M Morjean
email: morjean@ganil.fr

A Chbihi

M Dasgupta

A Drouart

J D Frankland

M O Frégeau

D J Hinde

D Jacquet

L Nalpas

M Pârlog

C Simenel

L Tassan-Got

E Williams

Long lifetime components in the decay of excited super-heavy nuclei

Long lifetime components in the decay of excited super-heavy nuclei

Introduction

In reactions between two very heavy nuclei, the tiny cross-sections associated with evaporation residue detection makes it very difficult to demonstrate the formation by fusion of super-heavy nuclei (atomic numbers Z > 110). Even if compound nuclei are formed, they decay dominantly by fission, symmetric or possibly asymmetric in mass. Therefore, experimental fusion cross-sections can only be reached through fission fragment detection. However, the distinction between fusion followed by fission (fusion-fission) and faster non-equilibrium processes (often called quasi-fission) is very tricky because the fission fragments and fission-like fragments from quasi-fission can be quite similar in mass, atomic number and energy [1,2,3].

In most of the experimental work, the discrimination between fusion-fission and quasi-fission reactions is somewhat arbitrary, based on considerations of the mass symmetry in the exit channel (ignoring thus any possible asymmetric fission) or the width of the mass and energy distributions. In fact, the objective difference between quasi-fission and fusion-fission is the reaction time [3,4,5]. After the fusion step, the nucleons are trapped within a potential pocket, and the composite system needs time to find its way to scission. By contrast, in quasi-fission reactions the nucleons of the system are not trapped and a very fast separation into two fission-like fragments takes place. Therefore, the most reliable experimental criterion that can be used to discriminate between fusionfission and quasi-fission reactions is the reaction time. In the following, we shall first present recent 1 reaction time measurements through mass-angular correlations, highlighting fast processes leading to fission-like fragments in reactions between very heavy nuclei. We shall then present results of experiments [5,6] in which long lifetime components (τ > 10 -18 s) characteristic of fusion reactions were observed, associated with part of the reactions leading to fission-like fragment production.

Reaction time from mass-angle distributions

The correlation between the mass of the reaction products and their emission angle can be used to highlight binary reactions in which the sticking time is shorter or of the same order as the rotational period of the composite system [3,4]. For such fast reactions, the angular distributions measured as a function of the mass asymmetry in the exit channel present maxima that can be linked to lifetimes of the composite system, assuming for each mass asymmetry a single fast component in the reaction time distribution. By contrast, for fusion-fission reactions associated with very long lifetimes, the composite system lives much more than one rotational period and flat angular distributions are expected.

Figure 1: Correlation between the center-of-mass deflection angle (θ CM ) and the mass ratio (M R ).

Mass-angle distributions for heavy systems have been recently studied at the Heavy Ion Accelerator Facility at the Australian National University. In these experiments [4,7], the two coincident fragments from binary reactions were measured by large area position sensitive multiwire proportional counters allowing the determination of the detection angle and of the mass asymmetry in the exit channel. Figure 1 presents, for the heaviest system presently studied at an energy above the Coulomb barrier, 48 Ti + 238 U at 289 MeV (E/B ∼ 1.1), the cross-section

d 2 σ
dθCM ×dMR as a function of the center-of-mass scattering angle θ CM and of M R , the ratio of fragment mass to compound nucleus mass. The minimum cross-sections observed for symmetric splittings at any angle confirm for this system the vanishing of symmetric fission already inferred from studies on lighter systems [3,4,7]. For mass asymmetric splittings, the angular distributions clearly present maxima between 30 and 90 • for M R < 0.5 (90 and 150 • for M R > 0.5), indicating reactions lasting less than the ro-tational period. Assuming reaction time distributions with a single fast component, most probable reaction times t react 10 -20 s can be inferred, even for the longest quasi-fission reactions associated with the largest deflection angles with respect to grazing trajectories. These quasi-fission times are in good agreement with fully microscopic quantum calculations [8]. Nevertheless, it must be stressed that, for mass asymmetries 0.3 < M R < 0.4 (0.6 <M R < 0.7), a region possibly populated by mass asymmetric fission, the angular distributions are quite broad. It seems therefore difficult to exclude, in addition to the fast component associated with quasi-fission reactions, long lifetime components characteristic of fusion followed by asymmetric fission.

