Matthieu Hillairet
email: matthieu.hillairet@umontpellier.fr

H Él

AND Ène Mathis
email: helene.mathis@univ-nantes.fr

Nicolas Seguin
email: nicolas.seguin@univ-rennes1.fr

ANALYSIS OF COMPRESSIBLE BUBBLY FLOWS. PART II : DERIVATION OF A MACROSCOPIC MODEL

Keywords: Homogenization, two-phase flows, compressible Navier-Stokes equations, Cauchy theory 2020 MCS. 76T05, 76T10, 35Q30

This paper is the second of the series of two papers, which focuses on the derivation of an averaged 1D model for compressible bubbly flows. For this, we start from a microscopic description of the interactions between a large but finite number of small bubbles with a surrounding compressible fluid. This microscopic model has been derived and analysed in the first paper. In the present one, provided physical parameters scale according to the number of bubbles, we prove that solutions to the microscopic model exist on a timespan independent of the number of bubbles. Considering then that we have a large number of bubbles, we propose a construction of the macroscopic variables and derive the averaged system satisfied by these quantities. Our method is based on a compactness approach in a strong-solution setting. In the last section, we propose the derivation of the Williams-Boltzmann equation corresponding to our setting.

) of (Q 1 ) Strict version (Q 2 ) of (Q 2 ) Strict version (Q 3 ) of (Q 3 ) Strict version (Q 4 ) of (Q 4 ) Strict version (Q 5 ) of (Q 5 ) Appendix B.

Analysis of the density equation References 1. Introduction

The present work represents a straight continuation of a series of articles which proposes to justify the construction of multiphase flow models. The structure of multiphase flow models can be derived formally by applying standard conservation principles [START_REF] Drew | Theory of multicomponent fluids[END_REF][START_REF] Embid | Mathematical analysis of a two-phase continuum mixture theory[END_REF][START_REF] Gavrilyuk | The structure of pressure relaxation terms: the one-velocity case[END_REF][START_REF] Ishii | Thermo-fluid dynamics of two-phase flow[END_REF]. However this procedure leaves aside key-terms that have to be related to mechanical/thermodynamical unknowns via state laws. To this end, a sharp description of the interactions between phases is required. Classical methods are based on averaging operators whose range of validity is still to be investigated. Furthermore, the action of these averaging operators on nonlinear quantities requires further modelling assumptions. From the analytical standpoint, the computations we provide herein follow previous analysis of the first author notably in collaboration with D. Bresch [START_REF] Bresch | Multi-fluid models including compressible fluids[END_REF][START_REF] Bresch | Note on the derivation of multi-component flow systems[END_REF][START_REF] Bresch | A compressible multifluid system with new physical relaxation terms[END_REF][START_REF] Hillairet | On Baer-Nunziato multiphase flow models[END_REF] complementing previous approaches in [START_REF] Amosov | On the quasi-averaging of a system of equations of the one-dimensional motion of a viscous heat-conducting gas with rapidly oscillating data[END_REF][START_REF] Hillairet | Propagation of density-oscillations in solutions to the barotropic compressible Navier-Stokes system[END_REF][START_REF] Plotnikov | Compressible Navier-Stokes equations[END_REF]. In these references, one-velocity Baer-Nunziato-like models are derived for multiphase fluids. These computations are based on the remark that, if the interfaces act as a "perfect" transducer (no mass transfer, perfect transfer of mechanical stress), combining the different phases equations yields a global one-fluid equation. Deriving multiphase flow models then reduces to a thorough analysis of highly-oscillatory solutions to the one-fluid equation. A particular analytical framework of mixed-regularity (smooth velocity with discontinuous densities [START_REF] Desjardins | Regularity of weak solutions of the compressible isentropic Navier-Stokes equations[END_REF][START_REF] Hoff | Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data[END_REF][START_REF] Serre | Variations de grande amplitude pour la densité d'un fluide visqueux compressible[END_REF]) is identified in [START_REF] Bresch | A multi-fluid compressible system as the limit of weak solutions of the isentropic compressible Navier-Stokes equations[END_REF] to make this approach fully rigorous. However, this approach is restricted to an ideal case (see [START_REF] Bresch | Physical relaxation terms for compressible two-phase systems[END_REF] for further investigations in this context). The aim of this paper is to tackle the derivation of averaged models in presence of jumps at interfaces. Starting from an original microscopic model (that is derived in the first paper [START_REF] Hillairet | Analysis of compressible bubbly flows. Part I: Construction of a microscopic model[END_REF]) in which the two phases are fully separated, we derive a 1D averaged compressible bubbly-flow model by performing space averaging operators.

The averaged model reads as follows. It is set on the container Ω = (-1, 1) filled with a gas/fluid mixture. The averaged variables are the void fractions ᾱf,g ∈ [0, 1], the mean densities ρf,g ∈ [0, ∞), a bubble phase covolume 1 fg ∈ [0, ∞) and the mixture velocity ū ∈ R. It reads:

(1)

              
∂ t (ᾱ g fg ) + ∂ x (ᾱ g fg ū) = 0, ∂ t (ᾱ f ρf ) + ∂ x ( ᾱf ρf ū) = 0, ∂ t (ᾱ g ρg ) + ∂ x (ᾱ g ρg ū) = 0,

∂ t ᾱf + ū∂ x ᾱf = RT , ∂ t (ρū) + ∂ x (ρū 2 ) = ∂ x Σ, on (0, T ) × Ω,
with the compatibility conditions:

(2) ᾱf + ᾱg = 1, ρ = ᾱf ρf + ᾱg ρg , and where the mixture stress tensor writes

(3) Σ = µ g µ f ᾱf µ g + ᾱg µ f ∂ x ū - ᾱf µ f p f (ρ f ) + ᾱg µ g p g (ρ g ) -γs ᾱg µ g fg ,
while the void fraction relaxation term reads:

(4) RT = ᾱg ᾱf ᾱf µ g + ᾱg µ f (µ g -µ f )∂ x ū + (p f (ρ f ) -p g (ρ g )) -γs fg .
In these latter identities appear the constants µ f , µ g > 0 (resp. the functions p f , p g ) representing the fluid and bubble viscosities (resp. the fluid and gas pressure laws). The constant γs > 0 represents the surface tension.

The system (1)-( 2) complemented with the state laws (3)-( 4) is obtained starting from the following microscopic model, where the two phases are disjoint and their interactions only appear through the interfaces. Again, the two-phase flow is posed in the one-dimensional domain Ω = (-1, 1), filled by a liquid (the fluid, or the continuous phase, indexed by f ) and bubbles (the gas, 1 The denomination covolume may be misleading here. In classical thermodynamic, the term covolume refers to the specific volume. Here the quantity fg is linked to the volume of the gaseous phase. In 3D configurations, it would be related to the interfacial area.

or the dispersed phase, indexed by g). The N bubbles are described by their centers c k and their radii R k , so that the k-th bubble is

B k = (x - k , x + k ), x ± k = c k ± R k , ∀ k = 1, .
. . , N. The fluid domain is

F = Ω \ N k=1 B k .
For later use, we also introduce the fluid intervals [START_REF] Bresch | A compressible multifluid system with new physical relaxation terms[END_REF] F k = (x + k , x - k+1 ) for k = 0, . . . , N setting x + 0 = -1 and x - N +1 = 1. The fluid is supposed to be compressible and viscous, so that it is governed by the 1D compressible Navier-Stokes system, posed in F: [START_REF] Drew | Theory of multicomponent fluids[END_REF] where ρ f is the density, u f the velocity and Σ f the stress tensor of the fluid. Moreover, µ f > 0 is the shear viscosity and p f is an isentropic pressure law for the fluid:

∂ t ρ f + ∂ x (ρ f u f ) = 0, ( 6
)
∂ t (ρ f u f ) + ∂ x (ρ f u 2 f ) = ∂ x Σ f , (7)
Σ f = µ f ∂ x u f -p f (ρ f ),
p f (ρ f ) = κ f ρ γ f f ,
where κ f > 0 and γ f > 1 stands for the adiabatic exponent. We assume that the fluid is present at the boundary of the domain Ω, where no-slip boundary conditions are imposed: [START_REF] Embid | Mathematical analysis of a two-phase continuum mixture theory[END_REF] u f (t, ±1) = 0.

Equations for bubble kinematics and dynamics are proposed in [START_REF] Hillairet | Analysis of compressible bubbly flows. Part I: Construction of a microscopic model[END_REF]. Therein, the derivation is based on the assumption that the bubbles are made of a compressible viscous fluid with an infinite shear viscosity (compared to the volumic viscosity) and that their spherical shapes are preserved (in three dimensions). This yields first that the continuity of the velocity at the interfaces reads: [START_REF] Feireisl | Homogenization and singular limits for the complete Navier-Stokes-Fourier system[END_REF] u f (t, x ± k (t)) = ċk (t) ± Ṙk (t) for k = 1, . . . , N. In addition, imposing that the jump of the stress tensor at the interfaces is due to the surface tension, one obtains the following system for the dynamics of a bubble:

m k ck (t) = Σ f (t, x + k ) -Σ f (t, x - k ), (11) m k 3 Rk (t) = Σ f (t, x - k ) + Σ f (t, x + k ) -2Σ k (t), (12)
Σ k = µ g Ṙk R k -p g (ρ k ) - F s 2 , ( 13
)
where µ g > 0 is the volumic viscosity of the gas, and m k and ρ k are the mass and the density of the bubble, linked by m k = 2R k ρ k . As a consequence of mass conservation in bubbles, the masses m k do not depend on time. The term F s denotes the force due to the surface tension and writes F s = γ s /R k , γ s being the surface tension. In order to simplify the analysis, we assume an isothermal equation of state in the bubbles, so that [START_REF] Hillairet | Analysis of compressible bubbly flows. Part I: Construction of a microscopic model[END_REF] π k := p g (ρ k ) +

F s 2 = (a g ) 2 m k + γ s /2 R k = κ k R k ,
where a g > 0 is the sound speed of the gas. The last form of π k will be used mainly for the analysis of the model, while the first form will be useful to interpret the various terms appearing in equations, notably those due to surface tension. In particular, computing surface tension effects in the microscopic system involves the quantity 1/(2R k ) that corresponds to the covolume of bubble B k in our 1D setting. We point out that the system (6)-( 13) is not integrable and, specifically, does not yield any particular value for the fluid velocity-field u f . We are then not in the Rayleigh-Plesset regime where the bubble equations ( 11)-( 12) reduce to ordinary differential equations in terms of (c k , R k ) and an asymptotic pressure [START_REF] Wang | Effective equations for sound and void wave propagation in bubbly fluids[END_REF]. We refer the reader to the companion paper [START_REF] Hillairet | Analysis of compressible bubbly flows. Part I: Construction of a microscopic model[END_REF] for more details on the derivation of ( 6)-( 13) and the analysis of the associated Cauchy problem. Yet, we shall explain in further details the construction of solutions in the next section.

The main result of this paper is to show that, starting from solutions to ( 6)-( 13) we obtain (1)-( 4) by letting the number N of bubbles go to infinity in case: [START_REF] Hoff | Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data[END_REF]

m k ∼ N -1 , R k ∼ N -1 , |F k | ∼ N -1 γ s ∼ N -1 ,
with the other parameters being fixed. One key-difficulty in the proof is that the target system (1)-( 4) is highly nonlinear. Specifically, products between volume fractions and other (fluid or gas) unknowns are ubiquitous. To obtain such nonlinear terms, it appears that strong convergences of densities or gas covolume in sufficiently smooth spaces are necessary. Hence, with this approach, we face two key-difficulties:

• to prove that the scaling regime [START_REF] Hoff | Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data[END_REF] holds on a timespan independent of the number N of bubbles, • to define the macroscopic unknowns and especially, the fluid and gas densities ρf , ρg and the gas covolume fg . The first item in this list is the content of the topic of the next section. Therein, we consider initial data that are constructed as follows. Firstly, we fix fluid initial data (ρ 0 f , u 0 f ) ∈ H 1 (Ω) × H 1 0 (Ω) that are thus defined globally on Ω. We assume further that they are far from vacuum. Secondly, we fix initial distributions of centers/radii (c 0 k , R 0 k ) k=1,...,N such that (15) holds. We complement then the microscopic system ( 6)-( 13) with initial conditions so that the initial bubble velocities match the velocities prescribed by the fluid on the boundaries. This reads:

c k (0) = c 0 k R k (0) = R 0 k , for k = 1, . . . , N , (16)
u(0, •) = u 0 f ρ(0, •) = ρ 0 f , on F 0 , (17)
and

ċ0 k = u 0 f (c 0 k + R 0 k ) + u 0 f (c 0 k -R 0 k ) 2 , for k = 1, . . . , N, (18)
Ṙ0 k = u 0 f (c 0 k + R 0 k ) -u 0 f (c 0 k -R 0 k ) 2 , for k = 1, . . . , N. (19)
The main result of Section 2 is then that there exists a classical solution to ( 6)-( 13) on a timespan that depends only on fluid initial data and the parameters quantifying initially assumption [START_REF] Hoff | Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data[END_REF]. To obtain this result, we combine classical energy and regularity estimates for Navier Stokes equations. We remind that, in this strategy, one classically uses extra regularity thanks to the form of the stress tensor Σ f . However, such regularity estimates should depend on the geometry (and then on N ). To overcome this difficulty, we propose to consider suitable extensions of Σ f (resp. Σ g ) on the complementary gas (resp. fluid) domain. In this way, the extension is defined on a fixed domain and the regularity gain is independent of the geometry. We point out here that contrary to the classical approach in the topic of homogenization of multidimensional compressible Navier Stokes equations in perforated domains [START_REF] Lu | Homogenization of the compressible Navier-Stokes equations in domains with very tiny holes[END_REF][START_REF] Feireisl | Homogenization and singular limits for the complete Navier-Stokes-Fourier system[END_REF], our construction takes advantage of the information on the moments on ∂B k of the fluid stress tensor that are provided by the bubble equations.

