Elle Tanguy
email: gaelle.tanguy@inrae.fr

Eric Beaucher

Anne Dolivet

Ali Kerjouh

Marie-Bernadette Maillard

Pascaline Hamon

Thomas Croguennec

Behaviour of minerals during the vacuum concentration of dairy ultrafiltration permeates Ga €

Whey concentration using evaporators results in deposit of matter on the hot surfaces of the evaporation tubes. The mass deposited during concentration depends mainly on the mineral composition of the whey, which varies considerably depending on the production process. We investigated the behaviour of ultrafiltration permeates (UFP) during concentration and the susceptibility of their mineral fraction to precipitation. UFP with differing mineral compositions were produced by acidification of milk with different acids to pH values between 4.6 and 6.3, followed by ultrafiltration. The results revealed either no precipitation or the precipitation of calcium salts (calcium phosphate, calcium citrate) in the concentrates. The behaviour of mineral fraction during concentration could be assessed by considering: (i) the nature and quantities of cations and anions in the concentrates, (ii) the anionic forms present at the pH of the concentrates and their association constants with calcium, and (iii) the solubility limit of calcium salts.

Introduction

Whey, the serum phase of milk, is a co-product of cheese and casein manufacture in the dairy industry. It is the liquid fraction that remains after the enzymatic coagulation of milk, its microbial fermentation or the addition of mineral/organic acid [START_REF] Smithers | Whey and whey proteinsdfrom 'gutter-to-gold[END_REF][START_REF] Zall | Sources and composition of whey and permeate[END_REF]. In addition, 96% of whey comes from cheesemaking manufacture and the rest from that of casein [START_REF] Ganju | A review on approaches for efficient recovery of whey proteins from dairy industry effluents[END_REF]. In the dairy industry it is usual to classify whey in two major categories depending on the manufacturing process; namely sweet whey and acid whey. Sweet whey is produced during the manufacture of enzymatic-type cheeses such as Cheddar and Swiss cheese whereas acid whey is generated during that of acidcoagulated dairy products such as fresh cheese, cottage cheese and strained yoghurt [START_REF] Ganju | A review on approaches for efficient recovery of whey proteins from dairy industry effluents[END_REF][START_REF] Schmidt | Effect of processing on whey-protein functionality[END_REF].

Whey contains a large quantity of water, lactose, nitrogen compounds (soluble proteins, peptides, amino acids) and varying amounts of minerals, particularly calcium, phosphate, sodium and potassium [START_REF] Gernigon | Effects of the addition of various minerals, proteins and salts of organic acids on the principal steps of alpha-lactose monohydrate crystallisation[END_REF][START_REF] Schmidt | Effect of processing on whey-protein functionality[END_REF]. The composition of whey can vary considerably and mainly depends on seasonal variations and the cheese manufacturing process [START_REF] Schmidt | Effect of processing on whey-protein functionality[END_REF]. Compared with sweet whey, acid whey has a much lower pH; it contains less protein and lactose but has a higher mineral content, particularly of calcium, phosphate and lactic acid [START_REF] Schmidt | Effect of processing on whey-protein functionality[END_REF][START_REF] Schuck | S echage des lactos erums et d eriv es : Rôle du lactose et de la dynamique de l'eau[END_REF][START_REF] Zall | Sources and composition of whey and permeate[END_REF]. This results in significant differences in the sensory, nutritional and technological properties of whey but also influences the processing of whey streams and the quality of the resulting products [START_REF] Pearce | Whey processing[END_REF].

Sweet whey is now efficiently processed into a variety of food ingredients (beverage powders, bakery items, ice creams); for example, sweet whey is converted into non-sticky free flowing powders using concentration by vacuum evaporation up to a 500e600 g kg À1 dry matter (DM) content, lactose crystallisation in tanks and subsequent spray drying [START_REF] De Wit | Lecturer's handbook on whey and whey products[END_REF]. By contrast, acid whey is considered a difficult product to process, mainly because of its high mineral content, low lactose content and markedly variable composition [START_REF] Saulnier | Organic and mineral composition from industrial acid wheys analysed by capillary electrophoresis[END_REF][START_REF] Schuck | S echage des lactos erums et d eriv es : Rôle du lactose et de la dynamique de l'eau[END_REF]. It has been evidenced during the manufacturing of acid whey powders that the high lactic acid and calcium contents hinder the correct crystallisation of lactose, which remains in its amorphous form. This in turn affects the spray-drying operation and the properties of the resulting powder [START_REF] Wijayasinghe | Lactose behaviour in the presence of lactic acid and calcium[END_REF].

Whey is concentrated using vacuum evaporation, either to reduce storage and transportation costs (concentration up to 300 g kg À1 DM) or to ease further processing such as lactose crystallisation and spray-drying (concentration from 250 to 600 g kg À1 DM, depending on the composition of the concentrate) [START_REF] Pearce | Whey processing[END_REF]. This operation is carried out in falling-film evaporators. The equipment consists of a bundle of vertical tubes in which a thin film of product flows down onto the inner tube surfaces in indirect contact with the heating medium (primary vapour). During vacuum concentration, the product components undergo numerous changes that in turn affect the biochemical, physical and rheological properties of the product [START_REF] Singh | Aspects of protein in milk powder manufacture[END_REF][START_REF] Walstra | Dairy science and technology[END_REF]. The behaviour of the mineral fraction is difficult to predict because phenomena with sometimes opposite effects occur in the products. For instance, the solubility limit of some salts, such as calcium phosphate and calcium citrate, may be exceeded during increasing ion concentration. The precipitation of calcium salts releases H þ that lowers the pH of the concentrated solution. In the meantime, the greater ion concentration of the concentrates increases the ionic strength which in turn reduces ion activity coefficients, increasing salt solubility [START_REF] Walstra | Dairy science and technology[END_REF].

One undesirable effect of salt precipitation is the deposition of matter on the hot surfaces of falling film evaporators. This phenomenon, defined as fouling, is a major problem in the dairy industry as it affects both product quality and processing efficiency. Fouling causes a reduction in heat transfer coefficients and requires more frequent cleaning of the evaporators. It generates product losses and increases both operating costs and environmental impact [START_REF] Daufin | Equipment fouling in the dairy appplication : Problem and pretreatment[END_REF][START_REF] De | Impact and control of fouling in milk processing[END_REF]. Unlike heatexchangers, few studies have focused on the fouling of fallingfilm evaporators, but it is generally agreed that the deposit composition in falling-film evaporators differs from that seen in heat-exchangers [START_REF] Jeurnink | The cleaning of heat exchangers and evaporators after processing milk or whey[END_REF][START_REF] Morison | Reduction of fouling in falling-film evaporators by design[END_REF]. [START_REF] Jeurnink | The cleaning of heat exchangers and evaporators after processing milk or whey[END_REF] evidenced the compositional differences of the deposits in heat-exchangers and falling-film evaporators at an industrial scale during the concentration of milk and whey. Further, they showed that the main components in the deposit were proteins and calcium phosphate after whey concentration from 50 to 280 g kg À1 DM, while it was calcium citrate after whey concentration from 280 to 550 g kg À1 DM. Similarly, Vavrusova, Johansen, Garcia, and Skibsted (2017a) evidenced the predominance of calcium citrate tetrahydrate in the deposit after whey concentration. Calcium phosphate was also present, but to a lesser extent. Lastly, [START_REF] Kessler | Multistage evaporation and water vapour recompression with special emphasis on high dry matter content, product losses, cleaning and energy savings[END_REF] showed that the composition of the deposit was closely dependent on the pH of acid whey; he stated that the proportion of minerals in the deposits would be greater at high pH values and high mineral concentrations. [START_REF] Tanguy | Calcium phosphate precipitation during concentration by vacuum evaporation of milk ultrafiltrate and microfiltrate[END_REF][START_REF] Tanguy | Calcium citrate insolubilization drives the fouling of falling film evaporators during the concentration of hydrochloric acid whey[END_REF] also conducted experiments at the pilot scale to determine the nature of the fouling deposit formed during the concentration of milk ultrafiltrate, milk microfiltrate, hydrochloric acid whey and lactic acid whey. Briefly, these studies showed that (i) calcium phosphate was the main component in the deposit when concentrating milk ultrafiltrate, (ii) proteins reduced the amount of deposit when concentrating milk microfiltrate, and (iii) the deposit formed when concentrating hydrochloric acid whey was composed of calcium citrate whereas no deposition was observed when concentrating lactic acid whey.

To improve our understanding of the fouling process and the subsequent cleaning of evaporators, it would be beneficial to generalise the previous results and predict how the mineral fraction of whey will behave during concentration. However, it would be relatively difficult to start using industrial whey because of its variable composition in minerals and other components. A potential strategy was therefore to modify the mineral composition of the same starting raw material.

During this study, we therefore investigated the degree to which the mineral composition and initial pH influenced the nature and amount of precipitate formed when concentrating different milk permeates, this precipitate being likely to deposit on the inner surface of tubes in evaporators. Permeates with varying mineral compositions and pH were prepared by the ultrafiltration of nonacidified (pH 6.7; control milk) and acidified milk at pH values of between 6.3 and 4.6. The type of acid used for milk acidification (hydrochloric, citric and lactic) and the final pH of the permeates were chosen to prepare products that would simulate the mineral composition of the whey produced during cheese manufacture (hard-type or soft-type cheeses) and acid casein production. However, it should be stressed that these ultrafiltration permeates differed substantially from real whey in terms of their higher content in citrate (not consumed by bacteria) and lower level of proteins (removed using ultrafiltration). The permeates were then concentrated at DM contents ranging from 45 to 500 g kg À1 . The mineral composition of the concentrates and the distribution of the minerals between the soluble phase and precipitate were then determined to understand the behaviour of the mineral fraction during concentration.

Materials and methods

Preparation of milk permeates

The processing scheme to prepare milk permeates (Fig. 1) was defined in line with the work of [START_REF] Mekmene | Physicochemical characterisation of calcium phosphates prepared from milk ultrafiltrates : Effect of the mineral composition[END_REF]. Thermised skimmed milk was divided into six batches; five of these were acidified at 4 C under agitation with lactic, hydrochloric or citric acid (VWR, Fontenay-sous-Bois, France) up to pH values representative of the whey drained during the manufacture of cheese and acid caseins. Preliminary tests were conducted to determine the quantity of acid required to reach the target pH during milk acidification. The concentrations of acid solutions were then adjusted so that the milk samples would be diluted similarly (dilution factor of 2.5%, w/w), the control milk at pH 6.7 being diluted with demineralised water. Acidified milk samples were kept overnight at 4 C.

Non-acidified milk and acidified milk were warmed and stored at 25 C for 2 h to reach mineral equilibrium before ultrafiltration. The milk was coarsely filtered to remove any large flocs and then ultrafiltered at 25 C through tubular ceramic membranes (total membrane area of 2 Â 0.36 m 2 ; cut-off 15 kDa, Membralox, EP1960, Pall, Bazet, France) set on a TIA pilot (Boll ene, France). The permeates were collected, filtered using a 0.45 mm PES membrane sterile filter unit (Nalgene, Thermo Fischer Scientific, Waltham, USA) and stored at 4 C in 1-L bottles.

Vacuum concentration of permeates

The permeates were concentrated using rotary evaporators (Hei-VAP Value Digital, Heidolph Instruments, Schwabach, Germany) at an absolute pressure of between 10.1 and 16.5 kPa, which corresponds to evaporation temperatures within the range 46e56 C. The rotation speed of the 1-L evaporation flask was 120 rpm. The permeates were concentrated at four different concentration factors (CF) ranging from 3.1 to 12.7. For each CF, the initial volume of permeate in the evaporation flask was adjusted to obtain about 100 mL of concentrate at the end of the experiment for further analysis. The experiments were performed in duplicate, more especially during two experimental series referred to as set #1 and set #2. A sample of permeate before concentration (named CF1) was saved from each series. In total, concentration of the permeates was studied at 10 experimental points, except P-AL3. In the latter case, only six points were available because four samples were contaminated and not appropriate for experimentation. The duration of the experiments ranged from 1 to 4 h depending of the expected CF. Images of the concentrates were taken on the day of the experiment, up to 3 h after the end of the concentration step.