Long lifetime components

The reaction time t react 10 -20 s inferred from the measured correlations between mass and angle for capture reactions (either complete fusion followed by fission or quasi-fission reactions) in the super-heavy nucleus domain clearly demonstrates the presence of non equilibrium processes. However, it does not rule out longer reaction times for a part of the events. Two different experimental techniques, which both present the advantage of being independent of any nuclear model, have been applied to detect long lifetime components (τ > 10 -18 s), characteristic of fusion reactions: the blocking technique in single crystals (section 3.1) and the X-ray fluorescence technique (section 3.2).

Long lifetime components from the blocking technique in single crystals

Three different systems have been studied in reverse kinematics with the blocking technique [5]:

208 Pb + Ge at 6.2 MeV/u, 238 U + Ni at 6.6 MeV/u and 238 U + Ge at 6.1 MeV/u, possibly leading to compound nuclei with Z CN = 114 , 120 and 124, respectively. All the coincident charged products were detected and identified by INDRA [9], a highly efficient detector array covering a solid angle close to 4π sr. In addition, the blocking patterns were measured for all the fragments detected around 20 • . The reaction mechanism analysis performed with INDRA shows that, for the two heaviest systems, the reactions are always binary when one of the fragments is detected with 70 Z 85 (only 2 fragments with Z > 6 in the exit channel). Furthermore, the sum of the atomic numbers of these two heavy fragments is precisely equal to the total number of protons in the system, as shown for example by figure 2 corresponding to the detection of a fission fragment with 70 Z 80 in the 238 U + Ni system. In addition, these reactions are associated with negligible multiplicities of lighter charged products (Z < 6). The detection of a fragment with 70 Z 85 provides thus us with an efficient selection of capture reactions.

Thermal vibrations of the atoms of the single crystals used as targets imply (see for example [10] and references therein) that all reactions lasting less than about 10 -18 s lead to the same value of χ min , the relative yield of fragments detected in the precise direction of the crystal axes. An increase of χ min for capture reactions is thus straightforward evidence for fusion-fission. For the two heaviest systems studied, a significant χ min increase (with respect to the one measured for either deep-inelastic or quasi-elastic reactions that gives us a reference for fast processes) was observed for fission fragments with 70 Z 85, indicating the formation of compound nuclei with Z = 120 and 124. A minimum proportion of 10% of fusion-fission reactions was directly inferred from the χ min increase for the detected fragments. By contrast, no long lifetime components could be evidenced for Z CN = 114, possibly due to the compound nucleus neutron number being much lower than the one usually predicted for the nearest shell closure (N = 184).

Long lifetime components from the X-ray fluorescence technique

During a collision between two heavy ions, vacancies are created in the inner electronic shells of the unified atom [11,[START_REF] Reinhard | Treatise on Heavy-Ion Science[END_REF]. The vacancies are thereafter filled by transitions of electrons from outer electronic shells, giving rise to X-ray fluorescence (in the case of very heavy atoms, the fluorescence quantum yield can be accurately estimated as 1.0). Considering the independent lifetimes of the electron vacancies and of the compound nucleus, sizable probabilities of X-ray fluorescence from the unified atom can only be observed when the compound nucleus lifetime is at least of the same order of magnitude as the vacancy lifetime. Nuclear lifetimes of excited uranium nuclei have been estimated with the fluorescence technique in good agreement [START_REF] Molitoris | [END_REF]14] with those inferred from the blocking technique in single crystals [15]. Since the lifetime of a vacancy in the K shell of a superheavy atom is of the order of 10 -18 s [16], the multiplicity of X K rays with an energy characteristic of the unified atom provides us with a sensitive probe for long lifetime components.

Unlike the blocking technique that requires good quality single crystals as targets, the X-ray fluorescence technique can be in principle applied to any combination of projectile and target nuclei, giving thus access to investigations of a broader Z range. Furthermore, it makes possible the use of isotopically enriched targets in order to study the long lifetime component production as a function of the compound nucleus isospin. However, this technique had never been previously used in the super-heavy atom domain. Therefore, the system 238 U + 64 Ni at 6.6 MeV/A, leading to 302 120 X compound nuclei, has been chosen for a first experiment [6], providing us both with a test of the fluorescence technique and a cross-check with the crystal blocking results.

