# Asymptotic Derivation of Multicomponent Compressible Flows with Heat
Conduction and Mass Diffusion

Stefanos Georgiadis  Computer, Electrical and Mathematical Science and
Engineering Division, King Abdullah University of Science and Technology
(KAUST), Thuwal 23955-6900, Saudi Arabia and Institute for Analysis and
Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße
8–10, 1040 Wien, Austria<EMAIL_ADDRESS>and  Athanasios E.
Tzavaras  Computer, Electrical and Mathematical Science and Engineering
Division, King Abdullah University of Science and Technology (KAUST), Thuwal
23955-6900, Saudi Arabia<EMAIL_ADDRESS>
###### Abstract.

A Type-I model of a multicomponent system of fluids with non-constant
temperature is derived as the high-friction limit of a Type-II model via a
Chapman-Enskog expansion. The asymptotic model is shown to fit into the
general theory of hyperbolic-parabolic systems, by exploiting the entropy
structure inherited through the asymptotic procedure. Finally, by deriving the
relative entropy identity for the Type-I model, two convergence results for
smooth solutions are presented, from the system with mass-diffusion and heat
conduction to the corresponding system without mass-diffusion but including
heat conduction and to its hyperbolic counterpart.

###### Key words and phrases:

Multicomponent systems, Euler flows, Non-isothermal model, Chapman-Enskog
expansion, Hyperbolic-parabolic, Relative entropy, Bott-Duffin inverse.

###### 2020 Mathematics Subject Classification:

35Q35, 76M45, 76N15, 76T30, 80A17.

##  1\. Introduction

Multicomponent systems of fluids occur often in nature and in industry: the
Earth’s atmosphere consists of nitrogen, oxygen, argon, carbon dioxide and
small amounts of other gases. Natural gas is made of gaseous hydrocarbons such
as methane, ethane, propane. The widespread presence of multicomponent fluids
suggests the importance in understanding their modeling and being able to
predict their behavior.

Depending on the phenomenon and the application, models might vary containing
more or less modeling detail. While one might prefer the more detailed model,
it is conceivable that modeling detail might not be available or that it
cannot be experimentally measured. There should be a balance: a theory must be
detailed enough to sufficiently describe a phenomenon, but not so detailed
that significant phenomena are hard to comprehend [10]. Modeling of
multicomponent fluids is a well-developed subject and the reader is referred
to [23, 24, 10, 12, 28, 2] for various approaches.

In a multicomponent theory with primitive variables mass density, velocity and
temperature, one distinguishes among three classes of models: In a Type-I
model, each component is described by its own mass density, but the components
move with a common velocity and have a common temperature. In Type-II models,
each component is described by its own mass density and velocity, but the
components have a common temperature. Type-III models (which are not
considered here) are described via the individual densities, velocities and
temperatures of each component. For information regarding Type-III models we
refer to [28], while for a short discussion on motivations and reasons to
employ each model we refer to [2].

It is useful to have systematic ways of passing from a detailed theory to a
less detailed one, which is mathematically easier to handle and experimentally
easier to measure. Previous works trying to pass from a Type-II to a Type-I
model are already known, for example in [2] in which the authors investigate
the reduction of a non-isothermal model using an entropy invariant method, or
in [16], in which the reduction is done using asymptotic methods, but for the
isothermal case and for a simple mixture (see Appendix B). The first objective
of this paper is to derive a Type-I model using the asymptotic method of [16],
but for the non-isothermal, non-simple mixture model treated in [2].