The second key-difficulty of our approach is tackled in Section 3. Once solutions to (6)-( 13) are constructed on a time-interval that does not depend on N, we consider the behavior of these solutions for large N . In particular, we look for definitions of the unknowns that are involved in the macroscopic system (1)- [START_REF] Bresch | Note on the derivation of multi-component flow systems[END_REF]. Volumic fractions as well as global velocity-fields are obtained classically by considering indicator functions or suitably extended vector-fields (see Proposition 13 and Proposition 15). However, the issue is more involved when going to density and covolume unknowns. Indeed, at the discrete level, fluid density and bubble density, for instance, are defined a priori on dispersed subdomains only. This cannot yield convergence with sufficient regularity. To get better convergence results, we decide to construct suitable extensions. For this we proceed in two steps. Firstly, we ensure that the initial conditions for ( 6)-( 13) enable to define smooth extended densities and covolume (see Proposition 12). Then, we propagate this regularity with a well-chosen extended flow (see Proposition 14 and Proposition 17). With this construction at-hand, the derivation of ( 1)-( 4) is plain sailing.

In our construction, we start from initial data for the macroscopic system and define a sequence of initial conditions for the microscopic system that are compatible with the scaling [START_REF] Hoff | Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data[END_REF] and enable to construct extended densities. It turns out that this requires further assumption on initial data that we explain now. We recall that initial data for the macroscopic system consists in:

• initial fluid and gas densities : ρ0 f , ρ0 g , • initial fluid and gas void fractions ᾱ0 f , ᾱ0 g

• an initial velocity of the two-phase mixture ū0 ,

• an initial gas covolume f 0 g . It is worth noting that all these functions are defined for x in Ω, since both phases are no longer separated at the macroscopic scale. We shall remain at the regularity level of classical solution and require that all these initial conditions are H 1 (Ω). For our construction, we require that initial densities and volumic fraction satisfy:

ρ min ≤ min(ρ 0 f , ρ0 g ) (20) α min ≤ min(ᾱ 0 f , ᾱ0 g ) ᾱ0 g + ᾱ0 f = 1 (21)
for some strictly positive constants ρ min , α min . The first condition means that we are away from void. The second one expresses that there is a mixture of both phases everywhere in Ω. Note that the second conditions implie simultaneously that

(22) max( ᾱ0 g L ∞ (Ω) , ᾱ0 f L ∞ (Ω) ) ≤ 1 -α min .
Concerning, f 0 g , we will require that:

f min ≤ f 0 g , ᾱ0 g f 0 g ∈ P(Ω), (23)
where f min is a strictly positive constant. To explain these latter conditions, we point out that in the 1D case the covolume of bubbles is proportional to the inverse radius. So f min is a bound from above on the initial radius of bubbles and, since we expect ᾱ0 g f 0 g to be the limit of the indicator function of bubble domains multiplied by the inverse radius of bubbles, a straightforward computations yields that it is a positive function whose total mass is 1, hence a probability density.

Though we propose a converse interpretation to the classical one, the multiphase system we consider in this paper enters the family of sprays as studied by Williams in [21, Section 11]. With this standpoint, a classical tool to analyze the behavior of the dispersed phase is the so-called "Williams-Boltzmann" equation which describes the time-evolution of the particle-distribution function of the dispersed phase. In the last section of this paper, we derive what would be the equivalent equation in our setting. It is worth to mention that this is no supplementary equation but simply a rephrasing of the bubble-gas equation that we derived previously. In particular, herein the bubble-gas velocities are correlated to their position and drag forces are at equilibrium. We do neither have collision or creation of bubbles. Hence, the only term to be taken into account is the "evaporation" term which should be understood as compression/expansion term herein in our compressible setting.

In brief, the outline of the paper is as follows. In the next section, we prove that solutions to the microscopic system (6)- [START_REF] Hillairet | On Baer-Nunziato multiphase flow models[END_REF] with well-prepared initial data do exist on a timespan independent of N , see Theorem 1. In Section 3 and Section 4, we tackle the asymptotics of these solutions when N → ∞. In the last section, we discuss an alternative approach based on using particledistribution functions for the bubbles. In appendices, we provide some technical computations involved in the construction of solutions to the microscopic model.

Local Cauchy theory for the microscopic system

In this section, we forget temporarily our homogenization goal. We focus on the microscopic model ( 6)- [START_REF] Hillairet | On Baer-Nunziato multiphase flow models[END_REF] in the scaling [START_REF] Hoff | Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data[END_REF] and we address the existence of solutions with lifespan independent of the number N of bubbles provided initial data are constructed as in ( 16)- [START_REF] Serre | Variations de grande amplitude pour la densité d'un fluide visqueux compressible[END_REF]. In particular, we fix (ρ 0 f , u 0 f ) ∈ H 1 (Ω) × H 1 0 (Ω) throughout the section. We assume these global fluid data satisfy:

(24) 2ρ ∞ ≤ ρ 0 f ≤ ρ ∞ /2 on Ω for some pair (ρ ∞ , ρ ∞ ) ∈ (0, ∞) 2 .
To make precise our main result, we start by giving a quantified version of assumption [START_REF] Hoff | Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data[END_REF] that we assume to hold initially. Firstly, we fix that bubbles characteristics enjoy the property:

(IC 0 ) M ∞ ≤ N m k , N κ k ≤ (M ∞ ) -1 , k = 1, . . . , N , (IC 1 ) 2d ∞ ≤ N R 0 k ≤ (2d ∞ ) -1 , k = 1, . . . , N , (IC 2 ) 2d ∞ ≤ N |F 0 k | ≤ (2d ∞ ) -1
, k = 0, . . . , N , Here M ∞ , d ∞ are strictly positive constants independent of N. We recall the convention (5) for the definition of F 0 k (adapted to notations for initial data). Their union constitutes the initial fluid domain F 0 . The physical parameters (µ f , µ g ) and pressure laws are fixed independent of N . With these conventions, the main result of this section reads: Theorem 1. Let initial conditions to (6)-( 13) be constructed as in ( 16)- [START_REF] Serre | Variations de grande amplitude pour la densité d'un fluide visqueux compressible[END_REF]. Assume further that parameters (m k , κ k ) k=1,...,N and initial bubble distributions (c 0 k , R 0 k ) k∈{1,...,N } satisfy (IC 0 )-(IC 2 ). Then, there exists T ∞ > 0 depending only on

(25) M ∞ , d ∞ , ρ ∞ , ρ∞ , u 0 f H 1 (Ω) , ρ 0 f H 1 (Ω)
, such that there exists a solution to (6)-( 13) on (0, T ∞ ). What remains of this section is devoted to the proof of this theorem. From now on, we pick a family of physical parameters and bubble centers/radii satisfying the assumptions of Theorem 1 and we construct initial data for (6)- [START_REF] Hillairet | On Baer-Nunziato multiphase flow models[END_REF].

In the companion paper [START_REF] Hillairet | Analysis of compressible bubbly flows. Part I: Construction of a microscopic model[END_REF], we prove local-in-time existence and uniqueness of classical solutions to the Cauchy problem associated with (6)- [START_REF] Hillairet | On Baer-Nunziato multiphase flow models[END_REF]. In this moving-domain setting, classical solution means broadly that:

• the motion of the bubbles is

H 2 (0, T ) (i.e. (c k , R k ) ∈ H 2 (0, T )), • u is H 1 t L 2 x and L 2 t H 2 x in the fluid domain, • ρ is H 1 t,
x in the fluid domain. Existence and uniqueness of solutions on a lifespan (0, T 0 ) is obtained for initial data such that

• there is no overlap of the bubbles,

• initial fluid data are H 1 in the fluid domain with strictly positive density,

• initial fluid and bubble velocities match at interfaces (so that ( 18)-( 19) hold true). It is also worth noting that the time T 0 is uniform in data satisfying uniform bounds from below for the distance between bubbles, the minimal radius of bubbles, the minimum density and also the size of initial fluid velocity and density in H 1 -spaces. We refer to [START_REF] Hillairet | Analysis of compressible bubbly flows. Part I: Construction of a microscopic model[END_REF] for more precise and quantitative statements. So, under the assumptions of Theorem 1, the local-in-time existence result of [START_REF] Hillairet | Analysis of compressible bubbly flows. Part I: Construction of a microscopic model[END_REF] yields a solution on a time-interval (0, T 0 ) that depends on the list of parameters (25) but also on N. To rule out this dependency, we construct T ∞ such that as long as t < T ∞ the solution (ρ f (t, •), u f (t, •), (c k (t), R k (t), ċk (t), Ṙk (t)) k∈{1,...,N } ) yields an initial condition that is compatible with the Cauchy theory of [START_REF] Hillairet | Analysis of compressible bubbly flows. Part I: Construction of a microscopic model[END_REF] with an associated existence time independent of t. We emphasize that any classical solution does not allow overlap of the bubbles and ensures identity ( 18)-( 19) is satisfied at any time. Controlling the existence time associated with the value of the solution at time t -considered as an initial data -reduces to obtaining uniform H 1 bound for the velocity field and for the density, uniform bound from above and from below on the fluid density, the radius of the bubbles and the length of fluid segments.

Our approach relies on a suitable combination of energy and regularity estimates for the coupled system (6)- [START_REF] Hillairet | On Baer-Nunziato multiphase flow models[END_REF]. So, we recall in the next sections the classical estimates that are associated with (6)- [START_REF] Hillairet | On Baer-Nunziato multiphase flow models[END_REF]. We will pay special attention to obtain estimates independent on N. This will be particularly challenging for regularity estimates. In particular, we shall study the regularity of fluid velocity-fields that can be gained through the integrability of the stress tensor by working on extensions of fluid unknowns on bubble domains and conversely. A tricky part of the proof is that we can obtain these sharp bounds under the condition that we have already a priori bounds. So, we implement a continuation argument. This continuation argument is explained in the last part of the section. However, the extensive proof is rather long and technical. Hence, the last subsection reduces to a roadmap of the proof that is detailed further in Appendix A.

Classical estimates. We introduce the function q

f : [0, ∞) → [0, ∞) defined by (26) q f (s)s -q f (s) = p f (s),
which is conjugate of the fluid pressure. In other words, the function q represents the volumic internal energy of the fluid. Considering an isentropic pressure law, it yields

q f (s) = a f s γ f γ f -1 .
We can now state the total energy equation. In the bracket of the statement below, the first term is the total energy of the fluid, while the second and the third terms respectively are the kinetic energy and the internal energy of the bubbles.

Proposition 2. For any reference radius

R ref > 0, it holds (27) d dt F ρ f |u f | 2 2 + q f (ρ f ) dx + N k=1 m k | ċk | 2 2 + | Ṙk | 2 6 -2 N k=1 κ k ln R k R ref + F µ f |∂ x u f | 2 dx + 2µ g N k=1 | Ṙk | 2 |R k | = 0.
Proof. First let multiply the Navier-Stokes equation ( 7) by the velocity u f and integrate over the fluid domain F. Using the mass conservation equation ( 6), it yields (28)

F ρ f (∂ t u f + u f ∂ x u f ) u f dx = F u f ∂ x Σ f dx.
Since the mass conservation (6) gives d dt

x - k+1 x + k ρ f |u f | 2 2 dx = x - k+1 x + k ρ f (∂ t u f + u f ∂ x u f )u f dx,
one obtains, using an integration by part of the right-hand side,

d dt F ρ f |u f | 2 2 dx = T 1 -T 2 -T 3 ,
with

T 1 = N k=0 Σ f (x - k+1 )u f (x - k+1 ) -Σ f (x + k )u f (x + k ), T 2 = F µ f |∂ x u f | 2 dx, T 3 = - F p f (ρ f )∂ x u f dx,
where the terms T 2 and T 3 come from the definition (8) of the stress Σ f .

Using the boundary conditions ( 9) and, after, the continuity of the velocities at the droplet interfaces [START_REF] Feireisl | Homogenization and singular limits for the complete Navier-Stokes-Fourier system[END_REF], the term T 1 can be rewritten as

T 1 = - N k=1 Σ f (x + k )u f (x + k ) -Σ f (x - k )u f (x - k ) = - N k=1 ċk Σ f (x + k ) -Σ f (x - k ) + Ṙk Σ f (x + k ) + Σ f (x - k ) .
Finally the droplets motion equations ( 11)-( 12) and the definition of the droplet pressure law ( 14) yield (whatever the value of R ref > 0):

T 1 = - d dt N k=1 m k | ċk | 2 2 + m k 3 | Ṙk | 2 2 -2 N k=1 κ k ln R k R ref -2µ g N k=1 | Ṙk | 2 R k .
We now turn to the term T 3 . By the definition (26) of the function q f , and by the mass conservation equation ( 6), it holds

∂ t q f (ρ f ) + ∂ x (q f (ρ f )u f ) = -p f (ρ f )∂ x u f .
Because the fluid domain evolves with the velocity u f , T 3 can be recovered

d dt F q(ρ f )dx = - F p f (ρ f )∂ x u f dx = T 3 .
One deduces the final estimate (27) combining the terms T 1 , T 2 and T 3 .

In the regime of initial data specified in this section, we obtain the following corollary:

Corollary 3. If initial data are constructed as in ( 16)-( 19) and satisfy (IC 0 )-(IC 1 )-(IC 2 ), there exists a constant E 0 depending only on the list of parameters (25) such that any classical solution to (6)-( 13) on some time-interval [0, T ] satisfies:

(29) F ρ f |u f | 2 2 + q(ρ f ) dx + 1 2 N k=1 m k | ċk | 2 + 1 3 | Ṙk | 2 -2 N k=1 κ k ln(d ∞ N R k ) ≤ E 0 ,
on (0, T ) with, denoting by ln + the positive part of the ln:

(30)

T 0 F µ f |∂ x u f | 2 dx + µ g N k=1 | Ṙk | 2 R k dt ≤ E 0 + 2 max [0,T ] N k=1 κ k ln + (d ∞ N R k ).
Proof. To obtain these inequalities, we integrate (27) with R ref = 1/d ∞ N and remark that all the terms on the left-hand side are positive but:

N k=1 κ k ln(d ∞ N R k ).
We obtain then the inequalities ( 29) and (30) with:

E 0 := F 0 ρ 0 f |u 0 f | 2 2 + q f (ρ 0 f ) dx + N k=1 | ċ0 k | 2 2 + | Ṙ0 k | 2 6 -2 N k=1 κ k ln(R 0 k d ∞ N ).
The first term in E 0 is clearly controlled by u 0 f L 2 and ρ∞ . As for the second term, the velocity continuity ( 18)- [START_REF] Serre | Variations de grande amplitude pour la densité d'un fluide visqueux compressible[END_REF] gives

| ċ0 k | + | Ṙ0 k | ≤ 2 u 0 f L ∞ (Ω) , ∀k = 1, . . . , N.
Then, with (IC 0 ), we obtain:

N k=1 m k | ċ0 k | 2 + 1 3 | Ṙ0 k | 2 ≤ 4 M ∞ u 0 f 2 L ∞ (Ω) ,
and, with a classical Sobolev embedding, this part is again controlled by M ∞ and u 0 f H 1 0 (Ω) . Now using the bound (IC 1 ) on the initial radii, it holds

2d 2 ∞ ≤ R 0 k N d ∞ ≤ 1 2 , so that - N k=1 κ k ln(d ∞ N R 0 k ) ≤ | ln(2d 2 ∞ )| M ∞ .
This concludes the proof.

