# Physical Relaxation Terms for Compressible Two-Phase Systems

Didier Bresch, Cosmin Burtea, Frédéric Lagoutière LAMA – UMR5127 CNRS, Bat. Le
Chablais, Campus Scientifique, 73376 Le Bourget du Lac, France, Email:
<EMAIL_ADDRESS>Research of D.B. was partially supported by the
SingFlows project, grant ANR-18-CE40-0027Université de Paris, Institut de
Mathématiques de Jussieu-Paris Rive Gauche (UMR 7586), F-75205, Paris, France,
Email<EMAIL_ADDRESS>Lyon, Université Claude Bernard Lyon 1,
CNRS UMR5208, Institut Camille Jordan, F.–69622 cedex, France, Email:
<EMAIL_ADDRESS>
###### Abstract

In this note, we propose the first mathematical derivation of a macroscopic
Baer-Nunziato type system for compressible two-phase flows allowing two
pressure state laws depending on the different phases. By doing so, we extend
the results obtained by the first author and M. Hillairet [Annales ENS (2019)]
to cover this important physical situation. A relaxation term in the mass
fraction equation is obtained without closure assumptions contrarily to
theoretical-physics literature dedicated to mixture theory, see for instance
“Thermo-Fluid dynamics of Two Fluid Flows” by M. Ishii. The relaxation
parameter is linked to the viscosities of the different fluids (which may be
small for applications) and the relaxed quantity is linked to the laws chosen
at interfaces of the two-fluid system at a mesoscale. As this paper is
intended for a large audience, we start with two formal arguments leading to
the effective system. This provides formal procedures which could be useful
for people working in environmental studies or industrial applications to
understand how mixture models may be derived. Then we propose two mathematical
proofs: One with a continuous approach (Hoff solutions for compressible NS
equations with pressure depending on two transported quantities and the
associated two-scale limit) and the other with a semi-discrete approach (ODE
deduced from the discretization and its continuous limit). Finally, owing to
the later approach, we describe some numerical experiments by comparing
mesoscopic discretization and macroscopic discretization. This shows that
theoretical proofs may be helpful in this topic to design appropriate
numerical schemes.

Keywords: Homogenization, Compressible Navier-Stokes, Multi-fluid systems,
Defect measures, Hoff solutions, Two-Scale limit, Numerical schemes.

Résumé

Dans cette note, nous étendons les résultats obtenus par le premier auteur et
M. Hillairet [Annales ENS (2019)] pour couvrir des situations physiques
importantes: Plus précisément, nous proposons la première justification
mathématique d’un système de type Baer-Nunziato pour écoulements biphasés
compressibles permettant deux lois d’état de pression dépendant des
différentes phases et un terme de relaxation qui s’obtient sans hypothèse de
fermeture contrairement aux ouvrages physiques dédiés à la théorie des
mélanges comme le livre «Thermo-Fluid dynamic of Two Fluid Flows »par M.
Ishii. La grandeur du paramètre de relaxation est liée aux viscosités des
différents fluides (qui peuvent être petites) et la quantité relaxée est liée
aux lois choisies aux interfaces du système bi-fluide à l’échelle
mésoscopique. Comme cet article est destiné à un large public, nous commençons
par deux arguments formels menant au système limite: Cela fournit des
procédures formelles qui pourraient être utiles pour d’autres disciplines, par
exemple, dans des études environnementales ou des applications industrielles.
Nous proposons ensuite deux preuves mathématiques: l’une avec une approche
continue (solutions à la Hoff d’équations de Navier-Stokes compressible avec
pression dépendant de deux quantités transportées et sa limite à deux
échelles) et l’autre avec une approche semi-discrète (Système d’ODE obtenu de
la discrétisation et sa limite continue). Enfin, grâce à ces approches, nous
présentons des illustrations numériques comparant la discrétisation
mésoscopique et la discrétisation macroscopique. Ceci montre que les approches
théoriques peuvent permettre sur ce sujet de définir des schémas numériques
appropriés.

###### Contents

  1. 1 Introduction.
  2. 2 Mathematical models and objectives.
    1. 2.1 The mesoscale system
  3. 3 Formal asymptotics
    1. 3.1 How to derive the relaxed equation (8)1 without closure assumptions?
      1. 3.1.1 Continuous level – Two scale analysis
      2. 3.1.2 Discrete level – scale analysis
    2. 3.2 How to derive the effective viscosity and the effective pressure?
  4. 4 Mathematical results
    1. 4.1 Continuous approach – Hoff solutions and its two-scale limit
    2. 4.2 The semi-discrete approach – ODEs and its continuous limit
  5. 5 Numerical illustrations
    1. 5.1 Mesoscopic discretization
    2. 5.2 Macroscopic discretization
    3. 5.3 Experiments
      1. 5.3.1 Case with equal viscosities
      2. 5.3.2 Case with different viscosities

Justification d’un terme de relaxation physique pour systèmes bi-phases.

Dedicated to the memory of Andro Mikelić

## Version française abrégée

Dans cette note, on propose de présenter l’obtention rigoureuse, en dimension
un d’espace, d’un modèle de mélange à deux phases à une seule vitesse avec
deux lois de pression différentes suivant la phase. Ce travail fait suite à un
travail du premier auteur avec M. Hillairet (voir [6]) sur la justification de
modèles de type Baer-Nunziato avec une pression commune aux deux phases. Il
demande, pour une généralisation à deux pressions, de nouveaux résultats
d’existence de solutions à la Hoff sur Navier-Stokes compressible avec
pression dépendant de deux quantités satisfaisant chacune une équation de
transport. Il est ensuite nécessaire d’introduire un paramètre
d’homogénéisation et de justifier une asympotique à deux échelles vers un
modèle cinétique sous-jacent: L’obtention du modèle limite bi-phases provient
alors de la caractërisation des mesures de défauts sous hypothèse initiale.

Plus précisemment, on considère deux fluides compressibles (dont on connaît la
loi d’état pour chacun) régis par les équations de Navier-Stokes avec chacun
leur viscosité supposée constante. En considérant que le mélange de ces
fluides est la limite de situations où les fluides sont séparés par des
interfaces (approche multi-fluides à l’échelle mesoscopique) mais à une
échelle $\varepsilon$ de plus en plus fine, on obtient un système vérifié par
la limite $\varepsilon\to 0$, pour lequel on a une formule pour calculer la
pression du mélange, ainsi qu’une équation pour la fraction volumique de
chaque constituant avec un terme de relaxation totalement justifié faisant
apparaître les différentes lois de pressions. Nous nous focalisons dans cette
note sur le cas à deux composants mais le résultat se généralise au cas à
plusieurs composant sans complication.

Pour commencer la note, nous présentons en section 2 les modèles mathématiques
qu’ils soient mésoscopique ou macroscopique ainsi que les objectifs de preuves
pour lier le modèle macroscopique bi-phases (8) au système mesoscopique bi-
fluides (5)–(7) avec les données initiales (4). Nous présentons ensuite en
section 3 deux approches formelles différentes (continue ou discrète)
permettant d’obtenir le modèle limite. Nous commençons par une approche
continue au travers d’une analyse à deux échelles de type WKB. Nous continuons
ensuite par une approche discrète qui serait plus proche de ce que nous
pouvons rencontrer dans le cadre d’applications. Ces résultats précisent les
calculs formels que l’on peut trouver par exemple dans [20], [19], [1] en
évitant l’hypothèse de fermeture formelle pour le terme de relaxation. Cette
partie peut avoir un réel intérêt pour le lecteur intéressé par une
compréhension formelle de l’obtention du modèle de mélange à partir du modèle
mésoscopique avec interfaces.

Nous proposons ensuite dans la section 4 deux approches théoriques de
justification mathématiques totalement complémentaires: I) Une approche
d’homogénéisation sur le modèle mesoscopique sous sa forme continue où un
théorème de type existence de solutions à la Hoff (pour NS compressible avec
pression dépendant de deux paramètres transportés) et contrôle uniforme en le
petit paramètre d’homogénéisation est nécessaire pour le passage à la limite.
II) Une approche d’homogénéisation sous un angle plutôt discret qui consiste à
considérer le système semi-discrétisé associé, d’en déduire un théorème
d’existence locale pour le système d’ODE, de montrer un contrôle uniforme de
versions discrètes des contrôles à la Hoff puis de passer à la limite vers le
modèle continu justifiant ainsi une analyse formelle par un des auteurs dans
[17]. Cette partie peut intéresser le lecteur motivé par une justification
rigoureuse au travers d’une approche continue ou d’une approche discrétisée.

Nous terminons ensuite par la section 5 où nous montrons comment, grâce aux
résultats théoriques, nous pouvons définir de bons schémas numériques et nous
illustrons le tout par des résultats de simulation. Cette note est écrite pour
présenter aux lecteurs, qu’ils soient mathématiciens, physiciens ou ingénieurs
un nouveau cadre permettant d’aborder formellement, mathématiquement et
numériquement l’obtention de certains modèles multiphases à partir de modèles
multi-fluides mesoscopiques. Ce travail doit être vu comme une partie de la
monographie complète [4] sur les systèmes multiphasiques avec deux pressions
que les auteurs avec M. Hillairet se proposent de rédiger. Cette monographie
rappellera les travaux précédents théoriques (de [8], [5], [6], [6], [3],
[12], [21], [22] par exemple) ainsi que les travaux de modélisation et aspects
numériques importants ([20], [10], [19], [1], [9] et références contenues par
exemple). Elle donnera également le détail des preuves et discutera quelques
extensions possibles.

##  1 Introduction.

In this note, we propose to present the rigorous justification, in one space
dimension, of a single velocity two-phase mixing model with two different
pressure laws depending on each phase. This work follows a work by the first
author with M. Hillairet (see [6]) on the justification of Baer-Nunziato type
models with a pressure common to both phases. This asks, for a generalization
to two possible different pressure state laws, new results of existence of
solutions à la Hoff on compressible Navier-Stokes with pressure depending on
two densities each satisfying a transport equation. It is then necessary to
introduce a homogenization parameter and to justify a two scales asymptotic
towards an underlying kinetic model: The obtaining of the two-phase limit
model then comes from the characterization of the defect measures under
initial hypothesis. More precisely, we consider two compressible fluids (of
which we know the pressure state law for each) governed by the Navier–Stokes
equations, each with their constant viscosity that we can write as a single
compressible system with a pressure dependent on the averaged-density and a
color function that is used to distinguish the fluid phase, both of which are
advected by the flow. By considering that the mixture of these fluids is the
limit of situations where the fluids are separated by interfaces (multi-fluid
approach at the mesoscopic scale) but at an $\varepsilon$ scale more and more
fine, we obtain a system verified by the limit $\varepsilon\to 0$, for which
we have a formula to calculate the pressure of the mixture, as well as an
equation for the volume fraction of each component with a completely justified
relaxation term showing the different laws of pressures. We focus in this note
on the two-component case but the result generalizes to the multi-component
case.

To start the note, we present in section 2 the mathematical models whether
they are mesoscopic or macroscopic as well as the proof objectives to link the
two-phase macroscopic model (8) to the bi-fluid mesoscopic system (5) – (7)
with the initial data (4). We then present in section 3 two different formal
approaches (continuous or discrete) allowing to obtain the limit model. We
start with a continuous approach through a two-scale analysis of WKB type. We
then continue with a discrete approach which would be closer to what we may
encounter in applications. These results render mathematically rigorous the
formal computations that can be found for example in [20], [19], [1] by
avoiding the formal closure hypothesis for the relaxation term. This part may
have a real interest for the reader interested in a formal understanding of
obtaining the mixture model from the mesoscopic model with interfaces.

