E Simondon
email: esther.simondon@cea.fr

P.-F Giroux

L Chaffron

A Fitch

P Castany

T Gloriant

Keywords: Pyrochlore oxide, Mechanical milling, nanostructure, X-ray diffraction (XRD), Synchrotron radiation GRAPHICAL

Mechanical synthesis of nanostructured Y 2 Ti 2 O 7 pyrochlore oxides

ARTICLE 1. INTRODUCTION

Y 2 Ti 2 O 7 oxides are part of a large family of A 2 B 2 O 7 oxides displaying pyrochlore structure, where A and B are metal cations. The fully ordered pyrochlore structure presents a cubic symmetry (Space Group: Fd m) and can be described in terms of a superstructure of the ideal defect fluorite 3 structure (cubic, S.G.: Fm m) with twice the cell constant (a ≈ 10 Å) [START_REF] Subramanian | Oxide pyrochlores -A review[END_REF].

3

In this study, Y 2 Ti 2 O 7 pyrochlore oxides are produced and examined to be directly introduced in Oxide-Dispersion-Strengthened (ODS) steel during manufacturing process. The presence of a dense dispersion of this highly stable oxide under irradiation in the ferritic matrix improves mechanical strength and traps irradiation defects [START_REF] Klueh | Oxide dispersionstrengthened steels: A comparison of some commercial and experimental alloys[END_REF] [START_REF] Yamamoto | The transport and fate of helium in nanostructured ferritic alloys at fusion relevant He/dpa ratios and dpa rates[END_REF]. Y 2 Ti 2 O 7 is usually produced by a standard solid state process. This method requires repeated cycles of ball-milling and heat treatments above 1000°C, involving an important growth and aggregation of particles [START_REF] Wang | Ion irradiation of rareearth-and yttrium-titanate-pyrochlores[END_REF] [START_REF] Hayward | Phase Separation during the Topotactic Reduction of the Pyrochlore Y 2 Ti 2 O 7[END_REF]. Among the solid state processes, high-energy ball milling has revealed to be an efficient method to produce complex oxide nanopowders [START_REF] Zyryanov | Mechanochemical synthesis of complex oxides[END_REF] [7] [START_REF] Fu | Low-temperature synthesis of Mg4Nb2O9 nanopowders by high-energy ball-milling method[END_REF]. It has been demonstrated that the process involving repeated ball-powder collision events during the milling could induce mechanochemical solid-state reactions in the milled powder [START_REF] Calos | A Crystallographic Contribution to the Mechanism of a Mechanically Induced Solid State Reaction[END_REF] [START_REF] Takacs | Solid state reactions induced by ball milling[END_REF]. The repeated welding and fracturing of powder particles increases the area of contact between the reactant powder particles and allows fresh surfaces to come into contact repeatedly. As a consequence, the reaction that normally requires high temperatures to facilitate the diffusion occurs at room temperature during mechanical milling and nanosized particles can be obtained [START_REF] Suryanarayana | Mechanical alloying and milling[END_REF]. Y 2 Ti 2 O 7 pyrochlore powders have recently been produced by a simple mechanosynthesis technique [START_REF] Moreno | Direct synthesis of A 2 (Ti (1 -y) Zr y ) 2 O 7 (A = Gd 3+ , Y 3+ ) solid solutions by ball milling constituent oxides[END_REF] [13] [START_REF] Singh | Preparation of Y 2 Ti 2 O 7 pyrochlore using highenergy ball milling and their structural, thermal and conducting properties[END_REF]. Moreno et al. [START_REF] Moreno | Direct synthesis of A 2 (Ti (1 -y) Zr y ) 2 O 7 (A = Gd 3+ , Y 3+ ) solid solutions by ball milling constituent oxides[END_REF] showed that a milling of a stoichiometric mixture of Y 2 O 3 and TiO 2 powders for 19 h in a zirconia planetary ball mill leads to obtaining a single-phased powder. Fuentes et al. [START_REF] Fuentes | Synthesis of disordered pyrochlores, A 2 Ti 2 O 7 (A=Y, Gd and Dy), by mechanical milling of constituent oxides[END_REF] and Singh et al. [START_REF] Singh | Preparation of Y 2 Ti 2 O 7 pyrochlore using highenergy ball milling and their structural, thermal and conducting properties[END_REF] showed that the produced as-milled powders were composed by partially disordered phases, which are intermediate between the ordered pyrochlore and disordered fluorite structures. This partially disordered pyrochlore structure transforms into the ordered form around 800°C, mainly due to oxygen re-arrangements [START_REF] Fuentes | Synthesis of disordered pyrochlores, A 2 Ti 2 O 7 (A=Y, Gd and Dy), by mechanical milling of constituent oxides[END_REF].

The aim of this contribution is to propose a mechanochemical method to synthesize highly pure Y 2 Ti 2 O 7 powders with an ordered pyrochlore type of structure and nanosized crystallites. After a first step of milling of the initial constituents, a short annealing allows the transformation into the fully ordered pyrochlore phase while limiting the crystallites growth.

MATERIELS AND METHODS

Initial powders used in this work were nano-sized TiO 2 (Degussa, 99,5%) and Y 2 O 3 (Sigma-Aldrich, 99,9%) powders. They were pelletized into an 8-mm-diameter die under 3 t mass and then submitted to a primary annealing for 10 h under vacuum (10 -6 mbar at 550°C and 1000°C respectively, in order to remove adsorbed H 2 O). Appropriate amount of the as-treated constituent oxides was selected (Y 2 O 3 /2TiO 2 ) to reach a complete transformation in Y 2 Ti 2 O 7 . Batches of 3 g were ball milled in a modified FRITSCH Pulverisette-0 mill with tungsten carbide vial and ball under a static vacuum (10 -6 mbar) for 120 h with an intensity of 1400 m.s -2 [START_REF] Chen | Ball-milling-induced amorphization in NiXZrY compounds: A parametic study[END_REF]. Heat treatments on the as-milled powders were performed under vacuum (10 -6 mbar) at 900°C with a 10°C.min -1 temperature ramp followed by an immediate natural cooling. From the first heat treatment of the initial powders to the harvesting of milled and annealed final powders, the samples were at all time handled in a controlled atmosphere (argon or vacuum) in order to prevent any interaction with air. The samples were characterized by X-ray powder diffraction, using a BRUKER D8 diffractometer with Ni-filtered CuK α1 radiation (λ = 1.5406 Å). The diffraction patterns of the samples were measured over the angular range 13 -70° (2θ) with step size 0.02° and scan speed 0.24°.min -1 . The samples were also examined by thermal analysis in a SETARAM TAG 24 using a typical sample mass of 100 mg and a heating rate of 10°C.min -1 in argon or air. In situ XRD spectra were collected on beam line ID22 (using λ=0.495992 Å wavelength) at the European Synchrotron Radiation Facility (ESRF, Grenoble, France). The scans were performed at 10°.min -1 from 5 to 20° (2θ) of the central detector, with a 0.5 mm silica-glass capillary, 10 µm wall thickness, spun axially at 787 rpm. The temperature ramp of 10 K.min -1 was achieved via a hot-air blower. Scanning electron microscopy (SEM) was carried out on a HIROX SH-3500MB microscope working at 20 kV. Transmission electron microscopy (TEM) and selected area electron diffraction (SAED) were performed on a JEOL 2100 working at 200 keV.