We proceed with a second classical regularity estimate:

Proposition 4.
The following identity holds

(31) d dt F µ f |∂ x u f | 2 2 -p f (ρ f )∂ x u f dx + N k=1 µ g | Ṙk | 2 R k -2κ k Ṙk R k + F ρ f |∂ t u f + u f ∂ x u f | 2 dx + N k=1 m k |c k | 2 + 1 3 | Rk | 2 = F p f (ρ f )ρ f |∂ x u f | 2 -µ f (∂ x u f ) 3 2 dx + N k=1 2κ k | Ṙk | 2 R 2 k -µ g | Ṙk | 3 R 2 k .
Proof. Multiplying the momentum equation ( 7) by ∂ t u f + u f ∂ x u f and integrating over the fluid domain F yield (32)

F ρ f |∂ t u f + u f ∂ x u f | 2 dx = F (∂ t u f + u f ∂ x u f )∂ x Σ f dx = T 4 -T 5 ,
with

T 4 = N k=0 Σ f (x - k+1 )(∂ t u f + u f ∂ x u f )(x - k+1 ) -Σ f (x + k )(∂ t u f + u f ∂ x u f )(x + k ), T 5 = F Σ f ∂ x (∂ t u f + u f ∂ x u f )dx.
The boundary term T 4 can be simplified by using the interface conditions [START_REF] Feireisl | Homogenization and singular limits for the complete Navier-Stokes-Fourier system[END_REF],

d dt ( ċk ± Ṙk ) = d dt u f (x ± k ) = (∂ t u f + u f ∂ x u f ) (x ± k ).
Then one obtains

T 4 = N k=0 Σ f (x - k+1 )(c k+1 -Rk+1 ) -Σ f (x + k )(c k + Rk ).
The boundary conditions (9) allow to reorganize the sum, and using the droplet equations of motion ( 11)-( 12) and the droplet pressure law ( 14), we have successively

T 4 = - N k=1 Σ f (x + k )(c k + Rk ) -Σ f (x - k )(c k -Rk ) = - N k=1 ck (Σ f (x + k ) -Σ f (x - k )) + Rk (Σ f (x + k ) + Σ f (x - k )) = - N k=1 m k |c k | 2 + 1 3 | Rk | 2 + 2 Rk µ g Ṙk R k - κ k R k = - N k=1 m k |c k | 2 + 1 3 | Rk | 2 + d dt µ g | Ṙk | 2 R k -2κ k Ṙk R k + N k=1 2κ k | Ṙk | 2 R 2 k -µ g ( Ṙk ) 3 R 2 k .
We now turn to the volumic term T 5 . Developing the term T 5 gives (33)

T 5 = T 6 + F µ f (∂ x u f ) 3 dx -T 7 - F p f (ρ f )|∂ x u f | 2 dx,
with

T 6 = F µ f ∂ x u f (∂ t (∂ x u f ) + u f ∂ x (∂ x u f )) dx, T 7 = F p f (ρ f ) (∂ t (∂ x u f ) + u f ∂ x (∂ x u f )) dx.
These two terms can be handled by classical manipulations, providing

d dt F µ f |∂ x u f | 2 2 dx = T 6 + F µ f (∂ x u f ) 3 2 , d dt F p f (ρ f )∂ x u f dx = T 7 - F (p f (ρ f ) -p f (ρ f )ρ f )|∂ x u f | 2 dx.
As a result,

T 5 = d dt F µ f |∂ x u f | 2 2 -p f (ρ f )∂ x u f dx + µ f F (∂ x u f ) 3 2 dx + F p (ρ f )ρ f |∂ x u f | 2 dx.
Finally plugging the expressions of T 4 and T 5 into (32) gives the expected result.

In the regime of initial data specified in this section, we obtain the following corollary:

Corollary 5. If initial data are constructed as in ( 16)-( 19) and satisfy (IC 0 )-(IC 1 )-(IC 2 ), there exists a constant E 1 depending only on the list of parameters (25) such that any classical solution to (6)-( 13) on some time-interval [0, T ] satisfies:

(34) sup [0,T ] F µ f |∂ x u f | 2 2 dx + µ g N k=1 | Ṙk | 2 R k + T 0 F ρ f |∂ t u f + u f ∂ x u f | 2 dx + N k=1 m k (|c k | 2 + | Rk | 2 ) ≤ sup [0,T ] 2 N k=1 κ k | Ṙk | R k + F p f (ρ f )|∂ x u f |dx + T 0 F µ f |∂ x u f | 3 2 dx + T 0 N k=1 2κ k | Ṙk | 2 R 2 k + µ g | Ṙk | 3 R 2 k + E 1 .
Proof. Integrating identity (31) given in Proposition 4 between 0 and t ≤ T , rejecting all nonsigned term on the right-hand side that we bound then by putting absolute values, it yields:

F µ f |∂ x u f | 2 2 dx + µ g N k=1 | Ṙk | 2 R k + t 0 F ρ f |∂ t u f + u f ∂ x u f | 2 dx + N k=1 m k (|c k | 2 + | Rk | 2 ) + T 0 F κ f γ f ρ γ f f |∂ x u f | 2 dx ≤ 2 N k=1 κ k | Ṙk | R k + F p f (ρ f )|∂ x u f |dx + t 0 F µ f |∂ x u f | 3 2 dx + t 0 N k=1 2κ k | Ṙk | 2 R 2 k + µ g | Ṙk | 3 R 2 k + F 0 µ f |∂ x u 0 f | 2 2 dx + N k=1 µ g | Ṙ0 k | 2 R 0 k +2κ k | Ṙ0 k | R 0 k .
To obtain the expected result, it remains to drop the last term in the left-hand side which is positive and to bound the last term on the right-hand side by a constant E 1 with the expected dependencies. For this, we note that the first integral in this last term clearly depends on u 0 f H 1 0 (Ω) . Concerning the first term in the sum, the continuity of the velocity field [START_REF] Serre | Variations de grande amplitude pour la densité d'un fluide visqueux compressible[END_REF] rewrites for any k :

Ṙ0 k = 1 2 B 0 k ∂ x u 0 f (s)ds, so that | Ṙ0 k | ≤ 1 2 R 0 k B 0 k |∂ x u 0 f (s)| 2 ds 1/2
.

As a consequence it holds

N k=1 | Ṙ0 k | 2 R 0 k ≤ 1 2 ∪B 0 k |∂ x u 0 f (s)| 2 ds ≤ u 0 f 2 H 1 (Ω) .
Finally, the last term in the sum is bounded by using that κ k scales like 1/N. Indeed, applying (IC 0 ) with (IC 1 ) we have:

κ k R 0 k ≤ M ∞ √ 2d ∞ 1 √ N ∀ k = 1, . . . , N,
and then, with the above bound on | Ṙ0 k |/ R 0 k , we obtain:

N k=1 κ k | Ṙ0 k | R 0 k ≤ M ∞ √ 8d ∞ ∪B 0 k |∂ x u 0 f | 2 1 2
.

This ends the proof.

2.2.

Extended stress-tensor estimates. In order to obtain regularity estimates on the fluid velocity field, a classical way is to use the stress tensor. However Σ f is only defined on the fluid domain F, so that estimates on this stress tensor depend on the geometric properties of F, in particular the number of bubbles. In order to remove this dependency, we define new stress tensors for the fluid and for the gas phase, extended to the full domain Ω:

(35) Σf =    Σ f , in F, Σ f (x - k ) + Σ f (x + k ) 2 - Σ f (x - k ) -Σ f (x + k ) 2R k (x -c k ), in B k , k = 1, . . . , N, and
(36) Σg =              Σ k , in B k , k = 1, . . . , N, Σ N , in F N , Σ 0 , in F 0 , Σ k + Σ k+1 -Σ k x - k+1 -x + k (x -x + k ), in F k , k = 1, . . . , N -1.
Observe that these two stress tensors are continuous at each interface x ± k . We analyze here the properties of these extensions, when Σ f obeys further the continuity properties adapted from ( 11)-( 12)- [START_REF] Hillairet | On Baer-Nunziato multiphase flow models[END_REF]. Namely:

m k ck = Σ f (x + k ) -Σ f (x - k ), (37) m k 3 Rk = Σ f (x - k ) + Σ f (x + k ) -2Σ k , (38)
Σ k = µ g Ṙk R k -p g (ρ k ) -F s /2, (39)
In the stationary analysis of this subsection, these latter identities may stand for definitions of ck and Rk . These quantities will be related to the dynamical problem afterwards. Proposition 6. Assume that Σ f ∈ H 1 (F) satisfies (37)-(38) with Σ k defined by (39). Then Σf ∈ H 1 (Ω) and there exists a constant C 0 > 0 such that

(40) Σf H 1 (Ω) ≤ C 0 Σ f 2 H 1 (F ) + N k=1 (m k ) 2 | Rk | 2 + |c k | 2 R k + N k=1 µ 2 g | Ṙk | 2 R k + κ 2 k R k 1 2
.

Proof. By continuity of Σf at the interfaces,

Σf 2 H 1 (Ω) = Σ f 2 H 1 (F ) + N k=1 Σf 2 H 1 (B k ) .
We just have to study the H 1 norm of Σf on a bubble B k . The L 2 norm of Σ f can be bounded as follows:

Σf 2 L 2 (B k ) = B k Σ f (x - k ) + Σ f (x + k ) 2 2 + Σ f (x - k ) -Σ f (x + k ) 2R k 2 |x -c k | 2 dx = Σ f (x - k ) + Σ f (x + k ) 2 2 2R k + Σ f (x - k ) -Σ f (x + k ) 2R k 2 2R 3 k 3 = Σ f (x - k ) + Σ f (x + k ) 2 R k 2 + Σ f (x - k ) -Σ f (x + k ) 2 R k 6 .
On the other hand,

∂ x Σf 2 L 2 (B k ) = B k Σ f (x - k ) -Σ f (x + k ) 2R k 2 dx = Σ f (x - k ) -Σ f (x + k ) 2R k 2 2R k = |Σ f (x - k ) -Σ f (x + k )| 2 2R k .
We now gather the two estimates, and obtain Σf 2

H 1 (B k ) = |Σ f (x - k ) -Σ f (x + k )| 2 2R k + Σ f (x - k ) + Σ f (x + k ) 2 R k 2 + Σ f (x - k ) -Σ f (x + k ) 2 R k 6 .
Using the equations of motion of the droplets [START_REF] Gavrilyuk | The structure of pressure relaxation terms: the one-velocity case[END_REF] and the definition (13) of the stress tensor Σ k , one gets

Σf 2 H 1 (B k ) ≤ m 2 k |c k | 2 1 2R k + R k 6 + m k 3 Rk + 2 µ g Ṙk R k - κ k R k 2 R k 2 ≤ m 2 k |c k | 2 1 2R k + R k 6 + 2 9 m 2 k | Rk | 2 R k + 8µ 2 g | Ṙk | 2 R k + 8 κ 2 k R k Finally, this gives the estimate Σf 2 H 1 (Ω) ≤ 8 Σ f 2 H 1 (F ) + N k=1 (m k ) 2 | Rk | 2 R k + |c k | 2 1 R k + R k + N k=1 µ 2 g | Ṙk | 2 R k + κ 2 k R k 1 2
, which leads to the desired result since R k < 1.

From the above inequality we deduce the following L ∞ -bound in case Σ f is a viscous stress tensor: [START_REF] Drew | Theory of multicomponent fluids[END_REF]. Then, there exists C 1 > 0 such that

Proposition 7. Assume that Σ f ∈ H 1 (F) satisfies (37)-(38) with Σ k defined by (39). Assume further that Σ f is related to (ρ f , u f ) ∈ H 1 (F) × H 2 (F) via
(41) ∂ x u f L ∞ (F ) ≤ C 1 µ f Σf H 1 (Ω) + p f (ρ f ) L ∞ (F ) .
Proof. In the fluid domain, the stress tensor writes

Σ f = µ f ∂ x u f -p f , which gives ∂ x u f = 1 µ f (Σ f -p f (ρ f )).
Hence one has

∂ x u f L ∞ (F ) ≤ 1 µ f ( Σ f L ∞ (F ) + p f (ρ f ) L ∞ (F ) ).
The definition of global tensor Σf gives then

Σ f L ∞ (F ) ≤ Σf L ∞ (Ω) .
The H 1 (Ω) ⊂ L ∞ (Ω) embedding allows to conclude the proof.