We then propose in the section 4 two completely complementary theoretical
approaches to mathematical justification: I) A homogenization approach on the
mesoscopic model under its continuous form where a theorem of type existence
of solutions à la Hoff (for compressible Navier-Stokes with pressure depending
on two transported parameters) and uniform control in the small parameter is
necessary for the passage to the limit. II) A homogenization approach from a
rather numerical analysis-angle which consists in considering the associated
semi-discrete system, hence to deduce a local existence theory for the ODE
system, to show a uniform control of discrete versions of the energy estimates
à la Hoff then to pass to the limit towards the continuous model thus
justifying a formal analysis by one of the authors in [17]. This part is
oriented towards the reader interested in a rigorous justification through a
continuous approach or a discrete approach.

Owing to the theoretical results, in the last Section, we show how to define
appropriate numerical schemes and we present the results of some numerical
experiments which better illustrate our results. This note is addressed to a
wide audience and aims at presenting a new framework allowing to approach
formally, mathematically and numerically the derivation of some multiphase
models with different pressure laws from mesoscopic descriptions.

This work should be seen as an introduction to the complete monograph [4] that
the authors propose to write in collaboration with M. Hillairet. This
monograph will recall previous important theoretical results (for instance
[3], [21], [22], [12], [8], [5], [6], [6] for example) as well as results
dealing with mathematical modelling and important numerical aspects ([20],
[10], [19], [1], [9] and references therein for example). It will also give
full details of the proofs and discuss some possible extensions.

##  2 Mathematical models and objectives.

###  2.1 The mesoscale system

In this part, we detail the process towards the mathematical justification of
the Baer-Nunziato model for multiphase mixture with a physical relaxation
term.

Our main assumption is to consider a two-fluid mixture (multiphase fluid) as
the limit of a sequence of situations where the two fluids are separated by
sharps interfaces (this configuration is sometimes referred as a multifluid
material). At a physical level this translates as follows: we assume that it
is possible to zoom-in in the mixture at a very fine scale, which is referred
to as being the mesoscopic scale.We assume that at this scale the two
constituents are separated which allows us to model their behaviour with the
well-known Navier-Stokes system plus an equation describing the evolution of
the interface. In order to recover a macroscopic effective model, we have to
zoom-out which mathematically is translated as a propagation of oscillations
problem and its averaging.

Then, our second assumption is that the multifluid is governed by a Navier-
Stokes type system where the viscosity and the pressure depend on the fluid,
dependence that can be modelled by introducing a binary color parameter, which
takes its values depending on the phase. More precisely, we consider a mixture
of two compressible fluids, which will be referred in the following as $+$ and
$-$ occupying a domain $\Omega$. In the following, physical quantities that
characterize the phase $+/-$ will be explicitly denoted by a $+/-$ lower
script.

Let us denote by $\Omega_{+}\left(t\right)$ the volume occupied by fluid $+$
and by $\Omega_{-}\left(t\right)$ the volume occupied by fluid $-$ at time
$t$. We assume that initially we have
$\Omega_{+}\left(0\right)\cap\Omega_{-}\left(0\right)=\emptyset$ and
$\Omega_{+}\left(0\right)\cup\Omega_{-}\left(0\right)=\Omega$. Lest us denote
by $c(t,\cdot)$ the characteristic function of $\Omega_{+}\left(0\right)$ (and
note that thus $c$ is also the volume fraction or the mass fraction of fluid
$+$). Let us denote $\rho_{+}$ and $\rho_{-}$ the densities of the two phases.
The densities are initially defined in $\Omega_{+}\left(t\right)$ respectively
in $\Omega_{-}\left(t\right)$. We extend $\rho_{+}$ and $\rho_{-}$ by $0$ in
$\Omega$ and slightly abusing the notation we will still call them
$\rho_{\pm}$. We see that owing to the separation hypothesis and the
definition of $c$ we have that

$\rho=c\rho_{+}+\left(1-c\right)\rho_{-}$ (1)

and

$\rho_{+}=c\rho\text{, }\rho_{-}=\left(1-c\right)\rho.$ (2)

a.e. in $\Omega$.

We assume that fluid $+$ has a constant viscosity $\mu_{+}>0$ and fluid $-$
has a constant viscosity $\mu_{-}>0$, and denote by $\mu(c)$ the viscosity of
the multifluid:

$\mu(c)=c\mu_{+}+(1-c)\mu_{-}.$

We assuming that both fluids are barotropic fluids with pressure laws
$p_{+}(\rho)$ and $p_{-}(\rho)$, and denote by $p(c,\rho)$ the pressure law of
the multifluid:

$p(c,\rho)=c\,p_{+}(\rho)+(1-c)p_{-}(\rho).$

We denote by $u(t,x)$ the velocity at time $t$ and position $x$. The above
notations and definitions lead to model the multifluid Cauchy problem with the
following mesoscopic system:

$\left\\{\begin{array}[c]{l}\partial_{t}c+u\,\partial_{x}c=0\hbox{ with
}c\,(1-c)=0,\\\ \partial_{t}\rho+\partial_{x}(\rho u)=0,\\\ \partial_{t}(\rho
u)+\partial_{x}(\rho
u^{2})-\partial_{x}(\mu(c)\partial_{x}u)+\partial_{x}p(c,\rho)=0,\\\
\mu(c)=(c\,\mu_{+}+(1-c)\mu_{-}),\qquad
p(c,\rho)=c\,p_{+}(\rho)+(1-c)p_{-}(\rho).\\\ \rho_{|t=0}=\rho_{0},\qquad
c_{|t=0}=c_{0}\hbox{ with }c_{0}(1-c_{0})=0,\qquad
u_{|t=0}=u_{0},\end{array}\right.$ (3)

Let take a moment to resume what has been done up to the present. We made a
number of hypothesis that allowed us to propose a system of equations
governing the evolution of a mixture at a mesoscopic scale. Although very
interesting at a mathematical level this system is very unlikely to be of use
in practical applications. Thus, the necessity to un-zoom back at macrospic
scale comes naturally. Using similar systems as
$\left(\text{\ref{System}}\right)$, performing a formal procedure of averaging
and assuming some closure assumptions, several calculations may be encountered
in physics-papers such as [11], [15] or modelling or numerical papers such as
[20], [1]. We describe in the following lines how to obtain mathematically
equations at a macroscopic scale. The key observation from [7, 8] is that when
un-zooming, an observer will witness rapid oscillations between the zones
occupied by the two phases. Thus, a multi-phase fluid can be represented by
the limit of solutions that widely oscillate in space. More precisely,
following the formalism introduced in [6], [7], we obtain a macroscopic
effective system for multiphase fluids by introducing a parameter
$\varepsilon$ describing the oscillation scale. The idea is to consider a
sequence of solutions of $\left(\text{\ref{System}}\right)$ generated by a
sequence of initial data widely oscillating in space and to analyse its
limiting behaviour. A typical example of such initial data is given by

$c_{0}^{\varepsilon}(x)=c_{0}(x/\varepsilon),\qquad\rho_{0}^{\varepsilon}(x)=c_{0}(x/\varepsilon)\rho_{+,0}(x)+(1-c_{0}(x/\varepsilon))\rho_{-,0}(x)$
(4)

where $c_{0}:\Omega\rightarrow\\{0,1\\}$ is a fixed profile and $\rho_{0,+}$,
$\rho_{0,-}$ are bounded initial data and where the initial velocity field
$u_{0}^{\varepsilon}(x)=u_{0}(x)\in H^{1}(\Omega)$. Thus, one considers
$(c^{\varepsilon},\rho^{\varepsilon},u^{\varepsilon})_{\varepsilon>0}$ a
sequence of solutions (in the sense of Hoff) of the bifluid system mentioned
previously i.e.

$\left\\{\begin{array}[c]{l}\partial_{t}c^{\varepsilon}+u^{\varepsilon}\partial_{x}c^{\varepsilon}=0,\\\
\partial_{t}\rho^{\varepsilon}+\partial_{x}(\rho^{\varepsilon}u^{\varepsilon})=0,\\\
\partial_{t}(\rho^{\varepsilon}u^{\varepsilon})+\partial_{x}(\rho^{\varepsilon}{u^{\varepsilon}}^{2})-\partial_{x}(\mu(c^{\varepsilon})\partial_{x}u^{\varepsilon})+\partial_{x}p(c^{\varepsilon},\rho^{\varepsilon})=0,\\\
\mu(c^{\varepsilon})=(c^{\varepsilon}\mu_{+}+(1-c^{\varepsilon})\mu_{-}),\\\
p(c^{\varepsilon},\rho^{\varepsilon})=c^{\varepsilon}p_{+}(\rho^{\varepsilon})+(1-c^{\varepsilon})p_{-}(\rho^{\varepsilon}),\end{array}\right.$
(5)

with

$\left\\{\begin{array}[c]{l}c^{\varepsilon}|_{t=0}=c_{0}^{\varepsilon}\mbox{
such that }c_{0}^{\varepsilon}(1-c_{0}^{\varepsilon})=0,\\\
\rho^{\varepsilon}|_{t=0}=\rho_{0}^{\varepsilon},\\\
u^{\varepsilon}|_{t=0}=u_{0}^{\varepsilon},\end{array}\right.$ (6)

where the initial data are supposed to oscillate wildly in space. The property
$c_{0}^{\varepsilon}(1-c_{0}^{\varepsilon})=0$ is important because it implies
that $c^{\varepsilon}(1-c^{\varepsilon})=0$ at any time and that thus
guarantees that it is legal to compute the pressure as

$p(c^{\varepsilon},\rho^{\varepsilon})=c^{\varepsilon}p_{+}(\rho^{\varepsilon})+(1-c^{\varepsilon})p_{-}(\rho^{\varepsilon}),$
(7)

the phases are “pure” at any point of $\Omega^{\varepsilon}$. The macroscopic
bi-phase model is then derived letting $\varepsilon$ tend to $0$ and computing
the limit system. We prove that, if the Cauchy data (6) converges weakly to
$(\alpha_{0},\rho_{0},u_{0})$ (note that the property
$\alpha_{0}(1-\alpha_{0})$ is of course lost), then, up to subsequence
$(c^{\varepsilon},\rho^{\varepsilon},u^{\varepsilon})$ converges to
$(\alpha,\rho,u)$ such that

$\left\\{\begin{array}[c]{l}\partial_{t}\alpha+u\,\partial_{x}\alpha=\dfrac{\alpha\left(1-\alpha\right)}{\alpha\mu_{-}+(1-\alpha)\mu_{+}}(F_{\pm}-F_{\mp}),\\\
\partial_{t}\rho+\partial_{x}(\rho u)=0,\\\ \partial_{t}(\rho
u)+\partial_{x}(\rho
u^{2})-\partial_{x}(\mu_{\mathrm{eff}}\partial_{x}u)+\partial_{x}p_{eff}=0,\\\
\rho=\alpha\rho_{+}+(1-\alpha)\rho_{-},\\\
\mu_{\mathrm{eff}}=\dfrac{\mu_{+}\mu_{-}}{\alpha\mu_{-}+(1-\alpha)\mu_{+}},\\\
p_{\mathrm{eff}}=\dfrac{\alpha
p_{+}(\rho_{+})\mu_{+}+(1-\alpha)p_{-}(\rho_{-})\mu_{-}}{\alpha\mu_{-}+(1-\alpha)\mu_{+}},\\\
F_{\pm}=-\mu_{\pm}\partial_{x}u+p_{\pm}(\rho_{\pm}).\end{array}\right.$ (8)

Observe that the resulting homogeneized system contains only physical
quantities such has the two viscosities, the two pressure laws and the volume
fraction. We did not suppose a closure assumption with a relaxing parameter as
usually in the physical literature.