The transitions from outer electronic shells to the K shell of 302 120 X atoms have been calculated [17,[START_REF] Desclaux | Mcdfgme, a multiconfiguration dirac fock and general matrix elements program[END_REF] with a multi-configuration-Dirac-Fock (MCDF) approach [START_REF] Desclaux | [END_REF]20]. Only 3 main X K transitions are predicted, shown in figure 3, yellow lines, for an ion with a charge 1+ for different nuclear lifetimes (the lines are broadened by the Weisskopf effect resulting from the finite nuclear lifetime [21] and by the Doppler effect due to the experimental set-up). For nuclear lifetimes associated with quasi-fission reactions (τ nucl ∼ 10 -20 s) [3], the 3 lines are so broad that the very weak fluorescence yield only gives rise to a continuous background. For longer nuclear lifetimes, 3 lines can still be observed, but the fluorescence probability is reduced by roughly a factor 30 between 10 - The Doppler effect associated with the experimental set-up described in section 3.2 is taken into account for the yellow and blue curves. 10 -19 s. It must be stressed however that, in fusion reactions, the atoms are actually formed with broad distributions of charge states and electronic configurations giving rise to slight shifts in the transition energies. Therefore, the 3 lines possibly detected in coincidence with fission fragments merge, even for the longest nuclear lifetimes (figure 3, top), in a single peak at an energy around 190 keV, with a width of about 50 keV (blue lines in figure 3).

During the experiment, the fission fragments were detected between 16 • and 70 • by telescopes (ionization chambers followed by double-sided silicon strip detectors). Coincident photons were detected by 3 planar germanium detectors, operated under vacuum and covering a solid angle Ω ≈ 0.8 sr . The 3 detectors were located at the same polar angle (θ = 127 • ), but at 3 different azimuthal angles (φ = 30, 150 and 270 • with respect to a vertical plane perpendicular to the beam direction). The photon energy spectrum measured by the germanium detector located at φ = 270 • in coincidence with fission-like fragments (fragments detected with 35 Z 90) is presented in the top left panel of figure 4 and in its bottom left panel after background subtraction. Two peaks are clearly seen around 150 and 190 keV. The peak at 150 keV is precisely located at the energy observed in singles measurements (as well as at the energy observed in the random coincidence spectra) for the 158.8 keV γ transition from the decay of the first rotational band of uranium nuclei (the energy of the γ-ray is shifted by the Doppler effect towards lower energy due to the backward angle of the germanium detectors). After correction for random coincidences (right panels in figure 4), the 150 keV peak is strongly suppressed, confirming the random aspect of these coincidences with fission-like fragments. A peak at 200 keV, arising from the 211 keV γ-transition of the same cascade decay as the 158.8 keV one, is also observed in the singles and random spectra. The broad peak observed in the left panel of fig 3 around 190 keV contains therefore random coincidences with 200 keV γ-rays. However, the peak measured at 190 keV in coincidence with fission fragments is much broader than the random ones at 150 and 200 keV (Γ ∼ 50 keV for the 190 keV peak and Γ ∼ 8 keV for the 200 keV one). Furthermore, since the 211 keV transition feeds the 158.8 keV one during the decay cascade, the peak observed in single and random spectra at 200 keV is much smaller than the one at 150 keV. Therefore, the probability measured for random coincidences with 200 keV γ-rays is much smaller than the one with the 150keV γ-rays. Consequently the peak at 190 keV is not eliminated by the random correction, as shown by the right panels of figure 4, and the good suppression of the more probable 150 keV peak ensures that the 190 keV peak after random correction corresponds to true coincidences with fission-like fragments. Such fission-like fragments can arise either from quasi-fission reactions or from uranium fission or from compound nucleus fission.