One can note here the gain of working with an extended stress tensor. Indeed, the constant C 1 we obtain in the previous proposition is independent of the position of the particles and their radius. This would not be a priori the case if we wanted to control ∂ x u by Σ f only. Nevertheless, in (40) we introduced on the right-hand side negative powers of R k that we shall control independently. To this end, we performed a symmetric construction with the bubble stress-tensor Σ g and we provide now a corresponding proposition: Proposition 8. Assume that Σ f and (Σ k ) k=1,...,N are related via (39). Then Σg ∈ H 1 (Ω) and there exists a constant C 2 > 0 such that

(42) Σg H 1 (Ω) ≤ C 2   Σf 2 H 1 (Ω) + 1 min k∈{0,...,N } |F k | N k=1 (m k ) 2 (| Rk | 2 + |c k | 2 )   1/2 .
Proof. By straightforward calculations, the definition of Σg yields Σg

2 H 1 (Ω) ≤ N k=1 2R k |Σ k | 2 + |Σ 0 | 2 |x - 1 -x + 0 | + |Σ N | 2 |x - N +1 -x + N | + N -1 k=1 |Σ k+1 -Σ k | 2 |x - k+1 -x + k | + 2|Σ k | 2 |x - k+1 -x + k | + 2 3 |Σ k+1 -Σ k | 2 |x - k+1 -x + k | ≤ C N k=1 R k |Σ k | 2 + N k=0 |F k ||Σ k | 2 + N k=1 |Σ k+1 -Σ k | 2 |x - k+1 -x + k |
where C is a positive constant, since the length of the bubbles and of the fluid parts are bounded. Summing equations ( 11) and ( 12) leads to

(43) Σ k = Σ f (x + k ) - m k 2 ck + Rk 3 .
We deduce the following estimates, with some constant C > 0,

|Σ k | ≤ Σf L ∞ (B k ) + m k C |c k | + | Rk | , |Σ k+1 -Σ k | ≤ |x + k+1 -x + k | 1/2 ∂ x Σf L 2 (x + k ,x + k+1 ) + m k 2 | Rk | + |c k | + m k+1 2 | Rk+1 | + |c k+1 | .
One can now go back to the estimate on Σg . Noting the relation:

N k=0 |F k | + N k=1 2R k = |Ω|, the embedding H 1 (Ω) ⊂ L ∞ (Ω) implies the expected result.
As for the fluid stress tensor, we deduce from the previous computation a control on the (Σ k ) k=1,...,N by applying again the embedding H 1 (Ω) ⊂ L ∞ (Ω): Corollary 9. Under the same assumptions as in Proposition 8, there holds:

(44) max k=1,...,N -1 µ g Ṙk R k - κ k R k ≤ C 2 Σf 2 H 1 (Ω) + 1 min k∈{0,...,N } |F k | N k=1 (m k ) 2 | Rk | 2 + |c k | 2 1/2
. This latter corollary shall enable to control the radius of the bubble from below, preventing from collapse.

Proof of Theorem 1.

We combine now the computations of the previous section to construct a solution on a time-interval independent of the number N of bubbles. For this, we show that the following bounds can be continued:

(Q 1 ) d ∞ ≤ N R k ≤ (d ∞ ) -1 , k = 1, . . . , N , (Q 2 ) d ∞ ≤ N |F k | ≤ (d ∞ ) -1 , k = 1, . . . , N , (Q 3 ) ρ ∞ ≤ ρ f ≤ ρ∞ on F(t)
and, introducing a sufficiently large K > 0 :

(Q 4 ) F µ f |∂ x u f | 2 2 dx + µ g N k=1 | Ṙk | 2 R k ≤ K, (Q 5 ) t 0 Σf 2 H 1 (Ω) + Σg 2 H 1 (Ω) + N k=1 m k | Rk | 2 + |c k | 2 ds ≤ K.
We keep the convention here that tildas represent extended stress tensors as constructed in the previous subsection. We prove that, if K is chosen sufficiently large wrt the list of parameters (25), then we have such estimates on a time interval (0, T ) that depends only on the same list of parameters (25) (possibly via K).

Technically, we apply a continuation argument based on the a priori assumption that the solution exists. The precise statement is the following proposition in which we denote (Q i ) i=1,...,5 the estimates corresponding to the above (Q i ) i=1,...,5 where large inequalities are replaced with strict inequalities. Tacitly, all constants that are introduced in the following proposition may depend on the list of parameters (25).

Proposition 10. There exists K ∞ > 0 such that, for any K > K ∞ there exists T ∞ [K] > 0 for which the following statement holds: if

T ≤ T ∞ [K] and ((ρ f , u f ), (c k , R k ) k=1,...,N ) is a classical solution to (6)-(13) on (0, T ) satisfying (Q 1 )-(Q 5 ) then it satisfies also (Q 1 )-(Q 5 ).
The proof of Proposition 10 is the content of Appendix A. We explain here how it implies Theorem 1. For this, given K > 0 we introduce:

I := {T ∈ (0, ∞) s.t.
the unique classical solution exists on (0, T ) and satisfies (Q 1 )-(Q 5 )}.

Firstly, thanks to the local-in-time existence result, there exists T 0 depending on N such that we have a classical solution on (0, T 0 ). Indeed, for such a solution the radius R k and c k are continuous in time. Since we assume initially (IC 1 )-(IC 2 ) (resp. (24)) we have that, up to restrict T 0 , this solution satisfies (Q 1 )-(Q 2 ) (resp. (Q 3 )) on [0, T 0 ]. Similarly, we remark that the quantities on the left-hand side of (Q 4 )-(Q 5 ) are continuous time-dependent functions of the classical solution. Since the left-hand side of (Q 4 ) is controlled initially by u 0 f H 1 0 (Ω) and parameters involved in (25) (see the proof of Corollary 5), there exists K 0 sufficiently large depending only on the list of parameters (25) such that we can enforce (Q 4 )-(Q 5 ) on [0, T 0 ] also whatever the value of K > K 0 .

Let fix now K = max(K 0 , K ∞ ) with K ∞ given by Proposition 10 and denote

T ∞ = T ∞ [K].
By the previous arguments, we have that [0, T 0 ] ⊂ I. We show now that [0, T ∞ ] ⊂ I which shall end the proof. By restriction, ) shall be satisfied on a slightly longer interval by continuity. To extend the solution, we note that (Q 1 )-(Q 2 ) (resp. (Q 3 )) entail "a minimum distance between" and "a minimum radius of" bubbles (resp. strictly positive distance to vacuum) on [0, T ]. Inequality (Q 4 ) also ensures a (uniform) bound from above for u f H 1 (F ) on [0, T ]. By Proposition 29 of Appendix B we have also a uniform bound for ρ f H 1 (F ) (up to take T ∞ smaller). We can then apply the local-in-time existence result with initial data ((ρ f (T , •), u f (T , •)), (c k (T ), R k (T )) k=1,...,N ) for T arbitrary close to T. This yields a solution on some time-interval ∆T (independent of T , given the uniform bound above). By concatenation, we obtain a solution on (0, T +∆T ) where T +∆T > T for a well-chosen T .

I ∩ [0, T ∞ ] is a closed subinterval of [0, T ∞ ] containing [0, T 0 ]. Let us prove that I ∩ [0, T ∞ ] is open (in [0, T ∞ ]). Indeed, assume [0, T ] is a strict subinterval of [0, T ∞ ] in I,
To conclude this section, we mention that the proof above entails that we have the following corollary to Theorem 1: Corollary 11. The unique classical solution to (6)-( 13) on [0, T ∞ ] satisfies the bounds (Q 1 )-(Q 2 ) (resp. (Q 3 )) with d ∞ corresponding to (IC 1 )-(IC 2 ) (res. ρ ∞ , ρ∞ corresponding to (24)) and (Q 4 )-(Q 5 ) with K ∞ depending on the list of parameters (25).

Construction of macroscopic unknowns

In this section, we detail the construction of the unknowns for the macroscopic model starting from a sequence of solutions to the microscopic model with increasing number of gas bubbles. The full justification of the system (1)-( 4) is postponed to the next section. From now on, we fix initial data (ρ 0 f , ρ0 g , ū0 , ᾱ0 f , ᾱ0 g , f 0 g ) for the macroscopic model. All these quantities are H 1 (Ω) functions. We assume further that they fulfill conditions ( 20)-( 21)-( 23).

The framework identified in the previous section must be adapted for homogenization purpose. For instance, given a N -bubble solution the gas unknowns at-hand are a priori the discrete set of center/radius/mass (c k , R k , m k ) k=1,...,N . From them, we can reconstruct a (functional) density and a covolume by defining for instance:

(45) f (N ) g := N k=1 1 2N R k 1 B k ρ (N ) g := N k=1 m k 2R k 1 B k .
However, these reconstructed functions experience O(1) jumps through bubble/fluid interfaces and might not have sufficient regularity to perform the homogenization process. To gain regularity, we shall propagate an initial regularity through a well-chosen evolution equation (which extends the one satisfied by f

(N ) g , ρ (N ) g
on the B k ). However, this requires to be able to construct regular initial covolume and density (with uniform bounds in terms of N ). This is obtained with the following proposition:

Proposition 12. Under the assumption that the initial data fulfill the conditions (20)-( 21 ) of the associated reconstructed covolumes and densities such that:

• ( f (N ),0 g , ρ(N),0 g ) is bounded in H 1 (Ω) • for arbitrary β ∈ C 1 ([0, ∞) × [0, ∞))
there holds:

β(ρ (N ),0 g , f (N ),0 g )1 Ω\ F (N ),0 ᾱ0 g β(ρ 0 g , f 0 g ) in D (Ω). Proof.
Up to a localizing argument, we give a proof in the case:

(1 -α min ) f 0 g L ∞ (Ω) < f min := inf Ω f 0 g .
To construct our gas bubble, we note that ᾱ0 g f 0 g is a probability density on Ω. Then, we might construct the associated cumulative distribution function:

F g (x) = x -1 ᾱ0 g (x) f 0 g (x)dx.
With assumptions ( 20)-( 21)-( 23), this is a C 1 one-to-one mapping Ω → [0, 1] with F g ≥ α min f min on Ω. We set then:

(46) c 0 k := F -1 g k N + 1 , R 0 k := 1 2N [ fg (c 0 k )] -1 m k := 2R 0 k ρ0 g (c 0 k ) for k = 1, . . . , N.
Considering the bounds from above and from below for F g , we obtain that:

1 N + 1 1 (1 -α min ) f 0 g L ∞ (Ω) ≤ c 0 k+1 -c 0 k ≤ 1 N + 1 1 α min f min while 1 2N f 0 g L ∞ (Ω) ≤ R 0 k ≤ 1 2N 1 f min .
In particular [START_REF] Williams | Combustion Theory[END_REF] for N large. Finally, we have:

|F 0 k | = (c 0 k+1 -R 0 k+1 ) -(c 0 k + R 0 k ) ≥ 1 N N/(N + 1) (1 -α min ) f 0 g L ∞ (Ω) - 1 f min . ≤ c 0 k+1 -c 0 k ≤ 1 N 1 α min f min where N/((N + 1)(1 -α min )) f 0 g L ∞ (Ω) -1/f min > 0 by
1 N ρ min f 0 g L ∞ (Ω) ≤ m k ≤ 1 N ρ0 g L ∞ (Ω)
f min .

Item i) is satisfied. For item ii), we remark that the reconstructed densities and covolumes read:

f (N ),0 g := N k=1 1 2N R 0 k 1 B 0 k ρ (N ),0 g := N k=1 m k 2R 0 k 1 B 0 k .
We recall that we denote

B 0 k = (x - k , x + k ) where x ± k = c 0 k ± R 0 k (and x + 0 = -1, x - N +1 =1
). At this point, we note that by item i), we have:

min k∈{0,...,N } |x - k+1 -x + k | ≥ 1 2d ∞ N .
Consequently, for k = 2, . . . , N -1. we can construct a piecewise affine function ψ 0 k wich satisfies ψ 0 k = 1 on B 0 k , that vanishes in x - k+1 and x + k-1 and further away from B 0 k . For k = 1 and k = N we define similarly ψ 0 1 and ψ 0 N up to the condition that ψ 0 1 is constant equal to 1 between -1 and B 0 1 (resp. ψ 0 N is constant equal to 1 between B 0 N and 1). Then, we set:

f (N ),0 g := N k=1 1 2N R 0 k ψ 0 k ρ(N),0 g := N k=1 m k 2R 0 k ψ 0 k .
By standard computations, we have for instance:

f (N ),0 g 2 L 2 (Ω) ≤ N k=1 1 N 2 |R 0 k | 2 ψ 0 k 2 L 2 (Ω) 1 N N k=1 1 N 2 |R 0 k | 2 f 0 g 2 L ∞ (Ω)
where the first inequality on the second line involves a constant depending on d ∞ . We also derive using that ψ k+1 = 1 -ψ k on Supp(ψ k ) ∩ Supp(ψ k+1 ) :

∂ x f (N ),0 g 2 L 2 (Ω) N -1 k=1 1 N R 0 k+1 - 1 N R 0 k 2 N N -1 k=1 N c 0 k+1 c 0 k ∂ x f 0 g (z)dz 2 ∂ x f 0 g 2 L 2 (Ω) .
In these computations, we use extensively the definitions (46) and also that |B 0 k | and |F 0 k | are both of size O(1/N ). Similar arguments yield that:

ρ(N),0 g 2 H 1 (Ω) ρ0 g 2 H 1 (Ω) .

Finally, for arbitrary

β ∈ C 1 ([0, ∞) × [0, ∞)) and ϕ ∈ C ∞ c (Ω), we have: Ω β(ρ (N ),0 g , f (N ),0 g )1 Ω\ F (N ),0 ϕdx = N k=1 B 0 k β(ρ g (c 0 k ), fg (c 0 k ))ϕ(x)dx = N k=1 2R 0 k β(ρ g (c 0 k ), fg (c 0 k ))ϕ(c 0 k ) + O(1/N ) ∂ x ϕ L ∞ (Ω) = 1 N N k=1 β(ρ g (c 0 k ), fg (c 0 k )) f 0 g (c 0 k ) ϕ(c 0 k ) + O(1/N ) ∂ x ϕ L ∞ (Ω)
At this point, we remark that, by construction, we have that

1 N N k=1 δ c 0 k ᾱ0 g f 0 g in P(Ω). Since t → β(ρ 0 g (t), f 0 g (t))/ f 0 g (t)
is continuous on Ω we infer that:

lim N →∞ Ω β(ρ (N ),0 g , f (N ),0 g )1 Ω\ F (N ),0 ϕdx = Ω β(ρ g , f 0 g )ᾱ 0 g ϕdx.
This concludes the proof.