To this end, consider the Type-II system of equations for multicomponent
fluids:

$\partial_{t}\rho_{i}+\textnormal{div}(\rho_{i}v_{i})=0$ (1.1)
$\partial_{t}(\rho_{i}v_{i})+\textnormal{div}(\rho_{i}v_{i}\otimes
v_{i})=\rho_{i}b_{i}-\rho_{i}\nabla\mu_{i}-\frac{1}{\theta}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})\nabla\theta-\frac{\theta}{\epsilon}\sum_{j\not=i}b_{ij}\rho_{i}\rho_{j}(u_{i}-u_{j})$
(1.2) $\begin{split}&\partial_{t}\left(\rho
e+\sum_{i}\frac{1}{2}\rho_{i}v_{i}^{2}\right)+\textnormal{div}\left(\left(\rho
e+\sum_{i}\frac{1}{2}\rho_{i}v_{i}^{2}\right)v\right)\\\
&\quad=\textnormal{div}\left(\kappa\nabla\theta-\sum_{i}(\rho_{i}e_{i}+p_{i}+\frac{1}{2}\rho_{i}v_{i}^{2})u_{i}\right)-\textnormal{div}(pv)+\rho
b\cdot v+\rho r+\sum_{i}\rho_{i}b_{i}\cdot u_{i}\end{split}$ (1.3)

Equations (1.1) are the partial mass balances and (1.2) the partial momentum
balances of the $n$ components of the fluid, while (1.3) is the balance of
total energy of the mixture. The index $i\in\\{1,\dots,n\\}$ refers to the
$i$-th component of the fluid. The prime variables are the mass densities
$\rho_{i}$, the partial velocities $v_{i}$ and the temperature of the mixture
$\theta$. We define the total mass $\rho=\sum_{i}\rho_{i}$, the barycentric
velocity of the mixture $v:=\frac{1}{\rho}\sum_{i}\rho_{i}v_{i}$ and the
diffusional velocities $u_{i}:=v_{i}-v$; the latter satisfy
$\sum_{i}\rho_{i}u_{i}=0$. Moreover, we have the following thermodynamic
quantities: the chemical potentials $\mu_{i}$, the partial pressures $p_{i}$,
the specific internal energies $e_{i}$ and we define the total pressure
$p=\sum_{i}p_{i}$ and the thermal energy $\rho e=\sum_{i}\rho_{i}e_{i}$.
Furthermore, $b_{i}$ stands for the body forces acting on the $i$-th component
and $r_{i}$ for the heat supply due to radiation, where $\rho
b=\sum_{i}\rho_{i}b_{i}$ is the total body force and $\rho
r=\sum_{i}\rho_{i}r_{i}$ the total heat supply due to radiation. Finally,
$\kappa=\kappa(\rho_{1},\dots,\rho_{n},\theta)\geq 0$ is the heat conductivity
and $b_{ij}=b_{ij}(\rho_{i},\rho_{j},\theta)$ are strictly positive and
symmetric coefficients that model the interaction between the $i$-th and
$j$-th components with a strength that is measured by $\epsilon>0$.

The system is complemented by a set of constitutive relations, which read

$\rho\psi=\rho\psi(\rho_{1},\dots,\rho_{n},\theta)$ (1.4) $\rho\psi=\rho
e-\rho\eta\theta$ (1.5) $(\rho\psi)_{\rho_{i}}=\mu_{i}$ (1.6)
$(\rho\psi)_{\theta}=-\rho\eta$ (1.7)

where $\rho\psi$ is the Helmholtz free-energy and $\rho\eta$ the entropy,
along with the Gibbs-Duhem relation, for determining the total pressure:

$\rho\psi+p=\sum_{i}\rho_{i}\mu_{i}$ (1.8)

as explained in the two appendices. The above type-II model is proposed in
[2]. The format of equations (1.4)-(1.7) is motivated by the usual
considerations of equilibrium thermodynamics, while the model is consistent
with the Clausius-Duhem inequality. For the reader’s convenience we present an
outline of its thermodynamic structure in Appendices A and B, while for
further elaboration and detail we refer to [10, 12, 2].