The multiplicities of the 190 keV photons for different bins in detected atomic numbers Z are presented in table 1. The maximum multiplicity is reached for 70 Z 79. Since this Z selection is only associated with capture reactions (see discussion in section 3.1 and figure 2), the coincident photons are emitted by the composite system or by its fission(-like) fragments. Emission from a fission(-like) fragment should be associated with significantly different Doppler shifted energies measured at φ = 30 and 270 • (δE ∼ 20 keV), whereas emission from the composite system should lead to identical energies, due to the symmetry of the detection set-up with respect to the beam axis. The energy spectra measured for 70 Z 85 at φ = 30 and 270 • are presented in the upper part of figure 5. The difference of the two spectra (normalized to the surface of the peak between 175 and 225 keV) is shown in the lower part of the figure. No statistically significant difference can be seen between the spectra which is a clear evidence for emission from the composite system. Therefore, considering its energy and its width predicted by MCDF calculations, the peak at 190 keV must be associated to X K fluorescence from Z=120 atoms. The X K multiplicity M XK ∼ 0.1 measured for 70 Z 79 is very high. It is indeed of the same order than the K-vacancy creation probability that can be inferred [6] from the one measured in coincidences with elastically scattered projectiles, P elast K ∼ 0.27 (elastic scattering and fusion reactions correspond to similar atomic impact parameters). Therefore, M XK can only be taken into account considering in the reaction time distribution sizable proportions of long lifetime components with τ 10 -18 s. Assuming an exponential reaction time distribution, at least 50% of the capture reactions associated with this Z bin would correspond to fusion-fission reactions. Considering the isotopically enriched target used for the X-ray fluorescence experiment, these conclusions are in good agreement with the ones inferred from the blocking technique in single crystals.

Conclusions

Reaction time measurements give access to unique pieces of information on the reaction mechanisms involved between two very heavy ions at energies slightly above the fusion barrier. Mass-angle correlations highlight fast non-equilibrium reactions and suggest a vanishing of the mass-symmetric fission cross-section for the heaviest systems. In a complementary approach, long lifetime components observed for two very heavy systems testify to mass asymmetric fission following fusion. However, fusion has been evidenced for these systems in experiments in which the fission fragments were detected backward of the grazing angle, whereas most of the cross-section associated with capture reactions was located inside the grazing angle. Therefore, experiments allowing the measurement of long lifetime component probabilities over broad angular and mass ranges are now highly desirable in order to determine cross-sections for fusion as well as for non-equilibrium processes.

Figure 2 :

 2 Figure 2: Sum Z 1 +Z 2 of the atomic numbers of the two coincident fission-like fragments for 70 Z 1 80.

Figure 3 :

 3 Figure3: Dominant X K lines for different nuclear lifetimes for Z= 120 atoms in a charge state 1+ (yellow curves) and for a more realistic electronic structure distribution (blue curves, see text). The Doppler effect associated with the experimental set-up described in section 3.2 is taken into account for the yellow and blue curves.

Figure 4 :

 4 Figure 4: Energy spectrum of photons in coincidence with fission fragments (upper left panel); Spectrum with background subtraction (lower left panel) ; Spectrum with random coincidence correction(upper right panel); spectrum with random coincidences correction and background subtraction (lower right panel)

  the peak surface)

Figure 5 :

 5 Figure 5: Upper panel: Energy spectra, normalized to the area of the peak between 175 and 225 keV, of photons detected at φ = 30 • (green) and 270 • (blue). Lower panel: Difference of the two spectra.

Table 1 :

 1 Multiplicity of photons with energy between 175 and 225 keV in coincidence with a fission fragment with atomic number between Z min and Z max . Z min Z max Multiplicity

	35	49	0.064 ± 0.002
	50	65	0.052 ± 0.002
	70	79	0.102 ± 0.004
	80	90	0.067 ± 0.004


F R Éd

Éric Coquel
email: frederic.coquel@cmap.polytechnique.fr

Jean-Marc H Érard
email: jean-marc.herard@edf.fr

Khaled Saleh
email: khaled.saleh@irsn.fr

AND Nicolas Seguin
email: nicolas.seguin@upmc.fr

TWO PROPERTIES OF TWO-VELOCITY TWO-PRESSURE MODELS FOR TWO-PHASE FLOWS

Keywords: Two-phase flows, entropy, symmetrizable system AMS subject classifications. 76T05, 35L60, 35F55

We study a class of models of compressible two-phase flows. This class, which includes the Baer-Nunziato model, is based on the assumption that each phase is described by its own pressure, velocity and temperature and on the use of void fractions obtained from averaging process. These models are nonconservative and non-strictly hyperbolic. We prove that the mixture entropy is non-strictly convex and that the system admits a symmetric form.