Below, we pick a sequence of initial bubble distribution (c

(N ),0 k , R (N ),0 k
) k=1,...,N and masses (m (N ) k ) k=1,...,N given by Proposition 12. For any N ∈ N, assuming the fluid initial data is associated with ρ0 f , ū0 , we construct initial data for the microscopic system like in ( 16)- [START_REF] Serre | Variations de grande amplitude pour la densité d'un fluide visqueux compressible[END_REF]. We have then that the initial data match the assumptions of Theorem 1 and we obtain a solution (ρ

(N ) f , u (N ) f , (c (N ) k , R (N ) k ) k∈{1,...,N } )
that is defined on a time-span [0, T ] which does not depend on N. This creates a sequence of solutions indexed by N whose asymptotic behavior (when N → ∞) is analyzed in the remaining sections.

Firstly, Corollary 11 entails that we have uniform bounds on [0, T ] in the form of (75)-( 76) with a right-hand side E 0 independent of N , and that (Q 1 )-(Q 5 ) hold also with a constant K independent of N . In passing, we point out that all the bounds that are derived in Appendix A and Appendix B are available since they are obtained under the sole assumptions that initial data are of the form ( 16)- [START_REF] Serre | Variations de grande amplitude pour la densité d'un fluide visqueux compressible[END_REF] and that the bounds (Q 1 )-(Q 5 ) hold true. Below we denote ũ(N) the "mixture" velocity-field meaning that

(47) ũ(N) =              u (N ) f , on F (N ) , u f (x -,(N ) k ) + u f (x +,(N ) k ) 2 - u f (x -,(N ) k ) -u f (x +,(N ) k ) 2R (N ) k (x -c k ), on B (N ) k , k = 1, . . . , N.
Note that the restriction of ũ(N) on the bubbles boils down to

(48) ũ(N) (•, x) = ċ(N) k + Ṙ(N) k R (N ) k (x -c (N ) k ) on B (N ) k .
In what remains of this section, we introduce functions describing the different species and the mixture and we analyse their possible convergences. Since we use mostly compactness argument below, all convergence results must be understood "up to the extraction of a subsequence that we do not relabel." 3.1. Fluid unknowns. In [START_REF] Amosov | On the quasi-averaging of a system of equations of the one-dimensional motion of a viscous heat-conducting gas with rapidly oscillating data[END_REF], the fluid behavior is encoded through its "volumic fraction" ᾱf and its density ρf . We recover such quantities from microscopic counterparts. We start with the following construction of the volumic fraction:

Proposition 13. Let χ (N ) = 1 F (N ) . It satisfies (49) ∂ t χ (N ) + ũ(N) ∂ x χ (N ) = 0, on (0, T ) × Ω, χ (N ) (0, .) = 1 F (N ),0 .
Moreover, there exists ᾱf ∈ L ∞ ((0, T ) × Ω), called the volumic fraction of the fluid, such that, up to the extraction of a subsequence, (50)

χ (N ) ᾱf in L ∞ ((0, T ) × Ω) -w * and 0 ≤ ᾱf ≤ 1 -2d 2 ∞ /3 a.e.
Proof. Since the fluid domain F (N ) is transported by the velocity field ũ(N) , (49) holds. The convergence result is straightforward since the sequence χ (N ) is nonnegative and bounded in L ∞ ((0, T ) × Ω). The limit is obviosuly positive. The only crucial information is the bound from above. For this, we remark that under (Q 1 )-(Q 2 ), any sequence of two bubble+fluid intervals has at most length 3/(d ∞ N ). Hence, for large N , any segment in Ω of length contains at least N d ∞ /3 -2 such sequences in which the volumic proportion of gas-bubbles is at least

2 d 2 ∞ /3 + O(1/N ).
The fluid part of this segment is then asymptotically less than (1 -2d 2 ∞ /3).

We point out that a strictly bound from below for ᾱf is also true with similar arguments. We dot not state this bound here since it will not help in the sequel. For constructing the macroscopic density, we choose to extend at first the microscopic density by "filling" the bubbles in a sufficiently smooth manner. To this end, we take advantage of the fact that ρ 0 f is initially defined (and sufficiently regular) on the whole Ω. So, we introduce ρ(N) f as the unique solution to:

(51)

       ∂ t ρ(N) f + ũ(N) ∂ x ρ(N) f = - ρ(N) f µ f Σ(N) f + p f (ρ (N ) f ) , on (0, T ) × Ω, ρ(N) f (0, .) = ρ0 f , on Ω, where Σ(N) f is defined from Σ f by (35).
Proposition 14. There exists a time T 0 < T , independent of N , such that the Cauchy problem (51) admits a unique solution ρ(N)

f ∈ C([0, T 0 ] × Ω).
Moreover, there exists ρf ∈ L 2 ((0, T 0 ) × Ω) called the density of the fluid such that, up to the extraction of a subsequence,

ρ(N) f -→ ρf in L 2 ((0, T 0 ) × Ω) when N → +∞.
Proof. The well-posedness of the Cauchy problem (51) is guaranteed by the method of characteristics, since ũ(N) belongs to L 2 ((0, T ); W 1,∞ (Ω)).

The result of convergence is an application of the Aubin-Lions lemma. One has to check that:

• (ρ (N ) f ) N bounded in L 2 ((0, T ); H 1 (Ω)), • (∂ t ρ(N) f ) N bounded in L 2 ((0, T ); L 2 (Ω)
). For the first item, we apply Proposition 29 in Appendix B which yields that, up to restrict to some time-interval [0, T 0 ] ⊂ [0, T ] we have that ρ(N) f satisfies a uniform bound in L ∞ ((0, T ); H 1 (Ω)). As for the second item, using directly Equation (51), a uniform estimate can be obtained:

∂ t ρ(N) f L 2 ((0,T )×Ω) ≤ C 0 ũ(N) L 2 ((0,T );H 1 (Ω)) ∂ x ρ(N) f L ∞ ((0,T );L 2 ((Ω)) + ρ(N) f L ∞ ((0,T )×Ω) µ f Σ(N) f L 2 ((0,T );H 1 (Ω)) + √ T ρ(N) f γ L ∞ ((0,T )×Ω)
,

where C 0 depends only on the parameters of the problem independent of N . Here again, the right-hand side is uniformly bounded with respect to N , so that the Aubin-Lions lemma can be applied to deduce the existence of the limit ρf stated in the proposition.

To illustrate again that our choice for ρ(N)

f is rigorously adapted, we mention that, on the fluid domain F (N ) , the definition of the fluid tensor [START_REF] Drew | Theory of multicomponent fluids[END_REF] gives

1 µ f Σ(N) f + p f (ρ (N ) f ) = ∂ x u (N ) f .
Moreover, u

(N ) f and ũ(N) coincide on F (N ) , and the density ρ(N)

f is also solution of ( 52)

   ∂ t ρ(N) f + ∂ x (ρ (N ) f u (N ) f ) = 0, on (0, T ) × F N , ρ(N) f (0, .) = ρ0 f .
As a consequence, the fluid density ρ

(N ) f = ρ(N) f , on (0, T ) × F (N ) . 3.2.
Mixture unknowns. We proceed with the construction of unknowns that are involved in composite equations: a mixture velocity, a mixture density and a mixture stress tensor.

The mixture velocity is deduced from the reconstructed velocity ũ(N) defined by (47):

Proposition 15. There exists ū ∈ L 2 ((0, T ); L 2 (Ω)) such that, up to the extraction of a subse- quence, ũ(N) → ū in L 2 ((0, T ); L 2 (Ω)) when N → +∞.
Proof. This result is an application of the Aubin-Lions lemma again. From ( 75) and ( 76), the sequence (ũ (N ) ) is bounded in L 2 ((0, T ); H 1 (Ω)). It remains to prove a uniform bound for (∂ t ũ(N) ) N in L 2 ((0, T ); L 2 (Ω)). By [START_REF] Desjardins | Regularity of weak solutions of the compressible isentropic Navier-Stokes equations[END_REF] and (48), the time derivative of the velocity reads:

∂ t ũ(N) =        -u (N ) f ∂ x u (N ) f - 1 ρ (N ) f ∂ x Σ (N ) f on F, ck + Rk R k -( Ṙk ) 2 R 2 k (x -c k ) -ċk Ṙk R k on B k ,
(note that some exponents (N ) have been removed to lighten the notations). Since the velocity ũ(N) is continuous through the interfaces c k ± R k , one has, in D ((0, T ) × Ω),

∂ t ũ(N) = -u (N ) f ∂ x u (N ) f - 1 ρ (N ) f ∂ x Σ (N ) f 1 F + N k=1 ck + Rk R k - ( Ṙk ) 2 R 2 k (x -c k ) -ċk Ṙk R k 1 B k .
We now take the L 2 norm:

∂ t ũ(N) 2 L 2 (Ω) ≤ ũ(N) 2 L ∞ (Ω) ∂ x ũ(N) 2 L 2 (Ω) + 1 |ρ ∞ | 2 ∂ x Σ(N) f 2 L 2 (Ω) + 2 N k=1 R k (c k ) 2 + R k ( Rk ) 2 + ( Ṙk ) 4 R k + ( ċk Ṙk ) 2 R k ≤ C ũ(N) 4 H 1 (Ω) + 1 |ρ ∞ | 2 Σ(N) f 2 H 1 (Ω) + 2 1 d ∞ M ∞ N k=1 m k (c k ) 2 + ( Rk ) 2 + 2 N k=1 1 R k ( Ṙk ) 4 + ( ċk Ṙk ) 2
by (IC 0 ) and (Q 1 ). Time-integrals of the two first terms on the right-hand side are bounded by (Q 4 ) and (Q 5 ) respectively. The third is controlled using (Q 5 ). Moreover, by (IC 0 ), (Q 1 ), and then by (75), the last term can be bounded this way:

T 0 N k=1 ( Ṙk ) 2 R k ( Ṙk ) 2 + ( ċk ) 2 dt ≤ 1 d ∞ M ∞ T 0 max k=1,...,N ( Ṙk ) 2 R 2 k N k=1 m k ( Ṙk ) 2 + ( ċk ) 2 dt ≤ 2E 0 d ∞ M ∞ T 0 max k=1,...,N ( Ṙk ) 2 R 2 k dt.
The last right-hand side is finally bounded by using Lemma 27. This concludes the proof of the assumptions of the Aubin-Lions lemma, leading to the convergence of the sequence (ũ

(N ) ) N in L 2 ((0, T ); L 2 (Ω)).
We focus now on the mixture density. For this, we construct the global density ρ (N ) : ( 54)

ρ (N ) = ρ (N ) f 1 F (N ) + N k=1 ρ (N ) k 1 B k ,
where ρ

(N ) k = m (N ) k /(2R (N )
k ) is the bubble density that we reconstruct from the bubble mass and radius. Notice that the global density ρ (N ) belongs to L ∞ ((0, T ) × Ω), and satisfies a classical mass conservation law (the proof is left to the reader):

(55) ∂ t ρ (N ) + ∂ x (ρ (N ) ũ(N) f ) = 0, in (0, T ) × Ω.
To conclude, we address the asymptotic behavior of extended stresses. This is the content of the following proposition: Proposition 16. There exist Σf and Σg in L 2 ((0, T ); H 1 (Ω)) such that, up to the extraction of a subsequence, Σ(N)

f Σf Σ(N) g Σg in L 2 ((0, T ); H 1 (Ω)) when N → +∞.
Proof. The estimate (Q 5 ) ensures that the sequences Σ(N) f and Σ(N) g are both bounded in the space L 2 ((0, T ); H 1 (Ω)). Hence they are relatively compact in L 2 ((0, T ); H 1 (Ω)) endowed with the weak topology, and the result follows.

Bubble unknowns.

We mention first that the indicator of the bubble domains reads 1χ (N ) . Similarly to Proposition 13 we obtain that it converges weakly to some ᾱg satisfying also 0 ≤ ᾱg ≤ 1 a.e.. Since 1 = ᾱg + ᾱf , Proposition 13 entails further that ᾱg ≥ 2d 2 ∞ /3. For our analysis, we need a sufficiently strong (pointwise) convergence of bubble density ρ (N ) g and covolume f (N ) g as defined in (45). Yet, these quantities are defined only partially on subsets depending on N. To overcome this difficulty, we note that both quantities satisfy the same continuity equation:

(56) ∂ t ρ (N ) g + ∂ x (ρ (N ) g ũ(N) ) = 0, ∂ t f (N ) g + ∂ x (f (N ) g ũ(N) ) = 0, in D ((0, T ) × Ω).
We used here in particular that m (N ) k is time-independent and that the bubbles follow the flow associated with the extended velocity. We propose then to reproduce the same method we used in the case of fluid unknowns (see Proposition 14). We remark that, on B k , there holds:

(57) ∂ x ũ(N) = 1 µ g Σ (N ) k + κ k R (N ) k .
We recall that, on the right-hand side, the first term is the restriction to B k of the extended stress tensor Σ(N) g . As for the last term, we wish to extract the contribution of the pressure and the contribution of the surface tension for modelling reason (even though keeping the current form would not change the remark in progress). So we rewrite:

κ k R (N ) k = p g (ρ (N ) k ) + γS 2N R (N ) k .
Here, the second term could be related artificially to a density, but it is usually related to a "covolume" and is treated independently. Actually, this is the reason motivating the introduction of the unknown f (N ) g

. We use now this novel writing of the term ∂ x ũ(N) to see that (ρ

(N ) g , f (N ) g
) is the restriction of a pair (ρ

(N ) g , f (N ) g
) solution to:

(58) ∂ t ρ(N) g f (N ) g + ũ(N) ∂ x ρ(N) g f (N ) g = - 1 µ g ρ(N) g f (N ) g Σ(N) g + p g (ρ (N ) g ) + γs f (N ) g , on (0, T ) × Ω.
We can then use the stability properties of this latter equation to yield the following proposition:

Proposition 17. There exists a time T 0 < T (independent of N ) and sequences (ρ

(N ) g , f (N ) g ) ∈ C([0, T 0 ]; H 1 (Ω))
satisfying the following properties:

• there holds ρ

(N ) g = ρ(N) g and f (N ) g = f (N ) g on B k for all k = 1, . . . , N, • there exists (ρ g , fg ) ∈ L 2 ((0, T 0 ) × Ω) 2 such that, up to the extraction of a subsequence, (ρ (N ) g , f (N ) g ) -→ (ρ g , fg ) in L 2 ((0, T 0 ) × Ω) 2 when N → +∞.
Proof. We recall that the initial bubble distribution (c

(N ) k , R (N )
k ) k=1,...,N is obtained by applying Proposition 12 so that they are associated with a sequence of initial density/covolume ρ(N),0 and that converge weakly in H 1 (Ω). Hence, we complement (58) with initial condition

ρ(N) g (0, •) = ρ (N ),0 g f (N ) g (0, •) = f (N ),0 g on Ω. ( 59
)
The result is then proved following exactly the same steps as in the proof of Proposition 14, since (Q 5 ) involves similar controls on Σ(N) f and Σ(N) g .