RESULTS

Fig. 1(a) shows the evolution of the XRD pattern of the Y 2 O 3 + 2TiO 2 mixture before and after the milling in a high energy ball mill. As a reference, the XRD pattern of the starting mixture (Fig. 1(a)(i)) presents peaks from both cubic Y 2 O 3 (ICDD card 00-41-1105) and TiO 2 in the form of rutile (01-089-0553) and anatase (00-021-1272). The XRD diffraction pattern of the ball-milled powder (Fig. 1(a)(ii)) indicates that the powder is mostly amorphous-like with broad peaks of low intensities and contains a crystalline fraction indicated by a few narrow peaks of low intensity. The decrease in intensity and increase in peak broadening is the result of, on the one hand, a considerable refinement of particle and crystallite size and, one the other hand, the introduction of a large number of structural defects in the constituent oxides. The diffraction pattern is then decomposed into a baseline (Fig. 1(a SEM observations reveal that after the milling, the powder is characterized by a typical morphology of ball-milled ceramic samples composed by sub-micronic agglomerated particles of irregular shape [START_REF] Bégin-Colin | Kinetics and Mechanisms of Phase Transformations Induced by Ball-Milling in Anatase TiO 2[END_REF] (Fig. 1(b)). They are themselves constituted of nanoparticles agglomerates (Fig. 1 (c)). The microstructure of this milled powder was characterized by TEM and SAED. Figs. 2(a) and(b) present observation zones at the edge of two different agglomerates, in bright (Fig. 2(a)) or dark field (Fig. 2(b)) with the associated SAED patterns. In most observed agglomerates, even the diffraction contrast in the micrograph fails to reveal any crystallite within the powder particle, as illustrated in Fig. 2(b). The corresponding SAED diagram presents a very diffuse ring along with some scattered bright spots. The diffuse maxima observed can be explained as a result of small domain size together with poor crystallinity at the atomic scale, or with an amorphous structure. The bright spots correspond to a few crystallites of various natures such as Y 2 O 3 or TiO 2 . The powder is then composed mainly of an amorphous phase along with a few un-milled crystallites of remaining initial constituents, which is consistent with the broad XRD pattern seen above (Fig. 1(a)). As illustrated in Fig. 2(b), some crystallites can be visible and the associated SAED patterns are typical of polycrystalline materials (diffraction rings instead of well-defined spots, for all crystallites have random orientations). The interplanar distances correspond to the main reflections (111), ( 220) and (311) of the fluorite structure. This phenomenon can be explained by the reaction occurring between the constituents during the milling. This mechanically activated reaction is induced by the combined action of shearing and temperature local rise during impacts, which facilitates atomic mobility and initiates transformations that would usually take place during a posterior heat treatment. The milled powder is then not fully crystallized pyrochlore and corresponds to an amorphous phase of Y 2 Ti 2 O 7 , along with some small crystallites of fluorite-like structure as well as some remaining crystallites of the initial constituents. Thermodifferential analyses on the milled powders revealed an exothermic event at about 855°C (Fig. 3(a)). The main peak is narrow around this temperature and is followed by a much smaller peak on a larger range of temperature (880°C-1000°C). As a comparison, the un-milled initial mixture does not show any peak in DTA spectra. Moreover, the exothermic transformation is found to be nonreversible. To highlight the origin of the exothermic event identified by DTA, the milled powder was characterized by XRD under Synchrotron radiation at ESRF Grenoble. Fig. 3(b) presents XRD patterns of the milled powder at several temperatures during the heat treatment from bottom to top: heating from 20°C to 925°C and then cooling to 500°C. Fig. 3(c) looks more closely on the patterns run between 790°C and 830°C while heating. The XRD pattern of the sample remains nearly identical to the initial milled powder from room temperature to about 800°C. Starting from 800°C, characteristic diffracted peaks of a fluorite-like phase start to appear, and their number and intensity increase with rising temperature (Fig. 3(c)). The pyrochlore structure can be considered as a superstructure of an anion deficient fluorite-like atomic arrangement with twice the cell constant. Therefore, its diffraction pattern is composed by (i) a set of strong intensities characteristic of the underlying fluorite-type average structure and (ii) an additional set of superstructure reflections with intensities depending on several factors (degree of ordering, difference in average scattering factors of the elements involved, distribution of oxygen vacancies, etc…). For Y 2 Ti 2 O 7 , according to the comparison between ICDD card 04-015-3376 for pyrochlore and 04-008-1806 for fluorite, the main superstructure visible peaks correspond to (111) and (331). There are labeled with a star on Fig. 3(b) and are only visible above the 900°C temperature patterns. As previous observations demonstrate that fully ordered pyrochlore structure Y 2 Ti 2 O 7 is synthetized with an annealing treatment at more than 850°C, the powder was heat treated at 900°C in a pyrolysis oven under vacuum with a 10°C.min -1 heating ramp and immediate natural cooling to prevent crystallite growth. The XRD pattern of the obtained powder is presented in Fig. 4(a). As a reference, the as-milled powder XRD pattern before annealing is also presented. After the annealing, the XRD pattern matches the one expected for pyrochlore-type Y 2 Ti 2 O 7 (ICDD card 04-015-3376) and no extra peaks are observed. In particular, Y 2 O 3 and TiO 2 peaks are not visible anymore, which is proof of a total transformation. The superstructure peaks of pyrochlore at 2θ=15.18° and 2θ=38.84° are clearly visible. These observations confirm the results obtained with Synchrotron experiments. TEM observations have been performed on several observation zones on the heat treated powder, showing nanocristallites on the range of 50 to 80 nm (Fig. 4(b)). SAED patterns enable to identify the pyrochlore Y 2 Ti 2 O 7 structure (Fig. 4(c)), confirming the XRD results. From XRD pattern shown in Fig. 4(a), the average crystallite size of sample calculated using Scherrer's equation is 50.6 nm [START_REF] Patterson | The Scherrer Formula for X-Ray Particle Size Determination[END_REF], which is consistent with the TEM observations in Fig. 4(b).

DISCUSSION

Milling a stoechiometric mixture of Y 2 O 3 and TiO 2 produces a mostly amorphous-like mixture of Y 2 Ti 2 O 7 containing small crystallites of both initial constituents and fluorite-like structures. Previous studies have showed that, after 9h of milling with different conditions (type of miller, milling media, atmosphere…), the XRD pattern of the milled powder is composed only by the peaks of a pyrochlorelike structure [START_REF] Fuentes | Synthesis of disordered pyrochlores, A 2 Ti 2 O 7 (A=Y, Gd and Dy), by mechanical milling of constituent oxides[END_REF] [12] [START_REF] Singh | Preparation of Y 2 Ti 2 O 7 pyrochlore using highenergy ball milling and their structural, thermal and conducting properties[END_REF]. The mechanically activated chemical reaction between Y 2 O 3 and TiO 2 proceeds through an initial step of particle and crystallite size reduction of the starting oxides, together with the polymorphic transformation from cubic to monoclinic Y 2 O 3 . The formation of the pyrochlore Y 2 Ti 2 O 7 seems to take place as a second step from an amorphous matrix of these oxides [START_REF] Fuentes | Synthesis of disordered pyrochlores, A 2 Ti 2 O 7 (A=Y, Gd and Dy), by mechanical milling of constituent oxides[END_REF].

The selected milling conditions of the present study do not enable to the occurrence of the crystallization of the milled powder from an amorphous form to a pyrochlore-like structure directly in the mill, even after a long time run (240 h). This phenomenon could be explained by the use of tungsten carbide as a milling media instead of zirconia. Several studies have showed that compared to other milling medias (stainless steel and tungsten carbide), zirconia induces faster reactions [START_REF] Yen | Influence of Powder Composition and Milling Media on the Formation of Molybdenum Disilicide by a Mechanically Induced Self-Propagating Reaction[END_REF] [19] [START_REF] Begin-Colin | Influence of the nature of milling media on phase transformations induced by grinding in some oxides[END_REF]. Because of its rather low thermal conductivity, the heat produced during milling using the zirconia milling media is assumed to be effectively stored, and the ignition temperature can be reached within a shorter time. However, zirconia is less resistant than tungsten carbide and can introduce a pollution from 2 to 4% in the powder [START_REF] Fuentes | Synthesis of disordered pyrochlores, A 2 Ti 2 O 7 (A=Y, Gd and Dy), by mechanical milling of constituent oxides[END_REF]. The increase in Zr content in the powder increases disorder [START_REF] Moreno | Direct synthesis of A 2 (Ti (1 -y) Zr y ) 2 O 7 (A = Gd 3+ , Y 3+ ) solid solutions by ball milling constituent oxides[END_REF], which is not in line with the synthesis of a stable and pure powder of Y 2 Ti 2 O 7 oxide with a pyrochlore structure of this study.