In the sequel, we present two approaches in order to derive solutions of the
bifluid system and to obtain at the limit the same macroscopic equations
namely

  * •

A so called continuous approach which follows the ideas introduced by D.
Bresch and M. Hillairet in [6], [7], [8]. This consist in using the techniques
introduced by D. Hoff in order to construct solutions of the bifluid system.
The main novelty here is the presence of the new unknow $c$ allowing to take
into account different pressure laws. The bifluid system is a compressible
Navier-Stokes system with a viscosity given in terms of $c$ and a pressure law
depending on $c$ and $\rho$.

  * •

A semi-discretized approach which will render rigorous the result by one of
the authors (see [17]). This approach is interesting because it allows to
numerically compute solutions for the macrospic mixture model
$\left(\text{\ref{TwoP}}\right)$ with a numerical scheme designed on the
mesoscopic system $\left(\text{\ref{meso}}\right)$.

In Section 3, we show that we may obtain the system
$\left(\text{\ref{TwoP}}\right)$ from the system
$\left(\text{\ref{meso}}\right)$ first by a formal two-scale analysis at a
continuous level and then by a scale analysis at a discrete level . In Section
4, we present the mathematical results justifying the derivation from the
bifluid description to the biphase system: continuous approach and then semi-
discrete approach. In the last section 5, we illustrate the results stated in
the paper. More precisely, we design two numerical schemes: one to approximate
the mesoscopic system, that is to say System (5) with a Cauchy datum (6), and
one to approximate the macroscopic system (8) with any Cauchy datum.

##  3 Formal asymptotics

###  3.1 How to derive the relaxed equation (8)1 without closure assumptions?

####  3.1.1 Continuous level – Two scale analysis

In the homogenization process, two-scale asymptotic is a natural tool (see for
instance [2], [21] and [22] and references cited therein). Let us recall
formally, how it may be used in the compressible setting using some
ellipticity properties of the effective flux and using the renormalized
approach for the transport equation. Such formal computation has been
performed in [8] but for the reader’s convenience we rewrite here such
calculation. In the bifluid setting, to get the asympotic system we formally
assume the following ansatz

$c\,(t,x)=c\,(t,\frac{t}{\varepsilon},x,\frac{x}{\varepsilon})$
$\rho(t,x)=c\,(t,\frac{t}{\varepsilon},x,\frac{x}{\varepsilon})\rho_{+}^{\varepsilon}(t,x)+\bigl{(}1-c\,(t,\frac{t}{\varepsilon},x,\frac{x}{\varepsilon})\bigr{)}\rho_{-}^{\varepsilon}(t,x)$
$u(t,x)=u_{0}(t,\frac{t}{\varepsilon},x,\frac{x}{\varepsilon})+\varepsilon
u_{1}(t,\frac{t}{\varepsilon},x,\frac{x}{\varepsilon})+\varepsilon^{2}u_{2}(t,\frac{t}{\varepsilon},x,\frac{x}{\varepsilon})+O(\varepsilon^{3})$

assuming

$\rho_{\pm}^{\varepsilon}(t,x)=\rho^{0}_{\pm}(t,x)+O(\varepsilon),\qquad
c(t,\tau,x,y)\in\\{0,1\\}\hbox{ \it a.e. }.$

Plugging these informations in the bifluid system, we first get

$\partial_{\tau}c+u_{0}\partial_{y}c=0$ (9)

and

$\partial_{t}c+u_{0}\partial_{x}c+u_{1}\partial_{y}c=0.$ (10)

The first equation provides the behavior of $c$ on a cell. This equation is
compatible with the assumption that $c$ is an indicator function. Averaging
with respect to the fast variable $y$ the second equation, we get the
following equation

$\partial_{t}\alpha+\overline{u_{0}\partial_{x}c}=-\overline{u_{1}\partial_{y}c}$
(11)

with $\alpha=\overline{c}$ where we denote temporarily with the bar average
with respect to $y$ on a cell. Let us now quickly recall the different steps.
Plugging now the ansatz in the momentum equation we get

$\partial_{y}((c\mu_{+}+(1-c)\mu_{-})\partial_{y}u_{0})=0.$

Multiplying this equation and integrate with respect to the $(t,\tau,x,y)$ and
using that

$c\mu_{+}+(1-c)\mu_{-}\geq\min(\mu_{+},\mu_{-})>0$

we get that

$\partial_{y}u_{0}=0$

and therefore coming back to the equation on $c$ and averaging with respect to
$y$ that $\alpha$ does not depend on $\tau$. Looking now the main part of the
momentum equation, we get

$\rho^{0}(\partial_{\tau}u_{0}+u_{0}\partial_{y}u_{0})+\partial_{y}p^{0}=\partial_{y}((c\mu_{+}+(1-c)\mu_{-})\partial_{y}u_{1})\\\
+\partial_{x}((c\mu_{+}+(1-c)\mu_{-})\partial_{y}u_{0})+\partial_{y}((c\mu_{+}+(1-c)\mu_{-})\partial_{x}u_{0}).$

Multiplying by $\partial_{\tau}u_{0}$ and integrating with respect to the
$(t,\tau,x,y)$, we get that $\partial_{\tau}u_{0}=0.$ Thus we get

$\partial_{y}((c\mu_{+}+(1-c)\mu_{-})\partial_{y}u_{1})-\partial_{y}p^{0}=-\partial_{y}((c\mu_{+}+(1-c)\mu_{-})\partial_{x}u_{0}).$

Denoting $\mu=c\mu_{+}+(1-c)\mu^{-}$, this gives

$\mu\partial_{y}u_{1}-p^{0}=\overline{\mu\partial_{y}u_{1}}-\overline{p^{0}}+(\overline{\mu}-\mu)\partial_{x}u^{0}.$

Note that using the expression of $\mu$ and the equation of $\alpha$, we get

$\mu\partial_{y}u_{1}=(p^{0}-\overline{p^{0}})-(\mu-\overline{\mu})\partial_{x}u_{0}+(\mu_{+}-\mu_{-})(\partial_{t}\alpha+u_{0}\partial_{x}\alpha)$

which may be rewritten as

$c\,\partial_{y}u_{1}=\frac{c}{\mu}\Bigl{(}(p^{0}-\overline{p^{0}})-(\mu-\overline{\mu})\partial_{x}u_{0}+(\mu_{+}-\mu_{-})(\partial_{t}\alpha+u_{0}\partial_{x}\alpha)\Bigr{)}$

and therefore after calculation

$\overline{c\partial_{y}u^{1}}=\frac{\alpha(1-\alpha)}{\mu_{+}}\bigl{(}(p_{+}(\rho^{0}_{+})-p_{-}(\rho_{-}^{0}))-\partial_{x}u_{0}(\mu_{+}-\mu_{-})+\alpha(1-\frac{\mu_{-}}{\mu_{+}})(\partial_{t}+u_{0}\partial_{x})\alpha$

using that

$\overline{u^{1}\partial_{y}c}=-\overline{c\partial_{y}u^{1}}$

and inserting in (11), we get after simple calculations

$\partial_{t}\alpha+u_{0}\partial_{x}\alpha=\frac{\alpha(1-\alpha)}{\alpha\mu_{-}+(1-\alpha)\mu_{+}}\Bigl{(}(p_{+}(\rho_{+}^{0})-p_{-}(\rho_{-}^{0}))-\partial_{x}u_{0}(\mu_{+}-\mu_{-})\Bigr{)}$

####  3.1.2 Discrete level – scale analysis

In this section we propose a formal procedure to derive the mixture model and
we explain with very simple arguments why the volume fraction should satisfy
the equation

$\partial_{t}\alpha+u\partial_{x}\alpha=\frac{\alpha(1-\alpha)}{\mu}(p_{+}-p_{-}).$

We denote by $D_{t}\alpha$ the Lagrangian time derivative of $\alpha$:

$D_{t}\alpha=\partial_{t}\alpha+u\partial_{x}\alpha$

(this derivative will also be denoted $\dot{\alpha_{+}}$ in the rest of the
paper). Consider a situation where the fluids are separated (say, at a small
scale $\varepsilon$), and a point $x(t)\in{\mathbb{T}}$ at an interface
between fluid $+$ on its right and fluid $-$ on its left, for any time $t$.
Denote by $x_{+}(t)$ the center of the zone of pure fluid $+$ on the right of
$x(t)$, by $x_{-}(t)$ the center of the zone of pure fluid $-$ on the left of
$x(t)$, and

$\varepsilon_{+}(t)=x_{+}(t)-x(t),\varepsilon_{-}(t)=x(t)-x_{-}(t)$

which are supposed to be small.

We define $\alpha(t)$ by

$\alpha(t)=\varepsilon_{+}(t)/(\varepsilon_{+}(t)+\varepsilon_{-}(t)).$

Indeed this quantity represents the local (at point $x(t)$) volume fraction of
fluid $+$. Obviously one has

$D_{t}\varepsilon_{+}(t)=u(t,x_{+}(t))-u(t,x(t))$

and

$D_{t}(\varepsilon_{+}+\varepsilon_{-})(t)=u(t,x_{+}(t))-u(t,x_{-}(t)).$

This allows to write

$D_{t}\alpha(t)=\frac{(\varepsilon_{+}+\varepsilon_{-})D_{t}\varepsilon_{+}-\varepsilon_{+}D_{t}(\varepsilon_{+}+\varepsilon_{-})}{(\varepsilon_{+}+\varepsilon_{-})^{2}}\\\
=\frac{\varepsilon_{-}(u(t,x_{+}(t))-u(t,x(t)))-\varepsilon_{+}(u(t,x(t))-u(t,x_{-}(t)))}{(\varepsilon_{+}+\varepsilon_{-})^{2}}$
(12)

The regularity of the solution is expected to be the following: at any time
$t$, the pressure and the space derivative of the velocity should be
continuous in space in each pure region (namely, in
$(x_{-}-\varepsilon_{-},x_{-}+\varepsilon_{-})$ and in
$(x_{+}-\varepsilon_{+},x_{+}+\varepsilon_{+})$), but not at the point $x(t)$.
At this point, what is expected is that the effective flux
$p-\mu\partial_{x}u$ is continuous (and this continuity in space stands for
the law of reciprocal forces of Newton). In the case where the two viscosity
coefficients are equal, the formal computation is straightforward. Thus we
propose to begin by assuming this equality, and to obtain the general law for
$\alpha$ in a second stage.

  * •

Case where $\mu_{+}=\mu_{-}=\mu$
The continuity of the effective flux together with the regularity on pure
zones expresses as

$p_{-}(t)-\mu\frac{u(t,x(t))-u(t,x_{-}(t))}{\varepsilon_{-}}=p_{+}(t)-\mu\frac{u(t,x_{+}(t))-u(t,x(t))}{\varepsilon_{+}}\\\
+r(\varepsilon_{-}+\varepsilon_{+}),$

where $p_{\pm}(t)$ denotes $p_{\pm}(\rho(t,x_{\pm}(t)))$ and $r$ is a function
such that $r(x)\rightarrow 0$ as $x\rightarrow 0^{+}$. This rewrites

$p_{+}(t)-p_{-}(t)=\mu\frac{\varepsilon_{-}(u(t,x_{+}(t))-u(t,x(t)))-\varepsilon_{+}(u(t,x(t))-u(t,x_{-}(t)))}{\varepsilon_{+}\varepsilon_{-}}\\\
+r(\varepsilon_{-}+\varepsilon_{+}),$

and, thanks to (12) and letting $\varepsilon_{\pm}$ go to $0$,

$p_{+}(t)-p_{-}(t)=\mu\frac{(\varepsilon_{+}+\varepsilon_{-})^{2}}{\varepsilon_{+}\varepsilon_{-}}D_{t}\alpha_{+}=\frac{\mu}{\alpha_{+}(1-\alpha_{+})}D_{t}\alpha_{+},$

which is exactly what is stated in this paper.