3.4. Two technical lemmas. We close this section by providing two crucial results which allow to pass to the limit in some nonlinear terms. The procedure we apply here is similar to the construction in [START_REF] Bresch | Physical relaxation terms for compressible two-phase systems[END_REF].

Let b ∈ C 1 ([0, 1] × R + × R + ) and consider the sequence (60) b (N ) (t, x) = b(χ (N ) (t, x), ρ (N ) (t, x), f (N ) g (t, x)), ∀(t, x) ∈ (0, T ) × Ω,
where ρ (N ) is defined by (54) and f (N ) g by (45).

Proposition 18. There exists b ∈ L ∞ ((0, T ) × Ω) such that, up to a subsequence, b (N ) b, in L ∞ ((0, T ) × Ω) -w when N → +∞.
This limit verifies the following identity, for almost every

(t, x) ∈ (0, T ) × Ω, (61) b
= b(1, ρf , 0) ᾱf + b(0, ρg , fg )ᾱ g .
Proof. By definition, we have:

b (N ) = b(1, ρ(N) f , 0)χ (N ) + b 0, ρ(N) g , f (N ) g (1 -χ (N ) )
The strong convergence of ρ(N)

f (resp. ρ(N) g and f (N ) g
), see Proposition 14 (resp. Proposition 17) and the weak convergence of χ (N ) (Proposition 13) ensure that the first term converges weakly towards b(1, ρf , 0) ᾱf and the second one to b(0, ρg , fg )ᾱ g .

In the following result, the term Σ(N) denotes either Σ(N)

f or Σ(N) g .
Proposition 19. Assume that Σ(N) converges weakly in L 2 ((0, T ); H 1 (Ω)), and denote by Σ its limit. Then for all

b ∈ C 1 ([0, 1] × R + × R + ), it holds Σ(N) b (N ) Σb , in D ((0, T ) × Ω) when N → +∞.
Proof. This result is a variant of so-called "compensated compactness" lemma. We can reproduce here the proof of [START_REF] Bresch | A compressible multifluid system with new physical relaxation terms[END_REF]Lemma 10] up to adapt the definition of the operator ∂ -1

x on mean free functions.

Derivation of a macroscopic model

Thanks to the results of the previous section, we are now in position to address the limit N → +∞ for the microscopic model ( 6)- [START_REF] Hillairet | On Baer-Nunziato multiphase flow models[END_REF]. Based on the previous definitions of macroscopic unknowns, we derive successively the various equations of (1). This is the content of the following theorem.

Theorem 20. Let ρf , ᾱf , ᾱg , ρg , ū be as constructed in the previous section. Then, we have that (ᾱ f , ρf , ᾱg , ρg , fg , ū) is a solution to (1)-( 2)-( 3)-( 4) on (0, T ) with initial condition on Ω:

ᾱf (0, •) = ᾱ0 f ᾱg (0, •) = ᾱ0 f ρf (0, •) = ρ0 f ᾱg ρg (0, •) = ᾱ0 g ρ0 g ū(0, •) = ū0 ᾱg fg (0, •) = ᾱ0 g f 0 g
What remains of this section is devoted to the proof of this theorem. Our first result provides the limit equation for the limit b associated with an abstract choice of b.

Proposition 21. Let b ∈ C 1 ([0, 1] × R + × R + ) and define b 1,f (z, ξ, ν) = (∂ 2 b(z, ξ, ν)ξ + ∂ 3 b(z, ξ, ν)ν -b(z, ξ, ν))z, b 1,g (z, ξ, ν) = (∂ 2 b(z, ξ, ν)ξ + ∂ 3 b(z, ξ, ν)ν -b(z, ξ, ν))(1 -z), b 2,f (z, ξ, ν) = (∂ 2 b(z, ξ, ν)ξ + ∂ 3 b(z, ξ, ν)ν -b(z, ξ, ν))zp f (ξ), b 2,g (z, ξ, ν) = (∂ 2 b(z, ξ, ν)ξ + ∂ 3 b(z, ξ, ν)ν -b(z, ξ, ν))(1 -z)(p g (ξ) + γs ν).
Then, the limit b defined in Proposition 18 satisfies the equation

(62)    ∂ t b + ∂ x (ū b) + 1 µ f b1,f Σf + b2,f + 1 µ g b1,g Σg + b2,g = 0 b(0, •) = ᾱ0 f b(1, ρ0 f , 0) + ᾱ0 g b(0, ρ0 g , f 0 g ) Proof. Let us compute for arbitrary N ∈ N ∂ t b(χ (N ) , ρ (N ) , f (N ) g ) = ∂ 1 b (N ) ∂ t χ (N ) + ∂ 2 b (N ) ∂ t ρ (N ) + ∂ 3 b (N ) ∂ t f (N ) g = -∂ 1 b (N ) ũ(N) ∂ x χ (N ) -∂ 2 b (N ) ∂ x (ρ (N ) ũ(N) ) -∂ 3 b (N ) ∂ x (f (N ) g ũ(N) )
by ( 49), (55) and (56). As a result, we obtain:

(63) ∂ t b(χ (N ) , ρ (N ) , f (N ) g ) + ∂ x (b(χ (N ) , ρ (N ) , f (N ) g )ũ (N ) ) + ∂ 2 b(χ (N ) , ρ (N ) , f (N ) g )ρ (N ) + ∂ 3 b(χ (N ) , ρ (N ) , f (N ) g )f (N ) g -b(χ (N ) , ρ (N ) , f (N ) g ) ∂ x ũ(N) = 0,
in D ((0, T )×Ω). In this equation, due to the weak convergence of b (N ) and the strong convergence of ũ(N) f , respectively stated in Propositions 18 and 15, it holds that:

b (N ) b, u (N ) f b(χ (N ) , ρ (N ) , f (N ) g ) ū b, in D ((0, T ) × Ω).
Then, we rewrite:

∂ 2 b(χ (N ) , ρ (N ) , f (N ) g )ρ (N ) + ∂ 3 b(χ (N ) , ρ (N ) , f (N ) g )f (N ) g -b (N ) ∂ x u (N ) f = 1 µ f b (N ) 1,f Σ(N) f + b (N ) 2,f + 1 µ g b (N ) 1,g Σ(N) g + b (N ) 2,g
.

The weak convergence stated in Proposition 19 allows to pass to the limit the right-hand side, leading to

∂ 2 b(χ (N ) , ρ (N ) , f (N ) g )ρ (N ) + ∂ 3 b(χ (N ) , ρ (N ) , f (N ) g )f (N ) g -b (N ) ∂ x u (N ) f 1 µ f b1,f Σf + b2,f + 1 µ g b1,g Σg + b2,g ,
where the terms b1,f , b1,g , b2,f , and b2,g are defined as in Proposition 18. This provides Equation (62) for b. Finally, we have initially

b (N ) (0, •) = χ (N ),0 b(1, ρ 0 f , 0) + (1 -χ (N ),0 )b(0, ρ(N),0 g , f (N ),0 g )
and we are in position to apply Proposition 12 to pass to the limit in this identity when N → ∞.

Let us recall that the link between the limit b and the function b is provided in Proposition 18. According to the choice of b, different relevant macroscopic equations can be obtained.

Corollary 22. The volumic fractions satisfy the following equations

(64)    ∂ t ᾱf + ∂ x (ᾱ f ū) = ᾱf µ f Σf + p f (ρ f ) , ᾱf (0, •) = ᾱ0 f ᾱf + ᾱg = 1
The covolume unkwnown fg satisfies the conservation equation:

∂ t ( ᾱg fg ) + ∂ x ( ᾱg fg ū) = 0, ᾱg (0, •) fg (0, •) = ᾱ0 g f 0 g . The mass conservation laws of both phases read

∂ t (ᾱ f ρf ) + ∂ x (ᾱ f ρf ū) = 0, ᾱf (0, •)ρ f (0, •) = ᾱ0 f ρ0 f (65) ∂ t (ᾱ g ρg ) + ∂ x ( ᾱg ρg ū) = 0, ᾱg (0, •)ρ g (0, •) = ᾱ0
1,f (1, r) = -1, b 1,g (1, r) = 0, b 2,f (1, r) = -p f (r), b 2,g (1, r) = 0, b 1,f (0, r) = 0, b 1,g (0, r) = 0, b 2,f (0, r) = 0, b 2
(∂ t (ρ (N ) f u (N ) f ) + ∂ x (ρ (N ) f |u (N ) f | 2 ))wdxdt = T 0 F (N ) (t) ∂ x Σ (N ) f wdxdt.
Since the fluid domain F (N ) (t) is transported with the velocity u

(N )
f , an integration by part in time of the left-hand side gives

T 0 F (N ) (t) (∂ t (ρ (N ) f u (N ) f ) + ∂ x (ρ (N ) f |u (N ) f | 2 ))wdxdt = - T 0 F (N ) (t) ρ (N ) f u (N ) f (∂ t w + u (N ) f ∂ x w)dxdt.
The right-hand side is handled by an integration by part in space. Reorganising the boundary terms yields (we omit time dependencies for simplicity):

T 0 F (N ) (t) ∂ x Σ (N ) f wdxdt = - T 0 N k=1 (Σ (N ) f (x + k )w(x + k ) -Σ (N ) f (x - k )w(x - k ))dt - T 0 F (N ) Σ (N ) f ∂ x wdxdt.
We now focus on the boundary terms. For k = 1, . . . , N , one has

Σ (N ) f (x + k )w(x + k ) -Σ (N ) f (x - k )w(x - k ) = (Σ (N ) f (x + k ) -Σ (N ) f (x - k ))w(c k ) + (Σ (N ) f (x + k ) + Σ (N ) f (x - k ))R k ∂ x w(c k ) + O( Σ (N ) f L ∞ (Ω) R 2 k w C 2 ).
From the bubbles equations ( 11) and ( 12), one deduces

Σ (N ) f (x + k )w(x + k ) -Σ (N ) f (x - k )w(x - k ) = m k ck w(t, c k ) + m k 3 Rk + 2Σ k R k ∂ x w(t, c k ) + O( Σ (N ) f L ∞ (Ω) R 2 k w C 2 ).
The term involving the stress tensor can be rewritten as follows

2Σ k R k ∂ x w(t, c k ) = B k Σ (N ) g ∂ x wdx + O( Σ (N ) g L ∞ (Ω) R 2 k w C 2 ).
Therefore, one has

- T 0 N k=1 (Σ (N ) f (x + k )w(x + k ) -Σ (N ) f (x - k )w(x - k ))dt = - T 0 N k=1 m k ck w(t, c k ) + m k 3 Rk R k ∂ x w(t, c k ) + B k Σ (N ) g ∂ x wdx + O(( Σ (N ) f L ∞ (Ω) + Σ (N ) g L ∞ (Ω) )R 2 k w C 2 ) dt.
An integration by part in time gives

- T 0 N k=1 (Σ (N ) f (x + k )w(x + k ) -Σ (N ) f (x - k )w(x - k ))dt = T 0 N k=1 m k | ċk | 2 ∂ x w(t, c k ) + 1 3 | Ṙk | 2 ∂ x w(t, c k ) + 1 3 Ṙk R k ċk ∂ xx w(t, c k ) dt + T 0 N k=1 m k ċk ∂ t w(t, c k ) + 1 3 Ṙk R k ∂ xt w(t, c k ) dt - T 0 Ω\F (N ) Σ (N ) g ∂ x wdxdt + O ( Σ (N ) f L 2 ((0,T ),H 1 (Ω)) + Σ (N ) g L 2 ((0,T ),H 1 (Ω)) ) √ T w C 2 max [0,T ] N k=1 R 2 k = T 0 N k=1 m k | ċk | 2 ∂ x w(t, c k ) + 1 3 | Ṙk | 2 ∂ x w(t, c k ) + ċk ∂ t w(t, c k ) dt - T 0 Ω\F (N ) Σ (N ) g ∂ x wdxdt + O ( Σ (N ) f L 2 ((0,T ),H 1 (Ω)) + Σ (N ) g L 2 ((0,T ),H 1 (Ω)) ) √ T w C 2 (d ∞ N ) -1 + (M ∞ N ) -1 2 w C 2 T E 0 .
where we applied (IC 0 ) and ( 75) with (Q 1 ) to yield the last term in the last inequality. On the bubble B k , it holds

B k ρ k ũ(N) (∂ t w + ũ(N) ∂ x w)dx = B k m k 2R k ũ(N) (∂ t w + ũ(N) ∂ x w)dx = m k ċk ∂ t w(c k ) + m k (| ċk | 2 + 1 3 | Ṙk | 2 )∂ x w(c k ) + O m k w C 2 (1 + | ċk | + | Ṙk |)(| ċk | + | Ṙk |)|R k | .
Gathering the fluid and gas expressions yields

- T 0 Ω ρ (N ) ũ(N) (∂ t w + ũ(N) ∂ x w)dxdt = - T 0 Ω (χ (N ) Σ(N) f + (1 -χ (N ) ) Σ(N) g )∂ x wdxdt + O(N -1/2 ).
Using the strong convergence of ũ(N) and the weak convergence of ρ (N ) , obtained by Proposition 18 with b(z, ξ, ν) = ξ, the left-hand side tends to

- T 0 Ω
ρū(∂ t w + ū∂ x w)dxdt.