In addition to system (1.1)-(1.3), consider also the system

$\displaystyle\partial_{t}\rho_{i}+\textnormal{div}(\rho_{i}v)$
$\displaystyle=-\textnormal{div}(\epsilon\rho_{i}\tilde{u}_{i})$ (1.9)
$\displaystyle\partial_{t}(\rho v)+\textnormal{div}(\rho v\otimes v)$
$\displaystyle=\rho b-\nabla p$ (1.10)
$\displaystyle\begin{split}\partial_{t}\left(\rho e+\frac{1}{2}\rho
v^{2}\right)+\textnormal{div}\left((\rho e+\frac{1}{2}\rho
v^{2})v\right)&=\textnormal{div}\left(\kappa\nabla\theta-\epsilon\sum_{i}(\rho_{i}e_{i}+p_{i})\tilde{u}_{i}\right)\\\
&\quad-\textnormal{div}(pv)+\rho r+\rho b\cdot v+\sum_{i}\rho_{i}b_{i}\cdot
u_{i}\end{split}$ (1.11)

where $u_{i}=\epsilon\tilde{u}_{i}$ is determined by solving the constrained
algebraic system of Maxwell-Stefan type

$\displaystyle-\sum_{j\not=i}b_{ij}\theta\rho_{i}\rho_{j}(u_{i}-u_{j})$
$\displaystyle=\epsilon\left(-\frac{\rho_{i}}{\rho}\nabla
p+\rho_{i}\theta\nabla\frac{\mu_{i}}{\theta}-\theta(\rho_{i}e_{i}+p_{i})\nabla\frac{1}{\theta}\right)$
(1.12) $\displaystyle\sum_{i}\rho_{i}u_{i}$ $\displaystyle=0$

The system (1.9)-(1.12) forms a Type-I model, with the same notation as before
and the same constitutive relations. The mathematical structure of
multicomponent systems falls into the general realm of hyperbolic-parabolic
systems [13, 4] and follows the general framework originated by Kawashima
[20]. One objective here is to study their structure from the perspective of
relative entropy identities and the general theory in [7].

Analytical results on multi-component flows extend in various directions. We
refer to [1, 19, 17] for existence and uniqueness studies of strong and/or
weak solutions for Maxwell-Stefan systems of mass diffusion, situations that
involve no mean flow and pertain to the general area of parabolic systems. By
contrast, situations that involve mean flow lead to questions in the realm of
hyperbolic or hyperbolic-parabolic systems. There are available analyses for
isothermal viscous flows of multicomponent systems [3] and even for
multicomponent compressible Euler flows [26, 25, 27, 30]. For analyses of non-
isothermal multicomponent systems that include effects of heat-conduction we
refer to [22, 31, 21, 5]. The above works concern the mathematical structure
of multi-component systems and existence of solutions for steady and dynamic
problems. There has been recent interest in the convergence from compressible
multicomponent Euler equations in the high-friction limit to Maxwell-Stefan
systems, a problem pertaining to the subject of relaxation approximations. A
number of studies have appeared regarding isothermal flows [4, 14, 16, 25]
achieving in the limit the classical Maxwell-Stefan system [4], or (for more
general chemical potentials) porous media variants or even fourth order
diffusions [16, 25].

The focus of the present work is on the system (1.9)-(1.12) modeling non-
isothermal multi-component flows that include the effects of mass-diffusion
and heat conduction but no viscous effects. As is typical for Maxwell-Stefan
systems, a key difficulty arises from the inversion of (1.12). First, the
Chapman-Enskog analysis from the isothermal case [14, 16] is extended to the
non-isothermal case thus obtaining an asymptotic derivation of the Type-I
model from the Type-II model (1.1)-(1.3). Next, system (1.1)-(1.3) is
complemented with its corresponding entropy identity (2.51) and, by performing
an asymptotic analysis to the entropy identity, we obtain the entropy identity
(2.57) for the system (1.9)-(1.12). This analysis indicates how the emerging
system inherits the dissipative structure of the original system.

The second step is to verify that (1.9)-(1.12) fits into the general framework
of systems of hyperbolic-parabolic type as introduced in [20] and generalized
in [7]. There are two aspects to this question: (i) to address the connection
between the ”mathematical entropy” and the thermodynamic structure of the
model and (ii) to identify the dissipative structure of mass-diffusion and
heat conduction. The latter connects to the issue of inversion of the
constrained algebraic system (1.12). This is overcome by using the notion of
the Bott-Duffin inverse of a matrix which was brought to the subject of
Maxwell-Stefan systems in [17]; the latter provides an important ingredient
for inverting the algebraic system and comparing the entropic structure of the
Type-I model with the usual entropic structure of hyperbolic-parabolic systems
in the format discussed in [20, 7].