Characterisation of permeates and concentrates

For each sample, the pH was measured and the soluble phase separated from the precipitated phase (method described below) immediately after vacuum concentration in the rotavapors. In addition, the concentrates and their respective soluble phases were diluted with demineralised water to a dry matter content comparable with that of the initial permeate to enable quantitative determination of the ash, total protein and total ion contents.

pH

The pH of permeates and concentrates was measured at 25 C using a pH meter (HI 9024 microcomputer pH meter, Hanna Instruments, Romania). All measurements were made in duplicate. The experimental error was ±0.02 units.

Dry matter content

The dry matter content of permeates and concentrates was determined as described in IDF standard 21B (ISO-IDF, 1987). Briefly, five grams of product were mixed with sand in a capsule and dried at 102 ± 2 C in an oven for 7 h. The weight loss of the capsule before and after drying was taken as the amount of water it had contained. This analysis was performed in duplicate. The experimental error was ±5%.

Total nitrogen and ash contents

The total nitrogen content of permeates and concentrates was determined using the Kjeldhal method as described in IDF standard 20B (ISO-IDF, 1993). The nitrogen content was converted to the protein content using a conversion factor of 6.38. Further, the noncasein and non-protein nitrogen contents (respectively NCN and NPN contents) of the thermised skimmed milk were determined using the same method. The NCN corresponds to the soluble fraction at pH 4.6 and was determined after the acid precipitation of casein micelles at pH 4.6 using a 10% acetic acid solution. The NPN corresponds to the soluble fraction after the acid precipitation of milk proteins using 12% trichloroacetic acid. The experimental errors were ±1% for the total nitrogen content and ±2% for the NCN and NPN contents.

An RP-HPLC method was used to quantify a-lactalbumin and blactoglobulin in the permeates. The method was adapted from (Resmini, Pellegrino, Adreini, & Prati, 1989) using a PLRP-S column (polystyrene divinylbenzene gel, 30 nm, 8 mm, 150 Â 2.1 mm; Agilent Technologies, Varian, Les Ulis, France).

The ash content was determined as described in IDF standard 27B (ISO-IDF, 1964). Ten grams of product were incinerated at 550 C in an oven for 5 h. The analysis was performed in duplicate. The experimental error was ±5%.

Concentrations of cations and anions

The contents in cations (calcium, magnesium, sodium, potassium) and anions (lactate, chloride, inorganic phosphate, citrate) were determined in the permeates and concentrates and their respective soluble phases. The soluble phases of the permeates and concentrates were obtained by ultrafiltration on an analytical membrane (molecular cut-off 10 kg mol À1 ; Vivaspin, Palaiseau, France) at a centrifugation rate of 1800Âg at 20 C for 1.5 h.

The cations were quantified using inductively coupled plasmaatomic emission spectroscopy (ICP-AES) (iCAP 7200 Duo, Thermo Fisher Scientific, Waltham, USA) and the samples were diluted with a 2% (w/w) HNO 3 solution before analysis. The anions were quantified using a 930 Compact IC Flex ion chromatography system (Metrohm, Herisau, Swiss) that included a Dionex IonPacAS11 column (4 Â 250 mm) and a conductivity detector. The following conditions were used at 30 C and 1 mL min À1 : gradient elution with 0.3e34 mM NaOH solution for 24 min and column reequilibration for 15 min with 0.3 mM NaOH. Ion contents resulted from two independent samplings and each sample was analysed in duplicate. The experimental errors of ICP-AES and ion chromatography were 2% and 1%, respectively. Changes to total and soluble ion contents during concentration were plotted on the same graph as a function of the concentrate DM content. The difference between the total and soluble ion contents corresponded to ions that precipitated in the concentrate. The experimental total ion content of the concentrate was compared with a calculated total ion content, defined as the product of the concentration factor by the experimental total ion content of the corresponding permeate.

Determination of the composition of precipitates

The precipitate composition in the concentrates with higher CFs was determined indirectly from the experimental composition of the concentrate and the corresponding supernatant obtained using centrifugation (1800Âg; 20 C; 20 min) in 50-mL plastic centrifuge tubes. Both products were characterised in terms of their DM, TN, ash and total ion contents using the methods described previously. In addition, the lactose content was quantified using a ICS-5000þ Dionex ion chromatography system (Thermo Electron SA, Courtaboeuf, France) that included a CarboPac PA210-4 mm column (2 Â 150 mm) and amperometric detector. The following conditions were used at 30 C and 0.2 mL min À1 : isocratic elution with 13 mM KOH for 32 min followed by flush out with 100 mM KOH for 23 min and column re-equilibration for 10 min with 13 mM KOH. A solution containing lactose (Sigma) at 0.1, 0.25, 0.5, 1, 2.5, 5, 10, 20 and 40 mg L À1 was used as a standard. The analytical results enabled an estimate of the percentage of precipitate in the different concentrates, as well as their composition.

Results and discussion

Composition of permeates

The composition of the permeates is given in Table 1. The mineral composition of the permeate resulting from milk ultrafiltration (PUF) was close to the lower end of the values commonly reported [START_REF] Gaucheron | Determination of anions of milk by ion chromatography[END_REF][START_REF] Gra€ | Effects of pH and ionic-strength on distribution of mineral salts in milk[END_REF][START_REF] Walstra | Dairy science and technology[END_REF]. Similarly, the mineral compositions in PUF, P-ACl, P-AL2 and P-AL3 were within the range of those reported by [START_REF] Mekmene | Physicochemical characterisation of calcium phosphates prepared from milk ultrafiltrates : Effect of the mineral composition[END_REF].

The acidification of milk prior to ultrafiltration led to the solubilisation of calcium and inorganic phosphate (and small quantities of magnesium and citrate) associated with casein micelles, and their subsequent transfer to the soluble phase of milk [START_REF] Gaucheron | The minerals of milk[END_REF]. As a consequence, the calcium and inorganic phosphate contents (magnesium and citrate contents to a lesser extent) in the permeates increased as the pH decreased.

As expected, the acidification of milk with citric, hydrochloric or lactic acid induced higher citrate, chloride and lactate contents in the resulting permeates (P-ACit, P-ACl, P-AL1 to P-AL3, respectively).

The TN content of permeates was composed of the non-protein content of milk (1.60 g kg À1 ) and some remaining soluble proteins.

b-lactoglobulin and a-lactoglobulin levels were within the range 0.141e1.318 g kg À1 and 0e0.429 g kg À1 , respectively. Both proteins were found in smaller quantities in P-AL2 (pH 5.25) compared with the other permeates. At this pH, milk was flocculated and more drastic sieving was required to recover the soluble phase, thus explaining the greater retention of protein. By contrast, b-lactoglobulin and a-lactoglobulin were found at higher levels in P-ACl and P-AL3, indicating that the ultrafiltration membranes became more leaky at low pH values (4.64 ± 0.02).

Evolution of the pH of concentrates during concentration

The pH of the concentrates fell during concentration (Fig. 2) due to the rise in ion concentrations that favoured the precipitation of poorly soluble salts, especially calcium phosphate. [START_REF] Gao | Modelling ion composition in simulated milk ultrafiltrate (SMUF). I : Influence of calcium phosphate precipitation[END_REF] had suggested this as the main reason for the fall in pH in simulated milk ultrafiltrate (SMUF) after adding CaCl 2 (i.e., a higher content in free Ca 2þ ). The added Ca 2þ interacted with HPO 4 2À to form amorphous CaHPO 4 .2H 2 O that was converted into its crystalline form, and HPO 4 2À was regenerated by the deprotonation of H 2 PO 4 causing a reduction in pH.

In concentrates in which calcium phosphate precipitation did not occur e or only at a high CF e the fall in pH during concentration might have been related to the formation of other calcium salts such as calcium citrate that precipitated and led to the deprotonation of HCit 2À , as well as an increase in ionic strength that subsequently lowered the ion activity coefficients. Lastly, even if P-AL3 and P-ACl permeates had the same initial pH (around 4.6), the pH decrease in P-AL3 was limited throughout concentration when compared with P-ACl. This might have been caused by less precipitation of calcium salts and the presence of lactate which contributed to the buffering capacity of the concentrates. 4.3 ± 0.1 5.2 ± 0.1 6.7 ± 0.1 4.4 ± 0.4 6.0 ± 0.1 6.5 ± 0.0 TN (g kg À1 )

1.9 ± 0.1 1.9 ± 0.1 3.3 ± 0.1 1.9 ± 0.1 1.5 ± 0.1 3.1 ± 0.0 a-Lactalbumin (g kg

Aspect of concentrates and the coarse composition of resulting precipitates

The concentrates became increasingly turbid as the concentration rose and a precipitate appeared gradually at the bottom of the sample bottles (Fig. 3). Although the initiation of the precipitation is time-dependent, and particularly close to the solubility limit, and the observed precipitates were probably not complete when the images were captured, we noted that the CF at which the precipitate appeared was dependent on the composition of the permeate. For instance, a precipitate appeared from CF 3.5 in PUF concentrates whereas it was observed later in P-AL3 concentrates (from CF 7.1).

The composition of the precipitates that formed in the concentrates at the highest CF was determined by analysing the global composition of the concentrates and corresponding supernatants after centrifugation. The main drawback of this indirect method was that it produced some inconsistent results, such as negative concentrations and dry matter contents over 1000 g kg À1 . However, it provided the coarse composition of the precipitates and evidenced the nature of the ions that precipitated in the concentrates.

Although the concentrates had variable CF values ranging from 8.6 to 10.7, it appeared that the P-AL1 concentrate contained much more precipitate than the others (Table 2), at 24.7 g per 100 g of P-AL1, whereas PUF, P-ACl and P-ACit contained only 8.9, 6.0 and 10.7 g of precipitate, respectively. In all precipitates at the end of concentration, the principal component was lactose which accounted for more than 80 g per 100 g precipitate. The higher level of lactose in the P-AL1 precipitate indicated that the physicochemical conditions in this concentrate favoured lactose crystallisation when compared with other concentrates such as PUF and those resulting from milk acidification using hydrochloric acid and citric acid. Lactate ions may be the promoting agent of lactose crystallisation. This hypothesis had been confirmed in the work of [START_REF] Gernigon | Effects of the addition of various minerals, proteins and salts of organic acids on the principal steps of alpha-lactose monohydrate crystallisation[END_REF] and [START_REF] Smart | Effect of selected compounds on the rate of alactose monohydrate crystallization, crystal yield and quality[END_REF] who evidenced the accelerating effect of lactate on lactose crystallisation, and in particular on crystal growth.

Apart from lactose, the nature and quantity of the ions precipitated depended on the type of permeate. The main mineral components in the PUF precipitate were calcium and phosphate (52 and 26 mmol per 100 g precipitate, respectively) (Table 2). Citrate was also present but to a lesser degree. Unlike the PUF concentrate, calcium and citrate were the main compounds found in P-ACit (150 and 88 mmol 100 g À1 , respectively) whereas phosphate was seen in smaller quantities. Lastly, calcium and citrate remained the main compounds in P-ACl precipitates (127 and 50 mmol 100 g À1 , respectively) but quite a large quantity of phosphate was present (27 mmol 100 g À1 ). The mineral fraction in the P-AL1 concentrate was very low with an ash content equal to 0.5 g 100 g À1 precipitate. It contained some calcium, phosphate and citrate (17, 13 and 3 mmol 100 g À1 ), but only 2 mmol 100 g À1 of lactate were evidenced.

Follow-up of the soluble ion contents during concentration showed that some of them remained soluble whatever the permeate and CF. This was the case for monovalent ions (Na þ , K þ and Cl À ) that existed mainly as free ions in the soluble phase of permeates and remained soluble in the concentrates, even in the most concentrated (Table 3). Further, although Mg 2þ has a strong affinity for citrate and phosphate ions, we could not find any evidence of magnesium insolubilisation during concentration. On the contrary, an increasing part of the soluble calcium, citrate and phosphate ions became insoluble during concentration (Figs. 4 and5) but the nature of the precipitate, and the CF at which precipitation appeared, depended on the type of permeate.