In this study, the heat treatment of the milled powder enables to synthesize the fully crystallized pyrochlore phase from a mostly amorphous structure. According to DTA, crystallization occurs around 855°C as revealed by a sharp exothermic event. However, previous calorimetry studies of ball-milled Y 2 Ti 2 O 7 attribute this exothermic event to atomic ordering [START_REF] Fuentes | Synthesis of disordered pyrochlores, A 2 Ti 2 O 7 (A=Y, Gd and Dy), by mechanical milling of constituent oxides[END_REF] [START_REF] Singh | Preparation of Y 2 Ti 2 O 7 pyrochlore using highenergy ball milling and their structural, thermal and conducting properties[END_REF]. During heating, the partially disordered pyrochlore phase (which is an intermediate between the ordered pyrochlore and disordered fluorite) forms the pyrochlore lattice. In the present study, the ball-milled sample is composed by an amorphous powder. As a consequence, the sharp exothermical peak could be associated to crystallization of the Y-Ti-O oxides and the second broader peak to rearrangement and redistribution of cations and oxygen ions. Measurements performed in Synchrotron show a crystallization of a fluoritelike structure around 800°C, and then formation of the pyrochlore superstructure around 900°C. This is consistent with the previous observation and interpretation of DTA. The difference between the crystallization temperatures measured by Synchrotron and DTA analysis can be explained by slight differences in experimental conditions (atmosphere, thermocouple position, measurements sensitivity…).

CONCLUSION

In order to form a nanosized powder of Y 2 Ti 2 O 7 oxide with pyrochlore structure, ceramic powders of Y 2 O 3 and TiO 2 were milled in a high energy ball mill for 120 h. The obtained ball-milled powder is composed by sub-micronic agglomerated particles of irregular shape, themselves constituted of nanoparticle agglomerates. The XRD analysis reveals the obtained milled-powder to be a stoichiometric amorphous mixture of Y 2 Ti 2 O 7 containing a small quantity of crystallized fluorite structure and initial constituents. The Synchrotron and DTA study enabled to observe the crystallization of the amorphous mixture into a fluorite-like phase around 800°C followed by a rearrangement forming the pyrochlore structure around 900°C with rising temperature. After the cooling, the sample is pure and composed of nanostructured pyrochlore powder.

  )(iii)) representing the amorphous component and the deconvolved signal (Fig. 2(a)(iv)) representing the crystalline fraction. On the baseline pattern (Fig. 1(a)(iii)), the main broad peak roughly correspond to the (222) main reflections of Y 2 Ti 2 O 7 at 2θ = 30.64°. On the deconvolved signal (Fig. 1(a)(iv)), the main reflections of anatase TiO 2 and cubic Y 2 O 3 are evident, revealing the presence of residual initial constituents.

Figure 1 :

 1 Figure 1 : (a) X-ray diffraction patterns of the starting mixture (i) and the ball-milled powder (ii). The diffractogram (ii) is decomposed in a baseline representing the amorphous component (iii) and the deconvolved signal representing crystalline fractions (iv). (b) and (c) are SEM micrographs of the ball-milled powder at different magnifications.

Figure 2 :

 2 Figure 2 : TEM micrographs of milled powder, associated SAED patterns are shown as insets

Figure 3 :

 3 Figure 3 : (a) Differential thermal analysis (10°C/min) of the initial mixture and of the as-milled powder. In situ XRD of the as-milled powder (b) heating from 20°C to 925°C and cooling to 500°C from bottom to top, (c) zoom on heating from 790°C to 830°C. Labelled peaks are pyrochlorelike structure, and the superstructure peaks are labelled with a star.

Figure 4 :

 4 Figure 4 : (a) X-ray diffraction patterns of ball-milled powder before and after annealing at 900°C. (b) TEM micrograph of powder after annealing at 900°C and (c) SAED pattern of the same sample

ACKOWLEDGEMENT

This work was supported by CEA in the framework of the MACNA project. The authors would like to thank Patrick Bonnaillie for the SEM characterizations.


Billel Guelmame

S T Éphane

Junca And

Didier Clamond

REGULARIZING EFFECT FOR CONSERVATION LAWS WITH A LIPSCHITZ CONVEX FLUX

Keywords: AMS Classification: 35L65, 35B65, 35L67, 26A45, (46E30) Scalar conservation laws, entropy solution, strictly convex flux, discontinuous velocity, wave front tracking, smoothing effect, BV Φ spaces

, while in the present paper the flux is only locally Lipschitz. Therefore, the wave velocity can be dicontinuous and the one-sided Oleinik inequality is lost. This inequality is usually the fundamental tool to get a sharp regularizing effect for the entropy solution. We modify the wave velocity in order to get an Oleinik inequality useful for the wave front tracking algorithm. Then, we prove that the unique entropy solution belongs to a generalized BV space, BV Φ .

Introduction

This paper is about the regularization effect on the unique entropy solution of the scalar hyperbolic conservation law

u t + f (u) x = 0, u(0, x) = u 0 (x), M = u 0 ∞ . (1)
Date: December 4, 2018.

The initial datum u 0 belongs to L ∞ (R, R). In [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF], f is a nonlinear convex flux on the whole real line, thence f is Lipschitz on [-M, M ]. The regularity of u for positive time t depends on the nonlinearity of f on [-M, M ]. (For a linear flux, the solution is nothing but a translation of the initial datum with a constant speed, so no regularity is enforced by the equation [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF].) To obtain a smoothing effect, the following Tartar condition [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF] is needed:

There are no non-trivial interval where f is affine.

Here, the flux being nonlinear and convex on R, it is strictly convex and thus it necessarily satisfies the condition [START_REF] Bianchini | On the Structure of L ∞ L-Entropy Solutions to Scalar Conservation Laws in One-Space Dimension[END_REF].

In [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF][START_REF] Oleinik | Discontinuous solutions of non-linear differential equations[END_REF], both Lax and Oleinik prove that for an uniform convex flux f such that f > c > 0 for some constant c (as, e.g., for the Burgers equation), the solution u(t, •) is in BV loc , for all time t > 0. (Definitions of the various BV spaces, spaces of functions of bounded variation, can be found below and also in [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF][START_REF] Musielak | On generalized variations (I)[END_REF].) This result is no longer true for flatter fluxes [START_REF] Cheng | The space BV is not enough for hyperbolic conservation laws[END_REF], such as f (u) = |u| 3 and f (u) = u 4 . The solution regularities in SBV and Sobolev spaces are obtained in [START_REF] Ghoshal | Finer regularity of an entropy solution for 1-d scalar conservation laws with non uniform convex flux[END_REF][START_REF] Jabin | Some regularizing methods for transport equations and the regularity of solutions to scalar conservation laws[END_REF]. To obtain more information on the regularity of u, generalized BV spaces, BV s and BV Φ , are needed. The regularity in those spaces implies the right regularity in Sobolev spaces, as well as the left and right traces for shock waves. For smooth fluxes with a polynomial degeneracy (e.g., f (u) = |u| 3 , f (u) = u 4 ), the solution u(t, •) belongs to BV s loc in space for t > 0 (see the last paragraph of section 2 and [START_REF] Bourdarias | Fractional BV spaces and applications to scalar conservation laws[END_REF][START_REF] Castelli | Fractional spaces and conservation laws[END_REF], and see [START_REF] Bianchini | On the Structure of L ∞ L-Entropy Solutions to Scalar Conservation Laws in One-Space Dimension[END_REF][START_REF] Marconi | Regularity estimates for scalar conservation laws in one space dimension[END_REF] for non-convex fluxes). This kind of regularity is still true for a C 1 convex flux in a bigger generalized BV space, u(t, •) ∈ BV Φ loc , t > 0 with a convex function Φ depending on the nonlinearity of f [START_REF] Castelli | Smoothing effect in BV for entropy solutions of scalar conservation laws[END_REF][START_REF] Marconi | Regularity estimates for scalar conservation laws in one space dimension[END_REF].