Third, we derive a relative entropy identity for (1.9)-(1.12) using the
methodological approach of relative entropy [8, 18, 7]. We compute the
relative entropy identity (4.23) which monitors the evolution of (4.18) and is
remarkable in its simplicity. This identity is, in turn, used in order to
prove convergence from strong solutions of the system (1.9)-(1.12) to strong
solutions of heat-conducting multicomponent Euler system when the mass-
diffusivity $\epsilon$ tends to zero. Also to prove convergence to smooth
solutions of multi-component adiabatic Euler flows when both heat conductivity
$\kappa$ and mass diffusivity $\epsilon$ tend to zero.

The outline of this work is as follows. In section 2, we derive (1.9)-(1.12)
as the high-friction limit of system (1.1)-(1.3), by doing a Chapman-Enskog
expansion, and present the asymptotic derivation of the entropy identity
(2.57). In section 3, we verify that (1.9)-(1.12) fits into the general
framework of systems of hyperbolic-parabolic type [20, 7] and show that the
system without mass diffusion and heat conduction, obtained by setting
$\epsilon=\kappa=0$, is hyperbolic, by computing its wave speeds. In section
4, using the methodological approach of relative entropy [8, 18, 7], we derive
the relative entropy identity (4.23), which is used to prove convergence from
strong solutions of (1.9)-(1.12) to strong solutions of the heat-conducting
multicomponent Euler flows as $\epsilon\to 0$ or to adiabatic multicomponent
Euler flows when both $\epsilon,\kappa\to 0$. We conclude with four
appendices: the first two appendices present the natural modeling framework of
multicomponent Type-II models (1.1)-(1.3), outline its thermodynamical
structure and the results of [2] on the consistency of the model with the
Clausius-Duhem inequality; the third appendix provides the computations for
deriving the entropy equation of system (1.1)-(1.3), while the fourth one the
computation for the relative entropy identity.

##  2\. Chapman-Enskog Expansion of the Type-II Model

The goal of the asymptotic procedure is to obtain from (1.1)-(1.3) a Type-I
model via a Chapman-Enskog expansion that will provide an approximation of the
initial system of order $\epsilon^{2}$. The resulting system will consist of
$n$ partial momentum balances, a single (total) momentum balance and one total
energy balance, thus the name Type-I. It will also be complemented by the same
constraint $\sum_{i}\rho_{i}u_{i}=0$ (at least up to order
$\mathcal{O}(\epsilon^{2})$) and a linear system for determining the
diffusional velocities $u_{i}$.

The resulting system should contain no partial velocities. Using the
definition of the diffusional velocities $u_{i}=v_{i}-v$ we perform the change
of variables $(v_{1},\dots,v_{n})\mapsto(v,u_{1},\dots,u_{n})$. Then system
(1.1)-(1.3) reads:

$\partial_{t}\rho_{i}+\textnormal{div}(\rho_{i}u_{i}+\rho_{i}v)=0$ (2.1)
$\begin{split}\partial_{t}\left(\rho_{i}(u_{i}+v)\right)+\textnormal{div}(\rho_{i}(u_{i}+v)\otimes(u_{i}+v))=\rho_{i}b_{i}&-\rho_{i}\nabla\mu_{i}-\frac{1}{\theta}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})\nabla\theta\\\
&-\frac{\theta}{\epsilon}\sum_{j\not=i}b_{ij}\rho_{i}\rho_{j}(u_{i}-u_{j})\end{split}$
(2.2) $\begin{split}\partial_{t}\Big{(}\rho
e&+\sum_{i}\frac{1}{2}\rho_{i}(u_{i}+v)^{2}\Big{)}+\textnormal{div}\left(\Big{(}\rho
e+\sum_{i}\frac{1}{2}\rho_{i}(u_{i}+v)^{2}\Big{)}v\right)=-\textnormal{div}(pv)+\rho
b\cdot v\\\
&+\textnormal{div}\left(\kappa\nabla\theta-\sum_{i}(\rho_{i}e_{i}+p_{i}+\frac{1}{2}\rho_{i}(u_{i}+v)^{2})u_{i}\right)+\rho
r+\sum_{i}\rho_{i}b_{i}\cdot u_{i}\end{split}$ (2.3)