PUF concentration

During PUF concentration, some calcium phosphate precipitated first and at a level higher than 400 g kg À1 DM, some citrate contributed to the precipitate (Fig. 4). In the soluble phase of milk at pH 6.6e6.7, calcium is mainly present as a soluble complex with citrate, as free ionic Ca 2þ and to a lesser extent associated with inorganic phosphate and chloride [START_REF] Gaucheron | The minerals of milk[END_REF]. It should be noted that calcium phosphate (which has a solubility limit of 0.6 mmol L À1 ) was at the edge of precipitating. In addition, the amount of phosphate ions, in the ionic form HPO 4 2À , was almost ten times that of free citrate ion in PUF at pH 6.6e6.7. Their contents have been estimated at 2.65 and 0.26 mM, respectively [START_REF] Gaucheron | The minerals of milk[END_REF]. These conditions were favourable to calcium phosphate precipitation in PUF concentrates. At higher concentrations, the solubility limit of calcium citrate might in turn be exceeded and its precipitation might add to that of calcium phosphate. Our results

were in line with those of a previous experimental study carried out at pilot-scale by [START_REF] Tanguy | Calcium phosphate precipitation during concentration by vacuum evaporation of milk ultrafiltrate and microfiltrate[END_REF]. They also agreed well with theoretical calculations performed using MilkSalt GLM®, a software developed by Mekmene, Le Gra€ et, andGaucheron (2009, 2010) that predicts ion distribution in milk, where calculations are based on the affinities between cations (calcium, sodium, potassium and magnesium) and anions (inorganic phosphate, chloride, citrate and lactate) and calcium phosphate saturation. The input data are the total ion contents and the pH value. For instance, for the more concentrated PUF at CF 10.4, the experimental concentrations of calcium and phosphate precipitated were equal to 62 and 38 mM, which were close to the theoretical values determined by the software, at 62 and 34 mM, respectively. Finally, these results could be related to the work by [START_REF] Garcia | Supersaturation of calcium citrate as a mechanism behind enhanced availability of calcium phosphates by presence of citrate[END_REF] who studied the dynamic exchange between calcium phosphate and calcium citrate from a equimolar (0.10 mol L À1 ) mixture of calcium chloride, sodium citrate and sodium phosphate. In that case, the time-dependent pH fell from 6.3 to 4.0 over 48 h due to calcium phosphate precipitation. With this decrease in the pH, a slow precipitation of calcium citrate (Ca 3 Cit 2 ) started. Moreover, the calcium phosphate precipitate formed initially gradually solubilised with a decreasing pH. The precipitation of calcium citrate in the PUF concentrate at a high CF and low pH was in agreement with this explanation.

P-ACit concentration

The experimental results indicated that only calcium citrate salt was formed during the concentration of P-ACit. Moreover, calcium citrate precipitation was observed early in the concentration process, as from about 150 to 250 g kg À1 DM (Fig. 4). Finally, large quantities of calcium and citrate ions were precipitated at high dry matter contents (about 80% of calcium and 75% of citrate ions at around 500 g kg À1 DM). On the other hand, the evolution of total and soluble phosphate ion contents indicated that all phosphate ions remained soluble throughout the concentration process. A large quantity of citrate ions (11.9 mmol kg À1 compared with a mean value of 6.7 mmol kg À1 in other permeates) was added during milk acidification with citric acid. Meanwhile, the pH of the milk fell from 6.8 to 5.7, leading to the dissolution of micellar calcium phosphate and a subsequent increase in the concentrations of calcium and phosphate ions in the soluble phase. As a result, the initial concentrations of calcium, phosphate and citrate ions in the P-ACit permeate reached 18, 15 and 12 mM. The results obtained during P-ACit concentration could be related to the works of Rice, Barber, O'Connor, Stevens, and Kentish (2010) who studied calcium speciation and calcium salt precipitation in a dairy ultrafiltration permeate using both experimental data and theoretical calculations. In a predicted equimolar (2 mM) calcium-phosphate-citrate system, they showed that calcium speciation preferentially formed calcium citrate salts whereas few calcium phosphate salts developed. The model predicted that calcium ions were mainly free (65%) in the complex form of CaCit À (30%) and CaHCit (5%) at pH 5.7. At a lower pH, the percentage of CaHCit and free Ca 2þ increased whereas that of CaCit À decreased. In P-ACit, the addition of citrate inhibited calcium phosphate precipitation due to the strong chelating capacity of citrate with calcium and the resulting formation of CaCit À , the ionic calcium available for calcium phosphate precipitation being reduced. It also induced a shift of the saturation ratios of calcium phosphate salts, HAP (hydroxyapatite) and OCP (octacalcium phosphate) to higher pH values and further reduced the likelihood of calcium phosphate precipitation. Rice et al 2010 added that in real dairy ultrafiltration permeates, other compounds such as magnesium might form a magnesium phosphate complex, reducing the phosphate ions available for calcium phosphate precipitation.

Several authors have stated that the rate of calcium citrate precipitation was slow [START_REF] Garcia | Supersaturation of calcium citrate as a mechanism behind enhanced availability of calcium phosphates by presence of citrate[END_REF][START_REF] Liu | Hydrates of calcium citrate and their interconversion in relation to calcium bioaccessibility[END_REF]Vavrusova, Garcia, Danielsen & Skibsted, 2017b). [START_REF] Garcia | Supersaturation of calcium citrate as a mechanism behind enhanced availability of calcium phosphates by presence of citrate[END_REF] mentioned that the solutions remained clear for hours before the precipitation of calcium citrate became visible, and even if the solution was supersaturated. According to our results, the precipitation of calcium citrate was quite rapid since P-ACit concentration up to the higher CF lasted for only 4 h. This precipitation could be explained by the considerable supersaturation of citrate and calcium ions (Vavrusova et al., 2017b) and/or a facilitated nucleation by "impurities" such as the lactose crystals in the P-ACit concentrate.

P-ACl concentration

The results obtained during P-ACl concentration were quite similar to those obtained during P-ACit concentration: they evidenced mainly calcium citrate precipitation and only some calcium phosphate precipitation at high DM contents (Fig. 4). In both cases, calcium citrate precipitation was seen from a citrate content close to 40 mmol kg À1 but due to lower citrate content in P-ACl than in P-ACit, it was observed later during P-ACl concentration, as from about 300 g kg À1 DM rather than 150e200 g kg À1 for P-ACit concentration. It should be noted that on the graph showing the evolution of total and soluble citrate ion contents as a function of the DM content, it appeared that the threshold from which precipitation started was shifted slightly upwards by about 40 g kg À1 DM in experimental set #2 compared with experimental set #1. This slight difference could be explained by the fact that the milk salt equilibrium constitutes a dynamic system which is sensitive to minor variations in conditions such as pH, temperature or sample heterogeneity. Moreover, for some salts such as calcium citrate, there is an induction period prior to precipitation [START_REF] Garcia | Supersaturation of calcium citrate as a mechanism behind enhanced availability of calcium phosphates by presence of citrate[END_REF].

According to the work by [START_REF] Rice | A theoretical and experimental analysis of calcium speciation and precipitation in dairy ultrafiltration permeate[END_REF], in the pH range 3.7e4.6, calcium would be mainly free (70e90%) in the complex form of CaHCit (17e28%) and CaCit À (0e12%). Only 1e2% of calcium ions would be under the calcium salt CaH 2 PO4. Even if HCit 2À has a lower affinity for calcium than Cit 3À (876 and 1.65 Â 10 5 M À1 , respectively, according to [START_REF] Holt | Calculation of the ion equilibria in milk diffusate and comparison with experiment[END_REF], HCit 2À sequestered ion calcium and reduced the part of ionic calcium available for calcium phosphate precipitation.

The results obtained during the present study were in accordance with those of a previous study carried out at the pilot scale [START_REF] Tanguy | Calcium citrate insolubilization drives the fouling of falling film evaporators during the concentration of hydrochloric acid whey[END_REF]: it was evidenced that the insoluble salt which formed in the concentrates during P-ACl concentration was calcium citrate. However, calcium citrate precipitation started at a lower DM content (around 150 g kg À1 ). The total calcium and citrate ion contents were close to 90 mmol kg À1 and 20 mmol kg À1 , respectively, but as mentioned previously, several parameters could affect salt solubility. Furthermore, the composition of the hydrochloric whey and the experimental conditions (pilot-scale versus labscale) were different in the two studies.

Concentration of P-AL1, P-AL2 and P-AL3

The experimental results regarding the concentration of P-AL1, P-AL2 and P-AL3 were compared with each other as the permeates differed in terms of the amount of lactic acid added and their resulting pH (pH 6.35, 5.25 and 4.63, respectively). As a consequence, larger quantities of lactate ions were present in P-AL3 and P-AL2 when compared with P-AL1. The same observation was also true for the total calcium and phosphate contents because of the pH decrease and the dissociation of micellar calcium phosphate. Meanwhile, the citrate content remained relatively constant and close to 6.5 mmol kg À1 in the three permeates. The evolution of total and soluble lactate ions during concentration were added to the graphs (Fig. 5) because lactate ions, by association with calcium, could form calcium lactate salt. The latter has a solubility limit of 4.08 g of anhydrous calcium lactate per 100 g of water at 25 C and 17.94 g at 60 C [START_REF] Kubantseva | Solubility of calcium lactate in aqueous solution[END_REF].

For P-AL1, the experimental results evidenced a predominant precipitation of calcium phosphate salt in the concentrates from the start of concentration. Citrate ions remained mostly soluble throughout the concentration process, except at a high dry matter content (over 350e400 g kg À1 DM). Lastly, lactate ions remained completely soluble in the concentrates within the range of the CF values studied. According to the MilkSalt GLM software and at pH 6.3, calcium ions existed mainly as a complex of CaCit À and as free Ca 2þ ions in the initial permeate. Only a very small part was associated with phosphate, chloride or lactate, the latter being mostly free ions. The determination of salt equilibria using the MilkSalt GLM software indicated that calcium phosphate was at the edge of precipitation. Then, as soon as P-AL1 was concentrated, calcium phosphate precipitated because the amount of phosphate ion in the form of HPO higher CF. At CF 10.8, the experimental results and the theoretical values provided by the software were quite similar. From the experiments, 95 mM of calcium and 52 mM of phosphate were precipitated, close to the theoretical values of 93 and 51 mM, respectively. Moreover, the software calculated a high saturation index for calcium citrate salt, which confirmed that some calcium citrate could be precipitated.

As for P-AL1, calcium phosphate precipitation occurred in the P-AL-2 concentrates but to a lesser extent and from a higher DM content (about 300 g kg À1 ). In experimental set #1, the evolution of the citrate ion content during concentration evidenced precipitation (i) from the same DM content as for calcium and phosphate ions, (ii) at a total citrate content close to 40 mmol kg À1 , the threshold previously evidenced during P-ACit and P-ACl concentrations and (iii) in large quantities as the proportion of insoluble citrate reached 54.9% in the concentrate at 452.4 g kg À1 DM. Surprisingly, the results for experimental set #2 showed a linear evolution of the soluble citrate ion content as a function of the DM content of concentrates. A supplementary analysis of a P-AL2 sample concentrated at 407.8 g kg À1 DM confirmed the findings regarding experimental set #2; the soluble citrate ion content of 51.6 mmol kg À1 reached a calculated total content of 57.4 mmol kg À1 . This discrepancy between experimental sets could be explained by the long induction period required for calcium citrate precipitation and/or the formation of lactose crystals during P-AL permeate concentration (see above); the latter constituted "impurities" that initiated the random crystallisation of calcium citrate salt in some samples.