In this paper, we show that this last result [START_REF] Castelli | Smoothing effect in BV for entropy solutions of scalar conservation laws[END_REF] remains true for all convex fluxes f on R satisfying the condition [START_REF] Bianchini | On the Structure of L ∞ L-Entropy Solutions to Scalar Conservation Laws in One-Space Dimension[END_REF], without requiring f to be in C 1 . Such a flux can appear in applications, such as in traffic flow model [START_REF] Whitham | Linear and nonlinear waves[END_REF] with a concave flux. If f is a strictly convex flux, then the (necessarily increasing) velocity a(u) = f (u) exists almost everywhere. The set of discontinuity of a is countable, the left and right limits a -(u) a + (u) existing everywhere. Thanks to the maximum principle, the entropy solution u takes values only in [-M, M ], hence a is bounded on [-M, M ]. In the case of C 1 convex fluxes, the simplest proof (see [START_REF] Castelli | Fractional spaces and conservation laws[END_REF] after [START_REF] Oleinik | Discontinuous solutions of non-linear differential equations[END_REF]) is based on the fundamental one-sided Oleinik inequality [START_REF] Hoff | The sharp form of Oleinik's entropy condition in several space variables[END_REF], a(u(t, x)) -a(u(t, y)) (x -y)/t a.e. x > y,

which implies that a(u) is a BV function and then u belongs to a BV Φ . Unfortunately, this inequality is no longer true for convex Lipschitz fluxes. Indeed, first, a(u) is not well defined because a is not continuous and, second, the Oleinik inequality is not true almost everywhere, as shown in the example 3.1 below. To our knowledge, the loss of the Oleinik inequality appears in the classical literature of conservation laws only in [START_REF] Hoff | The sharp form of Oleinik's entropy condition in several space variables[END_REF], for a piecewise linear flux. Note that, though not always true, the Oleinik inequality is true on a large subset of R + × R. We prove in this paper that this is enough to still obtain the smoothing effect in the right BV Φ space with a modified wave velocity and a wave front tracking algorithm for scalar conservation laws [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF].

The paper is organized as follows. In section 2, the function Φ is built to state the main theorem. The loss of the Oleinik inequality and the resulting difficulty to prove the main theorem 2.1 is discussed in section 3. The section 4 studies the approximate Riemann problem and a modified Oleinik inequality. The section 5 is devoted to obtaining a BV estimates on the modified velocity by the wave front tracking algorithm. The main result is proved in section 6.

The main result

In this section, definitions of weak entropy solutions and BV Φ spaces are recalled and the function Φ related to the smoothing effect is built. Then, the smoothing effect is stated in Theorem 2.1. Definition 2.1. u is called a weak solution of (1), if for all smooth functions θ with a compact support, i.e., for

θ ∈ D(R + × R) R R + [ u(t, x) θ t (t, x) + f (u(t, x)) θ x (t, x) ] dt dx + R u 0 (x) θ(0, x) dx = 0. ( 4
)
For a given u 0 ∈ L ∞ , the equation ( 4) has at least one weak solution [START_REF] Bressan | Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem[END_REF][START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF], the uniqueness being ensured by the Kruzkov entropy conditions: Definition 2.2 (Kruzkov entropy solution). A weak solution of (4) is called an entropy solution if for all positive θ ∈ D(R + * × R) and for all convex functions η ∈ C 1 and with F def = f (u)η (u) du (primes denoting the derivatives), the following inequality holds:

R R + [ η(u(t, x)) θ t (t, x) + F (u(t, x)) θ x (t, x) ] dt dx 0. ( 5
)
In addition, u has to belong to C 0 ([0, +∞[, L 1 loc (R)). The functional space BV Φ [START_REF] Musielak | On generalized variations (I)[END_REF] is defined as follow.

Definition 2.3. Let be Φ a convex function such that Φ(0) = 0 and Φ(h) > 0 for h > 0, the total Φ-variation of v on K ⊂ R is

TV Φ v {K} = sup p∈P n i=2 Φ(|v(x i ) -v(x i-1 )|) ( 6
)
where

P = {{x 1 , • • • , x n }, x 1 < • • • < x n } is the set of all subdivisions of K. The space BV Φ is defined by BV Φ = {v, ∃λ > 0, TV Φ (λv) < ∞}.
The BV Φ space, a generalization of the BV space, is the space of functions with generalized bounded variations. Our goal here is to construct the best convex function Φ, such that u is in BV Φ loc , that means choosing Φ to obtain the smallest space BV Φ in order to characterize the regularity of the entropy solution.

The one-sided Oleinik inequality is directly linked with the increasing variation of an entropy solution. Such variation is defined with y + = max(y, 0) as follow with the same notations as in the previous definition 2. [START_REF] Bourdarias | Fractional BV spaces and applications to scalar conservation laws[END_REF].

TV Φ+ v {K} = sup p∈P n i=2 Φ((v(x i ) -v(x i-1 )) + ) (7)
In order to build the function Φ, a definition of the generalized inverse of non decreasing functions is needed Definition 2.4. Let be g a non decreasing function from R to R, the generalized inverse of g is defined on g([-M, M ]) as following

g -1 (y) def = inf{x ∈ R, y g(x)}, (8)
Remark 2.1. It's obvious from the definition that

(g -1 • g)(x) x, ∀x. (9)
The usual properties of the generalized inverse can be found in [START_REF] Embrechts | A note on generalized inverses[END_REF].

Proposition 2.1 (see proposition 2.3 in [START_REF] Embrechts | A note on generalized inverses[END_REF]). Billel 1-If g is continuous then g -1 is a strictly increasing function and

(g • g -1 )(y) = y, ∀y (10)
2-If g is strictly increasing then g -1 is continuous and

(g -1 • g)(x) = x, ∀x, y (11)
Now, the function Φ will be built, as a consequence of the strict convexity of f , a = f is strictly increasing so its generalized inverse

b = a -1 , (12)
is continuous and non decreasing. The function b is constant on [a -(u), a + (u)] when a is discontinuous at u. Let be ω[b] the modulus of continuity of b, i.e.,

ω[b](h) def = sup |x-y| h x,y∈a([-M,M ]) |b(x) -b(y)|, (13)
and let be φ the generalized inverse of ω[b], i.e., φ(y) = inf{x ∈ R, y ω[b](x)}. We denote Φ the convex upper envelope of φ, that is related to the nonlinearity of the flux via the velocity. Let us write, in a concise way, the definition of Φ. This function is the key ingredient to define a suitable functional space describing the regularity of entropy solutions.

Definition 2.5 (Choice of Φ). Φ is the convex upper envelope of the generalized inverse of the modulus of continuity of the generalized inverse of the velocity,

Φ def = upper convex envelope of ω[a -1 ] -1 . ( 14
)
Remark 2.2. This definition generalizes the ones given in [START_REF] Bourdarias | Fractional BV spaces and applications to scalar conservation laws[END_REF][START_REF] Castelli | Smoothing effect in BV for entropy solutions of scalar conservation laws[END_REF] for a discontinuous velocity. This is the optimal choice for a flux with a power law degeneracy, as proved in e.g. [START_REF] Castelli | On the maximal smoothing effect for multidimensional scalar conservation laws[END_REF][START_REF] Ghoshal | On a conjecture by Lions-Perthame-Tadmor and non existence of the BV regularizing effect for scalar conservation laws[END_REF], for the convex power flux f (u) = |u| 1+p /(1 + p), p 1, Φ(u) = |u| p = |a(u)|. Indeed, when the velocity is convex for u > 0 and is an odd function, then Φ(u) = |a(u)| [START_REF] Castelli | Smoothing effect in BV for entropy solutions of scalar conservation laws[END_REF].

C w denoting the space of continuous functions with modulus of continuity w, the goal of the paper is to prove the Theorem: Theorem 2.1 (Regularising effect in BV Φ ). Let f be a strictly convex flux on R, u 0 ∈ L ∞ and u being the unique entropy solution of (1), then u(t, •) ∈ BV Φ loc , i.e., for all [α, β] ⊂ R,

TV Φ+ u(t, •){[α, β]} (β -α) t -1 , (15)
TV Φ u(t, •){[α, β]} 2 a(u 0 ) ∞ + (β -α) t -1 , (16)
Moreover, if u 0 is compactly supported, then there exists C > 0 such that

TV Φ u(t, •){R} C 1 + t -1 , (17)
and, in addition, for all τ > 0

u ∈ C wτ ]τ, +∞[, L 1 loc where w τ (y) = Φ -1 (C 1 + τ -1 y). ( 18
)
Inequality 15 is the natural way to recover the one-sided Oleinik inequality. When Φ is the identity function, so TV Φ u = TVu, the inequality 17 is the classical one for uniformly convex smooth flux. This regularity in time is proven in the last section of the present paper.