  * •

Case where $\mu_{+}\neq\mu_{-}$
In this general case, it is convenient to define the approximate space
derivatives of the velocity $d_{-}(t)$ and $d_{+}(t)$

$d_{-}(t)=\frac{u(t,x(t))-u(t,x_{-}(t))}{\varepsilon_{-}(t)},\quad
d_{+}(t)=\frac{u(t,x_{+}(t))-u(t,x(t))}{\varepsilon_{+}(t)}.$

Equipped with this, we can rewrite (12) as

$D_{t}\alpha(t)=\frac{\varepsilon_{-}\varepsilon_{+}}{(\varepsilon_{-}+\varepsilon_{+})^{2}}(d_{+}(t)-d_{-}(t)).$

We would like to express the limit, as $\varepsilon_{-}+\varepsilon_{+}$ tends
to $0$, of the right-hand side term as a function of the limit quantities.
Remark that $u$ is intended to converge strongly but $\partial_{x}u$ only
weakly, thus $d_{+}(t)$ and $d_{-}(t)$ are not approximations of
$\partial_{x}u(t,x(t))$: however
$\dfrac{\varepsilon_{-}}{\varepsilon_{-}+\varepsilon_{+}}d_{-}+\dfrac{\varepsilon_{+}}{\varepsilon_{-}+\varepsilon_{+}}d_{-}$
is intended to converge toward $\partial_{x}u$. The limit of the right-hand
side should be expressed as a function of the limit unknowns $\alpha_{+}$,
$\alpha_{-}$, $p_{+}$, $p_{-}$, $\partial_{x}u$… We already know that
$\dfrac{\varepsilon_{-}\varepsilon_{+}}{(\varepsilon_{-}+\varepsilon_{+})^{2}}$
converges to $\alpha_{+}\alpha_{-}$. It remains to treat the term
$d_{+}-d_{-}$. As $\mu_{+}d_{+}-\mu_{-}d_{-}$ is intended to converge to
$p_{+}-p_{-}$, it is quite natural to try to write

$d_{+}-d_{-}=a(\mu_{+}d_{+}-\mu_{-}d_{-})+(1-a\mu_{+})d_{+}-(1-a\mu_{-})d_{-}$

with $a\in\mathbb{R}$ such that there exists $b\in\mathbb{R}$ satisfying

$1-a\mu_{+}=b\alpha\quad\mbox{and}\quad 1-a\mu_{-}=-b(1-\alpha),$

in which case one would have

$d_{+}-d_{-}\longrightarrow_{\varepsilon_{-}+\varepsilon_{+}\rightarrow
0}=a(p_{+}-p-)+b\partial_{x}u.$

The linear system in $a$ and $b$ has a unique solution,
$a=\dfrac{1}{(1-\alpha)\mu_{+}+\alpha\mu_{-}}$ and
$b=\dfrac{\mu_{-}-\mu_{+}}{(1-\alpha)\mu_{+}+\alpha\mu_{-}}$, which finally
gives

$D_{t}\alpha_{+}=\frac{\alpha(1-\alpha)}{(1-\alpha)\mu_{+}+\alpha\mu_{-}}(p_{+}-p_{-}-(\mu_{+}-\mu_{-})\partial_{x}u),$

which is exactly the first equation in (8)

###  3.2 How to derive the effective viscosity and the effective pressure?

It is interesting to note that the effective viscosity is computed in the same
way as one computes the effective diffusion when considering homogenisation
for 1d elliptic equations. The main remark is to use for the compressible the
ellipticity of the effective flux $F=p(\rho)-\mu\partial_{x}u.$ Looking at the
order $0$ terms in the momentum equation and averaging with respect to $y$ we
get using that $u^{0}$ does not depend on $y$ that

$\rho^{0}(\partial_{t}u^{0}+u^{0}\partial_{x}u^{0})-\partial_{x}(\overline{\mu^{0}}\partial_{x}u^{0})-\partial_{x}(\overline{\mu^{0}\partial_{y}u^{1}})+\nabla\overline{p(\rho)}=0.$

It remains to use the expression of $\overline{\mu^{0}\partial_{y}u^{1}}$
found previously to get the homogeneized momentum equation related to
$\alpha_{\pm}$ and $u^{0}$.

##  4 Mathematical results

###  4.1 Continuous approach – Hoff solutions and its two-scale limit

First of all, we can adapt the techniques introduced by D. Hoff in order to
obtain the well possedness of system (3). More precisely, let us assume the
folllowing initial data conditions

$\left\\{\begin{array}[c]{l}c_{0}\in L^{\infty}({\mathbb{T}})\qquad\hbox{ such
that }\qquad 0\leq m=\inf c_{0}\leq c_{0}(x)\leq M=\sup c_{0}\qquad\hbox{\it
a.e. on }(0,1)\\\ \rho_{0}\in L^{\infty}({\mathbb{T}})\qquad\hbox{ such that
}\qquad 0<\inf\rho_{0}\leq\rho_{0}(x)\qquad\hbox{ a.e. on }(0,1)\\\
G(\rho_{0},c_{0})=\rho_{0}{\displaystyle\int_{1}^{\rho_{0}}}p(s,c_{0})/s^{2}<+\infty\\\
u_{0}\in H^{1}({\mathbb{T}})\end{array}\right.$ (13)

and let us assume that $p$ and $\mu$ satisfies for all
$(\rho,c)\in[0,+\infty]\times[0,1]$ the following:

$\left\\{\begin{array}[c]{l}p(\rho,c)\geq 0\quad\hbox{ and
}\quad\mu(c)\geq\mu_{\mathrm{min}}>0\\\ \exists C_{0}>0\hbox{ such that
}p(\rho,c)\leq C_{0}(\rho+G(\rho,c))\hbox{ where
}G(\rho,c)=\rho\int_{1}^{\rho}p(\rho,c)/s^{2}\,ds\\\
\rho\partial_{1}p(\rho,c)\in
L_{\mathrm{loc}}^{\infty}([0,+\infty]\times[0,1]).\end{array}\right.$ (14)

Then we have the following

###### Theorem 4.1

Consider two functions
$p\in\mathcal{C}^{1}\left([0,\infty)\times\left[m,M\right]\right)$ and
$\mu\in\mathcal{C}^{1}\left(\left[0,M\right]\right)$ verifying Hypothesis
(14). Let $\left(c_{0},\rho_{0},u_{0}\right)$ satisfy (13), then there exists
a unique weak solution $\left(c,\rho,u\right)$ of system (3) with intial data
$\left(c_{0},\rho_{0},u_{0}\right)$ with

$\begin{array}[c]{l}c,\rho\in
C(\mathbb{[}0,\infty\mathbb{)};L^{q}\left(\mathbb{T}^{1}\right))\hbox{ for all
}q<+\infty,\\\ u\in
L^{\infty}\left(\mathbb{[}0,\infty\mathbb{)};H^{1}(\mathbb{T}^{1})\right),\text{
}\partial_{x}u\in
L^{2}\left(\mathbb{[}0,\infty\mathbb{)}\times\mathbb{T}^{1}\right).\end{array}$

Moreover, for any $T>0$, there exists a constant $C\left(T\right)$ which
depends only on the norms of the initial data and $T$ such that the following
uniform bounds hold true:

$\displaystyle{\displaystyle\int_{0}^{1}}\frac{\rho
u^{2}}{2}+{\displaystyle\int_{0}^{1}}G\left(\rho,c\right)+{\displaystyle\int_{0}^{t}}{\displaystyle\int_{0}^{1}}\mu(c)(\partial_{x}u)^{2}\leq{\displaystyle\int_{0}^{1}}\frac{\rho_{0}u_{0}^{2}}{2}+{\displaystyle\int_{0}^{1}}G\left(\rho_{0},c_{0}\right),$
(15) $\displaystyle\inf\limits_{x\in\left[0,1\right]}c_{0}(x)\leq
c(t,x)\leq\sup\limits_{x\in\left[0,1\right]}c_{0}(x),$ (16) $\displaystyle
C\left(T\right)^{-1}\leq\rho\left(t,x\right)\leq C\left(T\right),$ (17)
$\displaystyle\frac{1}{2}{\displaystyle\int_{0}^{1}}\mu(c)(\partial_{x}u)^{2}+{\displaystyle\int_{0}^{t}}{\displaystyle\int_{0}^{1}}\rho\dot{u}^{2}\leq
C(T),$ (18)
$\displaystyle\frac{1}{2}{\displaystyle\int_{0}^{1}}\sigma\left(t\right)\rho\dot{u}^{2}+\frac{1}{2}{\displaystyle\int_{0}^{t}}{\displaystyle\int_{0}^{1}}\sigma\left(t\right)\mu(c)\left(\partial_{x}\dot{u}\right)^{2}\leq
C\left(T\right),$ (19)
$\displaystyle\left\|u\right\|_{L^{\infty}((0,1)\times\mathbb{T}^{1})}+\left\|u\right\|_{H^{1}((0,1)\times\mathbb{T}^{1})}\leq
C\left(T\right),$ (20)
$\displaystyle\sigma^{\frac{1}{2}}\left(t\right)\left\|(\partial_{t}u(t),\partial_{x}u(t))\right\|_{L^{\infty}(\mathbb{T})}\leq
C\left(T\right).$ (21)

for all $t\in\left[0,T\right]$ where
$\sigma\left(t\right)=\min\left\\{t,1\right\\}$.

###### Remark 1

It is interesting to note that the previous theorem includes general pressure
laws $p(\rho,c)$ and viscosity $\mu(c)$. It includes for instance the pressure
law $p(\rho,c)=\rho^{\gamma(c)}$ that depends on the density $\rho$ and the
fraction $c$ of each chemical/phase component. As described in [18], the
function $\gamma(c)$ depends on the constant heat capacity ratios of each
component of the multifluid, the pressure $p(\rho,c)$ effectively traces the
thermodynamic "signature" of mixing chemicals/phases in solution.