Based on MilkSalt GLM software, the predicted calcium speciation in initial P-AL2 at pH 5.25 indicated that calcium was mainly present as free ions (50%), and the other fraction was under soluble complexes, mainly CaCit À , CaLact À (16% and 18%, respectively) and to a lesser extent associated with inorganic phosphate (H 2 PO 4 -: 5.6%; HPO 2À : 2%), chloride (5%) and HCit 2À (2.4%). Calcium phosphate was below its solubility limit which explains why its precipitation was shifted to the higher CF in comparison with P-AL1. For the higher concentrated product at FC 12.7, the software indicated 119 mM calcium and 65 mM inorganic phosphate were precipitated, whereas the experimental results evidenced 116 mM and 84 mM, respectively. Further, the calculated saturation ratio for calcium citrate salt was evaluated at 54, which would indicate the likelihood of calcium citrate precipitation. Interestingly, the software also indicated that around 22% of total calcium was associated with lactate ions, which reduced the ionic calcium available and the likelihood of the precipitation of calcium citrate and calcium phosphate.

The experimental results showed that few calcium precipitates were formed throughout P-AL3 concentration and they only appeared at higher CF values. Based on the MilkSalt GLM software, the predicted calcium speciation in initial P-AL3 at pH 4.63 indicated that calcium existed mainly as free ions (50%), and as CaLact À (23%) and CaCit À (9.4%) soluble complexes. It was also associated to a lesser extent with H 2 PO 4 -(5.9%), HCit 2À (5.8%) and chloride (4.7%).

Several parameters contributed to the small quantity of calcium salts formed during P-AL3 concentration: (i) formation of the CaLact À complex that reduced the amount of ionic calcium available for potential associations with citrate and phosphate, (ii) the higher solubility of calcium lactate salt compared with calcium phosphate solubility, (iii) the high lactate content, (iv) the low citrate content compared with lactate and phosphate contents, and (v) the ionic form of phosphate within the range 4.63 to 4.23 (H 2 PO 4 -) that has a low affinity for calcium (12 M À1 [START_REF] Holt | Calculation of the ion equilibria in milk diffusate and comparison with experiment[END_REF]). The presence of lactate ions displaced the equilibrium in the calcium-phosphate-citrate system towards the formation of calcium lactate, thus inhibiting the precipitation of calcium citrate and calcium phosphate. For a comparison, [START_REF] Tanguy | Calcium citrate insolubilization drives the fouling of falling film evaporators during the concentration of hydrochloric acid whey[END_REF] performed the vacuum concentration of an industrial lactic acid whey using a pilot-scale falling-film evaporator and studied the evolution of total and soluble ion contents during concentration. The lactic acid whey was a by-product of industrial fresh cheese production, which was characterised by a large quantity of lactate ions (109.8 mmol kg À1 ) and an absence of citrate ions, which were consumed by lactic bacteria during cheese processing. The authors evidenced an absence of precipitation in the lactic acid whey concentrates and explained this result by the presence of lactate. Similarly, and in the context of producing lactic acid whey powders, [START_REF] Mimouni | Cristallisation du lactose et epaississement dans les lactos erums concentr es[END_REF] showed that the addition of sodium citrate to industrial lactic acid whey concentrates was a means to reduce the formation of calcium lactate crystals, which are responsible for a thickening of concentrates during lactose crystallisation.

Conclusion

This work investigated the influence of mineral composition and pH on the nature and amount of minerals that precipitate in concentrates during whey concentration. Dairy ultrafiltration permeates with variable mineral compositions and pH values were prepared by the ultrafiltration of non-acidified and acidified milk using different acids. Depending on the permeates considered and the CF, the results evidenced either some precipitation of calcium salts (calcium phosphate, calcium citrate) in the concentrates, or no precipitation. Calcium phosphate was the dominant mineral specie in the precipitate of concentrates in which the main ionic form of inorganic phosphate was HPO 4 2À . This was the case of concentrates obtained by the concentration of PUF at pH 6.7 and P-AL1 at pH 6.3. At a pH of around 5.0, the high affinity between calcium ions and Cit 3À , and the reduced affinity of phosphate for calcium, favoured the precipitation of calcium citrate. However, the degree of precipitation could be limited because insufficient Cit 3À was present to exceed the solubility limit of calcium citrate. Calcium citrate precipitation occurred extensively in concentrates resulting from the concentration of P-ACit at pH 5.6 due to the large quantity of citrate ions (both initial and added). It also took place in P-ACl concentrates, but at a higher CF because of the smaller quantities of citrate ions. Lastly, the large quantities of lactate ions in the P-AL2 and P-AL3 concentrates modified the ion equilibrium and particularly the calcium-citrate-phosphate system. It reduced the quantity of ionic calcium available for further association with citrate and phosphate ions, inhibiting the precipitation of calcium citrate and calcium phosphate. However, lactate ions promoted lactose crystallisation in the concentrates.

During permeate concentration, the affinity between calcium and the main anions (phosphate, citrate and lactate) and the amounts of each species determine the likelihood of the precipitation of salts. The behaviour of these ions during concentration can be assessed in light of: (i) the nature and quantities of ions available in the concentrates, (ii) the anionic forms present at the pH of the concentrates and their association constants with calcium, and (iii) the solubility limit of calcium salts. The next stage in this work will be to study the influence of the whey proteins present in industrial whey on calcium salt precipitation.

Fig. 1 .

 1 Fig.1. Processing scheme for the preparation of milk permeates from non acidified milk (pH 6.7) and milks acidified at different (pH 4.6e6.3) with different acids (citric, hydrochloric and lactic). In italics: wheys produced from the manufacture of cheese and acid caseins for which each model cheese whey is representative.

Fig. 2 .

 2 Fig. 2. Evolution of pH of concentrates during vacuum concentration of (a) PUF, (b) P-ACit, (c) P-ACl, (d) P-AL1, (e) P-AL2 and (f) P-AL3; , set #1 and , set #2, representing the 2 series of concentration experiments.

Fig. 4 .

 4 Fig. 4. Evolution of total and soluble calcium, inorganic phosphate and citrate contents during concentration of PUF, P-ACit and P-ACl permeates. Set #1 and set #2 represented the 2 series of concentration experiments. Dark and white diamonds are relative respectively to experimental total ( ) and soluble ( ) ion contents for set #1. Dark and white triangles are relative respectively to experimental total ( ) and soluble ( ) ion contents for set #2. Dotted line referred to calculated total ion content.

Fig. 5 .

 5 Fig.5. Evolution of total and soluble calcium, inorganic phosphate, citrate and lactate contents during concentration of P-AL1 P-AL2 and P-AL3 permeates. Set #1 and set #2 represented the 2 series of concentration experiments. Dark and white diamonds are relative respectively to experimental total ( ) and soluble ( ) ion contents for set #1. Dark and white triangles are relative respectively to experimental total ( ) and soluble ( ) ion contents for set #2. Dotted line refers to calculated total ion content.

Table 1

 1 Composition of permeates obtained from ultrafiltration of nonacidified and acidified milk. a

	Parameter	PUF	P-ACit	P-ACl	P-AL1	P-AL2	P-AL3
	pH	6.89 ± 0.01	5.72 ± 0.01	4.64 ± 0.00	6.34 ± 0.01	5.21 ± 0.04	4.63 ± 0.00
	DM (g kg À1 )	47.5 ± 0.3	43.2 ± 0.2	52.8 ± 0.4	48.2 ± 0.3	42.3 ± 0.2	54.4 ± 0.0
	Ash (g kg À1 )

Table 2

 2 Estimation of the composition of the precipitate deduced from the experimental analysis of the more concentrated permeates (set #2) and the corresponding supernatant after centrifugation. a Centrifugation was at 1800Âg, 20 min, 20 C; mass of precipitate was per 100 g of concentrate, composition of precipitate is given per 100 g of precipitate.

	Parameter	PUF	P-ACl	P-ACit	P-AL1
		CF 8.9	CF 8.6	CF 10.7	CF 10.6
	Mass of precipitate	8.9	6.0	10.7	24.7
	Composition of precipitate
	DM (g)	101.8	105.2	111.3	92.0
	TN (g)	0.6	7.2	3.6	3.0
	Ash (g)	4.4	10.6	9.5	0.5
	Lactose (g)	100.2	79.9	87.2	96.7
	Calcium (mmol)	52	127	150	17
	Lactate (mmol)	0	0	2	2
	Inorganic phosphate (mmol)	26	27	5	13
	Citrate (mmol)	3	50	88	3

a

Table 3

 3 Total and soluble sodium, potassium, chloride and magnesium ion contents in the more concentrated permeate concentrates.

	Ionic species	PUF concentrate	P-ACit concentrate	P-ACl concentrate	P-AL1 concentrate	P-AL2 concentrate	P-AL3 concentrate
	(mmol kg À1 )	CF 10.4		CF 12.5		CF 10.1		CF 10.8		CF 12.7		CF 9.0
		total	soluble	total	soluble	total	soluble	total	soluble	total	soluble	total	soluble
	Na	127	103	120	137	119	113	110	117	154	185	110	107
	K	344	341	399	403	375	357	371	368	443	488	339	333
	Cl	283	281	329	332	730	706	288	285	313	337	216	213
	Mg	25	22	35	32	41	38	28	26	41	43	34	33

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration of competing interest

None.


Annabelle Vernouillet

Charlotte Carré

Delphine Neff

Xavier Bourbon

Valérie Deydier

Nicolas Michau

Philippe Dillmann

C Carre

Corrosion of carbon-steel casing used in deep geological radioactive waste storage


Corentin Audiard

On the sharp regularity of solutions to hyperbolic boundary value problems

We prove some sharp regularity results for solutions of classical first order hyperbolic initial boundary value problems. Our two main improvements on the existing litterature are weaker regularity assumptions for the boundary data and regularity in fractional Sobolev spaces. This last point is specially interesting when the regularity index belongs to 1/2+N, as it involves nonlocal compatibility conditions.

Introduction

Everything in a toy model Consider the simplest hyperbolic initial boundary value problem (IBVP)    ∂ t u + ∂ x u(x, t) = 0, (x, t) ∈ (R + ) 2 u(x, 0) = u 0 (x), u(0, t) = g(t)

When (u 0 , g) ∈ L 2 (R + ) 2 , the solution is piecewise defined: u(x, t) = u 0 (x -t) for x -t ≥ 0, g(t -x) for x -t < 0, it belongs to C t L 2 . It is well known that the smoothness of (u 0 , g) is not enough to ensure the smoothness of u, compatibility conditions are required: for k ∈ N, u ∈ ∩ k j=0 C j t H k-j if and only if

(u 0 , g) ∈ (H k ) 2 and ∀ j ≤ k -1, u (j)
0 (0) = (-1) j g (j) (0).

These compatibility relations are trivial here due to the solution formula, but are more generally derived considering u (and its derivatives) at the corner x = t = 0, and writing

∂ α u| x=0 | t=0 = ∂ α u| t=0 | x=0 .
A basic rule of thumb being that any compatibility condition that makes sense should be true. For fractional regularity, not much changes except in the notoriously pathologic case s ≡ 1/2[Z]. Indeed even if there is no trace in H 1/2 (R + ), the gluing of two functions in H 1/2 (R + ) is not H 1/2 (R). The simplest way to see this is to consider the map

f ∈ L 2 (R) → f (•) -f (-•) ∈ L 2 (R + ). It is continuous L 2 (R) → L 2 (R + ) and H 1 → H 1 0 (R + ) hence H 1/2 (R) → [L 2 , H 1 0 ] 1/2 1 INTRODUCTION
2 by interpolation. The interpolated space is the famous Lions-Magenes space H 1/2 00 (R + ), and it is different (algebraically and topologically) from H 1/2 (R + ): by interpolation of Hardy's inequality, any function f ∈ H 1/2 00 (R + ) must satisfy

R + f 2 (x)
x dx < ∞, this is obviously not the case for functions merely in H 1/2 (R + ).

For the regularity of solutions of the BVP, this adds a "global" compatibility condition u ∈ C t H 1/2 ⇔ (u 0 , g) ∈ H 1/2 (R + ) and

R + |g(x) -u 0 (x)| 2 x dx < ∞.
Our aim here is to prove an analogous result for general hyperbolic boundary value problems.

