γ (g δ ) and integrating with respect to x yields abs γ (g δ ) dx -

1 -1 σ γ (g δ ) ∂ t (g δ ) dx + 1 -1 σ γ (g δ ) [∂ x (g δ ϕ δ ) + g δ ∂ x ϕ δ + u δ ∂ 2 x ϕ δ ] dx -r 1 -1 σ γ (g δ ) (1 -2 u δ ) g δ dx = δ 1 -1 σ γ (g δ ) ∂ 2 x g δ dx. ( 70
) Since 1 -1 σ γ (g δ ) ∂ x (g δ ϕ δ ) dx = -
1 -1 σ ′ γ (g δ ) ∂ x g δ g δ ϕ δ dx + 1 -1 σ γ (g δ ) g δ ∂ x ϕ δ dx + 1 -1 σ γ (g δ ) u δ ∂ 2 x ϕ δ dx -r 1 -1 σ γ (g δ ) (1 -2 u δ ) g δ dx = -δ 1 -1 (∂ x g δ ) 2 σ ′ γ (g δ ) dx ≤ 0. ( 71
)
The function f γ (z) = σ γ (z) z -abs γ (z) satisfies f ′ γ (z) = σ ′ γ (z) z and converges to 0 uniformly in z ∈ R. We integrate the second term in (71) by parts and we use the second equation in (59) in the fourth one to obtain d dt

1 -1 abs γ (g δ ) dx + 1 -1 f γ (g δ ) ∂ x ϕ δ dx + 1 -1 σ γ (g δ ) g δ ∂ x ϕ δ dx ≤ - 1 ε 1 -1 σ γ (g δ ) u δ g δ dx - 1 ε 1 -1 σ γ (g δ ) u δ ϕ δ dx + r 1 -1 σ γ (g δ ) (1 -2 u δ ) g δ dx. ≤ 1 ε ||u δ || 1 ||ϕ δ || ∞ + r 1 -1 |g δ | dx,
since u δ ≥ 0, 1 -2u δ ≤ 1 and 0 ≤ σ γ (g δ ) g δ ≤ |g δ |. Passing to the limit γ → 0 in the above inequality, the second term on the left-hand side vanishes. It follows from Lemma 6.2 and (60) that d dt

1 -1 |g δ | dx - 2 ε ||ϕ δ || ∞ + C 8 (T ) 1 -1 |g δ | dx ≤ C(T ) ε ||ϕ δ || ∞ + r 1 -1 |g δ | dx. (72)
Integrating (72) in time, and using (62) yield that there exists C 10 (T ) such that (68) holds.

As in the previous section, we have the following consequence of Lemma 6.3 and Lemma 6.4. Lemma 6.5. There is C 11 (T ) > 0 independent of δ such that Proof. Consider ψ ∈ W 1,2 (-1, 1) and t ∈ (0, T ). We have by the first equation in (59)

1 -1 ∂ t u δ ψ dx = 1 -1 (-δ ∂ x u δ ∂ x ψ + u δ ϕ δ ∂ x ψ + r u δ (1 -u δ ) ψ) dx ≤ δ ||∂ x ψ|| 2 ||∂ x u δ || 2 + ||∂ x ψ|| 2 ||u δ || ∞ ||ϕ δ || 2 + r ||u δ (1 -u δ )|| 2 ||ψ|| 2 .
Using Lemma 6.3, Lemma 6.5 and Lemma 6.6, we end up with

1 -1 ∂ t u δ ψ dx ≤ C(T ) ||ψ|| W 1,2 ,
and a duality argument gives

||∂ t u δ (t)|| (W 1,2 ) ′ ≤ C(T ), t ∈ [0, T ]
and the proof of Lemma 6.7 is complete.

Convergence

Proof of Theorem 2.2. Thanks to the previous analysis, the proof of Theorem 2.2 can then be done as that of Theorem 2.1.

Then ( 5 )

 5 has a unique global nonnegative solution u in the sense of Definition 3.1.

1 - 1 σ ′ γ (g δ ) ∂ x g δ g δ ϕ δ dx and δ 1 - 1 σ γ (g δ ) ∂ 2 x g δ dx = -δ 1 - 1 (

 111111 ∂ x g δ ) 2 σ ′ γ (g δ ) dx

Lemma 6 . 6 .Lemma 6 . 7 .

 6667 ||∂ x ϕ δ (t)|| ∞ + ||u δ (t)|| ∞ ≤ C 11 (T ) for all t ∈ [0, T ].(73) There is C 12 (T ) > 0 independent of δ such that||∂ x u δ (t)|| 2 ≤ C 12 (T ) for all t ∈ [0, T ].(74)Proof. Since (73) holds, we argue as in the proof of Lemma 5.6 to obtain (74). There is C 13 (T ) > 0 independent of δ such that||∂ t u δ (t)|| 2 (W 1,2 ) ′ ≤ C 13 (T ) for all t ∈ [0, T ].(75)

Acknowledgment

I thank Philippe Laurençot for valuable and fruitful discussions.