Let us now consider a sequence of initial data
$\left(\rho_{0}^{\varepsilon},c_{0}^{\varepsilon},u_{0}^{\varepsilon}\right)_{\varepsilon>0}$
that satisfies:

$\left\\{\begin{array}[c]{l}\rho_{0}^{\varepsilon}\in
L^{\infty}\left(\mathbb{T}^{1}\right)\hbox{ with
}0<\inf\limits_{n,x\in\mathbb{T}^{1}}\rho_{0}^{\varepsilon}\left(x\right)\leq\rho_{0}^{\varepsilon}\left(x\right)\leq\sup\limits_{n,x\in\mathbb{T}^{1}}\rho_{0}^{\varepsilon}\left(x\right)\leq
M<+\infty,\\\ c_{0}^{\varepsilon}\,(1-c_{0}^{\varepsilon})=0\qquad\text{ a.e.
on }\mathbb{T}^{1}\hbox{ with }\qquad c_{0}^{\varepsilon}\in[0,1],\\\
{\displaystyle\int_{0}^{1}}\left(c_{0}^{\varepsilon}p_{+}(\rho_{0}^{\varepsilon})+(1-c_{0}^{\varepsilon})p_{-}(\rho_{0}^{\varepsilon})\right)\leq
M,\\\ u_{0}^{\varepsilon}\in H^{1}\left(\mathbb{T}^{1}\right)\text{ such that
}\left\|u_{0}^{\varepsilon}\right\|_{H^{1}}\leq M\end{array}\right.$ (22)

with $M>0$ independent of $\varepsilon$. We note that these assumptions are
satisfied in particular for initial configurations as depicted in (4). The
bounds (22) allow us to conclude that there exists
$\left(\rho_{0},c_{0},u_{0}\right)\in
L^{\infty}\left(\mathbb{T}^{1}\right)\times
L^{\infty}\left(\mathbb{T}^{1}\right)\times H^{1}\left(\mathbb{T}^{1}\right)$
such that

$\rho_{0}^{\varepsilon}\rightharpoonup\rho_{0}\text{ in
}L^{\infty}\left(\mathbb{T}^{1}\right)-w\star,\quad
c_{0}^{\varepsilon}\rightharpoonup\alpha_{0}\text{ in
}L^{\infty}\left(\mathbb{T}^{1}\right)-w\star,\quad
u_{0}^{\varepsilon}\rightharpoonup u_{0}\text{ in
}H^{1}\left(\mathbb{T}^{1}\right).$

Furthermore, given $\varepsilon>0$ the initial data
$(\rho_{0}^{\varepsilon},c_{0}^{\varepsilon},u_{0}^{\varepsilon})_{\varepsilon>0}$
enters the scope of Theorem 4.1. So, we can associate to this initial data a
solution
$(\rho^{\varepsilon},c^{\varepsilon},u^{\varepsilon})_{\varepsilon>0}$ to
(22). Moreover, this sequence satisfies the following uniform bounds on any
interval $[0,T]$ independent of $\varepsilon$:

$\displaystyle{\displaystyle\int_{0}^{1}}\rho^{\varepsilon}(u^{\varepsilon})^{2}+{\displaystyle\int_{0}^{1}}\left(G(\rho^{\varepsilon},c^{\varepsilon})\right)+\mu{\displaystyle\int_{0}^{t}}{\displaystyle\int_{0}^{1}}(\partial_{x}u^{\varepsilon})^{2}\leq
C,$ (23) $\displaystyle
C\left(T\right)^{-1}\leq\rho^{\varepsilon}\left(t,x\right)\leq
C\left(T\right),$ (24)
$\displaystyle\frac{\mu}{2}{\displaystyle\int_{0}^{1}}(\partial_{x}u^{\varepsilon})^{2}+{\displaystyle\int_{0}^{t}}{\displaystyle\int_{0}^{1}}\rho^{\varepsilon}(\dot{u}^{\varepsilon})^{2}+\int_{0}^{t}\int_{0}^{1}\left|\partial_{x}(\mu\partial_{x}u^{\varepsilon}-p^{\varepsilon})\right|^{2}\leq
C(T),$ (25)
$\displaystyle\frac{1}{2}{\displaystyle\int_{0}^{1}}\sigma\left(t\right)\rho^{\varepsilon}(\dot{u}^{\varepsilon})^{2}+\frac{\mu}{2}{\displaystyle\int_{0}^{t}}{\displaystyle\int_{0}^{1}}\sigma\left(t\right)\left(\partial_{x}\dot{u}^{\varepsilon}\right)^{2}\leq
C\left(T\right),$ (26)
$\displaystyle\left\|u^{\varepsilon}\right\|_{L^{\infty}((0,1)\times\mathbb{T}^{1})}+\left\|u^{\varepsilon}\right\|_{H^{1}((0,1)\times\mathbb{T}^{1})}\leq
C\left(T\right),$ (27)
$\displaystyle\sigma^{\frac{1}{2}}\left(t\right)\left\|(\partial_{t}u^{\varepsilon}(t),\partial_{x}u^{\varepsilon}(t))\right\|_{L^{\infty}}\leq
C\left(T\right)$ (28)

with $\mu=\min(\mu_{+},\mu_{-}).$ Using the uniform bounds of
$\left(\text{\ref{ineg(2)H}}\right)$-$\left(\text{\ref{ineg(7)}}\right)$ we
conclude that

$\left\\{\begin{array}[c]{l}\rho{}^{\varepsilon}\rightharpoonup\rho{},\text{
}p{}(\rho{}^{\varepsilon},c^{\varepsilon})\rightharpoonup\Pi{}\text{ in
}L^{\infty}(\mathbb{R}_{+};L^{\infty}(\mathbb{T}^{1})),\\\
u^{\varepsilon}\rightharpoonup u\text{ in
}L^{\infty}(\mathbb{R}_{+};H^{1}(\mathbb{T}^{1})),\\\
Z^{n}:=\mu\partial_{x}u^{\varepsilon}-p(\rho^{\varepsilon},c^{\varepsilon})\rightharpoonup
Z^{\infty}:=\overline{\mu\partial_{x}u}-\Pi\text{ in
}L^{2}(\mathbb{R}_{+};H^{1}(\mathbb{T}^{1})).\end{array}\right.$ (29)

As explained previously, the density $\rho^{\varepsilon}$ and the parameter
$c^{\varepsilon}$ are expected to oscillate widely in space. For this reason,
it is hopeless to obtain stronger convergence on these sequences (and it would
not be a good news for our plan either) than in a weak $L^{p}$-setting. It
turns out that we may characterize weak*-limits of the form if we suppose that
they are given at time $0$. In the following lines we make rigourous the
previous assertion. On the other hand, we need to recover some properties of
the sequence $p(\rho^{\varepsilon},c^{\varepsilon})$ to compute a limit system
satisfied by $(\rho,u,\Pi).$ To this end, we associate to the sequence
$(\rho^{\varepsilon},c^{\varepsilon})_{\varepsilon}$ a sequence of measures on
the space $\mathbb{T}_{x}^{1}\times\mathbb{R}_{\xi}\times\mathbb{R}_{\eta}$
(here $\mathbb{R}_{\xi}$ must be understood as the range of the
$\rho^{\varepsilon}$ while $\mathbb{R}_{\eta}$ is the range of the
$c^{\varepsilon}$). Namely, given $n\geq 0$ and $t\geq 0$, we consider the
measure on $\mathbb{T}_{x}^{1}\times\mathbb{R}_{\xi}\times\mathbb{R}_{\eta}$
as defined by

$\left\langle\Theta{}^{\varepsilon}\left(t\right),b\right\rangle:\overset{def.}{=}\int_{\mathbb{T}^{1}}b\left(x,\rho{}^{\varepsilon}\left(t,x\right),c^{\varepsilon}(t,x)\right)dx,\qquad\forall\,b\in
C_{c}\left(\mathbb{T}_{x}^{1}\mathbb{\times
R}_{\xi}\times\mathbb{R}_{\eta}\right)$ (30)

We have the following proposition.

###### Proposition 4.1

For fixed $\varepsilon>0$ fixed there holds

$\Theta^{\varepsilon}\in
C_{w}([0,\infty);\mathcal{M_{+}(\mathbb{T}}_{x}^{1}\mathcal{\mathbb{\times
R}_{\xi}\times\mathbb{R}_{\eta})})$ (31)

with

$\operatorname{Supp}(\Theta^{\varepsilon}(t))\subset\mathbb{T}_{x}^{1}\times[C\left(t\right)^{-1},C\left(t\right)]\times[0,1]\quad\langle\Theta^{\varepsilon},1\rangle=1.\quad\forall\,t\geq
0,$ (32)

where $C\left(t\right)$ is given by (24).

Once these measures are constructed, the rigorous justification of system (8)
consists in the derivation of the kinetic limit and the characteirzation of
the measures family, it reduces to

###### Theorem 4.2

Up to the extraction of a subsequence, we have
$\Theta^{\varepsilon}\rightharpoonup\Theta$ in
$C_{w}([0,\infty);\mathcal{M_{+}(}\mathbb{T}_{x}^{1}\times\mathbb{R}_{\xi}\times\mathbb{R}_{\eta}))$
where $\Theta$ satisfies

$\partial_{t}\Theta+\partial_{x}\left(u\,\Theta{}\right)-\partial_{\xi}\left(\left(\dfrac{\xi
Z^{\infty}}{\mu{}\left(\eta\right)}+\dfrac{\xi
p{}(\xi,\eta)}{\mu{}\left(\eta\right)}\right)\Theta\right)-\left(\dfrac{Z^{\infty}}{\mu\left(\eta\right)}+\dfrac{p{}\left(\xi,\eta\right)}{\mu{}\left(\eta\right)}\right)\Theta=0$
(33)

with $(u,\Pi,Z^{\infty})$ as defined in (29). Moreover, if there exists
$(\alpha_{0},\rho_{+,0},\rho_{-,0})\in L^{\infty}(\mathbb{T}^{1})$ with
$\alpha_{0}\in\left[0,1\right]$ a.e. and such that

$\langle\Theta(0),b\rangle=\int_{\mathbb{T}^{1}}(\alpha_{0}(x)b\left(x,\rho_{+,0}(x),0\right)+(1-\alpha_{0}(x))b\left(x,\rho_{-,0}(x),1\right))dx,\\\
\quad\forall\,b\in
C(\mathbb{T}_{x}^{1}\times\mathbb{R}_{\xi}\times\mathbb{R}_{\eta})$ (34)

then there exists
$(\alpha,\rho_{+},\rho_{-})\in[L_{loc}^{\infty}([0,\infty);L^{\infty}(\mathbb{T}))\cap
C([0,\infty);L^{1}(\mathbb{T}))]^{4}$ such that, for any $t\geq 0,$
$\alpha\left(t\right)\in\left[0,1\right]$ a.e. and

$\langle\Theta(t),b\rangle=\int_{\mathbb{T}^{1}}(\alpha(t,x)b\left(x,\rho_{+,0}(t,x),0\right)+(1-\alpha(t,x))b\left(x,\rho_{-,0}(t,x),1\right))dx,\\\
\quad\forall\,b\in
C(\mathbb{T}_{x}^{1}\times\mathbb{R}_{\xi}\times\mathbb{R}_{\eta}),$ (35)

Furthermore, $\left(\alpha,\rho_{+},\rho_{-}\right)$ together with $u$
verifies the biphase Baer-Nunziato type system (8).

Remark. It is interesting to understand that Hoff’s solution allows the
density to be only bounded and the velocity to be sufficiently regular to
characterize its evolution in time.