Settings and results

Let Ω be a smooth open set of R d , we consider first order boundary value problems of the form

   Lu := (∂ t -d j=1 A j ∂ j )u = 0, (x, t) ∈ Ω × R + t , Bu| ∂Ω = g, (x, t) ∈ ∂Ω × R + t , u| t=0 = u 0 , x ∈ Ω.
(1.1) IBVP The index t in R + t has no meaning except to emphasize the time variable. The A j s are q × q matrices depending smoothly on (x, t), B is a smooth b×q matrix, b is the number of boundary conditions. For data (u 0 , g, f ) ∈ L 2 (Ω) × L 2 (∂Ω × R + t ) × L 2 (R + t × Ω), the well-posedness of such hyperbolic BVP has been obtained in a large variety of settings, that we will only shortly mention. After the pioneering results of Friedrichs Friedrichs [START_REF] Friedrichs | Symmetric positive linear differential equations[END_REF] for symmetric dissipative systems, Kreiss Kreiss [START_REF] Kreiss | Initial boundary value problems for hyperbolic systems[END_REF] proved the well-posedness of the BVP with zero initial data in the strictly hyperbolic case ( A j ξ j has only real eigenvalues of algebraic multiplicity one) under the now standard Kreiss-Lopatinskii condition on B. In Kreiss's framework, the case of L 2 initial data was then tackled by Rauch Rauch [START_REF] Rauch | L 2 is a continuable initial condition for Kreiss' mixed problems[END_REF]. Well-posedness of constantly hyperbolic BVP was later obtained by Métivier Met [START_REF] Métivier | The block structure condition for symmetric hyperbolic systems[END_REF] (zero initial data), the author then proved well-posedness with L 2 initial data Audiard1 [START_REF] Audiard | On mixed initial-boundary value problems for systems that are not strictly hyperbolic[END_REF]. A further generalization was obtained by Métivier Metsemigroupe [START_REF] Métivier | On the L 2 well posedness of hyperbolic initial boundary value problems[END_REF] for a new class of hyperbolic operators, larger than the constantly hyperbolic ones. He also gave a new proof, both more general and simpler, of well-posedness with L 2 initial data. For more references and results, in particular for characteristic BVP (that we do not consider here) the reader may refer to the book Benzoni3 [START_REF] Benzoni | Multidimensional hyperbolic partial differential equations[END_REF].

Let n be a normal on ∂Ω, the problem ( IBVP 1.1) is said to be noncharacteristic when A j n j is invertible on ∂Ω. For non characteristic boundary value problems, the main reference on the smoothness of solutions is the classical paper of Rauch and Massey RauchMassey [START_REF] Rauch | Differentiability of solutions to hyperbolic initial-boundary value problems[END_REF], where, under no specific assumption (except of course well-posedness), the authors prove that the solution of (

IBVP 1.1) belongs to ∩ k j=0 C j t (R + t , H k-j (Ω)) when (u 0 , g, f ) ∈ H k (Ω) × H k+1/2 (∂Ω × R + t ) × H k (Ω × R + t
) and satisfy natural compatibility conditions that we describe now. For conciseness, when there is no ambiguity we will usually denote

H k instead of H k (X), X = Ω, ∂Ω × R + t , Ω × R + t .
We denote A = A j ∂ j and define inductively v j the formal value of (∂ j t u)| t=0 by

v 0 = u 0 , v j+1 = (∂ j t ∂ t u)| t=0 = ∂ j t (Au + f )| t=0 = j l=0 j l (∂ l t A| t=0 )v j-l + ∂ j t f | t=0 . (1.2) taylor
The first order compatibility condition is Bv 0 | ∂Ω = g| t=0 and the generic compatibility condition of order j is Compatibility at order j :

∂ j-1 t g| t=0 = j-1 l=0 j -1 l (∂ l t B)v j-1-l | ∂Ω . (1.3) CCj Note that ( CCj 1.
3) makes sense as soon as (u 0 , g, f ) ∈ (H s ) 3 , s > j -1/2. If the smoothness of the data is j -1/2, j ∈ N * , we define a special compatibility condition : when Ω = R d-1 × R + , denote x = (x , y); the condition is Compatibility at order j -1/2:

∂ j-1 t g(x , t) - j-1 l=0 j -1 l (∂ l t B)v j-1-l (x , t) ∈ H 1/2 00 R d-1 × (R + ) . (1.4) CCj-
For general smooth Ω, (

CCj- 1.4
) is defined similarly through local maps and a partition of unity: near the boundary Ω is diffeomorphic to (a part of) R d-1 × R + thanks to some map Φ, one simply requires ( CCj 1.3) to stand for g(Φ(x , 0), t), (v l • Φ(x , t)) 0≤l≤j-1 . Note that due to Hardy's inequality, the j-th condition implies the condition of order j -1/2.

Definition 1.1. If s = k + θ, -1/2 < θ < 1/2, k ∈ N * , θ = 1/2,
we say that data (u 0 , g, f ) ∈ (H s ) 3 satisfy the compatibility conditions at order s when

( CCj 1.3) is satisfied for 1 ≤ j ≤ k. If s = k -1/2,
the compatibility conditions are satisfied at order s when

( CCj 1.3) is true for 1 ≤ j ≤ k -1 and ( CCj- 1.4) is true for j = k. A strong L 2 solution of ( IBVP 1.1) is a function u ∈ C t L 2 such
that there exists a sequence u n of smooth solutions of ( IBVP 1.1) with data (u 0,n , g n , f n ) that converge to (u 0 , g, f ) in L 2 , and for any T > 0, u -u n C([0,T ],L 2 ) → 0.

Assumptions We need the smoothness of Ω and the well-posedness of ( IBVP 1.1): assump1 1. ∂Ω is a smooth hypersurface with normal ν, parametrized by local maps (φ j (y )) 1≤j≤J , y ∈ R d-1 , and ϕ j (y , y d ) := φ j (y ) + y d ν(ϕ j (y )) are local diffeomorphisms

V j → U j , with ϕ j ((R d-1 × R + * ) ∩ V j )
⊂ Ω, and ∪ J j=1 U j ⊃ ∂Ω. We do not assume that the U j are bounded sets, but Dϕ j , Dϕ -1 j must be uniformly bounded, and d(Ω \ ∪Im(ϕ j ), ∂Ω) > 0.

1 INTRODUCTION 4
3. For data (u 0 , g, f ) ∈ (L 2 ) 3 , there exists a unique strong L 2 solution1 to ( IBVP 1.1) that satisfies the semi-group estimate for γ large enough

e -γ• u C([0,t],L 2 (Ω)) + √ γ|e -γ• u| ∂Ω | L 2 (∂Ω×[0,t]) u 0 L 2 (Ω) + |e -γ• g| L 2 (∂Ω×[0,T ]) + e -γ• f L 2 ([0,t]×Ω) √ γ . (1.5)
We use the convention that norms inside the domain are denoted • while norms on the boundary are denoted | • |.

We point out that a consequence of the semi-group estimate is the resolvent estimate: for γ large enough (larger than for (

semigroupe 1.5)) γ e -γt u 2 L 2 (Ω×R + t ) + |e -γt u| ∂Ω | 2 L 2 (∂Ω×R + t ) u 0 2 L 2 (Ω) + |e -γt g| 2 L 2 (∂Ω×R + t ) + e -γt f 2 L 2 γ . (1.6)
This is readily obtained by squaring ( semigroupe 1.5) for some fixed γ 0 , multiplication by e -2γt , γ > γ 0 and integration in t. Higher regularity versions of the resolvent and the semi-group estimates are a bit more delicate to state. We define weighted Sobolev spaces H s γ in section notations 2, the weighted resolvent estimate is then

γ u 2 H s γ + |u| ∂Ω | 2 H s γ u 0 2 H s (Ω) + |g| 2 H s γ + f 2 H s γ γ . (1.7)
The main point of this estimate is that it is sharp with respect to the parameter γ and allows to absorb commutators in a priori estimates. Moreover, it implies the following (simpler to read) estimate

e -γt u 2 H s (Ω×R + ) + |e -γt u| ∂Ω | 2 H s (∂Ω×R + ) u 0 2 H s (Ω) + |e -γt g| 2 H s (∂Ω×R + ) + e -γt f 2 H s (Ω×R + ) . (1.8)
We shall not need something as precise for the semi-group estimate:

let s = k + θ, k ∈ N, 0 < θ < 1, then k j=0 e -γt ∂ j t u 2 C(R + t ,H k-j+θ (Ω)) + |e -γt u| ∂Ω | 2 H s (Ω×R + t ) u 0 2 H s (Ω) + |e -γt g| 2 H s (∂Ω×[0,T ] + e -γt f 2 H s . (1.9)
Both estimates should be modified when s = k + 1/2, k ∈ N: it is necessary to add in the right hand side the H

1/2 00 norm of ∂ k t g -k 0 k l (∂ l t B)v k-1-l , see page cas1/2
15 for details. This is the (implicit) convention that we use in theorem mainth 1.3, we refer to the proof for more details. An interesting related feature is that the constant in can not be uniform in θ, it blows up as θ → 1/2 and the estimates are actually not true for θ = 1/2.

We can now state more precisely the regularity result of Rauch and Massey:

Theorem 1.2 ( RauchMassey [18]). If (u 0 , g, f ) ∈ H k (Ω) × H k+1/2 (∂Ω × R + t ) × H k (Ω × R + t )
satisfy the compatibility condition up to order k, the solution of

( IBVP 1.1) belongs to ∩ k j=0 C j t H k-j .
The only suboptimal part of the theorem is the regularity assumption on g. This is due to the fact that the theorem is deduced from the homogeneous case g = 0 with a lifting argument. It was already pointed out at the time by the authors that it could be improved (without proof), but quite unfortunately the result that remained in the litterature is the suboptimal one, see for example the reference book Benzoni3 [START_REF] Benzoni | Multidimensional hyperbolic partial differential equations[END_REF], and in somewhat different settings the lecture notes Met2 [START_REF] Métivier | Stability of multidimensional shocks[END_REF] or the interesting discussion in the introduction of lanigu [START_REF] Iguchi | Hyperbolic free boundary problems and applications to wave-structure interactions[END_REF]. Our result is that the same property holds with boundary data in H k instead of H k+1/2 , moreover we allow k to be any nonnegative real number rather than an integer.

mainth Theorem 1.3. Let s ∈ R + . If (u 0 , g, f ) ∈ H s (Ω) × H s (∂Ω × R + t ) × H s (Ω × R + t )
satisfy the compatibility condition up to order s, the solution of (

= k + θ, k ∈ N, 0 ≤ θ < 1, it satisfies ( semigreg 1.9).
The proof when s is an integer is quite similar to the original argument of Rauch and Massey, actually the fact that we handle directly nonzero boundary data leads to some slight simplifications due to the fact that it allows to avoid a reduction to the case where B is constant. The fractional case is essentially an interpolation argument, however it is not trivial due to the presence of the compatibility conditions. For example, in the model case described

earlier instead of interpolating [L 2 × L 2 , H 1 × H 1 ] θ one must identify [L 2 × L 2 , {(u 0 , g) ∈ H 1 (R + ) × H 1 (R + ) : u 0 (0) = g(0)}] θ .
The litterature on such problems is not very rich. Another related problem is the interpolation of Sobolev spaces with boundary conditions, that are in some sense between H s and H s 0 . This issue appeared quite long ago for elliptic equations on non smooth domains or parabolic problems, see e.g. the last section of Grisvard [START_REF] Grisvard | Caractérisation de quelques espaces d'interpolation[END_REF], sections 14-17 of chapter 4 in lionsmagenes2 [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] (where most of the identification problems were left open), or the more recent (and much more involved) book Amann [START_REF] Amann | Anisotropic function spaces and maximal regularity for parabolic problems[END_REF], in particular VIII.2.5. Due to the technicity of this last reference (anisotropic Besov spaces are studied), degenerate cases (in our settings s ∈ N + 1/2) are not considered. The Schrödinger equation on a domain and related interpolation problems were also studied by the author in Audiard7 [START_REF] Audiard | Global Strichartz estimates for the Schrödinger equation with non zero boundary conditions and applications[END_REF], where the natural spaces for the boundary data are Bourgain spaces.