Theorem 2.1 covers all previous results (with a different proof) on the smoothing effect for a strictly convex flux, the C 2 case being considered in [START_REF] Bourdarias | Fractional BV spaces and applications to scalar conservation laws[END_REF][START_REF] Castelli | Fractional spaces and conservation laws[END_REF][START_REF] Castelli | Oscillating waves and optimal smoothing effect for one-dimensional nonlinear scalar conservation laws[END_REF][START_REF] Castelli | On the maximal smoothing effect for multidimensional scalar conservation laws[END_REF][START_REF] Jabin | Some regularizing methods for transport equations and the regularity of solutions to scalar conservation laws[END_REF][START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF][START_REF] Marconi | Regularity estimates for scalar conservation laws in one space dimension[END_REF][START_REF] Oleinik | Discontinuous solutions of non-linear differential equations[END_REF] and the C 1 case being treated in [START_REF] Castelli | Smoothing effect in BV for entropy solutions of scalar conservation laws[END_REF]. All these proofs make use, directly or indirectly, of the Oleinik inequality (3). The proof of the theorem 2.1 here is necessarily more complicated due to the loss of the Oleinik inequality. This crucial point is discussed in details in the next section.

Notes on the Oleinik inequality for discontinuous wave speeds

The following example shows that the Oleinik inequality (3) is no longer true everywhere when a = f is not continuous on [-M, M ]. The Oleinik inequality requires the velocity is defined everywhere. For this purpose, the velocity can be defined everywhere as the mean of its left and right limits with a weight λ ∈

[0, 1], i.e., ā(x) = λ a + (x) + (1 -λ) a -(x).
Now, a key result about the Oleinik inequality for the proof of the theorem is stated with this mean velocity.

Let us consider the Riemann problem, that is a Cauchy problem with a piecewise constant datum u 0 (x) = u l for x < 0 and u 0 (x) = u r for x > 0. The Oleinik inequality is clearly true for a shock wave, but it is not always valid for a rarefaction wave. Proposition 3.1 (One-sided Oleinik inequality). For a Riemann problem producing a rarefaction wave -i.e.,

u l < u r -if x/t, y/t ∈ ā([-M, M ]) then the Oleinik inequality holds ā(u(t, x)) -ā(u(t, y)) (x -y)/t a.e. x > y. (19)
The set ā([-M, M ]) is not an interval since a is not continuous. This is a reason for the loss of the Oleinik inequality. Moreover, the solution is constant where a is not defined, as shown in the example 3.1 below. When the velocity is continuous this problem disappears, as proved in the next section.

Remark 3.1. The Oleinik inequality is true a.e. with a velocity chosen as

a -(u(t, x)) -a + (u(t, y)) (x -y)/t a.e. x > y. ( 20
)
But, it is less useful to get the BV estimate for this velocity. This is the key point to prove the BV Φ regularity of the entropy solution. It should also be noticed that the Oleinik inequality (20) can be invalid if the signs are exchanged.

From now on, we denote ξ = x/t and η = y/t for brevity.

Example 3.1. (The one-sided Oleinik inequality is not always valid) Consider

f (u) = u 2 + |u|, a(u) = 2u + sign(u) (21)
and u 0 (x) = sign(x). The entropy solution of Considering t > 0, the Oleinik inequality is not satisfied and ā(u(t, x)) -ā(u(t, y)) > (x -y)/t in the following cases

(1) is u(t, x) = U (ξ) with U (ξ) =            -1 ξ -3, 1 2 (ξ + 1) -3 ξ -1, 0 -1 ξ 1, 1 2 (ξ -1) 1 ξ 3, 1 3 ξ. ξ U x t u = 1 u = -1 u = 0 u = ξ-1 2 u = ξ+1 2 ξ = -3 ξ = -1 ξ = 1 ξ = 3
• if λ = 0 and -1 < η < 1 < ξ < 3; • if λ = 1 and -3 < η < -1 < ξ < 1; • if λ ∈]0, 1[ and ( 2λ -1 < η < 1 < ξ < 3 or -3 < η < -1 < ξ < 1 -2λ ).
Remark 3.2. The function b is the generalized inverse of ā for all λ, and b(ā(u)) = u.

Hereinafter, we take λ = 1/2 for the sake of simplicity, thus

ā(x) = 1 2 [ a + (x) + a -(x) ]. ( 22
)
Remark 3.3. If we change the flux f in the example 3.1 by

f (u) = u 2 + u + |u| and if α < 0 < β, then ā(u(t, β)) -ā(u(t, α)) 1
, where the inequality (15) remains true.

Remark 3.4. The converse of the proposition 3.1 is false in general. For instance, it is sufficient to take x, y in ] -t, 0[ or in ]0, t[ in the example 3.1.

At this stage, it is important to outline the main difficulties for proving the theorem 2.1. These difficulties result from the discontinuity of the velocity a(u) (yielding the loss of the Oleinik inequality). For instance, consider the case when a(u) is discontinuous at u = u # with the jump a (u # ) = a + (u # ) -a -(u # ) > 0. Let be u(x, t) a rarefaction wave, non-decreasing solution of the Riemann problem with initial data u 0 (x) = u l for x < 0 and u 0 (x) = u r for x > 0 with u l < u # < u r . The solution u(x, t) is flat, with value u = u # on the interval x ∈ [a -(u # ) × t; a + (u # ) × t], see figure 1. The length of this interval, at time t, is exactly

∆x = t × a (u # ).
At first sight, it seems a good case where the Oleinik inequality is actually an equality. However, it is not the case for the two following reasons.

First, u being constant on the flat part (with u ≡ u # ), there are of course no variations of u on this part, while there is a variation of a equals to the jump of a at u # . This shows that the variations of a are bad indicators of the variations of u. Usually, the total variation of u is controlled by the total variation of a(u) [START_REF] Castelli | Fractional spaces and conservation laws[END_REF].

Second, since u is constant on this part, the shock wave penetrates the flat part, reducing the length of this part, i.e. ∆x < t × a (u # ).1 In other words, the jump of a(u) at u # does not represent well the size of the flat part, which is problematic as already mentioned. As a consequence, the total variation of a(u) is not controlled in the present work. It is an important difference with the case of smooth fluxes, where a(u) is known to be in BV [START_REF] Cheng | A regularity theorem for a non-convex scalar conservation law[END_REF], at least C 2 , with precise assumptions and counter-examples given in [START_REF] Marconi | Regularity estimates for scalar conservation laws in one space dimension[END_REF]. Here, assuming only that the convex flux is Lipchitz, it is not clear whether a(u) belongs to BV or not.

If the velocity has only one discontinuity, it is easy to overcome this difficulty. Consider

a(u) = ã(u) + a(u # ) H(u -u # ),
where H is the Heaviside function; that is to say ã is the continuous part of the velocity a(u). The generalised inverses of a and ã having the same modulus of continuity, they define the same space BV Φ . Moreover, ã is easy to estimate in BV. Thus, the BV Φ regularity of u follows as in [START_REF] Castelli | Fractional spaces and conservation laws[END_REF]. This simple case shows again that the variation of a through its discontinuity is useless to capture the regularity of u. Moreover, for the example 3.1 above, the regularity of u is simply BV since ã corresponds to a Burgers flux.

Removing the discontinuity of a can be done only if a has finitely many points of discontinuity. This is not possible in general. Consider the example with the following velocity

a(u) = n 2 -n H(u -r n ),
where the sequence (r n ) takes all the values of rational numbers. This velocity corresponds to a strictly convex flux where the second derivatives has only an atomic part. Thus, u is regularised in some BV Φ . Removing all the discontinuities of a is not a good idea here since the corresponding flux is flat and does not correspond to any smoothing effect.

The way we solve this difficulty is to keep the Oleinik inequality by introducing a new velocity, called χ below. The total variation of χ can be estimate geometrically by the mean of characteristics through a wave front tracking algorithm. Moreover, the variation of χ corresponds exactly to the generalised variation of the entropy solution u.

In order to prove Theorem 2.1, the wave front tracking algorithm will be used. To do so, u 0 is approximated by a sequence of step functions and thus the Riemann problem for each sequence can be solved [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF]Lemma 3.1]. A Riemann problem with a discontinuous velocity is expounded in the next section.

Riemann problem

This section is devoted to the Riemann problem. The shock wave is solved as usual, but solving the rarefaction wave is more complicated. For this purpose, the flux f is approached by piecewise quadratic C 1 fluxes, in order to show that, for a rarefaction wave, the solution is given by b (x/t); this is the classic formula for smooth fluxes where b is the inverse of the velocity. We extend this formula when b is the generalized inverse of a discontinuous velocity. The second part deals with a piecewise linear flux f ε [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF], which gives a modified Oleinik inequality up to a small error.