###  4.2 The semi-discrete approach – ODEs and its continuous limit

This part corresponds in some sense to a mathematical justification of the
formal description indicated previous. We consider the following system of
ODEs:

$\left\\{\begin{array}[c]{l}\dot{x}_{j+\frac{1}{2}}=u_{j+\frac{1}{2}},\\\
\dot{c}_{j}=0,\\\ \dfrac{d}{dt}\left(\rho_{j}\Delta x_{j}\right)=0,\\\
\rho_{j+\frac{1}{2}}\Delta
x_{j+\frac{1}{2}}\dot{u}_{j+\frac{1}{2}}+p_{j+1}-p_{j}=\left\\{\mu\left(c_{j+1}\right)\dfrac{u_{j+\frac{3}{2}}-u_{j+\frac{1}{2}}}{\Delta
x_{j+1}}-\mu\left(c_{j}\right)\dfrac{u_{j+\frac{1}{2}}-u_{j-\frac{1}{2}}}{\Delta
x_{j}}\right\\},\\\ \Delta x_{j}=x_{j+\frac{1}{2}}-x_{j-\frac{1}{2}},\text{ \
}\\\ \Delta x_{j+\frac{1}{2}}=\dfrac{\Delta x_{j}+\Delta x_{j+1}}{2},\\\
\rho_{j+\frac{1}{2}}=\dfrac{\rho_{j}\Delta x_{j}+\rho_{j+1}\Delta
x_{j+1}}{\Delta x_{j}+\Delta x_{j+1}}.\end{array}\right.$ (36)

for all $j\in\overline{0,J-1}$ and

$\left\\{\begin{array}[c]{c}c_{0}=c_{J},\\\ \rho_{0}=\rho_{J},\\\
u_{\frac{1}{2}}=u_{J+\frac{1}{2}}.\end{array}\right.$ (37)

From the previous system of equations we also deduce that

$\left\\{\begin{array}[c]{c}\dot{\Delta}x_{j}=u_{j+\frac{1}{2}}-u_{j-\frac{1}{2}},\\\
\dfrac{d}{dt}\left(\rho_{j+\frac{1}{2}}\Delta
x_{j+\frac{1}{2}}\right)=0.\end{array}\right.$ (38)

System $\left(\text{\ref{semi_discrete_semi_lagrangian}}\right)$ is to be
completed with initial data
$\left(x_{j+\frac{1}{2}}^{0}\right)_{j=\overline{-1,J-1}}$,
$\left(c_{j}^{0}\right)_{j=\overline{0,J-1}}$,
$\left(\rho_{j}^{0}\right)_{j=\overline{0,J-1}}$,
$\left(u_{j+\frac{1}{2}}^{0}\right)_{j=\overline{0,J-1}}$ such that:

$\left\\{\begin{array}[c]{l}x_{-\frac{1}{2}}^{0}<\text{
}x_{\frac{1}{2}}^{0}<x_{\frac{3}{2}}^{0}<\cdots<x_{J-\frac{1}{2}}^{0},\\\
0<\underline{c^{0}}=\min\limits_{j\in\overline{0,J-1}}c_{j}^{0}\leq\overline{c^{0}}=\max\limits_{j\in\overline{0,J-1}}c_{j}^{0}<\infty,\\\
0<\underline{\rho^{0}}=\min\limits_{j\in\overline{0,J-1}}\rho_{j}^{0}\leq\overline{\rho^{0}}=\max\limits_{j\in\overline{0,J-1}}\rho_{j}^{0}<\infty,\\\
\left\|\left(u_{j+\frac{1}{2}}\right)_{j\in\overline{0,J-1}}\right\|_{\hat{H}_{J}^{1}}^{2}={\displaystyle\sum\limits_{j=0}^{J-1}}\left|u_{j+\frac{1}{2}}^{0}\right|^{2}\Delta
x_{j}^{0}+{\displaystyle\sum\limits_{j=0}^{J-1}}\left|\dfrac{u_{j+\frac{1}{2}}^{0}-u_{j-\frac{1}{2}}^{0}}{\Delta
x_{j}}\right|^{2}\Delta x_{j}^{0}<\infty.\end{array}\right.$ (H)

System $\left(\text{\ref{semi_discrete_semi_lagrangian}}\right)$ is a system
of ODEs which, owing to the fact that

$c_{j}\left(t\right)=c_{j}\left(0\right),\rho_{j}\left(t\right)=\dfrac{\rho_{j}\left(0\right)\Delta
x_{j}\left(0\right)}{x_{j+\frac{1}{2}}\left(t\right)-x_{j-\frac{1}{2}}\left(t\right)},\rho_{j+\frac{1}{2}}\left(t\right)\Delta
x_{j+\frac{1}{2}}\left(t\right)=\rho_{j+\frac{1}{2}}\left(0\right)\Delta
x_{j+\frac{1}{2}}\left(0\right),$

can be put in the form

$\left\\{\begin{array}[c]{l}\dfrac{dX}{dt}=U,\\\
\dfrac{dU}{dt}=F\left(X,U\right),\\\
\left(X,U\right)=\left(X_{0},U_{0}\right),\end{array}\right.$ (39)

with $X=\left(x_{j+\frac{1}{2}}\right)_{j=0,J-1}\in\mathbb{R}^{J}$ and
$U=\left(u_{j+\frac{1}{2}}\right)_{j=0,J-1}\in\mathbb{R}^{J}$ and
$F:D\rightarrow\mathbb{R}^{J}$ where

$D=\left\\{\left(X,U\right)\in\mathbb{R}^{J}\times\mathbb{R}^{J}:\text{
}x_{\frac{1}{2}}<x_{\frac{3}{2}}<\cdots<x_{J-\frac{1}{2}}\right\\}$

which is an open set of $\mathbb{R}^{J}\times\mathbb{R}^{J}$. Owing to the
fact that $F$ is $C^{\infty}$ on $D$ we obtain via the Cauchy-Lipschitz-Peano
theorem that for any initial data $\left(X_{0},U_{0}\right)\in D$, there
exists a unique maximal solution for $\left(\text{\ref{system_ode}}\right)$

$\left(X,U\right):[0,T_{\max})\rightarrow D$

with $T_{\max}=\infty$ or if $T_{\max}<\infty$ then

$\lim_{t\rightarrow
T_{\max}}\left(X\left(t\right),U\left(t\right)\right)\in\partial D$ (40)

As expected, system $\left(\text{\ref{semi_discrete_semi_lagrangian}}\right)$
shares a lot of properties with its continious version, in particular we can
find a number of apriori estimates related to the mass and energy conservation
which enable us to prove that $\left(\text{\ref{explosion}}\right)$ does not
occur. We are thus able to show that the local solutions can be extended to
global ones. The following energy functionals play a key role: the total mass
Consider the mass

$M\left(t\right)=\text{\
}{\displaystyle\sum\limits_{j=0}^{J-1}}\rho_{j}\left(t\right)\Delta
x_{j}\left(t\right)$ (41)

and basic energy functionals

$E\left(t\right)=\dfrac{1}{2}{\displaystyle\sum\limits_{j=0}^{J-1}}\rho_{j+\frac{1}{2}}\left(t\right)\left|u_{j+\frac{1}{2}}\right|^{2}\Delta
x_{j+\frac{1}{2}}\left(t\right)+{\displaystyle\sum\limits_{j=0}^{J-1}}H(\rho_{j}\left(t\right),c_{j}\left(t\right))\Delta
x_{j}\left(t\right)\\\
+{\displaystyle\int\limits_{0}^{t}}{\displaystyle\sum\limits_{j=0}^{J-1}}\mu\left(c_{j}\left(\tau\right)\right)\left|\dfrac{u_{j+\frac{1}{2}}\left(\tau\right)-u_{j-\frac{1}{2}}\left(\tau\right)}{\Delta
x_{j}\left(\tau\right)}\right|^{2}\Delta x_{j}\left(\tau\right).$ (42)

We formalize in the following theorem our first result.

###### Theorem 4.3

The system (36) along with initial data verifying hypothesis
$\left(\text{\ref{discrete_Initial_data}}\right)$ admits an unique solution
and, moreover, the following estimates hold true uniformly in $J$:

$\left\\{\begin{array}[c]{l}\dfrac{1}{C_{ini}^{1}\left(t\right)}\leq\rho_{j}\left(t\right)\leq
C_{ini}^{1}\left(t\right),\text{ for all }j\in\overline{0,J-1},\\\
\dfrac{1}{C_{ini}^{2}\left(t\right)}\Delta x_{j}^{0}\leq\Delta
x_{j}\left(t\right)\leq\Delta x_{j}^{0}C_{ini}^{2}\left(t\right),\text{ for
all }j\in\overline{0,J-1},\\\ \underline{c^{0}}\leq
c_{j}\left(t\right)\leq\overline{\text{ }c^{0}}\\\
M\left(t\right)=M_{0},\text{ }E\left(t\right)=E_{0}\end{array}\right.$

where $C_{ini}^{1}\left(\cdot\right)$, $C_{ini}^{2}\left(\cdot\right)$ and
$C_{ini}^{3}\left(\cdot\right)$ are strictly positive increasing continuous
functions that depend only on the initial data.

For any $J\in\mathbb{N}^{\ast}$, having constructed the functions
$\left(c_{j},\rho_{j},u_{j+\frac{1}{2}}\right)_{j\in\overline{0,J-1}}$ as
above we consider

$\hat{c}_{J},\hat{\rho}_{J},\hat{u}_{J}:[0,\infty)\times\mathbb{R\rightarrow
R}$

defined by

$\displaystyle\hat{c}_{J}\left(t,x\right)$
$\displaystyle=c_{j}\left(t\right)\text{ if
}x\in[x_{j-\frac{1}{2}}\left(t\right),x_{j+\frac{1}{2}}\left(t\right))\text{,}$
(43) $\displaystyle\hat{\rho}_{J}\left(t,x\right)$
$\displaystyle=\rho_{j}\left(t\right)\text{ if
}x\in[x_{j-\frac{1}{2}}\left(t\right),x_{j+\frac{1}{2}}\left(t\right))\text{,}$
(44) $\displaystyle\hat{u}_{J}\left(t,x\right)$
$\displaystyle=\frac{x-x_{j-\frac{1}{2}}}{\Delta
x_{j}}u_{j+\frac{1}{2}}\left(t\right)+\frac{x_{j+\frac{1}{2}}-x}{\Delta
x_{j}}u_{j-\frac{1}{2}}\left(t\right)\text{ if
}x\in[x_{j-\frac{1}{2}}\left(t\right),x_{j+\frac{1}{2}}\left(t\right))\text{,}$
(45) $\displaystyle\hat{Z}_{J}\left(t,x\right)$
$\displaystyle=\frac{x-x_{j}}{\Delta
x_{j+\frac{1}{2}}}\sigma_{j+1}+\frac{x_{j+1}-x}{\Delta
x_{j+\frac{1}{2}}}\sigma_{j}\text{ if }x\in[x_{j},x_{j+1}).$ (46)

where

$Z_{j}=\mu\left(c_{j}\right)\frac{u_{j+\frac{1}{2}}-u_{j-\frac{1}{2}}}{\Delta
x_{j}}-p\left(c_{j},\rho_{j}\right).$

First, we observe the following remarkable property which will be strongly use
to get the limit

###### Proposition 4.2

The functions $\left(\hat{c}_{J},\hat{\rho}_{J},\hat{u}_{J}\right)$ verify the
following transport equations

$\left\\{\begin{array}[c]{l}\partial_{t}\hat{c}_{J}+\hat{u}_{J}\partial_{x}\hat{c}_{J}=0,\\\
\partial_{t}\hat{\rho}_{J}+\partial_{x}\left(\hat{\rho}_{J}\hat{u}_{J}\right)=0,\end{array}\right.$
(47)

with initial data

$\left\\{\begin{array}[c]{l}\hat{\rho}_{J|t=0}=\hat{\rho}_{J}^{0},\\\
\hat{c}_{J|t=0}=\hat{c}_{J}^{0},\end{array}\right.$

in the sense of distributions.

Of course, the estimates announced in Theorem 4.3 can be used in order to
estimate various norms of the functions
$\left(\hat{c}_{J},\hat{\rho}_{J},\hat{u}_{J},\hat{Z}_{J}\right)$ as in the
Hoff solution proof.