The limit of the right-hand side is deduced from Proposition 19. One ends up with the desired momentum equation (67). It remains to close the system by determining relations between the tensors Σf and Σg and the other quantities. To do so, we prove that Σf and Σg are solutions of a 2 × 2 system. First observe that

∂ x u (N ) = χ (N ) Σ(N) f + p f (ρ (N ) f ) µ f + (1 -χ (N ) ) Σ(N) g + p g (ρ (N ) g ) + F s /2 µ g .
The different results of convergence given in Section 3, especially Proposition 19, allow to pass to the limit in both sides of the equation. In particular, in the right-hand side, the definition of the surface tension yields

(1 -χ (N ) ) F s 2 = N k=1 γs 2N R k 1 B k = γs f (N ) g (1 -χ (N ) ) γs ᾱg fg .
Eventually, it holds

∂ x ū = ᾱf µ f Σf + p f (ρ f ) + ᾱg µ g Σg + p g (ρ g ) + γs fg .
The second equation is obtained while studying the difference Σf -Σg . Using the definition (35) of the extended tensor Σf and the Newton laws [START_REF] Gavrilyuk | The structure of pressure relaxation terms: the one-velocity case[END_REF] and [START_REF] Hillairet | Propagation of density-oscillations in solutions to the barotropic compressible Navier-Stokes system[END_REF] for the bubbles, it holds

Σ(N) f = Σ f (x - k ) + Σ f (x + k ) 2 - Σ f (x - k ) -Σ f (x + k ) 2R (N ) k (c -c (N ) k ) = m k 6 Rk + Σ k + m k ck 2R k (x -c k ). Since Σ(N) g = Σ k on the bubbles domain B k , one has (1 -χ (N ) )( Σ(N) f -Σ(N) g ) = N k=1 m k 6 Rk + m k ck 2R k (x -c k ) 1 B k .
Proposition 19 applies to the left-hand side:

(

1 -χ (N ) )( Σ(N) f -Σ(N) g ) (1 -ᾱf )( Σf -Σg ),
in the sense of distributions. The right-hand side can be proved to tend to zero in L 2 ((0, T ) × Ω)) since

N k=1 m k 6 Rk + m k ck 2R k (x -c k ) 1 B k 2 L 2 (Ω) ≤ 1 2 N k=1 m k | Rk | + |c k | 1 B k 2 L 2 (Ω) ≤ N k=1 B k m 2 k (| Rk | 2 + |c k | 2 )dx ≤ 2 N 2 d ∞ M ∞ N k=1 m k (| Rk | 2 + |c k | 2 ) ≤ 2K ∞ N 2 d ∞ M ∞ ,
thanks to (IC 0 ), (Q 1 ) and (Q 5 ). Recalling the second part of (50), one recovers (69).

An alternative description of the bubble dynamics

In order to describe the dynamics of the bubbles, an alternative approach is to introduce the distribution function in position and (scaled) radius ( 70)

S (N ) t = 1 N N k=1 δ c k (t),N R k (t) ,
which is a measure on Ω × (0, ∞). According to (Q 1 ), one has supp(S

(N ) t ) ⊂ Ω × [d ∞ , 1/d ∞ ], ∀ t ∈ (0, T ).
Since Σg is uniformly bounded in L 2 ((0, T ), H 1 (Ω)) ⊂ L 2 ((0, T ), C 0,1/2 ( Ω)), the second term writes

S (N ) t , r Σ(N) g (x)β x ⊗ β r = 1 N N k=1 Σ(N) g (c k )N R k β x (c k )β r (N R k ) = 1 2 N k=1 B k Σ(N) g (x)β x (x)β r (N R k )dx + C β Σg H 1 (Ω) √ N = 1 2 Ω Σ(N) g (x)β x (x)b (N ) (x)dx + C β Σg H 1 (Ω) √ N where b (N ) is defined by Equation (60), with b(1, •, •) = 0, b(0, •, ν) = β r (1/(2ν)).
Indeed, this provides

b (N ) = 0 in F (N ) , β r (1/(2f k )) in B k .
Using successively Propositions 19, 18 and 25, we obtain

S (N ) t , r Σ(N) g (x)β x ⊗ β r -→ 1 2 Ω Σg (x) b(x)β x (x)dx = 1 2 Ω Σg (x) b(1, ρg , 0)ᾱ f + R + (2r)b(0, ρg , 1/(2r)) Sg,t (dr) β x (x)dx = 1 2 Ω Σg (x) R + (2r)β r (r) Sg,t (dr) β x (x)dx = Sg,t , r Σg β x ⊗ β r .
For the third term, we proceed similarly, defining

b(1, •, •) = 0, b(0, ξ, ν) = p g (ξ)β r (1/(2ν)), so that S (N ) t , rp g (ρ (N ) g )β x ⊗ β r -→ Sg,t , rp g (ρ g )β x ⊗ β r .
The convergence of the last term is nothing else but the convergence of S (N ) t .

Observe that ᾱg fg and ᾱg are respectively the zeroth and first moments of Sg . Their PDE's, see Corollary 22, can be deduced from the Equation (74).

Strict version (Q 2 ) of (Q 2 ). First, we remark that we can also adapt the previous proof to yield the following lemma: Lemma 28. There exists a constant C , depending only on K and the list of parameters (25), such that, for T < 1, there holds

T 0 ∂ x u f L ∞ (F ) dt ≤ C √ T .
Proof. We use the L ∞ bound on ∂ x u f , see (41), and the bound (Q 3 ) on the density ρ f . It holds, by integrating on (0, T ),

T 0 ∂ x u f L ∞ (F ) dt ≤ C µ f T 0 Σf H 1 (Ω) dt + T max ρ ∞ /2≤r≤2 ρ∞ p f (r).
Inequality (Q 5 ) on the stress tensor leads to

T 0 ∂ x u f L ∞ (F ) dt ≤ C µ f √ T K + T max ρ ∞ /2≤r≤2 ρ∞ p f (r),
which gives the expected bound for T < 1.

The continuity of the velocities [START_REF] Feireisl | Homogenization and singular limits for the complete Navier-Stokes-Fourier system[END_REF] implies then that

d dt (x - k+1 -x + k ) = u(x - k+1 ) -u(x + k ) ≤ ∂ x u f L ∞ (F ) |x - k+1 -x + k |.
From Lemma 28, we can choose T small (depending only on C ) such that there holds:

|F 0 k | 2 < |F k | < 2|F 0 k |,
on (0, T ), which leads to the desired estimate.

Strict version (Q 3 ) of (Q 3 ). Since the fluid density ρ f satisfies a continuity equation associated with the velocity u f on the fluid domains F k which are transported by the same velocity field u f , a classical estimate on (0, T ) provides

(78) ( min x∈F 0 ρ 0 f ) × exp - T 0 ∂ x u f L ∞ (F ) dt ≤ ρ f (t, x) ≤ (max x∈F 0 ρ 0 f ) × exp T 0 ∂ x u f L ∞ (F ) dt ,
Lemma 28 allows to bound the exponential terms in (78) for small time. Namely, for T small (depending on C ) it holds, on [0, T ],

ρ f (t, x) ∈ 1 2 min x∈F 0 ρ 0 f , 2 max x∈F 0 ρ 0 f .
Then the assumption (24) on the initial fluid density allows to deduce a strict version of estimate (Q 3 ).

Strict version (Q 4 ) of (Q 4 ). Applying (34), we obtain:

sup [0,T ] F µ f |∂ x u f | 2 2 dx + µ g N k=1 | Ṙk | 2 R k + T 0 F ρ f |∂ t u f + u f ∂ x u f | 2 dx + N k=1 m k (|c k | 2 + | Rk | 2 ) ≤ sup [0,T ] 2 N k=1 κ k | Ṙk | R k + F p f (ρ f )|∂ x u f |dx + T 0 F µ f |∂ x u f | 3 2 dx + T 0 N k=1 2κ k | Ṙk | 2 R 2 k + µ g | Ṙk | 3 R 2 k + E 1 ,
with a constant E 1 depending only on the list of parameters (25). To proceed, we detail now the controls of the five remaining terms on the right-hand side.

Concerning the first line in the right-hand side, the first term can be rewritten with (IC 0 ):

N k=0 κ k | Ṙk | R k ≤ 1 M ∞ N k=0 | Ṙk | √ R k 1 N √ R k .
The Cauchy-Schwarz inequality gives then

N k=0 κ k | Ṙk | R k ≤ 1 M ∞ √ µ g µ g N k=0 | Ṙk | 2 R k 1/2 N k=0 1 N 2 R k 1/2 .
The first parenthesis can be bounded by K using (Q 4 ) and the second one by 1/ √ d ∞ thanks to (Q 1 ) so that (79)

N k=0 κ k | Ṙk | R k ≤ 1 M ∞ √ µ g √ K √ d ∞ .
The control of the second pressure term relies on (Q 3 )-(Q 4 ) and a Cauchy-Schwarz inequality :

F p f (ρ f )|∂ x u f |dx ≤ √ 2 √ µ f F µ f |∂ x u f | 2 2 dx 1/2 max [ρ ∞ /2,2 ρ∞] p f √ 2, ≤ 2 √ µ f max [ρ ∞ /2,2 ρ∞] p f √ K. ( 80
)
As for the term on the second line in the right-hand side of (34), we decompose as follows

T 0 F µ f |∂ x u f | 3 2 dx ≤ sup [0,T ] F µ f |∂ x u f | 2 2 T 0 ∂ x u f L ∞ (F ) ,
where the first term can be bounded by K according to (Q 4 ). The second one is bounded using Lemma 28. It follows that (81)

T 0 F µ f |∂ x u f | 3 2 dx ≤ C √ T K.
We now turn to the first term on the third line. Applying a standard L ∞ -L 1 Hölder inequality allows to bound this term by

T 0 N k=1 κ k | Ṙk | 2 R 2 k ≤ 1 M ∞ µ g N max k∈{1,...,N } 1 R k L ∞ (0,T ) T 0 µ g N k=1 | Ṙk | 2 R k .
The L ∞ norm can be handled by the bound (Q 1 ) and the integral term by (76). It follows that (82)

T 0 N k=1 κ k | Ṙk | 2 R 2 k ≤ E 0 M ∞ µ g d ∞ .
It remains to bound the second term on the third line. In this respect, we decompose the nonlinear term

| Ṙk | 3 R 2 k = | Ṙk | R k | Ṙk | 3/2 R 3/4 k | Ṙk | 1/2 R 1/4 k
and apply a L ∞ -L 4/3 -L 4 Hölder inequality to yield:

N k=1 | Ṙk | 3 R 2 k ≤ max k∈{1,...,N } | Ṙk | R k N k=1 | Ṙk | 2 R k 3/4 N k=1 | Ṙk | 2 R k 1/4
.

Integrating over (0, T ) we obtain again with a L 2 -L 4 -L 4 Hölder inequality that:

T 0 N k=1 | Ṙk | 3 R 2 k ≤   T 0 max k∈{1,...,N } | Ṙk | R k 2   1/2   T 0 N k=1 | Ṙk | 2 R k 3   1/4 T 0 N k=1 | Ṙk | 2 R k 1/4
.

Corollary 9 and inequality (Q 1 ) allow to control the first term on the right-hand side. Indeed,

µ g max k∈{1,...,N } | Ṙk | R k ≤ 1 M ∞ d ∞ + C 1 Σf 2 H 1 (Ω) + 1 M ∞ d ∞ N k=1 m k (| Rk | 2 + |c k | 2 ) 1/2 .
Taking the L 2 -norm in time and applying a triangular inequality and (Q 5 ) provides

  T 0 max k∈{1,...,N } | Ṙk | R k 2   1/2 ≤ 1 µ g √ T 1 M ∞ d ∞ + C 1 K 1 + 1 M ∞ d ∞ .
As the second term is concerned, it holds

  T 0 N k=1 | Ṙk | 2 R k 3   1/4 ≤ T 1/4 sup [0,T ] N k=1 | Ṙk | 2 R k 3/4
, which can be handled thanks to (Q 4 ), leading to

  T 0 N k=1 | Ṙk | 2 R k 3   1/4 ≤ T 1/4 K µ g 3/4
. Now the bound (76) gives

T 0 N k=1 | Ṙk | 2 R k 1/4 ≤ E 0 µ g 1/4
.