Introduction

The modeling of compressible two-phase flows is a challenging task in Thermohydraulics. It is a crucial issue for many applications, for instance for water flows in some components of nuclear power plants such as the pressurized water reactors or the steam generators, especially in some specific situations, as the departure from nucleate boiling or the loss of coolant accident. When dealing with highly heterogeneous and disturbed flows, it is now commonly accepted that averaged models have to be considered. However, there is no consensus on the "good" model to use, especially when focusing on the two-fluid approach, where it is assumed that state variables within each phase (namely pressure, velocity and temperature) should not be confused. When restricting to the latter two-fluid framework, one can distinguish the Baer-Nunziato model [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF] among the huge literature on the modeling of two-phase flows, both from the mathematical and physical point (see for instance [START_REF] Bdzil | Two-phase flow modeling of a DDT in granular materials: acritical examination of modeling issues with micro-inertia[END_REF][START_REF] Embid | Mathematical analysis of a two-phase continuum mixture theory[END_REF][START_REF] Gavrilyuk | Mathematical and numerical modelling of two phase compressible flows with micro-inertia[END_REF][START_REF] Glimm | Two-phase flow modeling of a fluid mixing layer[END_REF][START_REF] Ransom | Hyperbolic two-pressure models for two-phase flows[END_REF] and references therein). Indeed, this system almost has the expected structure: if we only consider its convective part (i.e. the first order differential terms), its eigenvalues are always real and the associated eigenvectors are linearly independent except for some sonic cases (this is the resonance phenomenon). Even more, according to some choices of closure laws, it is possible to provide a wave-by-wave study, in spite of its nonconservative structure, and obtain some positive results on the solution of the associated Riemann problem [START_REF] Embid | Mathematical analysis of a two-phase continuum mixture theory[END_REF][START_REF] Coquel | Closure laws for a two-fluid two-pressure model[END_REF][START_REF] Gallouët | Numerical modeling of two-phase flows using the two-fluid two-pressure approach[END_REF].

In this work, we propose to investigate two properties which are crucial in the theory of nonlinear hyperbolic partial differential equations: the convexity of the entropy and the existence of a symmetric form. While such properties are very well understood for systems of conservation laws since Godunov [START_REF] Godunov | An interesting class of quasi-linear systems[END_REF] and Mock [START_REF] Mock | Systems of conservation laws of mixed type[END_REF], it remains an open question for nonconservative and non-strictly hyperbolic systems, such as the Baer-Nunziato model. Equipped with these properties, it is possible to pursue our study of the Baer-Nunziato models towards the Cauchy problem, which will be the subject of forthcoming works.

Actually we will discuss these two properties, not restricting to the exact Baer-Nunziato model, and we will consider a larger framework of two-phase flow models, introduced in [START_REF] Gallouët | Numerical modeling of two-phase flows using the two-fluid two-pressure approach[END_REF], that contains the former model [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF]. Note that such properties have been investigated in [START_REF] Forestier | Criterion of hyperbolicity for non-conservative quasilinear systems admitting a partially convex conservation law[END_REF] for a subclass of the systems we are interested in.

2. The two-velocity two-pressure models

In the models we study, each phase is described by its own density, velocity and pressure (or equivalently any other thermodynamical variable, such as the temperature). They are obtained after averaging Euler equations or Navier-Stokes equations and this averaging process introduces the notion of void fraction α(t, x) which represents the probability of presence (in the case of statistical average) of phase 1 at time t in position x. The resulting system is composed by a transport equation for the void fraction α with velocity v i , two equations of mass, momentum and total energy, and momentum and total energy equations include nonconservative terms due to the variations of the void fraction α. The main alternative class of models is based on the assumption of the equality of the two pressures, which replaces the transport equation on α. The main inconvenience is that these models are conditionally hyperbolic and their eigen-structure is very difficult to compute. As it is recalled in Proposition 2.1, the structure of the two-pressure models is more appealing, see for instance [START_REF] Gallouët | Numerical modeling of two-phase flows using the two-fluid two-pressure approach[END_REF], and our aim is to go further into the understanding of the structure of these models.