If u l > u r , the solution generates a shock with a speed given by the Rankine-Hugoniot relation s = f (u) / u , where u = u r -u l is the jump of u. The entropy solution is u(t, x) = u l for x < st and u(t, x) = u r for x > st.

The interesting case is u l < u r because the Oleinik inequality is not always true in this case.

The solution has a non decreasing rarefaction wave between x = a + (u l )t and x = a -(u r )t, given by the following proposition:

Proposition 4.1. Let be u the entropy solution of the Riemann problem with u l < u r . For ξ = x/t the solution is

u(t, x) =      u l ξ < a -(u l ), b (ξ) a -(u l ) < ξ < a + (u r ), u r ξ > a + (u r ). ( 24
)
Remark 4.1. A similar formula for systems with Lipschitz fluxes is given in [START_REF] Correia | Hyperbolic systems of conservation laws with Lipschitz continuous flux-functions: the Riemann problem[END_REF]Th. 3.3,p. 279]. Another formula is proposed by Bressan in [4, Problem 3, p. 120]. In Bressan's book, the result is given for all Lipschitz fluxes. One can take the upper convex envelop of the flux instead of the flux in the same formula, thanks to the Oleinik criteria for entropy solutions with general fluxes. Moreover, if the flux is strictly convex, then the solution is defined everywhere by formula [START_REF] Marconi | Regularity estimates for scalar conservation laws in one space dimension[END_REF]. Otherwise, it is defined a.e in [START_REF] Bressan | Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem[END_REF]. To be self contained, the proposition 4.1 has been added here with a short proof (see also [START_REF] Caginalp | Minimization Solutions to Conservation Laws with Non-smooth and Non-Strictly Convex Flux[END_REF][START_REF] Caginalp | Minimization Approach to Conservation Laws with Random Initial Conditions and Non-smooth and Non-Strictly Convex Flux[END_REF]).

For n ∈ N * , let be

v i = u l + (i/n)(u r -u l ) (i = 0, 1, • • • , n)
and let be a n the sequence of functions such that ∀i Proof.

a n (v i ) = ā(v i ), a n being linear on [v i , v i+1 ], so f n (u) = f (u l ) + u u l a n (v)
• For a given v such that a is continuous at v, We are now able to prove the Proposition 4.1. Proof. By definition of a n , a n (u l ) = ā(u l ) and a n (u r ) = ā(u r ). Since f n ∈ C 1 , then, from [START_REF] Bressan | Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem[END_REF][START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF], the exact entropy solution of u t + f n (u) x = 0 with the initial condition ( 23) is

v i = u l + i(u r -u l )/n and v i+1 are chosen such that v ∈ [v i , v i+1 ] (note that v i depends on n). By definition, a n (v) = n ā(v i+1 )-ā(v i ) ur-u l (v -v i ) + ā(v i ). Since |v -v i | ur-u l n and since a is continuous at v, then lim n→∞ a n (v) = a(v). Therefore, lim n→∞ f n (v) = f (v)
u n (t, x) =      u l ξ < ā(u l ), b n (ξ) ā(u l ) < ξ < ā(u r ), u r ā(u r ) < ξ. (25)
Taking the limit n → ∞, thanks to Lemma 4.1, u has an explicit continuous formula. Thus, u is a weak entropy solution satisfying (4) and [START_REF] Caginalp | Minimization Solutions to Conservation Laws with Non-smooth and Non-Strictly Convex Flux[END_REF]. Noting that b(ξ 25) can be replaced by a -(u l ) or by a + (u l ). Similarly, ā(u r ) can be replaced by a -(u r ) or a + (u r ), thus concluding the proof.

) = u l ∀ξ ∈ [a -(u l ), a + (u l )], ā(u l ) in (
Finally, the proposition 3.1 can be proved: Proof. Since x/t, y/t ∈ ā([-M, M ]) and since b is the generalized inverse of a, then ā (u(t, x)) = ā (b(x/t)) = x/t, ā (u(t, y)) = ā (b(y/t)) = y/t. Therefore, the Oleinik inequality is true. 4.2. Approximate Riemann solver. In the section 5 below, the wave front tracking algorithm is used. Therefore, a suitable Oleinik inequality is needed for the approximate solutions. In order to get this inequality, the flux is replaced by a suitable piecewise linear approximation.

For this purpose, a piecewise constant approximation of the velocity is used as in [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF]. A key point is to choose a discrete set of the value of the approximate solution u ε , taking into account the discontinuities of the velocity. Let be ε > 0 and let be

B = {c 0 = -M, c 1 , • • • , c p = M } (26)
a subdivision of the interval [-M, M ] including a too large jump of the velocity. For this purpose the subdivision is chosen such that c i < c i+1 , ∀i, and

a -(c i+1 ) -a + (c i ) 1 4 ε. ( 27
)
That means that the variation of a is small on ]c i , c i+1 [, ∀i, thus the big jumps of a are located at c i .

Remark 4.2. Due to the jumps of the velocity a which are not expected on a uniform grid, the subdivision 2 -n Z cannot be used as in [START_REF] Bressan | Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem[END_REF].

Remark 4.3. a + (u)-a -(u) 2 = a (u)
2 can be bigger than ε/4, thus, the condition (27) cannot be replaced by ā(c i+1 ) -ā(c i )

that if ε → 0, then c i+1 -c i → 0, ∀i, since c i+1 -c i = b a -(c i+1 ) -b a + (c i ) ω[b] a -(c i+1 ) -a + (c i ) ω[b]( 1 4 ε). ( 28
)
And ω[b] is continuous at 0, thanks to the Heine theorem.

As in [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF], the flux f is approximated by a continuous and piecewise linear flux. For this purpose, the approximate flux f ε is chosen as the continuous piecewise linear interpolation of f on the subdivision B,

f ε (c i ) = f (c i ) ∀i and f ε is linear on [c i , c i+1 ]. Its derivative a ε = f ε is piecewise constant.
Now, the approximate Riemann solver is expounded. Let be u l , u r ∈ B, and let be u ε the entropy solution of the Riemann problem u t + f ε (u) x = 0, with the initial data [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF].

If u l > u r , as the previous subsection, the solution generates a shock with the Rankine-

Hungoniot relation s = f ε (u) / u = f (u) / u (Notice that f = f ε on B). If u r > u l , let u l = c k , u r = c k , for a fixed t > 0, u ε (t, •
) is non decreasing and piecewise constant. Defining

s i = f ε (c i ) -f ε (c i-1 ) c i -c i-1 = f (c i ) -f (c i-1 ) c i -c i-1 = a ε (c) on (c i-1 , c i ). ( 29
)
The solution is given by u ε (t, x) = c i for x/t ∈]s i , s i+1 [ as in [START_REF] Dafermos | Polygonal approximations of solutions of the initial value problem for a conservation law[END_REF]. The curves of discontinuity in this case are called contact discontinuities. In fact those curves represent an approximation of a rarefaction wave, so in this paper we call them rarefaction curves. Since s i ā(c i ) s i+1 , then [START_REF] Rao | Theory of Orlicz spaces[END_REF] implies

u(t, ts i ) = b(s i ) b(ā(c i )) = c i b(s i+1 ) = u(t, ts i+1 ). ( 30
)
For i with k < i < k , the equation u(t, xi ) = c i has at least one solution since the exact solution u is non decreasing and continuous. Adding the condition ξi = xi /t ∈ ā([-M, M ]), this solution is unique, and xi = ā(c i )t so ξi = ā(c i ). Let be ξk , ξ+ k , ξk , ξk defined as

• ξk = ā(u l ), • ξ+ k = a + ε (u l ), • ξk = ā(u r ), • ξ- k = a - ε (u r ). Now, xk , x+ k , xk , x-
k are defined by the relation x = ξ t. By construction, the approximate solution at the point (t, xi ) equals the exact solution at the same point, i.e., u Summing up for all i, since

ε (t, xi ) = u(t, xi ) ∀i = k + 1, . . . , k -1. Since ξi ∈ ā([-M, M ]), then ā(u ε (t, xi+1 )) -ā(u ε (t, xi )) = ā(b(x i+1 /t)) -ā(b(x i /t)) = (x i+1 -x i )/t, ∀i = k+1, . . . , k -2.
a -(u k ) -a - ε (u k ) a -(u k ) -a + (u k -1 ) ε/4, and
a + ε (u k ) -a + (u k ) a -(u k+1 ) -a + (u k ) ε/4,
the error is smaller than ε/4 on each boundary term. Thus, the modified Oleinik inequality holds also on the whole interval as

ā(u ε (t, xk )) -ā(u ε (t, xk )) = (x k -xk )/t (31a) a -(u ε (t, x- k )) -ā(u ε (t, xk )) (x - k -xk )/t + ε/4, (31b) ā
(u ε (t, xk )) -a + (u ε (t, x+ k )) (x k -x+ k )/t + ε/4, (31c)
a -(u ε (t, x- k )) -a + (u ε (t, x+ k )) (x - k -x+ k )/t + ε/2. ( 31d
)
The value of u ε (t, xk ) is considered on the right of the curve of discontinuity, and the value of u ε (t, x+ k ) is considered on its left (see figure 2), i.e., u ε (t, x-

k ) = u k = u r , u ε (t, x+ k ) = u k = u l .
This proves the approximated Oleinik inequality for rarefaction waves.