subject to the constraint

$\sum_{i}\rho_{i}u_{i}=0$ (2.4)

###  2.1. Hilbert Expansion

Observe that, as $\epsilon\to 0$, we formally obtain the equation

$\displaystyle{-\theta\sum_{j\not=i}b_{ij}\rho_{i}\rho_{j}(u_{i}-u_{j})=0}\,.$

In [16] it was proved that the system

$-\sum_{j\not=i}B_{ij}(u_{i}-u_{j})=d_{i},~{}~{}i=1,\dots,n~{}~{}\textnormal{subject
to}~{}~{}\sum_{i}\rho_{i}u_{i}=0$ (2.5)

under the hypotheses:

(i) $d_{1},\dots,d_{n}\in\mathbb{R}^{3}$ satisfy $\sum_{i}d_{i}=0$

(ii) $\rho_{1},\dots,\rho_{n}>0$

(iii) $(B_{ij})\in\mathbb{R}^{n\times n}$ is a symmetric matrix with
$B_{ij}\geq 0$ for $i,j=1,\dots,n$ and

(iv) all solutions of the homogeneous system
$\sum_{j\not=i}B_{ij}(u_{i}-u_{j})=0$, $i=1,\dots,n$ lie in the space spanned
by $(1,\dots,1)\in\mathbb{R}^{n}$,
has the unique solution

$\displaystyle\rho_{i}u_{i}=-\sum_{j,k=1}^{n-1}(\delta_{ij}\rho_{i}$
$\displaystyle-\frac{\rho_{i}\rho_{j}}{\rho})\tau_{jk}^{-1}d_{k},~{}~{}i=1,\dots,n-1$
(2.6) $\displaystyle\rho_{n}u_{n}$
$\displaystyle=-\sum_{j=1}^{n-1}\rho_{j}u_{j}$

where $(\tau_{ij}^{-1})\in\mathbb{R}^{(n-1)\times(n-1)}$ is the inverse of a
regular submatrix, obtained by reordering the matrix
$(\tau_{ij})\in\mathbb{R}^{n\times n}$ of rank $n-1$ with coefficients

$\tau_{ij}=\delta_{ij}\sum_{k=1}^{n}B_{ik}-B_{ij}\,.$

Our case is simpler since the coefficients $b_{ij}$ are symmetric and strictly
positive. We also require that $\theta>0$, which means that hypotheses (iii)
and (iv) are automatically satisfied for $\rho_{i}>0$, $i=1,\dots,n$.

Next, introduce the Hilbert expansion

$\rho_{i}=\rho_{i}^{0}+\epsilon\rho_{i}^{1}+\epsilon^{2}\rho_{i}^{2}+\mathcal{O}(\epsilon^{3})$
(2.7) $u_{i}=u_{i}^{0}+\epsilon
u_{i}^{1}+\epsilon^{2}u_{i}^{2}+\mathcal{O}(\epsilon^{3})$ (2.8)
$v=v^{0}+\epsilon v^{1}+\mathcal{O}(\epsilon^{2})$ (2.9)
$\theta=\theta^{0}+\epsilon\theta^{1}+\epsilon^{2}\theta^{2}+\mathcal{O}(\epsilon^{3})$
(2.10)

Inserting this into $\rho=\sum_{i}\rho_{i}$ and the constraint (2.4) we obtain

$\rho=\rho^{0}+\epsilon\rho^{1}+\mathcal{O}(\epsilon^{2})$ (2.11)

where we set $\rho^{0}:=\sum_{i}\rho_{i}^{0}$ and
$\rho^{1}:=\sum_{i}\rho_{i}^{1}$ and