The first-order part of the models we focus on can be written under the form

∂ t u + ∂ x f (u) + c(u)∂ x α = 0 (2.1)
where

u T = (α, αu T 1 , (1 -α)u T 2 ), f (u) T = (0, αf 1 (u 1 ) T , (1 -α)f 2 (u 2 ) T ), u T k = (ρ k , ρ k u k , ρ k E k ), f k (u k ) T = (ρ k u k , ρ k u 2 k /2 + p k , u k (ρ k E k + p k )), c(u) T = (v i , 0, -p i , -v i p i , 0, p i , v i p i ),
with k = 1, 2. The notations are classical: α is the void fraction of phase 1 (and 1 -α the void fraction of phase 2), ρ k the density, u k the velocity, p k the pressure and E k the specific total energy of the phase k, with k = 1, 2. Besides, v i and p i , usually called the interfacial velocity and the interfacial pressure, are given functions of u. The total energies are defined by

E k = ε k + u 2 k 2 ,
where ε k denotes the specific internal energy of the phase k. We assume in the sequel that each phase admits an entropy s k , complying with

T k ds k = dε k + p k dτ k , (2.2)
noting T k the temperature and τ k = 1/ρ k the specific volume of the phase k. The knowledge of these entropies enables us to deduce the temperature and the pressure of each phase:

∂s k ∂τ k (τ k , ε k ) = p k T k (τ k , ε k ) and ∂s k ∂ε k (τ k , ε k ) = 1 T k (τ k , ε k ).
Besides, we assume that each entropy s k is a strictly concave function of τ k and ε k . Moreover, one can define the sound speeds c k by

ρ k (c k ) 2 = p k ρ k -ρ k (∂ ρ k ε k )(ρ k , p k ) (∂ p k ε k )(ρ k , p k ) -1 .
Let us recall the hyperbolicity property of system (2.1) for solutions in the set of admissible states

Ω = {u ∈ R 7 ; α ∈ (0, 1), ρ k > 0, ε k > 0, k = 1, 2}. (2.3)
Proposition 2.1. System (2.1) admits seven real eigenvalues on Ω:

v i , u k and u k ± c k , k = 1, 2.
The corresponding eigenvectors form a basis of R 7 as soon as

(u k -v i ) 2 = (c k ) 2 , k = 1, 2. (2.4)
This is called the non-resonance condition.

Non strict convexity of the mixture entropy

Let us introduce the entropy pair (S k , F k ) defined by

S k (u k ) = -ρ k s k and F k (u k ) = (S k (u k )) T f k (u k ), (3.1)
associated with the Euler systems

∂ t u k + ∂ x f k (u k ) = 0.
It is classical to define the entropy of the two-phase model (2.1) by Definition 3.1. The mixture entropy for system (2.1) is

S(u) = αS 1 (u 1 ) + (1 -α)S 2 (u 2 ), (3.2)
and the associated mixture entropy flux

F (u) = αF 1 (u 1 ) + (1 -α)F 2 (u 2 ). (3.3)
These definitions are classical and may lead to a conservative entropy inequality under some assumptions on p i , see [START_REF] Coquel | Closure laws for a two-fluid two-pressure model[END_REF][START_REF] Gallouët | Numerical modeling of two-phase flows using the two-fluid two-pressure approach[END_REF].

We state now the crucial property of the mixture entropy: Theorem 3.2. The mixture entropy S is a convex, but not strictly convex, function of u. Moreover, for any u = (α, αu 1 , (1 -α)u 2 ) ∈ Ω, the degeneracy manifold of S (u) is only due to variations with respect to the void fraction:

v ∈ D(u) = {v ∈ Ω; v = (β, βu 1 , (1 -β)u 2 )), β ∈ (0, 1) \ {α}} ⇐⇒ S(v) -S(u) = S (u) T (v -u).
Proof. With a slight abuse of notation, let us rewrite the mixture entropy S as a function of (α, αu 1 , (1 -α)u 2 ):

S(α, αu 1 , (1 -α)u 2 ) = αS 1 αu 1 α + (1 -α)S 2 (1 -α)u 2 1 -α .
Then, the Hessian matrix of S with respect to (α, αu 1 , (1 -α)u 2 ) has the form

S (u) =   A B T C T B 1 α S 1 (u 1 ) 0 C 0 1 1-α S 2 (u 2 )   with A = 1 α u T 1 S 1 (u 1 )u 1 + 1 1 -α u T 2 S 2 (u 2 )u 2 , B = - 1 α S 1 (u 1 )u 1 and C = 1 1 -α S 2 (u 2 )u 2 .
Let (a, b T , c T ) be a non-null vector of R 7 . Let us check that the Hessian S is positive as soon as S 1 and S 2 are positive. We have

(a, b T , c T ) S (u)   a b c   = a 2 A + aB T b + aC T c + ab T B + 1 α b T S 1 (u 1 )b + ac T C + 1 1 -α c T S 2 (u 2 )c.
Using the definitions of A, B and C we obtain