Here, the rarefaction wave has been approached by a sequence of step functions and satisfies the approximate Oleinik inequality [START_REF] Whitham | Linear and nonlinear waves[END_REF]. Usually, the Oleinik inequality gives that a(u) is in BV and then u in BV Φ [START_REF] Castelli | Fractional spaces and conservation laws[END_REF]. Unfortunately, a(u) is not well defined. Moreover, the modified Oleinik inequality [START_REF] Whitham | Linear and nonlinear waves[END_REF] does not imply that ā(u) is BV. The next section is devoted to define another velocity χ ∼ = a(u ε ) which can be controlled in BV with the wave front tracking algorithm and the restricted Oleinik inequality (31).

Wave front tracking algorithm

This section deals with the BV estimate of a velocity χ defined below. For that purpose, a BV + estimate is used. With (x) + def = max{x, 0}, the BV + space is defined by BV + def = {u, TV + u < ∞}, TV + being the positive total variation

TV + v def = sup p∈P n i=2 (v(x i ) -v(x i-1 )) + , (32)
where

P = {{x 1 , x 2 , • • • , x n }, x 1 < • • • < x n , 1 n} is the set of all subdivisions of R.
The function u 0 being bounded, we can assume that u 0 has a compact support to prove Theorem 2.1 (thanks to the finite speed of waves propagation).

Let be A positive such that supp(u 0 ) ⊂ [-A, A]. Let be h = A/2m and x i = -A + hi with i = 0, 1, • • • , m. The initial datum u 0 is approached by a sequence of step functions (u 0,m ) m taking values in B (see remark 4.4) and constant in ]x i , x i+1 [ as in [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF]Chapter XIV]. Consider the initial value problem

u t + f ε (u) x = 0, u(0, x) = u 0,m . (33)
The entropy solution of (33), u ε m , is piecewise constant [START_REF] Bressan | Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem[END_REF][START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF]. The problem (33) requires to solve m + 1 Riemann problems and the need to study the waves interactions.

Note that in the special case of figure 2, if there is a shock on the right of the rarefaction wave, that has the values u 4 on its left and u 5 < u 3 on its right, then the distance between the shock and the rarefaction becomes very small. There, the total variation of ā(u) is bigger than a (u 4 ), that is problematic because the total variation of ā(u) can not be controlled by the distance between the rarefaction wave and the shock wave. To avoid this problem, the velocity on the part u = u 4 is replaced by a -(u 4 ) instead of ā(u 4 ).

In the general case, a new velocity, χ ε m , is introduced. This velocity is defined by removing the jumps of a(u) on the boundaries of the rarefaction wave, if this wave is close to a shock. Consider t > 0 a fixed time and x ∈ R. If there is a shock on the left of the point (t, x) and a rarefaction on its right, then

χ ε m (t, x) = a + (u ε m (t, x)).
If there is a shock on the right of the point (t, x) and a rarefaction of its left, then

χ ε m (t, x) = a -(u ε m (t, x)). Otherwise χ ε m (t, x) = ā(u ε m (t, x)
). This definition avoids the problem mentioned above for the special case of figure 2. Note that in all the three cases, the solution u ε m can be obtained by u ε m (t, x) = b(χ ε m (t, x)), which is a key point to take the limit in section 6. Note also that if f ∈ C 1 , then χ ε m = a(u ε m ). The choice of the inequality (31) depends on the following cases:

• If two shocks appear on both sides of the rarefaction wave, then the inequality (31d) is used; • If a shock appears only on the left of the rarefaction wave, then the inequality (31c) is used; • If a shock appears only on the right of the rarefaction wave, then the inequality (31b) is used; • Else, the inequality (31a) is used. For t > 0, the positive total variation of χ ε m over a rarefaction is smaller than the length of the rarefaction divided by t, plus a small error [START_REF] Whitham | Linear and nonlinear waves[END_REF]. For a shock, the positive total variation is equal to zero.

Here, the positive total variation is estimated after wave interactions, as in Bressan's book [START_REF] Bressan | Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem[END_REF]Chap. 6,Prob. 6]. Let be u 1 , u 2 and u 3 the values of u ε m from the left to the right. The speed of the left jump is s 1 = fε(u 2 )-fε(u 1 ) u 2 -u 1 and the speed of the right jump is

s 2 = fε(u 2 )-fε(u 3 ) u 2 -u 3
. The rarefaction wave is replaced by a contact discontinuity since the flux is piecewise affine. All the possibilities for wave interactions are listed below.

(SS) Shock-shock interaction: u 3 < u 2 < u 1 . When two shocks collide they generate a new shock, and the positive total variation is always equal to zero. (RS) Rarefaction-shock interaction: u 3 u 1 < u 2 . the values u 1 and u 2 are consecutive in I, which implies that u 3 u 1 . We consider the two cases:

• u 1 = u 3 . After the interaction, the curves of discontinuity will disappear, and the positive total variation will be zero. • u 3 < u 1 . After the interaction, a shock will appear (see figure), the rarefaction fan will be smaller, and it will lose the curve of discontinuity on the right (between u 1 and u 2 ). The value of χ ε m will be changed from ā(u ε m (u 1 )) to a -(u ε m (u 1 )) (see the definition of χ ε m ), and also the choice of the points will be changed from ξ1 to ξ-1 , which makes the inequality (31) holds after the interaction. (SR) Shock-rarefaction interaction: u 2 < u 3 u 1 . This case can be treated exactly like the case Rarefaction-shock. (RR) Rarefaction-rarefaction interaction: u 1 < u 2 < u 3 . Two rarefactions cannot collide (even the points xi ). This case is impossible, because the convexity of f ε implies that s 1 < s 2 .

x s 0 s 1 s 2 ξ1 ξ- 1 u 0 u 1 u 2 u 3
Remark 5.1. In the case (RS) the new shock that appears can be very close to the rarefaction, which means that if the jump of the velocity a on u 1 is big enough, then, the positive total variation of ā(u ε m ) cannot be controlled by the length of the rarefaction wave. That is the reason of using the function χ ε m . In summary, the positive total variation of χ ε m and the number of rarefaction waves do not increase. Also, TV + χ ε m (t, •) is bounded by summing up the positive variation of all the rarefaction waves, thanks to the modified one-sided Oleinik inequality [START_REF] Whitham | Linear and nonlinear waves[END_REF]. For each rarefaction wave, this variation is related to the length of the interval at time t > 0 up to a small error ε/2. The sum of all lengths of the intervals cannot exceed the size of the support of the solution. Since the number of rarefaction waves is less than m, there are only m error terms of size less than ε/2. Thence

TV + χ ε m (t, •) L(t)/t + m ε/2, (34)
where

L(t) = 2A(t) and supp u ε m (t, •) ⊂ [-A(t), A(t)]. Recall that BV + ∩ L ∞ = BV since TVχ 2 (TV + χ + χ ∞ ).
The boundedness of the propagation velocity yields L(t) 2A + 2t a(u) ∞ . Then, taking the constant C = C(u 0 , f ) = max(4A, 6 a(u) ∞ ) > 0, which doesn't depend on ε and m, gives TVχ ε m (t, •))

C (1 + 1/t) + m ε. (35)

Compactness and regularity

This section is devoted to the proof of Theorem 2.1. For this purpose, an uniform estimate of the velocity χ ε m is obtained, which gives the compactness of the sequence (χ ε m ). To take the limit in (35) as m → ∞ and ε → 0, the parameters m can be chosen such that lim ε→0 m ε ε = 0.