Plan of the article Section 2 is devoted to notations and a brief reminder on interpolation. The proof of theorem mainth 1.3 is then organized in three sections : in section 3 we recall a standard smoothness result for the pure boundary value problem posed for t ∈ R, due to Tartakov. For completeness, we include a sketch of proof that follows an argument of the (unfortunately depleted) book ChaPi [START_REF] Chazarain | Introduction à la théorie des équations aux dérivées partielles linéaires[END_REF]. Theorem mainth 1.3 in the case s integer is proved in section 4. An important point is a basic lifting lemma which proves also useful for the general case. In section 5, smoothness is first proved for 0 ≤ s ≤ 1 with an interpolation argument, then for any s with a non trivial differentiation argument.

Ackowledgement This work was partially funded by the ANR project NABUCO, ANR-17-CE40-0025.

Notations and basic results

notations Basic notations Proofs are often reduced to the case Ω = R d-1 × R + . In such settings, we denote the variable x = (x , y) x ∈ R d-1 . The variables x , t are called tangential, while y is the normal variable. Partial differential operators acting on functions of (x, t) are written as ∂ α , α ∈ N d+1 , by convention α d+1 is the order of differentation in time. A multi-index, or a differential operator, is said to be tangential when

α d = 0. We denote [L 1 , L 2 ] = L 1 L 2 -L 2 L 1 the commutator between two linear operators.
Sobolev spaces Ω is assumed to be a smooth open set as in definition page assump1 3. The Sobolev spaces H s (Ω), are defined when s is an integer as

{u ∈ L 2 : u 2 H s = |α|≤s Ω |∂ α u| 2 dx < ∞}.
When s is not an integer, they are defined by (complex) interpolation, H s = [L 2 , H k ] s/k for any integer k larger than s. This definition does not depend on k.

The Sobolev spaces for functions defined on ∂Ω, Ω × R + t etc are defined in the same standard way.

H s 0 (Ω) is the closure of C ∞ c (Ω). We do have [L 2 , H 1 0 ] s = H s 0 for 0 < s < 1, except for s = 1/2, where H 1/2 0 = H 1/2 and [L 2 , H 1 0 ] 1/2 = H 1/2
00 is different algebraically and topologically from H 1/2 . It is a Banach space endowed with the norm

u 2 H 1/2 00 = u 2 H 1/2 + Ω |u(x)| 2 d(x) dx,
where d is the distance to ∂Ω (see lionsmagenes [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]). The only important fact, regularly used in the article, is that if X 0 , X 1 are Banach spaces, an operator T :

X 0 → L 2 , X 1 → H 1 0 maps [X 0 , X 1 ] 1/2 to H 1/2 00 . For example, u ∈ H s (R d ) → u(x , y) -u(x , -y) maps H 1/2 (R d ) to H 1/2 00 (R d-1 × R + ).
The weighted Sobolev spaces H s γ are defined as follows :

Definition 2.1. When s is a nonnegative integer we define H s γ (Ω × R + t ) as the the set of functions in L 2 such that the following norm is finite

u H s γ = |α|≤s e -γt ∂ α u L 2 .
When s is not an integer, H s γ is defined by complex interpolation : if k is an integer larger than s,

H s γ = [L 2 γ , H k γ ] s/k . H s γ (∂Ω × R + t ) is defined similarly.
When s is an integer, it is a straightforward consequence of Leibniz formula ∂ j t (e -γt u) = j i (-γ) i e -γt ∂ j-i t u that the H s γ norm is equivalent to e -γt u H s , though with constants that depend on γ, hence the H s γ spaces coincide algebraically and topologically with the set of functions such that e -γt u ∈ H s .

Traces Sobolev spaces on ∂Ω are defined with local maps. The trace operator is an isomorphism:

H s (Ω) → k<s-1/2 H s-1/2-k (∂Ω), u → (∂ k n u| ∂Ω ) k<s-1/2
, where ∂ n is the normal derivative on ∂Ω. For functions defined in H s (Ω × R + * ), the trace operator on ∂Ω × R + * and Ω × {0} is more subtle, the map

H s (Ω × R + * t ) → k<s-1/2 H s-1/2-k (∂Ω × R + * t ) × k<s-1/2 H s-1/2-k (Ω × {0}) , u → (∂ k n u| ∂Ω×R + * , ∂ k t u| Ω×{0} ) k<s-1/2 , (2.1) trace2 is continuous but not surjective: if s / ∈ N, local compatibility conditions between (g k , v k ) ∈ ( H s-1/2-k (∂Ω × R + * t )) × ( H s-1/2-k (Ω × {0}
)) are required as follows

∀ k + j < s -1, ∂ j t g k | t=0 = ∂ k n v j | ∂Ω , (2.2) compaclassiqu
(see lionsmagenes2 [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]). In the case s = 1, and Ω = R d-1 × R + * , surjectivity requires the global compatibility condition

v 0 (x , t) -g 0 (x , t) ∈ H 1/2 00 (∂Ω). ( 2

.3) compaclassiqu

This condition extends to smooth Ω, see the short comment after ( CCj-1.4). Provided such compatibility conditions are added, the trace map is a surjection and has a right inverse, this very well known fact will be proved later in the article in some basic cases where it is needed with more precise estimates.

Regularity for the pure boundary value problem

Consider the boundary value problem

   Lu = f, (x, t) ∈ Ω × R + t Bu| ∂Ω = g, u| t=0 = 0. (3.1) bvp
When g, f can be smoothly extended by 0 for t < 0, the smoothness of u is well known Tartakoff [START_REF] Tartakoff | Regularity of solutions to boundary value problems for first order systems[END_REF],

ChaPi [START_REF] Chazarain | Introduction à la théorie des équations aux dérivées partielles linéaires[END_REF]. The classical proof is done by first studying the pure boundary value problem posed on t ∈ R, the case t ∈ R + is then deduced by an extension by 0 for t < 0. We give here a minor variation of this argument that directly tackles ( bvp 3.1).

regBVP Proposition 3.1. Let k ∈ N. If the extension of f and g by 0 for t < 0 belongs to H k , then for γ large enough the solution of

( bvp 3.1) satisfies e -γt u ∈ H k (Ω × R + t ).
In particular, its belongs to H k (Ω × [0, T ]) for any T > 0.

Proof. The classical plan is to straighten the boundary through local maps, then use a tangential regularization. It is done by induction on k, it suffices to prove the final step where we assume u ∈ H k-1 (R d × R t ) and prove u ∈ H k .

We fix local maps ϕ j as in assumption assump1 1. Let (ψ j ) 0≤j≤J be a partition of unity associated to Ω ∪ (∪ j Im(ϕ j )). We denote the new variable y = (y , y d ), u j = (ψ j e -γt u) • ϕ j , and

u 0 = ψ 0 u, L j = ∂ t + γ + i k A k (D y ϕ j ) -1 ik (y) ∂ y i . For 1 ≤ j ≤ J, u j satisfies L j u j + ([ψ j , L]e -γt u) • ϕ j = e -γt (ψ j f ) • ϕ j := f j , (y , y d , t) ∈ R d-1 × R + × R + t , B(ϕ j (y , 0))u j (y , 0, t) = e -γt (ψ j g)(ϕ j (y , 0), t) := g j . (3.2) eqredresse
For simplicity we still denote B for B • ϕ j (•, 0). The regularization procedure was introduced by Hörmander

hormanderBVP [8]: for v ∈ L 2 (R p ), p ≥ 1, define v 2 H s,δ (R p ) = R d | v(ξ)| 2 (1 + |ξ| 2 ) s+1 1 + |δξ| 2 dξ -→ δ→0 v 2 H s+1 . Let ρ(x) ∈ C ∞ c (R p ), such that | ρ(ξ)| |ξ| m , m
> k and ρ does not cancel on a neighborhood outside 0 (such functions are easily constructed, for example using ∆ m/2 (ρ 0 (t)ρ (y )), m even). Define

ρ ε = ρ(•/ε)/ε d . It is an exercise in calculus that for 0 ≤ s ≤ k -1, an equivalent norm to • H s,δ -uniformly in δ-is v L 2 + 1 0 v * ρ ε 2 L 2 1 ε 2(s+1) (1 + δ 2 /ε 2 ) dε ε 1/2 ∼ v H s,δ . (3.

3) equivsobo

Friedrich's lemma can be generalized in such settings: for P a first order differential operator with smooth coefficients

1 0 [P, ρ ε * ]v 2 L 2 1 ε 2(s+1) (1 + δ 2 /ε 2 ) dε ε v 2 H s,δ . ( 3

.4) friedrichs

For details, we refer to ChaPi [START_REF] Chazarain | Introduction à la théorie des équations aux dérivées partielles linéaires[END_REF] chapter 2 section 6. We shall use tangential mollifiers ρ ε (x , t) for the functions u j , 1 ≤ j ≤ J, and full mollifiers ρ ε (x, t) for u 0 . Everything in ( eqredresse 3.2) is extended by 0 for t < 0. Note that due to the assumptions on f, g, the extensions of (f j , g j ) are still in H k . We apply ρ ε * to ( eqredresse 3.2) for 1 ≤ j ≤ J:

L j ρ ε * u j = ρ ε * f j -ρ ε * [ψ j , L j ]e -γt u • ϕ j -[ρ ε * , L j ]e -γt u j , B(ρ ε * u j )| y d =0 = ρ ε * g j -[ρ ε * , B]u j | y d =0 . (3.5) eqreg Since ρ ε * u j belongs to L 2 (R + , H ∞ (R d-1 × R t ) it is in H ∞ due to non-characteristicity, we can use the resolvent estimate ( resolv 1.6). γ ρ ε * u j 2 L 2 + |ρ ε * u j | 2 L 2 ρ ε * f j 2 L 2 + ρ ε * [ψ j , L j ]e -γt u • ϕ j 2 L 2 + [ρ ε * , L j ]u j 2 L 2 γ +|ρ ε * g j -[ρ ε * , B]u j | 2 L 2 .
Multiplying by ε -2k-1 1 + (δ/ε) 2 -1 , integrating in ε and using Friedrich's lemma we have

γ u j 2 L 2 H k-1,δ + |u j | 2 H k-1,δ f j 2 H k + [ψ j , L j ]e -γt u • ϕ j 2 L 2 H k-1,δ γ + g j 2 H k . ( 3

.6) estimtan

The commutator [ψ j , L j ] is the multiplication by a smooth matrix θ j . Due to the special structure of the local maps, ϕ -1 i • ϕ j has the form (ϕ i,j (y ), y d ) hence

θ j e -γt u • ϕ j = J 1 ψ i θ j u i (ϕ i,j (y ), y d ) + θ j u 0 • ϕ j .
Thanks to composition rules (in H s,δ , again see

ChaPi [5]), [ψ j , L j ]e -γt u • ϕ j L 2 H k-1,δ J i=1 u i L 2 H k-1,δ + u 0 H k-1,δ
For γ large enough, this can be absorbed in (the sum over j of) the left-hand side of ( estimtan 3.6):

J j=1 γ u j 2 L 2 H k-1,δ + |u j | 2 H k-1,δ J 1 f j 2 H k + u 0 2 H k-1,δ γ + J 1 |g j | 2 H k . (3.7) estimtan2
It seems "moral" that noncharacteristicity should imply the same bound for u j H k-1,δ , however the H k-1,δ norm is a non local norm for functions defined on R d × R t , hence such an assertion is not clear. Instead we first obtain interior estimates with similar, simpler computations

γ u 0 2 H k-1,δ f 0 2 H k + e -γt ψ 0 u 2 H k-1,δ γ , supp( ψ 0 ) ⊂ Ω, ψ 0 ≡ 1 on supp(ψ 0 ). ( 3

.8) interieur

Decomposing again ψ 0 u = J j=0 ψ 0 ψ j u, and following the same lines that led to (

estimtan2 3.7), J j=1 γ ψ 0 ψ j u • ϕ j 2 L 2 H k-1,δ J 1 f j 2 H k + u 0 2 H k-1,δ γ + J 1 |g j | 2 H k . (3.9)
A simple consequence of the definition of the H s,δ spaces is that for any tangential differential operator D of order 1 and s ≥ 1

Dv H s-2,δ 1 C v H s-1,δ + C v L 2 H s-1,δ . (3.10) transfert Now for j ≥ 1, each function ψ 0 ψ j u • ϕ j is compactly supported in R d-1 × R + * × R t ,
and on its support L j is (uniformly) non characteristic, so we may extend it by zero for y d < 0 and use ( transfert 3.10) to deduce

J j=1 γ ψ 0 ψ j u • ϕ j 2 H k-1,δ f j 2 H k + u 0 2 H k-1,δ γ + J 1 |g j | 2 H k + γ u 2 H k-1 γ . (3.11) interieur2
Note that the term γ e -γt u 2

H k-1 γ
is present due to the factor γ in the definition of L j . Thanks to the induction assumption, this lower order term is bounded by g 2

H k-1 γ + f 2 H k-1 γ . Putting together ( estimtan2 3.7), ( interieur 3.8), ( interieur2
3.11) we have J 1 u j 2 L 2 H k-1,δ + u 0 2 H k-1,δ + J 1 |u j | 2 H k-1,δ e -γt f 2 H k + |e -γt g| 2 H k .
Letting δ → 0 we have u j ∈ L 2 H k , 1 ≤ j ≤ J and u 0 ∈ H k . We conclude that u ∈ H k again thanks to the uniform non characteristicity.