(a, b T , c T ) S (u)   a b c   = 1 α (b -au 1 ) T S 1 (u 1 )(b -au 1 ) + 1 1 -α (c + au 2 ) T S 2 (u 2 )(c + au 2 ).
This right-hand side is clearly non-negative since S 1 and S 2 are strictly convex, which proves the non strict convexity of S. The case of degeneracy of S (u) corresponds to

(a, b T , c T ) S (u)   a b c   = 0 ⇐⇒ (a, b, c) = a(1, u 1 , -u 2 ). (3.4)
In other words, from a given state u = (α, αu 1 , (1 -α)u 2 ) ∈ Ω, the degeneracy manifold of S (u) is given by the set {u + (a, au 1 , -au 2 ), a ∈ R \ {0}} ∩ Ω. Now, introducing w = (a, au 1 , -au 2 ), let us remark that u + w = (α + a, (α + a)u 1 , (1α -a)u 2 ) and setting β = α + a, we recover the set D(u).

4.

The system is symmetrizable out of resonance Definition 4.1. The system (2.1) is said to be symmetrizable if there exists a C 1 -diffeomorphism from R 7 to R 7 ϕ : u → y, a symmetric positive definite matrix P (y) ∈ R 7×7 and a symmetric matrix Q(y) ∈ R 7×7 such that the smooth solutions of system (2.1) satisfy Proof. Let us define y = ϕ(u) := (α 2 , u 2 , p 2 , s 2 , u 1 , p 1 , s 1 ) t and introduce the partial masses m k = α k ρ k . One may check by classical manipulations that the smooth solutions of system (2.1) satisfy

P (y)∂ t y + Q(y)∂ x y = 0. ( 4
∂ t y + M (y)∂ x y = 0 where M =   v i 0 0 M 2α M 2 0 M 1α 0 M 1   , M k =   u k τ k 0 ρ k (c k ) 2 u k 0 0 0 u k   and M T kα = (-1) k p k -p i m k , M (2)
kα , M

kα , M

kα = (-1) k u k -v i α k ρ k (c k ) 2 + p i -p k ρ k (∂ p k ε k ) -1 , M (2)
kα = (-1) k (u k -v i ) p i -p k m k T k . (3)
The proof we provide here is constructive for the sake of understanding. We seek for a matrix of symmetrization P (y) of the form

P =   P α P T 2α P T 1α P 2α P 2 0 P 1α 0 P 1   , P k =   (ρ k c k ) 2 0 0 0 1 0 0 0 1  
and the associated symmetric convection matrix is Q(y) = P (y)M (y). We have to find P such that it is positive definite and that Q is symmetric. Let us first focus on this latter condition. We have

Q =   P α v i + P T 2α M 2α + P T 1α M 1α P T 2α M 2 P T 1α M 1 v i P 2α + P 2 M 2α P 2 M 2 0 v i P 1α + P 1 M 1α 0 P 1 M 1   .
This matrix is symmetric if and only if we have for k = 1, 2

(M T k -v i I)P kα = P k M kα , (4.2)
where I is the 3 × 3 identity matrix. Assume now that

δ k := (u k -v i ) 2 -(c k ) 2 = 0,
i.e. inequality (2.4) holds. As a consequence, the first two equations of system (4.2) can be solved:

P (1) kα P (2) kα = 1 δ k u k -v i -ρ k (c k ) 2 -τ k u k -v i (-1) k ρ k (c k ) 2 p k -pi α k M (2) kα . (4.3)
It remains to solve the third equation of system (4.2), which writes

(u k -v i )P (3)
kα = (-1) k (u k -v i ) p i -p k m k T k . (4.4)
Clearly, this equation admits a unique solution if u k = v i and infinitely many solutions if u k = v i . Therefore, if P kα is defined by (4.3) and (4.4), the matrices P and Q are symmetric.