More precisely, one has that

###### Theorem 4.4

Consider discrete initial data verifying the hypothesis (H) along with the
globally defined solution of the system of ODEs (36)–(37). Furthermore,
consider the functions
$\left(\hat{c}_{J},\hat{\rho}_{J},\hat{u}_{J},\hat{Z}_{J}\right)$ given by
(43)–(46). Then

$\left\\{\begin{array}[c]{l}\min\limits_{j\in\overline{0,J-1}}c_{j}^{0}\leq\hat{c}_{J}\left(t,x\right)\leq\max\limits_{j\in\overline{0,J-1}}c_{j}^{0},\\\
\dfrac{1}{C_{ini}^{1}\left(t\right)}\leq\hat{\rho}_{J}\left(t,x\right)\leq
C_{ini}^{1}\left(t\right),\\\
{\displaystyle\int_{0}^{1}}\hat{\rho}_{J}\left|\hat{u}_{J}\right|^{2}\left(t,x\right)dx+{\displaystyle\int_{0}^{1}}H\left(\hat{\rho}_{J}\left(t,x\right),\hat{c}_{J}\left(t,x\right)\right)dx+{\displaystyle\int_{0}^{t}}{\displaystyle\int_{0}^{1}}\mu\left(\hat{c}_{J}\left(\tau,x\right)\right)\left|\partial_{x}\hat{u}_{J}\left(\tau,x\right)\right|^{2}dxd\tau\leq
2E_{0},\\\
\left\|\partial_{x}\hat{u}_{J}\right\|_{L_{t,x}^{2}}+\left\|\partial_{t}\hat{u}_{J}\right\|_{L_{t,x}^{2}}+\min\left\\{1,t\right\\}\left\|\partial_{t}\hat{u}_{J}\left(t\right)\right\|_{L_{x}^{2}}\leq
C\left(t\right),\\\
{\displaystyle\int_{0}^{t}}{\displaystyle\int_{0}^{1}}\left|\partial_{x}\hat{\sigma}_{J}\left(\tau,x\right)\right|^{2}dxd\tau+\min\left\\{1,t\right\\}{\displaystyle\int_{0}^{1}}\left|\partial_{x}\hat{\sigma}_{J}\left(t,x\right)\right|^{2}dx+{\displaystyle\int_{0}^{t}}(\sup_{x\in\left[0,1\right]}\left|\partial_{x}\hat{\sigma}_{J}\left(\tau,x\right)\right|)^{\frac{4}{3}-}d\tau\leq
C\left(t\right)\\\
{\displaystyle\int_{0}^{t}}{\displaystyle\int_{0}^{1}}\sigma\left(\tau\right)\left|\partial_{t}\hat{\sigma}_{J}\left(\tau,x\right)\right|^{2}dxd\tau\leq
C\left(t\right).\end{array}\right.$ (48)

and basic energy functionals

For all $J\in\mathbb{N}^{\ast}$ and all $t\geq 0$, we consider the measure on
$\mathbb{T}_{x}^{1}\times\mathbb{R}_{\xi}\times\mathbb{R}_{\eta}$ defined by

$\left\langle\Theta{}^{J}\left(t\right),b\right\rangle:\overset{def.}{=}\int_{\mathbb{T}^{1}}b\left(x,\hat{\rho}{}^{J}\left(t,x\right),\hat{c}^{J}(t,x)\right)dx,\qquad\forall\,b\in
C_{c}\left(\mathbb{T}_{x}^{1}\mathbb{\times
R}_{\xi}\times\mathbb{R}_{\eta}\right)$ (49)

We may establish immediately similar results as in the previous section
namely:

###### Proposition 4.3

For fixed $J\in\mathbb{N}$ we have that

$\Theta^{J}\in
C_{w}([0,\infty);\mathcal{M_{+}(\mathbb{T}}_{x}^{1}\mathcal{\mathbb{\times
R}_{\xi}\times\mathbb{R}_{\eta})})$ (50)

with

$\operatorname{Supp}(\Theta^{J}(t))\subset\mathbb{T}_{x}^{1}\times[C_{ini}^{1}\left(t\right)^{-1},C_{ini}^{1}\left(t\right)]\times[0,1]\quad\langle\Theta^{J},1\rangle=1.\quad\forall\,t\geq
0,$ (51)

where $C_{ini}^{1}\left(t\right)$ is given by (48).

We can also pass to the limit in the kinetic system to get the following
result

###### Theorem 4.5

Up to the extraction of a subsequence, we have
$\Theta^{J}\rightharpoonup\Theta$ in
$C_{w}([0,\infty);\mathcal{M_{+}(}\mathbb{T}_{x}^{1}\times\mathbb{R}_{\xi}\times\mathbb{R}_{\eta}))$
where $\Theta$ satisfies

$\partial_{t}\Theta+\partial_{x}\left(u\,\Theta{}\right)-\partial_{\xi}\left(\left(\dfrac{\xi
Z^{\infty}}{\mu{}\left(\eta\right)}+\dfrac{\xi
p{}(\xi,\eta)}{\mu{}\left(\eta\right)}\right)\Theta\right)-\left(\dfrac{Z^{\infty}}{\mu\left(\eta\right)}+\dfrac{p{}\left(\xi,\eta\right)}{\mu{}\left(\eta\right)}\right)\Theta=0$
(52)

with $(u,\Pi,Z)$ as defined in (29). Moreover, if there exists
$(\alpha_{0},\rho_{+,0},\rho_{-,0})\in L^{\infty}(\mathbb{T}^{1})$ such that
$\alpha_{0}\in\left[0,1\right]$ a.e. and

$\langle\Theta(0),b\rangle=\int_{\mathbb{T}^{1}}(\alpha_{0}(x)b\left(x,\rho_{+,0}(x),0\right)+(1-\alpha_{0}(x))b\left(x,\rho_{-,0}(x),1\right))dx\\\
\quad\forall\,b\in
C(\mathbb{T}_{x}^{1}\times\mathbb{R}_{\xi}\times\mathbb{R}_{\eta}),$ (53)

then there exists
$(\alpha,\rho_{+},\rho_{-})\in[L_{loc}^{\infty}([0,\infty);L^{\infty}(\mathbb{T}))\cap
C([0,\infty);L^{1}(\mathbb{T}))]^{4}$ such that, for any $t\geq 0$ we have
$\alpha\left(t,\cdot\right)\in\left[0,1\right]$ a.e. and

$\langle\Theta(t),b\rangle=\int_{\mathbb{T}^{1}}(\alpha(t,x)b\left(x,\rho_{+}(t,x),0\right)+(1-\alpha(t,x))b\left(x,\rho_{-}(t,x),1\right))dx\\\
\quad\forall\,b\in
C(\mathbb{T}_{x}^{1}\times\mathbb{R}_{\xi}\times\mathbb{R}_{\eta}).$ (54)

Furthermore, $\left(\alpha,\rho_{+},\rho_{-}\right)$ together with $u$
verifies the biphase Baer-Nunziato system (8).

##  5 Numerical illustrations

In this last section we will illustrate the results stated in the paper. We
design two numerical schemes: one to approximate the mesoscopic system, that
is to say System (5) with a Cauchy datum (6), and one to approximate the
macroscopic system (8) with any Cauchy datum.

###  5.1 Mesoscopic discretization

The numerical scheme we design here consists in a “brute force" discretization
of System (5) where $c_{0}(1-c_{0})=0$.

As the fluids have to remain pure (not mixed) in every cell, because, for
modelling reasons, we want to use only the pure pressure laws (the mixture
pressure law being unknown at this stage), the length of each pure zone has to
be larger than a cell (and, more precisely, has to be large as an integer
number of cells). Here, in the numerical tests, we choose to consider a
numerical initial condition such that the fluid changes from one cell to the
other (but of course this is not a restriction). The problem to achieve the
aim here comes from the so-called numerical diffusion: the discretization of
the $\partial_{t}c+u\partial_{x}c=0$ with a stable scheme usually brings a
certain amount of diffusion, the effect of which being not to preserve the
important feature $c(t,\cdot)(1-c(t,\cdot))=0\quad a.e.$. In order to pass
over this phenomenon, we consider a Lagrangian, or pseudo-Lagrangian111It can
be called pseudo-Lagrangian because, although the solution is actually
expressed in the classical Euler variable, the scheme strongly uses the
Langrange formulation of the system.  scheme in which the cells follow the
fluid in its transport, namely in which the edges of every cell moves at the
fluid velocity. In this Lagrangian frame, the equation for the mass fraction
is $D_{t}c=0$ (recall that $D_{t}=\partial_{t}+u\partial_{x}$).

The spirit of the proposed scheme is the one of staggered schemes: it can be
seen as a modification of the schemes in [16] and [13], this modification
being that the present scheme is more explicit (precisely, the nonlinearity
are time-discretized in a backward Euler way) and that it is a pseudo-Lagrange
scheme. Staggered schemes are schemes in which different unknowns are
associated to different points or cells in the mesh (for example, the density
and the velocity, here). At last, this scheme is a time discretization of the
semi-discrete scheme (36) that was proposed to determine the limit macroscopic
system.

The discretization is the following. Let $J\in\mathbb{N}\setminus\\{0\\}$ be
the number of cells in $[0,1)$. Let $(x_{j-1/2}^{0})_{j=1}^{J}$ be the
collection of cell interface positions at time $0$. One assumes $0\leq
x_{j-1/2}^{0}<x_{j+1/2}^{0}<1$ for any $j=1,\dots,J-1$.

In order to take into account the fact that the problem under consideration is
posed on $\mathbb{T}$ in a simple manner, i.e. without taking care of the
cells and quantities on the boundary, we extend all the data over $\mathbb{R}$
and $\mathbb{Z}$ by periodicity.

The cells itselves are denoted by
$\omega_{j}^{0}=[x_{j-1/2}^{0},x_{j+1/2}^{0})$ for $j\in\mathbb{Z}$. We denote
by $\Delta x_{j}^{n}=x_{j+1/2}^{n}-x_{j-1/2}^{n}$ their length. The minimum
length of these cells is intended to be small (and to tend to $0$ as $J$ tends
to $\infty$ to reach convergence). We also will need the distance between two
centers of consecutive cells: $\Delta x_{j+1/2}^{n}=(\Delta x_{j}^{n}+\Delta
x_{j+1}^{n})/2$.