Smoothness of the IBVP: the integer case

We assume in this section that (u 0 , g, f ) ∈ (H k ) 3 satisfy the compatibility conditions ( CCj 1.3) up to order k, and we prove theorem mainth 1.3 in these settings. To prove that u ∈ ∩ k j=0 C j t H k-j , the strategy is to use the regularity for the pure boundary value problem by substracting an approximate solution (actually a Taylor expansion at t = 0) to u. For technical reasons, it is necessary to use much more regular data that satisfy compatibility conditions to higher order. The construction of such data requires the following lifting lemma that is also used in the next section.

releve Lemma 4.1. For m ∈ N, there exists a lifting map R m : H s (∂Ω) → H m+s+1/2 (∂Ω × R t ), continuous for any s > 0 such that

∂ m t R m g| t=0 = g, ∂ j t R m g| t=0 = 0, j < m + s, j = m -1, (4.1) trace and for r < m + 1/2, | R m | L 2 →H r << 1 is arbitrarily small.
Proof. Up to the use of local maps, the problem is reduced to ∂Ω = R d-1 , and to construct a lifting valued in

H m+s+1/2 (R d-1 × R t ). The variables are denoted (x , t). We choose χ ∈ C ∞ c (R) such that χ (k) (0) = 0, k = m, χ (m) (0) = 1.
We use the Fourier transform on R d-1 × R t and denote ξ the dual variable of x , τ the dual variable of t, and λ is a large parameter to fix later:

R m g = χ(τ /(λ ξ )) (λ ξ ) m+1 g(ξ), equivalently F x (R m (g)(ξ, y)) = χ(λt ξ ) λ m ξ m g(ξ), ξ = 1 + |ξ| 2 .
The trace relations ( trace 4.1) are obvious from the second formula. The H m+s+1/2 norm is easily bounded

R m g 2 H m+s+1/2 (R d ) = | χ(τ /(λ ξ ))| 2 | g| 2 (λ ξ ) 2(m+1) ( ξ 2 + τ 2 ) m+s+1/2 dξdτ = | χ(τ )| 2 | g| 2 (λ ξ ) 2(m+1) ( ξ 2 (1 + λ 2 τ 2 )) m+s+1/2 dξλ ξ dτ ≤ | g| 2 ξ 2s | χ(τ )| 2 (1 + λ 2 τ 2 ) m+s+1/2 λ 2m+1 dτ dξ. λ 2s g 2 H s .
With the same computation

R m g 2 H r ≤ | g| 2 (λ ξ ) 2(m-r)+1 | χ(τ )| 2 (1 + λ 2 τ 2 ) r dτ dξ g 2 L 2 λ 2(m-r)+1 .
It is therefore sufficient to choose λ large enough to ensure the smallness of R m L 2 →H r . compahaute Lemma 4.2 (Construction of smooth compatible data). Let k ≥ 0, (u 0 , g, f ) ∈ (H k ) 3 satisfying the compatibility conditions up to order k. For any m > k, there exists (u 0,n , g n , f n ) ∈ (H ∞ ) 3 satisfying the compatibility conditions up to order m, and such that

(u 0 , g, f ) -(u 0,n , g n , f n ) (H k ) 3 → 0.
Proof. By density of smooth functions, there exists a sequence (u 0,n , g n , f n ) ∈ (H ∞ ) 3 converging to (u 0 , g, f ) in (H k ) 3 . We denote v j,n the corresponding functions in ( taylor 1.2). For j ≥ 1 the "compatibility error" is defined as

ε j,n := ∂ j-1 t g n | t=0 - j-1 l=0 j l (∂ l t B)v j-1-l,n | ∂Ω .
Due to the compatiblity conditions and continuity of traces we have

∀ 1 ≤ j ≤ k, ε j,n H k-j+1/2 -→ n 0.
As a consequence, given a lifting operator R j-1 as in lemma releve 4.1, R j-1 ε j,n H k → n 0. For k < j ≤ m, ε j,n is not small in any Sobolev space, nevertheless from lemma releve 4.1 there exists a lifting R j-1,n such that R j-1,n ε j,n H k ≤ 1/n. We then define

g n := g n - m j=1 R j-1 (ε j,n ).
This choice ensures that compatibility conditions are satisfied by (u 0,n , g n , f n ) up to order m and g n -g H k → 0. 2) for smooth data (u 0,n , g n , f n ). We define the approximate solution

Proof of theorem

u app,n (x, t) = m-1 j=0 t j j! v j,n (x)χ(t), χ ∈ C ∞ c (R + ), χ ≡ 1 near 0.
We solve then

   Lw n = f n -Lu app,n , w n | t=0 = 0, Bw n = g n -Bu app,n ,
By construction, the data (0, g n -Bu app,n , f n -Lu app,n ) are smooth and it is easily seen that

∂ j t (g n -Bu app,n ) = 0, ∂ j t (f n -Lu app,n ) = 0, j ≤ k + 1 provided m ≥ k + 4.
Hence according to proposition regBVP 3.1, the solution w n belongs to H k+2 , this implies by Sobolev embedding w n ∈ ∩ k+1 j=0 C j t H k+1-j . Therefore u n := w n + u app,n is also in ∩ k+1 j=0 C j t H k+1-j , and it is a solution of ( IBVP 1.1) with data (u 0,n , g n , f n ). Using a differentiation argument similar to the proof of proposition regBVP 3.1, but much simpler since no regularization is needed, we see that u n satisfies semigreg 1.9:

k j=0 ∂ j t (e -γt u n ) C(R + ,H k-j (Ω)) + |e -γt u n | ∂Ω | H k u 0,n H k (Ω) + |e -γt g n | H k (∂Ω×[0,T ] + e -γt f n H k ,
as well as ( resolvreg 1.7). The same estimates, applied to u p -u q , (p, q) ∈ N 2 , shows that (u n ) is a Cauchy sequence in ∩ k j=0 C j t H k-j , but since (u n ) converges (in L 2 ) to the solution u of ( IBVP 1.1) with data (u 0 , g, f ), this ensures that u ∈ ∩ k j=0 C j t H k-j . The estimate ( resolvreg 1.7) is then an elementary differentiation argument : tangential regularity is obtained directly by differentiation (which is now legal) and use of the L 2 estimate, while normal regularity uses the non characteristicity.

Regularity for positive s

For ease of presentation, we only detail the case Ω = R d-1 × R + . The general case can be obtained by using a partition of unity as in the previous section. In this section, we follow the (non standard) convention that H s 0 is H 1/2 00 if s = 1/2. Under such settings, we can assume that A d is invertible and A -1 d is uniformly bounded. Furthermore since B : R p → R b has maximal rang b, there exists a smooth basis of Ker B (as a smooth vector bundle over the contractible space R d-1 × R + t ) that we denote (k 1 , • • • k p-b ). A basis (v j ) 1≤j≤b of (KerB) ⊥ is then obtained easily:

B =      B k t 1 . . . k t p-b      is an isomorphism R p → R p , we can choose v j = B -1 (e j ), 1 ≤ j ≤ b.
We remind that compatibility conditions of order s = k + θ, k ∈ N * , 0 < θ < 1 are defined as follows:

1. If θ < 1/2, then compatibility conditions ( 3. If θ = 1/2, compatibility conditions up to order k are satisfied and

R d-1 ∂ k-1 t g(x , y) - k-1 j=0 k -1 j (∂ j t B) (A k-1-j u 0 + B k-1-j f | t=0 ) |(x , y) 2 dy y < ∞.
The case 0 < s < 1 From the previous section, the map (u 0 , g, f ) → u solution of ( IBVP 1.1) is continuous

X 0 × L 2 := (L 2 ) 3 → C t L 2 and X 1 × H 1 := {(u 0 , g) ∈ (H 1 ) 2 : Bu 0 | ∂Ω = g| t=0 } × H 1 → C t H 1 ∩ C 1 t L 2 .
Let us define for 0 ≤ θ ≤ 1 X θ = (u 0 , g) ∈ (H θ ) 2 : the compatibility condition of order θ is satisfied , (note that compatibility conditions of order less than 3/2 do not involve f ).

Both the semi-group estimate ( semigreg 1.9) and the resolvent estimate ( resolvreg 1.7) follow from an interpolation argument if we can prove that X θ = [X 0 , X 1 ] θ .

(5.1) interpX More precisely, since the resolvent estimate implies for s = 0, 1

γ u 2 L 2 γ + u| ∂Ω 2 L 2 γ (u 0 , e -γt g) 2 X 0 + f 2 L 2 γ γ γ u 2 H 1 γ + u| ∂Ω 2 H 1 γ C(γ) (u 0 , e -γt g) 2 X 1 + f 2 H 1 γ γ ,
the interpolation identity ( interpX 5.1) implies

γ u 2 H θ γ + u| ∂Ω 2 H θ γ C (γ) (u 0 , e -γt g) 2 X θ + f 2 H θ γ γ .
(5.2) resolvsupersh (a better estimate would require to use weighted X θ spaces, a course that we chose not to follow).