$\sum_{i}\rho_{i}^{0}u_{i}^{0}+\epsilon\sum_{i}(\rho_{i}^{1}u_{i}^{0}+\rho_{i}^{0}u_{i}^{1})+\mathcal{O}(\epsilon^{2})=0$
(2.12)

Equating terms of the same order gives

$\sum_{i}\rho_{i}^{0}u_{i}^{0}=0~{}~{}\textnormal{and}~{}~{}\sum_{i}(\rho_{i}^{1}u_{i}^{0}+\rho_{i}^{0}u_{i}^{1})=0$
(2.13)

Next we insert the expansion (2.7)-(2.10) into system (2.1)-(2.3) and identify
terms of the same order:

(i) The terms at the order $\mathcal{O}(1/\epsilon)$ give:

$-\theta^{0}\sum_{j\not=i}b_{ij}\rho_{i}^{0}\rho_{j}^{0}(u_{i}^{0}-u_{j}^{0})=0$
(2.14)

For the expansion we assume $\rho^{0}_{i}>0$ and note that (2.14) is of the
form (2.5) with $d_{i}=0$ and $B_{ij}$ symmetric and strictly positive, which
implies that the null-space of the homogeneous system is
$\textnormal{span}\\{(1,\dots,1)\\}$. This, along with (2.6), gives
$u_{i}^{0}=0$ for $i\in\\{1,\dots,n\\}$. This is incorporated in the remaining
expansions.

For the expansion of $\frac{1}{\theta}$ we use the Taylor series
$\frac{1}{1+x}=1-x+x^{2}+\mathcal{O}(x^{3})$ and thus

$\frac{1}{\theta}=\frac{1}{\theta^{0}(1+\epsilon\frac{\theta^{1}}{\theta^{0}}+\epsilon^{2}\frac{\theta^{2}}{\theta^{0}}+\cdots)}=\frac{1}{\theta^{0}}-\epsilon\frac{\theta^{1}}{(\theta^{0})^{2}}+\epsilon^{2}\frac{(\theta^{1})^{2}-\theta^{2}\theta^{0}}{(\theta^{0})^{3}}+\mathcal{O}(\epsilon^{3})$
(2.15)

For the expansions of the thermodynamic functions we use

$\displaystyle e_{i}$
$\displaystyle=\hat{e}_{i}(\rho_{1}^{0}+\epsilon\rho_{1}^{1}+\mathcal{O}(\epsilon^{2}),...,\rho_{n}^{0}+\epsilon\rho_{n}^{1}+\mathcal{O}(\epsilon^{2}),\theta^{0}+\epsilon\theta^{1}+\mathcal{O}(\epsilon^{2}))$
(2.16) $\displaystyle=e_{i}^{0}+\epsilon
e_{i}^{1}+\epsilon^{2}e_{i}^{2}+\mathcal{O}(\epsilon^{3})$

where $e_{i}^{0},\,e_{i}^{1},\,e_{i}^{2},...$ are computed by using the Taylor
theorem, for instance