Let us now check that P is a positive definite matrix, that is to say for all non-null vector

(a, b T 2 , b T 1 ) of R 7 , we have (a, b T 2 , b T 1 ) P   a b 2 b 1   = a 2 P α + 2a(P T 2α b 2 + P T 1α b 1 ) + b T 2 P 2 b 2 + b T 1 P 1 b 1 > 0.
This is a polynomial of degree 2 in a and its discriminant is

∆ = 4 |P T 2α b 2 + P T 1α b 1 | 2 -P α (b T 2 P 2 b 2 + b T 1 P 1 b 1 ) = 4 (P -1/2 2 P 2α ) T b2 + (P -1/2 1 P 1α ) T b1 2 -P α (| b2 | 2 + | b1 | 2 ) = 4 |P -1/2 2 P 2α | 2 + |P -1/2 1 P 1α | 2 -P α (| b2 | 2 + | b1 | 2 ) -(P -1/2 2 P 2α ) T b1 -(P -1/2 1 P 1α ) T b2
2 where bk = P is its inverse). The discriminant ∆ is negative if

P α > |P -1/2 2 P 2α | 2 + |P -1/2 1 P 1α | 2 , (4.5)
which is realizable under the condition of non-resonance (2.4).

Consequences and further works

First of all, it is worth noting that Theorems 3.2 and 4.2 have been obtained for system (2.1) without any assumption on the definitions of v i and p i . Moreover, these results have been obtained for any admissible equations of state within each phase. It is then straightforward to extend these results to similar models (such as the barotropic Baer-Nunziato model, or the extended model introduced in [START_REF] Coquel | A class of two-fluid two-phase flow models[END_REF]) and it also seems to be a reasonable assumption for the model discussed in [START_REF] Gavrilyuk | Mathematical and numerical modelling of two phase compressible flows with micro-inertia[END_REF]. Moreover, we have restricted ourselves to the one-dimensional case, but since this model is invariant under frame rotation (under some natural conditions on v i and p i ), these properties are still verified in the multidimensional setting.

It is well-known for hyperbolic symmetric systems that the L ∞ norm of the spatial derivative of the solution may blow up in finite time. Here, the symmetric form (4.1) is admissible far from resonance. As a consequence, starting from a non-resonant solution, there exists a local-in-time smooth solution to the Cauchy problem. The final time corresponds to the smaller time for which the L ∞ norm of the spatial derivative of the solution blows up or for which the solution becomes resonant. This is a direct application of Kato's theorem [START_REF] Kato | The Cauchy problem for quasi-linear symmetric hyperbolic systems[END_REF]. Let us mention that the resonance phenomenon prevents us from proving a well-posedness result in a weaker setting (as entropy weak solutions with small total variation), since the Riemann problem is known to admit up to three solutions [START_REF] Isaacson | Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law[END_REF][START_REF] Goatin | The Riemann problem for a class of resonant hyperbolic systems of balance laws[END_REF].

Nonetheless, we must recall here that the full version of two-phase models also includes source terms which govern the trend of a flow to converge towards equilibrium: equality of pressures, velocities, temperatures and chemical potential. These respective equilibria correspond to mechanic, kinematic, thermal and thermodynamical equilibrium, and thus they tend to remove the occurrence of the resonance phenomenon, since condition (2.4) is expected to be satisfied when an equilibrium is not far from being reached. Such source terms are entropy-dissipative, that is to say that their contribution to the mixture entropy balance law is non-positive (see for instance [START_REF] Coquel | A class of two-fluid two-phase flow models[END_REF]). As a consequence, one may wonder if they may help to obtain a global-in-time solution to the Cauchy problem, following [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF] and [START_REF] Yong | Entropy and global existence for hyperbolic balance laws[END_REF]. Even if these two papers are dedicated to systems of conservation laws, the analysis relies on the use of the entropy (and equivalently in the conservation case, upon the symmetric form of the equations). Let us note that the crucial assumptions exhibited in [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF] and [START_REF] Yong | Entropy and global existence for hyperbolic balance laws[END_REF] are the following: entropy dissipative source terms and Kawashima-Shizuta structure.

It is well-known that the aforementioned source terms are entropy dissipative with respect to entropy (3.1). As far as the Kawashima-Shizuta structure is concerned, the situation is not so clear and it is probable that all the relaxation source terms will be needed. Moreover, the degeneracy of the Hessian of the entropy has to be carefully handled to obtain the global-in-time result. This work is under investigation.

Eventually, these properties may also be used for computational purposes.

Acknowledgments. The second author received partial support from the NEP-TUNE project, which benefits from financial support of CEA, EDF, AREVA-NP and IRSN. The last author is partially supported by the LRC Manon (Modélisation et Approximation Numérique Orientées pour l'énergie Nucléaire -CEA DM2S/LJLL).