Each time step of the scheme, given a discrete datum
$\left(x_{j-1/2}^{n},\rho_{j}^{n},c_{j}^{n},u_{j-1/2}^{n}\right)_{j\in\mathbb{Z}}$,
consists in defining appropriately $\Delta t^{n}>0$ and constructing
$\left(x_{j-1/2}^{n+1},\rho_{j}^{n+1},c_{j}^{n+1},u_{j-1/2}^{n+1}\right)_{j\in\mathbb{Z}}$
by the formula

$\left\\{\begin{array}[]{l}\displaystyle{}\rho_{j+1/2}^{n}=\frac{\Delta
x_{j}^{n}\rho_{j}^{n}+\Delta x_{j+1}^{n}\rho_{j+1}^{n}}{\Delta
x_{j}^{n}+\Delta x_{j+1}^{n}},\quad j\in\mathbb{Z},\\\
\displaystyle{}c_{j}^{n+1}=c_{j}^{n},\quad j\in\mathbb{Z},\\\
\displaystyle{}\rho_{j+1/2}^{n}\Delta
x_{j+1/2}^{n}u_{j+1/2}^{n+1}=\rho_{j+1/2}^{n}\Delta
x_{j+1/2}^{n}u_{j+1/2}^{n}-\Delta
t^{n}\left(p(c_{j+1}^{n},\rho_{j+1}^{n})-p(c_{j}^{n},\rho_{j}^{n})\right)\\\
\displaystyle{}\phantom{\displaystyle{}\rho_{j+1/2}^{n+1}\Delta
x_{j+1/2}^{n+1}u_{j+1/2}^{n+1}=}+\Delta
t^{n}\left(\mu(c_{j+1}^{n},\rho_{j+1}^{n})\frac{u_{j+3/2}^{n+1}-u_{j+1/2}^{n+1}}{\Delta
x_{j+1}^{n}}-\mu(c_{j}^{n},\rho_{j}^{n})\frac{u_{j+1/2}^{n+1}-u_{j-1/2}^{n+1}}{\Delta
x_{j}^{n}}\right),\quad j\in\mathbb{Z},\\\
x_{j+1/2}^{n+1}=x_{j+1/2}^{n}+\Delta t^{n}u_{j+1/2}^{n+1},\quad
j\in\mathbb{Z},\\\ \displaystyle{}\Delta
x_{j}^{n+1}=x_{j+1/2}^{n+1}-x_{j-1/2}^{n+1},\quad j\in\mathbb{Z},\\\
\displaystyle{}\Delta x_{j+1/2}^{n+1}=\frac{\Delta x_{j}^{n+1}+\Delta
x_{j+1}^{n+1}}{2},\quad j\in\mathbb{Z},\\\
\displaystyle{}\rho_{j}^{n+1}=\rho_{j}^{n}\frac{\Delta x_{j}^{n}}{\Delta
x_{j}^{n+1}},\quad j\in\mathbb{Z}.\\\ \end{array}\right.$ (55)

In the system above,

  * •

The first equation defines a density associated to the nodes $x_{j+1/2}^{n}$,
density that is used in the third equation,

  * •

The second equation is a (non-diffusive) discretization of $D_{t}c=0$,

  * •

The third equation is the discretization of $\partial_{t}\rho
u+\partial_{x}(\rho u^{2}+p)=\partial_{x}(\mu\partial_{x}u)$: indeed notice
that thanks to the last equation of the system, this third equation rewrites

$\displaystyle{}\rho_{j+1/2}^{n+1}\Delta
x_{j+1/2}^{n+1}u_{j+1/2}^{n+1}=\rho_{j+1/2}^{n}\Delta
x_{j+1/2}^{n}u_{j+1/2}^{n}-\Delta
t^{n}\left(p(c_{j+1}^{n},\rho_{j+1}^{n})-p(c_{j}^{n},\rho_{j}^{n})\right)\\\
\displaystyle{}+\Delta
t^{n}\left(\mu(c_{j+1}^{n},\rho_{j+1}^{n})\frac{u_{j+3/2}^{n+1}-u_{j+1/2}^{n+1}}{\Delta
x_{j+1}^{n}}-\mu(c_{j}^{n},\rho_{j}^{n})\frac{u_{j+1/2}^{n+1}-u_{j-1/2}^{n+1}}{\Delta
x_{j}^{n}}\right),\quad j\in\mathbb{Z},$

which is consistent with the partial differential equation,

  * •

The fourth equation is the translation of the mesh,

  * •

Fifth and sixth equations redefine quantities that are used in the scheme,

  * •

The last equation expresses the conservation of mass in a material volume
$\partial_{t}\rho+\partial_{x}\rho u=0$).

It is possible to prove that if the time step $\Delta t^{n}$ is sufficently
small, $x_{j-1/2}^{n}<x_{j+1/2}^{n}$ for all $j$ implies
$x_{j-1/2}^{n+1}<x_{j+1/2}^{n+1}$ for all $j$.

###  5.2 Macroscopic discretization

For the macroscopic homogenized system (8), we use the same type of scheme.
The only difference is that the volume fraction of fluid $+$ does not satisfy
$\alpha_{+}(1-\alpha_{+})=0$ but

$D_{t}\alpha=\frac{\alpha(1-\alpha)}{\alpha\mu_{-}+(1-\alpha)\mu_{+}}(p_{+}(\rho_{+})-p_{-}(\rho_{-})-(\mu_{+}-\mu_{-})\partial_{x}u).$

In the following we choose to discretize this equation in a forward Euler way
(but a backward Euler scheme has also been tested and validated):

$\alpha_{j}^{n+1}=\alpha_{j}^{n}+\Delta
t\frac{\alpha_{j}^{n}(1-\alpha_{j}^{n})}{\alpha_{j}^{n}\mu_{-}+(1-\alpha_{j}^{n})\mu_{+}}\left(p_{+}(\rho_{+,j}^{n})-p_{-}(\rho_{-,j}^{n})-(\mu_{+}-\mu_{-})\frac{u_{j+1/2}^{n+1}-u_{j-1/2}^{n+1}}{x_{j+1/2}^{n+1}-x_{j-1/2}^{n+1}}\right).$

All the other variables are approximated in a very standard and natural way.

###  5.3 Experiments

We propose two test-cases with $p_{+}(x)=x$ and $p_{-}(x)=x^{2}$. They are
associated with a Cauchy datum of Riemann type:

$\left\\{\begin{array}[]{l}\alpha_{0}(x)=1/2,\quad x\in\mathbb{T}_{x},\\\
\rho_{+}(x)=\rho_{-}(x)=\left\\{\begin{array}[]{l}1/8\mbox{ if
}x\in[0,1/4)\cup[3/4,1),\\\ 2\mbox{ if }x\in[1/4,3/4),\end{array}\right.\\\
u(x)=0,\quad x\in\mathbb{T}_{x},\\\ \end{array}\right.$

and we propose to compare the numerical solutions obtained at time $t=0.1$
with 1000 cells

  * •

with the homogenized scheme of Section 5.2,

  * •

and with the mesoscopic scheme of Section 5.1 by setting

$(\alpha_{j}^{0},\rho_{+,j}^{0},\rho_{-,j}^{0})=\left\\{\begin{array}[]{l}(1,\rho_{j}^{0},0)\mbox{
if }j\mbox{ is even},\\\ (0,0,\rho_{j}^{0})\mbox{ if }j\mbox{ is odd},\\\
\end{array}\right.$

and with a mesh with constant space step, which indeed corresponds in the weak
limit to $\alpha=1/2$. Note that the pressure is largely oscillating in this
initial condition for the mesoscopic system.

In the first test, we take $\mu_{+}=\mu_{-}=0.1$ while in the second one we
choose $\mu_{+}=0.1$ and $\mu_{-}=0.02$. Figures 1 to 5 allow to compare the
density, velocity, pressure and volume fraction. We observe a very good
agreement between the mesoscopic and the macroscopic results. Note that for
the mesoscopic computation, we consider that there is only one density and one
pressure, thus these quantities oscillate very fast (at the scale of the cell,
which is the scale of the mixture). We observe, especially on the zoom of the
density proposed by Figure 2, that these oscillations occur between two
functions that are very close to $\rho_{+}$ and $\rho_{-}$ computed by the
macroscopic scheme. With the mesoscopic scheme, the volume fraction of fluid
$+$ should oscillate between $0$ and $1$. In order to evaluate a volume
fraction of $+$ in the limit mixture, what we here (Figure 5) call
$\alpha_{j}^{n}$ is computed by

$\alpha_{j}^{n}=\frac{c_{j}(x_{j+1/2}^{n}-x_{j-1/2}^{n})+c_{j-1}(x_{j-1/2}^{n}-x_{j-3/2}^{n})/2+c_{j+1}(x_{j+3/2}^{n}-x_{j+1/2}^{n})/2}{x_{j+3/2}^{n}-x_{j-3/2}^{n}}$

(recall that $c_{j}$ is equal to $0$ or $1$ and does not depend on the time
index).

The organization and the comments for the case with different viscosities,
from Figure 6 to Figure 10, are the same.

####  5.3.1 Case with equal viscosities

$0$$0.5$$1$$1.5$$2$$2.5$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$\rho$$\rho_{+}$$\rho_{-}$$0$$0.5$$1$$1.5$$2$$2.5$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$\rho$

Figure 1: Densities. On the left, the $3$ densities of the mixture, on the
right, the density of the unmixed fluid.

$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$1.4$$0.16$$0.18$$0.2$$0.22$$0.24$macroscopic
density $\rho$$\rho_{+}$(macroscopic)$\rho_{-}$(macroscopic)mesoscopic density
$\rho$

Figure 2: Densities. Zoom of the preceding figures.

$-1.5$$-1$$-0.5$$0$$0.5$$1$$1.5$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$u$$-1.5$$-1$$-0.5$$0$$0.5$$1$$1.5$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$u$

Figure 3: Velocities. On the left, the velocity of the mixture, on the right,
the velocity of the unmixed fluid.

$0$$0.5$$1$$1.5$$2$$2.5$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$p$$p_{+}(\rho_{+})$$p_{-}(\rho_{-})$$0$$0.5$$1$$1.5$$2$$2.5$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$p$

Figure 4: Pressures. On the left, the $3$ pressures in the mixture, on the
right, the pressure in the unmixed fluid.

$0.38$$0.4$$0.42$$0.44$$0.46$$0.48$$0.5$$0.52$$0.54$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$\alpha_{+}$$0.3$$0.35$$0.4$$0.45$$0.5$$0.55$$0.6$$0.65$$0.7$$0.75$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$\alpha_{+}$

Figure 5: Volume fractions. On the left, the volume fraction $\alpha_{+}$ in
the mixture, on the right, the estimate of the volume fraction in the unmixed
fluid.

####  5.3.2 Case with different viscosities

$0$$0.5$$1$$1.5$$2$$2.5$$3$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$\rho$$\rho_{+}$$\rho_{-}$$0$$0.5$$1$$1.5$$2$$2.5$$3$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$\rho$

Figure 6: Densities. On the left, the $3$ densities of the mixture, on the
right, the density of the unmixed fluid.

$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$1.1$$0.12$$0.13$$0.14$$0.15$$0.16$$0.17$$0.18$$0.19$$0.2$macroscopic
density $\rho$$\rho_{+}$$\rho_{-}$mesoscopic density $\rho$

Figure 7: Densities. Zoom of the preceding figures.

$-1.5$$-1$$-0.5$$0$$0.5$$1$$1.5$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$u$$-1.5$$-1$$-0.5$$0$$0.5$$1$$1.5$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$u$

Figure 8: Velocities. On the left, the velocity of the mixture, on the right,
the velocity of the unmixed fluid.

$0$$0.5$$1$$1.5$$2$$2.5$$3$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$p$$p_{+}(\rho_{+})$$p_{-}(\rho_{-})$$0$$0.5$$1$$1.5$$2$$2.5$$3$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$p$

Figure 9: Pressures. On the left, the $3$ pressures in the mixture, on the
right, the pressure in the unmixed fluid.

$0.35$$0.4$$0.45$$0.5$$0.55$$0.6$$0.65$$0.7$$0.75$$0.8$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$\alpha_{+}$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$\alpha_{+}$

Figure 10: Volume fractions. On the left, the volume fraction $\alpha_{+}$ in
the mixture, on the right, the estimate of the volume fraction in the unmixed
fluid.

Acknowledgments: The authors want to thank M. Hillairet and L. Saint-Raymond
for several discussions and comments. D. Bresch and C. Burtea are partially
supported bySingFlows project, grant ANR-18-CE40-002.

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