To sum up, it finally yields (83)

T 0 N k=1 | Ṙk | 3 R 2 k ≤ 1 µ 2 g T 1/4 K 3/4 E 1/4 0 √ T 1 M ∞ d ∞ + C 1 K 1 + 1 M ∞ d ∞ .
Plugging (79)-( 80)-( 81)-( 82)-( 83) into (34), it yields (84) sup

[0,T ] F |∂ x u f | 2 2 dx + N k=1 µ g | Ṙ2 k | R k ≤ 2 √ K M ∞ µ g d ∞ + 1 √ µ f max [ρ ∞ /2,2 ρ∞] p f (r) √ K + C √ T K + E 0 M ∞ µ g d ∞ + 2 µ 2 g T 1/4 K 3/4 E 1/4 0 √ T 1 M ∞ d ∞ + C 1 K 1 + 1 M ∞ d ∞ + E 1 .
If one considers K > 1 and T < 1, defining

C 1 = 2 1 M ∞ µ g d ∞ + 1 √ µ f max [ρ ∞ /2,2 ρ∞] p f (r) , C 2 = 2 µ 2 g E 1/4 0 1 M ∞ d ∞ + C 1 1 + 1 M ∞ d ∞ ,
the previous inequality writes (85) sup

[0,T ] F |∂ x u f | 2 2 dx + N k=1 µ g | Ṙ2 k | R k ≤ C 1 √ K + E 0 M ∞ µ g d ∞ + C √ T K + T 1/4 K 5/2 C 2 + E 1 .
Introduce λ ∈ (0, 1/2) to be fixed later on and set:

K ∞ := 4|C 1 | 2 λ 2 + 1 M ∞ µ g d ∞ E 0 2λ + E 1 1 -2λ .
When K > K ∞ , the sum of the two first terms on the right-hand side of (85) are bounded by λK. Now taking T small enough, for instance T = min{(λ/2|C |) 2 , (λ(K 3/2 C 2 ) -1 /2) 4 }, the sum of the third and fourth term can be bounded by λK as well. Finally, the right-hand side is bounded according to sup

[0,T ] F |∂ x u f | 2 2 dx + N k=1 µ g | Ṙ2 k | R k ≤ E 1 + 2λK < K since λ < 1/2 and C 1 > 0.
Strict version (Q 5 ) of (Q 5 ). In order to prove this estimate, we can adjust with the parameter λ. First, thanks to Proposition 6 and to the bounds (Q 1 ) and (IC 0 ), there exists C > 0, depending in particular on M ∞ and d ∞ , such that (86)

T 0 Σf 2 H 1 (Ω) dt ≤ C 0 T 0 Σ f 2 H 1 (F ) + N k=1 m k | Rk | 2 + |c k | 2 + N k=1 µ 2 g | Ṙk | 2 R k + κ 2 k R k dt.
The second term of the right-hand side can be bounded with the help of (34) by bounding the right-hand side of (34) as in the previous analysis on (Q 4 ). This entails:

(87)

T 0 N k=1 m k | Rk | 2 + |c k | 2 dt ≤ E 1 + 2λK.
The third term is controlled using (76). The last term can be bounded by T /((M ∞ ) 2 d ∞ ). Therefore, this inequality becomes (88

) T 0 Σf 2 H 1 (Ω) dt ≤ C 0 T 0 Σ f 2 H 1 (F ) dt + (E 1 + 2λK) + µ g E 0 + T (M ∞ ) 2 d ∞ .
Let us now focus on the first term. We use the definition (8) of Σ f and the momentum equation ( 7) to write

T 0 Σ f 2 H 1 (F ) dt = T 0 µ f ∂ x u f -p f (ρ f ) 2 L 2 (F ) dt + T 0 ρ f (∂ t u f + u f ∂ x u f ) 2 L 2 (F ) dt.
Thanks to (76) and (Q 3 ), the first term can be bounded. For the second term, one has

T 0 ρ f (∂ t u f + u f ∂ x u f ) 2 L 2 (F ) dt ≤ ρ∞ T 0 F ρ f |∂ t u f + u f ∂ x u f | 2 dt,
and this right-hand side actually appears in (34) and thus, the previous estimate obtained to prove (Q 4 ) can be used. This provides

T 0 Σ f 2 H 1 (F ) dt ≤ 2µ f E 0 + 2T max [ρ ∞ , ρ∞] p f 2 + ρ∞ (E 1 + 2λK).
Gathering the previous estimates and after rearrangement,

T 0 Σf 2 H 1 (Ω) dt ≤ C 0 2µ f E 0 + (ρ ∞ + 1)E 1 + µ g E 0 + 2 max [ρ ∞ , ρ∞] p f 2 + 1 (M ∞ ) 2 d ∞
T + 2(ρ ∞ + 1)λK holds. Now starting from Proposition 8, a similar estimate can be proved for Σg . Then, using again (87), one finally have

T 0 Σf 2 H 1 (Ω) + Σg 2 H 1 (Ω) + m k | Rk | 2 + |c k | 2 dt ≤ C 1 + C 2 T + C 3 λK,
where C 1 , C 2 and C 3 are positive and independent of N , T , λ and K. To conclude, it suffices to choose λ sufficiently small so that λ ≤ (4C 3 ) -1 and K ≥ 4C 1 . We can then take T smaller if necessary so that C 2 T ≤ K/4.

with ρ 0 f ∈ H 1 (Ω). To extend this property, we construct again extensions ũf of fluid velocity-field u f and Σf of stress tensor Σ f with the same formula as in (47) and (35) respectively. We then construct ρf (89)

   ∂ t ρf + ũf ∂ x ρf = - ρf µ f
Σf + p f (ρ f ) , on (0, T ) × Ω, ρf (0, .) = ρ 0 f , on Ω, By (Q 5 ) with Proposition 41 for the fluid part and (Q 1 )-(Q 5 ) with Corollary 9 for the bubble part, we obtain that ũf ∈ L 2 (0, T ; W 1,∞ (Ω)) with Σf ∈ L 2 (0, T ; H 1 (Ω)). Consequently, we have a unique solution (89) which solves (6) on F. By uniqueness of the solution to [START_REF] Bresch | A multi-fluid compressible system as the limit of weak solutions of the isentropic compressible Navier-Stokes equations[END_REF] in the regularity class of classical solutions (see [START_REF] Hillairet | Analysis of compressible bubbly flows. Part I: Construction of a microscopic model[END_REF]), we have thus ρf = ρ f on F(t) for t ∈ (0, T ). So, our proof reduces to computing bounds for ρf . First, we prove that there exists T 0 ≤ T such that we can control ρf L ∞ (Ω) explicitly on (0, T 0 ). By the method of characteristics and the explicit value of p f :

ρf (t, .) L ∞ (Ω) ≤ ρ 0 f L ∞ (Ω) exp 1 µ f T 0 Σf L ∞ (Ω) + a f ρf (t, .) γ f L ∞ (Ω) dt .
The bound (Q 5 ) coupled with the embedding of H 1 (Ω) in L ∞ (Ω) allows to control the stress tensor norm by K. If ρf (t, .) L ∞ (Ω) ≤ 2 ρ 0 f (t, .) L ∞ (Ω) , it yields ρf (t, .)

L ∞ (Ω) ≤ ρ 0 f L ∞ (Ω) exp 1 µ f √ T K + 2a f T ρ 0 f (t, .) γ f L ∞ (Ω) .
By a standard continuation argument, we construct then a time-interval (0, T 0 ) depending only on K, a f , γ f and ρ 0 f (t, .) L ∞ (Ω) so that:

ρf (t, .) L ∞ (Ω) ≤ 2 ρ 0 f L ∞ (Ω)
for t < T 0 .

We focus now on ∂ x ρf . For this, we apply a space derivative to (89):

           ∂ t (∂ x ρf ) + ∂ x ũ∂ x ρf = - ρf µ f ∂ x Σf - 1 µ f Σf + p f (ρ f ) + ρf p f (ρ f ) ∂ x ρf
(∂ x ρf )(0, •) = ∂ x ρ 0 f . For simplicity, we denote from now on Y := ∂ x ρf . We multiply the previous equation by 2Y , leading to

∂ t (Y 2 ) + ∂ x (ũY 2 ) = -2Y ρf µ f ∂ x Σf -Y 2 A
where A denotes ∂ x ũ + 2 µ f Σf + κ f (γ f + 1)(ρ f ) γ f . Let first bound the right-hand side by a standard Cauchy-Schwarz/Minkowski inequality:

Ω -2Y ρf µ f ∂ x Σf -Y 2 A dx ≤ 1 µ f ρf ∂ x Σf 2 L 2 (Ω) + 1 µ f + A L ∞ (Ω) Y 2 L 2 (Ω) .
Going back to the PDE for Y 2 , the L 2 norm of ∂ x ρf can be bounded as

∂ x ρf 2 L 2 (Ω) ≤ ∂ x ρ 0 f 2 L 2 (Ω) + 1 µ f T 0 ρf ∂ x Σf 2 L 2 (Ω) dt × exp T µ f + T 0 ∂ x ũ L ∞ (Ω) + 2 µ f Σf L ∞ (Ω) + κ f (γ f + 1) ρf γ f L ∞ (Ω)
.

  then we can apply Proposition 10 and the solution satisfies (Q 1 )-(Q 5 ) on [0, T ]. It remains to show that we can continue the solution beyond [0, T ]. The inequalities (Q 1 )-(Q 5 ) being strict, the large inequalities (Q 1 )-(Q 5

()

  )-(23), there exist sequences of initial bubble center/radii ((c k=1,...,N ) N ∈N and masses (m (N ) k ) k=1,...,N so that: i) (IC 0 )-(IC 1 )-(IC 2 ) are satisfied with M ∞ and d ∞ independent of N , ii) there exist H 1 (Ω) extensions ( f (

  on the fluid domain F (N ) is the restriction of the global microscopic density ρ(N) f : (53) ρ

  By Proposition 21, it suffices to compute the different terms of Equation (62). In the first case, we consider b(z, ξ, ν) = z. It yields b = ᾱf and b

4 . 1 .

 41 ,g (0, r) = 0.Computing the associated limits, one recovers the first equation of (64). The second equation is true by construction. The equation on fg is obtained in the same way, taking b(z, ξ, ν) = ν. Finally the phasic mass conservation laws are derived using b(z, ξ, ν) = zξ and b(z, ξ, ν) = (1 -z)ξ respectively. Momentum equation and closure laws. We proceed with the derivation of the momentum equation.Proposition 23. Let ρ = ᾱf ρf + ᾱg ρg be the mixture density. The mixture momentum equation reads(67) ∂ t (ρū) + ∂ x (ρū 2 ) = ∂ x (ᾱ f Σf + ᾱg Σg ), with(68)∂ x ū = ᾱf µ f Σf + p f (ρ f ) + ᾱg µ g Σg + p g (ρ g ) +γs fg , and (69) Σf = Σg . Proof. Let us consider the momentum equation in the fluid domain and multiply it by a test function w ∈ C ∞ c ((0, T ) × Ω). It yields T 0 F (N ) (t)

Acknowledgement. The first author acknowledges support of the Institut Universitaire de France and project "SingFlows" ANR-grant number: ANR-18-CE40-0027. This paper was finished while M.H. was benifiting a "subside à savant" from Université Libre de Bruxelles. He would like to thank the mathematics department at ULB for its hospitality.

, ( Σ(N) g (x) + p g (ρ (N ) g ))r + γs /2 ∂ r β = 0.

Moreover, the sequence of applications t → S (N ) t is compact in C([0, T ];

)). As a consequence, there exists Sg ∈ C([0, T ]; P( Ω × [d ∞ , 1/d ∞ ])) such that, up to the extraction of a subsequence, S (N ) , β → Sg , β , in C([0, T ]),

Proof. Let us first prove that, for any

, β is uniformly equicontinuous. The definition of β (N ) and (70) enable to write

For legibility, we drop the exponent (N ) in c k and R k here and in what remains of the proof. By construction, c k and R k belong to H 2 (0, T ) and thus are in

Recall that, by Corollary 3, ũ(N) f is bounded in L 2 ((0, T ); H 1 (Ω)) and then in L 2 ((0, T ); C( Ω)). N ) is bounded in H 1 (0, T ) and then uniformly equicontinuous. The compactness result and the existence of Sg is then straightforward. It remains to check that S (N ) t verifies equation (70). This comes directly from (72) where the term Ṙk /R k is replaced using [START_REF] Hillairet | On Baer-Nunziato multiphase flow models[END_REF]:

Actually, the dependence of the measures Sg,t with respect to the space variable x can be precised:

In other words, we have:

One has for every t ∈ (0, T )

The second term of the right-hand side can be bounded by

and then tends to 0 when N → +∞ (see (Q 1 )). The first term can be written as

which provides a bounded sequence in L ∞ ((0, T ) × Ω), by (Q 1 ). Therefore, there exists Sβ ∈ L ∞ ((0, T ) × Ω) such that, up to the extraction of a subsequence,

Then, letting N → ∞ in the previous equality yields (73).

We obtain then the following limiting equation for Sg,t :

Proposition 26. The limit Sg,t defined in Proposition 24 satisfies the equation

Proof. To obtain a time-evolution PDE for Sg,t , we go back to Equation (71) with a tensorised test function β(x, r) = β x (x)β r (r), which writes

The first term of the right-hand side can be dealt using the strong convergence of (ũ (N ) ) N in L 2 ((0, T ), L 2 (Ω)). Since it is bounded in L 2 ((0, T ), H 1 (Ω)), the sequence (ũ (N ) ) N converges also in L 2 ((0, T ), C( Ω)) by interpolation. The weak convergence of (S (N ) t

) N in C([0, T ], P( Ω × R + )) together with this strong convergence gives

Appendix A. Proof of Proposition 10

In the whole section, we consider T > 0 and (ρ f , u f , (c k , R k ) k=1,...,N ) is classical solution to ( 6)-( 13) on (0, T ), satisfying (Q 1 )-(Q 5 ).

To start with, we recall that Corollary 3 applies. With (Q 1 ), these estimates yield:

with a constant E 0 depending only on the list of parameters (25).

Strict version (

as soon as the α, β and γ are nonnegative, it follows from Corollary 9, (IC 0 ) and the bounds (

, and then:

The last term can be bounded by √ K ∞ according to (Q 5 ). Integrating on the time interval (0, t), t < T , it yields

Considering a smaller time T , only depending on

Finally the Assumption (IC 1 ) on the initial radii leads to the desired estimate (Q 1 ). We note in passing that we obtained the following lemma:

Lemma 27. There exists a constant K depending on µ g , d ∞ , M ∞ , C 0 and K, such that:

The proof of this lemma is a straightforward application of (77) and is left to the reader.

Appendix B. Analysis of the density equation

This section is devoted to the proof of the following proposition:

Proposition 29. Assume that T > 0 and (ρ f , u f , (c k , R k ) k=1,...,N ) is a classical solution to (6)-( 13) on (0, T ) -complemented with initial conditions constructed as in ( 16)-( 19) -that satisfies (Q 1 )-(Q 5 ). Then, there exists strictly positive constants K 1 and T 1 depending only on the list of parameters (25) and K such that

Again, the main difficulty in obtaining this proposition is to make the constant K 1 independent of the parameter N. For this, we proceed as in Section 4 and interpret ρ f on F(t) as the trace of some global density defined on Ω. We notice here that, by assumption, we already have this property initially since we set ρ f (0, •) = ρ 0 f on F 0

All the terms can be controlled using (Q 3 ) and (Q 5 ), except

dt. This latter term can be bounded using lemmas 28 and 27 (corresponding respectively to the contributions of ∂ x ũ L ∞ (F ) and ∂ x ũ L ∞ (Ω\F ) ). Then, for a sufficiently small time T 1 ≤ T 0 ,

so that on (0, T 1 ) :

This completes the proof.