Proof of ( interpX 5.1) We extend B on Ω × R + t as B(x , y, t) = B(x , t), and consider the map u 0 → Bu 0 := u 0 . It is an isomorphism (H s (Ω)) p → (H s (Ω)) p , and the compatibility condition can be rewritten

Bu 0 | ∂Ω = g| t=0 ⇔ B B -1 Bu 0 | ∂Ω = g| t=0 ⇔ I b 0 u 0 | ∂Ω = g| t=0 , with u 0 = Bu 0 .
This transformation "diagonalizes" ( interpX 5.1), and we are reduced to determine

[L 2 × L 2 , H 1 × H 1 ] θ , and L 2 × L 2 , {(u 0 , g) ∈ H 1 × H 1 : u 0 | y=0 = g| t=0 } θ = [Y 0 , Y 1 ] θ ,
where u 0 and g are now scalar functions. Of course, it is well-known that [L 2 , H 1 ] θ = H θ , so the first case is immediate. In the second case, surprisingly, we were not able to find results in the litterature except in the simplest case θ < 1/2, which is in lionsmagenes2 [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] section 14.

nterpfacile Lemma 5.1. For θ < 1/2, [Y 0 , Y 1 ] θ = Y θ .
Proof. The following inclusions are clear :

H 1 0 × H 1 0 ⊂ Y 1 ⊂ H 1 (Ω) × H 1 (∂Ω × R + ). On the other hand, for θ < 1/2 we have [L 2 , H 1 0 ] θ = H θ ( lionsmagenes [ 12
], chapter 1 section 11), and we can conclude

H θ × H θ = [L 2 × L 2 , H 1 0 × H 1 0 ] θ ⊂ [Y 0 , Y 1 ] θ ⊂ [L 2 × L 2 , H 1 × H 1 ] θ = H θ × H θ . relevecoin Lemma 5.2. For 0 < θ ≤ 1, there exists an universal (independent of θ) operator R R : Y θ → H θ+1/2 (Ω × R + ), ∀ 0 < θ ≤ 1.
Proof. This is a result due to Grisvard Grisvard [START_REF] Grisvard | Caractérisation de quelques espaces d'interpolation[END_REF], for completeness we include a simple proof. Given (u 0 , g) ∈ (H θ ) 2 , from lemma

: g → R b (g) ∈ H θ+1/2 which is independent of θ. By construction, R b g| t=0 -u 0 ∈ H θ 0 . If θ = 1/2, we also notice R b g(x , y, 0) -u 0 (x , y) = R b g(x , y, 0) -g(x , y) H 1/2 00 by interpolation + g(x , y) -u 0 (x , y) H 1/2
The case 0 < θ < 1/2 With the same argument as in the integer case (note that the condition θ < 1/2 allows to use lemma releve 4.1), there exists regularized data (u 0,n , g n , f n ) ∈ (H k+1 ) 3 , converging to (u 0 , g, f ) that satisfy the compatibility conditions up to order k + 1. The corresponding solution u n belongs to ∩ k+1 0 C j t H k+1-j so that we may apply the resolvent estimate ( resolvreg 1.7) to ∂ α u n with s = θ, combined with basic trace estimates and the commutator estimate

[∂ α , L]u n H θ γ u H s γ : γ ∂ α u n 2 H θ γ u 0,n 2
H s + g n 2 H s γ + f n 2 H s γ + u n 2 H s γ γ .
Due to the boundary being non characteristic, we deduce as for the integer case (note that the fractional regularity gained here includes conormal regularity) for γ large enough only depending on s

γ u n 2 H s γ u 0,n 2
H s + g n 2 H s γ + f n 2 H s γ γ .
With the resolvent estimate available, the semi group estimate is now an immediate consequence of the case 0 < s < 1 applied to ( derBVP 5.4):

e -γt ∂ α u n 2 CtH θ u 0,n 2
H s (Ω) + f n 2 H s γ ([0,T ]×Ω) + [∂ α , L]u n 2 H θ γ + g n 2 H s γ u 0,n 2
H s (Ω) + f n H s γ ([0,T ]×Ω) + g n 2 H s γ .
Once more, normal regularity is then obtained thanks to the boundary being non characteristic.

Letting n → ∞, we deduce that e -γt u is in H s (R + × Ω) ∩ (∩ k j=0 C j (R + , H s-j (Ω)) and satisfies the semi group estimate and the resolvent estimate.

The case 1/2 < θ < 1 This can be done with exactly the same argument. Actually, the construction of regularized data (u 0,n , g n , f n ) ∈ (H k+1 ) 3 that satisfy compatibility conditions up to order k + 1 and converging to (u 0 , g, f ) in (H s ) 3 is even simpler. Indeed (u 0 , g, f ) satisfy compatibility conditions up to order k + 1, hence any regularization of (u 0 , g, f ) satisfies

∀ 1 ≤ j ≤ k + 1, ∂ j-1 t g n | t=0 - k-1 l=0 j l (∂ l t B)v j-1-l,n | ∂Ω :=ε j,n H s-j+1/2
-→ n 0, and it suffices to modify g n as g n -δ n where δ n is a function in H k+1 (∂Ω × R + t ) that satisfies for 1

≤ j ≤ k + 1, ∂ j-1 t δ n | t=0 = ε j,n cas1/2
The case θ = 1/2 When s = k + 1/2, the compatibility conditions are satisfied in particular up to order k. From the previous study, we have e -γt u ∈ (∩ k j=0 C j t H k+θ-j ) ∩ H k+θ for any θ < 1/2, with the estimate e -γt u (∩ k j=0 C j t H j+θ-j ) + e -γt u H k+θ ≤ C(θ) (u 0 , g, f ) (H s ) 3 .

Of course this is not enough to conclude, but the estimate can be sharpened: apply estimate ( Recall that the compatibility conditions at order j are are

∀ 1 ≤ j ≤ k, ∂ j-1 t g| t=0 - j-1 l=0 j l (∂ l t B)v j-1-l | ∂Ω = 0,
and at order k + 1/2

∂ k t g(x , t) - k l=0 k l (∂ l t B)v k-1-l (x , t) ∈ H 1/2 00 R d-1 × (R + ) .
As a consequence, for any j ≤ k + 1 and any

β ∈ N d-1 , |β| = k + 1 -j, ∂ β x ∂ j-1 t g(x , t) -∂ β x j-1 l=0 j -1 l (∂ l t B)v j-1-l (x , t) ∈ H 1/2 00 (R d-1 × R + ). ( 5

.5) compafrac

Furthermore, e -γt u ∈ H k (Ω × R + t ), hence for any multi-index of order k -1 e -γt ∂ α u| y=0 -e -γt ∂ α u| t=0 H 1/2 00 (R d-1 ×R + ) e -γt u H k (R d-1 ×(R + ) 2 ) .

(5.6) compagratuite

Now to make (

IBVP 1 . 1 )

 11 belongs for any T > 0 to H s (Ω × [0, T ]), satisfies estimate ( resolvreg 1.7) for γ large enough, and if s

mainth 1 . 3 (

 13 integer case) We follow the notations of lemma compahaute 4.2; v j,n are smooth functions defined by ( taylor 1.

CCj 1 . 3 )

 13 up to order k are satisfied. 2. If θ > 1/2, then compatibility conditions ( CCj 1.3) up to order k + 1 are satisfied.

releve 4 .

 4 1 there exists an opertor R b

  for θ < 1 and any tangential multi-index α ∈ N d , |α| = k, this reads γ ∂ α u 2 H θ γ L α u 0 + L α f | t=0 , e -γt (∂ α g + [B, ∂ α ]u| ∂Ω )

derBVP 5 . 4 )

 54 more explicit, let us write∂ α = ∂ j t ∂ β x , β ∈ N d-1 , |β| = k -j. Then ∂ α u| t=0 = ∂ β x v j ∈ H 1/2 (R d-1 × R + ), the compatibility condition of order 1/2 for ( derBVP 5.4) is thuse -γt (∂ α g + [B, ∂ α ]u| ∂Ω ) -B∂ β x v j ∈ H 1/2 00 (R d-1 × R + ).With basic computations, we now check that it is implied by (e -γt (∂ α g + [B, ∂ α ]u| ∂Ω ) -B∂ β x v j = e -γt ∂ α gt B)∂ j-l t u| ∂Ω + e -γt B∂ α u| ∂Ω -B∂ β x v j = e -γt ∂ α ge -γt ∂ j-l t u| ∂Ω -v j-l -∂ β x Be -γt ∂ j t u| ∂Ω + e -γt B∂ α u| ∂Ω = e -γt ∂ α gt B) e -γt ∂ j-l t u| ∂Ω -v j-l +[B, ∂ β x ]e -γt ∂ j t u| ∂Ω -[B, ∂ β x ]v j .

This assumption is somewhat too strong, as it is classical that in this framework, weak solutions are actually strong, see LaxPhillips[START_REF] Lax | Local boundary conditions for dissipative symmetric linear differential operators[END_REF].

If there exists an universal lifting R 0 : H θ 0 (Ω) → {u ∈ H θ+1/2 (Ω × R + )| u| ∂Ω = 0}, R can be defined as R(u 0 , g) = R b g + R 0 (u 0 -R b g| t=0 ) so we focus on the construction of R 0 . For u 0 ∈ H θ 0 (H 1/2 00 for θ = 1/2), we extend it as an odd function of y, I(u 0 ) defined on R d . The map I :

where χ is as in lemma

Proof. On one hand, the map (u 0 , g) → u 0 (x , y) -g(x , y) is continuous

.3) inclusion

On the other hand, from Lions-Peetre reiteration theorem, for any 0 < s, θ < 1

If we have for some

On the other hand, the case θ ≤ s is contained in lemma interpfacile 5.1. For any 0 < r < 1 we define the map

It is easily seen that Tr is continuous

we deduce by interpolation

Tr :

We observe now that the lifting R from lemma relevecoin 5.2 is a right inverse for Tr: for fixed 0 < s < 1/2 and any 0 < θ < 1, we have Tr

which was the required converse inclusion.

The case s > 1 We denote s = k + θ, 0 ≤ θ < 1. According to the integer case, we already have u ∈ ∩C k-j H j . For any tangential multi-index α of order k (that is,

(5.4) derBVP where L α , L α are differential operators of respective order α, α -1. Regularity will again be obtained by regularization of the data, we distinguish three cases:

For j ≤ k, due to the compatibility condition ( compafrac 5.5), the first line in the last equality is in H

1/2 00 . The H 1/2 00 norm of the second line is easily controlled by writing e -γt ∂ j-l t u| ∂Ω -v j-l = e -γt (∂ j-l t u| ∂Ω -v j-l ) + (1 -e -γt )v j-l , the first term can be bounded thanks to ( compagratuite 5.6) while for the second we simply use (1-e -γt )/t 1. The same argument is used for the third line. We deduce that for θ

Using non characteristicity, we recover

.

This estimate is uniform in θ < 1/2, we deduce that the same estimate holds for θ = 1/2. Finally we deduce that the semi group estimate is true with the same argument as for the end of the case 0 < θ < 1/2. This ends the proof of theorem mainth 1.3.


Alexis Delanoë

Charly Carrière

Philippe Dillmann

Delphine D Neff

C Martin

Olivier Bildstein

Jean-Eric Lartigue

S Gin

Glass-Iron-Claystone interactions: a study relying on characterizations and geochemical modeling

published or not. The documents may come

In France, Andra considers to dispose High-level waste in deep geological repository. The concept rely on disposal cells dug into the Callovo-Oxfordian claystone, where stainless steel canisters filled up with vitrified borosilicate glass would be disposed. Understanding the waste forms degradation in contact with groundwater is crucial to assess the release of radionuclides in the geosphere. Canisters corrosion would release iron into the groundwater, and therefore the Fe-containing solution would alters the glass.

The purpose of this work is to investigate the glass-iron-claystone interactions in conditions close to that expected in a geological repository. A mockup experiment composed of two SON68 glass coupons (one of them doped with 57 Fe and 29 Si) in contact with an iron piece surrounded by a Callovo-Oxfordian claystone cylinder was setup. The claystone was saturated with synthetic groundwater, under anoxic atmosphere. Two identical mockups were prepared and run in parallel at 50°C for 2.5 [1] and 6.1 years, respectively.

Post-mortem multi-scale characterizations were performed on the samples. They rely on SEM, EDS, Raman spectroscopy, LA-ICP-MS, Tof-SIMS, HRTEM, and STXM to identify the phases, investigate the corrosion products and alteration layers, and discuss the underlying mechanisms.

The release of Fe and Si into the groundwater lead to the formation of iron phyllosilicates into the glass alteration layer and iron corrosion products. Their precipitation on the glass surface prevents the formation of an amorphous silica passivating layer, keeping high glass alteration rates (around r0/4, where r0 represents the initial dissolution rate of the glass) which is far from the residual rate (r0/1550). Interestingly, two types of phyllosilicates were characterized in the iron corrosion products and the glass alteration layer: 10.4Å-phyllosilicates and 7.8Å-phyllosilicates, after 2.5 and 6.1 years respectively. These results suggest an evolution of the system between the two mockup. Aqueous parameters (pH, [Si], [Fe], …) might play a relevant role on phyllosilicates formation, glass alteration and iron corrosion. A modeling approach is under developement using Hytec and Crunch reactive transport codes to reproduce the phenomena observed and the geochemical evolution of the mockup.

  [1] C. Carrière and al. NPJ Materials Degradation, 5, 2021 (The fate of Si and Fe while nuclear glass alters with steel and clay)