(36)

We begin with an estimate of the velocity. The BV estimate of the velocity χ ε m proved in the previous section yields to Lip t L 1

x estimates. First, let t a fixed time belonging to

[T 1 , T 2 ] ⊂]0, +∞[. We have R χ ε m (t, x + h) -χ ε m (t, x) dx T V χ ε m (t, •)|h| C 1 + 1 T 1 + m ε |h|. (37)
Second, consider two different times T 1 < T 2 . χ ε m is piecewise constant and it has exactly the same curves of discontinuity of u ε m . These curves are Lipchitz and the speed of any curve cannot exceed k = a(u 0 ) ∞ . We suppose at first that there is no wave interaction between T 1 and T 2 . Then the domain [T 1 , T 2 ] × R can be divided as the following Using (37) within each small rectangle, gives

R χ ε m ( Tj , x) -χ ε m ( Tj+1 , x) dx K C 1 + 1 T 1 + m ε Tj+1 -Tj .
In general, there are many interactions, so [T 1 , T 2 ] = ∪ J j=0 [t j , t j+1 ], where t 0 = T 1 , t J = T 2 and the points t j , j = 1, • • • , J -1 are the instants of the interactions. Let be 0 < δ < 1 2 inf j (t j+1 -t j ). The inequality holds true for t ∈ [t j + δ, t j+1 -δ]. Taking δ -→ 0 and using that χ ε m (•, x) is continuous at t j for almost all x, the inequality

R χ ε m (T 1 , x) -χ ε m (T 2 , x) dx K C 1 + 1 T 1 + m ε T 1 -T 2 , (38)
follows. Notice that in the estimates (37) and (38), the term m ε is bounded by a constant, thanks to (36). These two inequalities and the uniform boundedness of χ ε m by k are the conditions of the classical compactness theorem A.8 in [START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF]. Hence, the sequence χ ε m converges up to a sub-sequence (if necessary) to some function χ in C([T 1 , T 2], L 1 loc )

χ = lim ε→0 χ ε m . (39)
Due to the lower semi-continuity of the total variation, we have χ ∈ BV and Lipchitz in time with value in L 1 loc in space. Thus, for any 0 < t, χ satisfies TV χ(t,

•) C 1 + 1 t , (40)
R χ(T 1 , x) -χ(T 2 , x) dx KC 1 + 1 T 1 |T 1 -T 2 | . (41)
Using that u ε m = b(χ ε m ), which also provides the compactness of the sequence of approximate solutions. Taking the limit m → ∞ in ( 4) and ( 5), gives that

u = b(χ) (42)
is an entropy solution of (1). The main theorem of [START_REF] Chen | Initial Layers and Uniqueness of Weak Entropy Solutions to Hyperbolic Conservation Laws[END_REF] (see also [START_REF] Panov | Existence of strong traces for generalized solutions of multidimensional scalar conservation laws[END_REF][START_REF] Panov | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF]) ensures that the initial datum is recovered. Then, u is the unique Kruzkov entropy solution with the initial datum u 0 .

Remark 6.1. Equality (42) means that χ = a(u) for a smooth velocity. Here, it is not necessarily true almost everywhere since the velocity a(u) can be discontinuous where u is constant.

Now, the BV Φ regularity of the entropy solution is proven. Let us first check that Φ is positive. The function b is not constant on the whole interval [-M, M ]. Then, ω(h) > 0 for h > 0. Thanks to Heine theorem, ω is continuous at 0, ensuring that φ(y) > 0 for y > 0. Φ the convex envelope of φ is then also strictly positive [START_REF] Castelli | Smoothing effect in BV for entropy solutions of scalar conservation laws[END_REF].

The BV Φ regularity of u is a direct consequence of the BV regularity of χ and the definition of Φ, which yield with (42), [START_REF] Cheng | A regularity theorem for a non-convex scalar conservation law[END_REF], and remark 2.1 to the following inequality for almost all t 1 , t 2 , x, y Proof of the inequality [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF]. The proof of Theorem 2.1 is done for u 0 compactly supported. In the general case, to prove the inequality [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF], the segment [α, β] is divided by a subdivision S = {α 0 = α, α 1 , • • • , α l = β} in order to separate rarefaction waves and shocks. Inequality [START_REF] Whitham | Linear and nonlinear waves[END_REF] implies that a -(u(t, α i+1 )) -a + (u(t, α i )) (α i+1 -α i ) t -1 + ε/4 .

Replacing χ(t 1 , x) -χ(t 2 , y) by a -(u(t 1 , x)) -a + (u(t 2 , y)) in the inequality (43), the result follows by restarting a similar proof, summing up for all i and taking the limit m → +∞. E-mail address: didier.clamond@unice.fr

Fig. 1 .

 1 Fig. 1. The solution of the problem.

4. 1 .

 1 The exact solution of the Riemann problem. The Riemann problem consists in solving u t + f (u) x = 0 with the initial condition u(0, x) = u l for x < 0 and u(0, x) = u r for x > 0.

  dv. For proving the proposition 4.1, we need the lemma 4.1:Lemma 4.1. For all (v, ξ) ∈ [-M, M ] × a([-M, M ]), the sequences f n (v) and b n (ξ) converge, respectively, to f (v) and b(ξ), and a n (v) → a(v) when n → ∞ if a is continuous at v. Moreover, the sequence (b n ) converges uniformly towards b on any bounded set.

  follows at once from the definition of f n . • For n ∈ N * and ξ ∈ a([-M, M ]), there exists i such that b(ξ) ∈ [v i , v i+1 ]. Moreover, b n being linear on [a n (v i ), a n (v i+1 )], |b n (ξ) -b(ξ)| v i+1 -v i = ur-u l n . In addition, b being continuous, a Dini's Lemma yields the uniform convergence of b n to b as n → ∞.

1 4Remark 4 . 4 .

 144 ε. Notice that in this case u necessarily belongs to B, because the velocity has a big discontinuity at u. The condition[START_REF] Panov | Existence of strong traces for generalized solutions of multidimensional scalar conservation laws[END_REF] is enough to show

4 u l = u 0 u r = u 4 Fig. 2 .

 442 Fig. 2. An example of a rarefaction wave with k = 0, k = 4.

Fig. 3 .

 3 Fig. 3. Interaction (RS).

T4 = T 2 T1 = T 1 T2T3Fig. 4 .

 214 Fig. 4. Decomposition of the domain, the blue lines are the curves of discontinuity.

Φ 1 x 1 p

 11 (|u(t 1 , x) -u(t 2 , y)|) = Φ(|b(χ(t 1 , x)) -b(χ(t 2 , y))|) φ(|b(χ(t 1 , x)) -b(χ(t 2 , y))|) φ ω |χ(t 1 , x) -χ(t 2 , y)| χ(t 1 , x) -χ(t 2 , y) .(43)The BV regularity of χ (40) and the inequality (43) show that u ∈ BV Φ . The Lip t L regularity of χ (41) and inequality (43) again implies thatR Φ u(t 1 , x) -u(t 2 , x) dx KC(1 + τ -1 )|t 1 -t 2 |. (44)This is an estimate in the Orlicz space L Φ , i.e., L Φ (R) denotes the set of measurable functionsf such that R Φ(|f (x)|)dx < ∞ [29].If Φ(u) = |u| p then the Lip (]τ, +∞[, L p loc ) estimate in[START_REF] Bourdarias | Fractional BV spaces and applications to scalar conservation laws[END_REF] is recovered. In general, Jensen inequality gives u ∈ C w (]τ, +∞[, L 1 loc ).

(

  Stéphane Junca) Université Côte d'Azur, CNRS UMR 7351, LJAD & Inria, Parc Valrose, F-06108 Nice cedex 2, France. E-mail address: stephane.junca@unice.fr (Didier Clamond) Université Côte d'Azur, CNRS UMR 7351, LJAD, Parc Valrose, F-06108 Nice cedex 2, France.

Note that it is the converse of the Oleinik inequality.