$\displaystyle e_{i}^{0}$
$\displaystyle=\hat{e}_{i}(\rho_{1}^{0},...,\rho_{n}^{0},\theta^{0})\,,\quad
e_{i}^{1}=\sum_{j=1}^{n}\frac{\partial\hat{e}_{i}}{\partial\rho_{j}}(\rho_{1}^{0},...,\rho_{n}^{0},\theta^{0})\,\rho_{j}^{1}+\frac{\partial\hat{e}_{i}}{\partial\theta}(\rho_{1}^{0},...,\rho_{n}^{0},\theta^{0})\,\theta^{1}\,,$
$\displaystyle e_{i}^{2}$
$\displaystyle=\sum_{j=1}^{n}\frac{\partial\hat{e}_{i}}{\partial\rho_{j}}(\rho^{0},\theta^{0})\,\rho_{j}^{2}+\frac{\partial\hat{e}_{i}}{\partial\theta}(\rho^{0},\theta^{0})\,\theta^{2}+\sum_{k,l=1}^{n}\frac{\partial^{2}\hat{e}_{i}}{\partial\rho_{k}\partial\rho_{l}}(\rho^{0},\theta^{0})\,\rho_{k}^{1}\rho_{l}^{1}$
$\displaystyle\quad+\sum_{k=1}^{n}\frac{\partial^{2}\hat{e}_{i}}{\partial\rho_{k}\partial\theta}(\rho^{0},\theta^{0})\,\rho_{k}^{1}\theta^{1}+\frac{\partial^{2}\hat{e}_{i}}{\partial\theta^{2}}(\rho^{0},\theta^{0})\,(\theta^{1})^{2}$

and so on. A similar expansion is used for the entropy

$\displaystyle\eta$
$\displaystyle=\hat{\eta}\left(\rho_{1}^{0}+\epsilon\rho_{1}^{1}+\mathcal{O}(\epsilon^{2}),...,\rho_{n}^{0}+\epsilon\rho_{n}^{1}+\mathcal{O}(\epsilon^{2}),\theta^{0}+\epsilon\theta^{1}+\mathcal{O}(\epsilon^{2})\right)$
(2.17)
$\displaystyle=\eta^{0}+\epsilon\eta^{1}+\epsilon^{2}\eta^{2}+\mathcal{O}(\epsilon^{3})$

as well as for the functions $e_{i}$, $p_{i}$ and $\kappa$ all of which only
depend on $(\rho_{1},...,\rho_{n})$ and $\theta$. We then obtain:

(ii) Terms of order $\mathcal{O}(1)$:

$\partial_{t}\rho_{i}^{0}+\textnormal{div}(\rho_{i}^{0}v^{0})=0$ (2.18)
$\begin{split}\partial_{t}(\rho_{i}^{0}v^{0})+\textnormal{div}(\rho_{i}^{0}v^{0}\otimes
v^{0})=\rho_{i}^{0}b_{i}^{0}-\rho_{i}^{0}\nabla\mu_{i}^{0}&-\frac{1}{\theta^{0}}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})\nabla\theta^{0}\\\
&-\theta^{0}\sum_{j\not=i}b_{ij}\rho_{i}^{0}\rho_{j}^{0}(u_{i}^{1}-u_{j}^{1})\end{split}$
(2.19)
$\begin{split}\partial_{t}\Big{(}\sum_{i}\rho_{i}^{0}e_{i}^{0}&+\sum_{i}\frac{1}{2}\rho_{i}^{0}(v^{0})^{2}\Big{)}+\textnormal{div}\left(\left(\sum_{i}\rho_{i}^{0}e_{i}^{0}+\sum_{i}\frac{1}{2}\rho_{i}^{0}(v^{0})^{2}\right)v^{0}\right)\\\
&=-\textnormal{div}(p^{0}v^{0})+\sum_{i}\rho_{i}^{0}r_{i}^{0}+\sum_{i}\rho_{i}^{0}b_{i}^{0}\cdot
v^{0}+\textnormal{div}\left(\kappa^{0}\nabla\theta^{0}\right)\end{split}$
(2.20)

(iii) Terms of order $\mathcal{O}(\epsilon)$:

$\partial_{t}\rho_{i}^{1}+\textnormal{div}(\rho_{i}^{0}(u_{i}^{1}+v^{1})+\rho_{i}^{1}v^{0})=0$
(2.21)
$\begin{split}&\partial_{t}(\rho_{i}^{1}v^{0}+\rho_{i}^{0}(u_{i}^{1}+v^{1}))+\textnormal{div}\Big{(}\rho_{i}^{1}v^{0}\otimes
v^{0}+\rho_{i}^{0}(u_{i}^{1}+v^{1})\otimes
v^{0}+\rho_{i}^{0}v^{0}\otimes(u_{i}^{1}+v^{1})\Big{)}\\\
&=\rho_{i}^{1}b_{i}^{0}+\rho_{i}^{0}b_{i}^{1}-\rho_{i}^{1}\nabla\mu_{i}^{0}-\rho_{i}^{0}\nabla\mu_{i}^{1}+\frac{\theta^{1}}{(\theta^{0})^{2}}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})\nabla\theta^{0}\\\
&-\frac{1}{\theta^{0}}(\rho_{i}^{1}e_{i}^{0}+\rho_{i}^{0}e_{i}^{1}+p_{i}^{1}-\rho_{i}^{1}\mu_{i}^{0}-\rho_{i}^{0}\mu_{i}^{1})\nabla\theta^{0}-\frac{1}{\theta^{0}}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})\nabla\theta^{1}\\\
&-\theta^{0}\sum_{j\not=i}b_{ij}\Big{\\{}(\rho_{i}^{1}\rho_{j}^{0}+\rho_{i}^{0}\rho_{j}^{1})(u_{i}^{1}-u_{j}^{1})+\rho_{i}^{0}\rho_{j}^{0}(u_{i}^{2}-u_{j}^{2})\Big{\\}}-\theta^{1}\sum_{j\not=i}b_{ij}\rho_{i}^{0}\rho_{j}^{0}(u_{i}^{1}-u_{j}^{1})\end{split}$
(2.22)
$\begin{split}&\partial_{t}\Big{(}\sum_{i}(\rho_{i}^{0}e_{i}^{1}+\rho_{i}^{1}e_{i}^{0})+\sum_{i}\frac{1}{2}\rho_{i}^{1}(v^{0})^{2}+\sum_{i}\frac{1}{2}\rho_{i}^{0}2v^{0}(u_{i}^{1}+v^{1})\Big{)}+\textnormal{div}\Big{(}\Big{(}\sum_{i}\rho_{i}^{0}e_{i}^{0}\\\
&+\sum_{i}\frac{1}{2}\rho_{i}^{0}(v^{0})^{2}\Big{)}v^{1}+\Big{(}\sum_{i}(\rho_{i}^{0}e_{i}^{1}+\rho_{i}^{1}e_{i}^{0})+\sum_{i}\frac{1}{2}\rho_{i}^{0}2v^{0}(u_{i}^{1}+v^{1})+\sum_{i}\frac{1}{2}\rho_{i}^{1}(v^{0})^{2}\Big{)}v^{0}\Big{)}\\\
&=-\textnormal{div}(p^{0}v^{1}+p^{1}v^{0})+\sum_{i}(\rho_{i}^{0}b_{i}^{0}\cdot
v^{1}+(\rho_{i}^{1}b_{i}^{0}+\rho_{i}^{0}b_{i}^{1})\cdot
v^{0})+\sum_{i}\rho_{i}^{0}b_{i}^{0}\cdot
u_{i}^{1}+\sum_{i}(\rho_{i}^{0}r_{i}^{1}+\rho_{i}^{1}r_{i}^{0})\\\
&+\textnormal{div}\left(\kappa^{0}\nabla\theta^{1}+\kappa^{1}\nabla\theta^{0}-\sum_{i}\left\\{(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0})+\frac{1}{2}\rho_{i}^{0}(v^{0})^{2}\right\\}u_{i}^{1}\right)\end{split}$
(2.23)

Summing (2.19) over $i$ and using the symmetry of $b_{ij}$ we obtain

$\partial_{t}(\sum_{i}\rho_{i}^{0}v^{0})+\textnormal{div}(\sum_{i}\rho_{i}^{0}v^{0}\otimes
v^{0})=\sum_{i}\rho_{i}^{0}b_{i}^{0}-\nabla p^{0}$ (2.24)

The reason behind the simplification of the right-hand side comes from the
thermodynamics relations of the problem. More precisely, we have:

$\begin{split}\sum_{i}\left(\rho_{i}b_{i}-\rho_{i}\nabla\mu_{i}-\frac{1}{\theta}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})\nabla\theta\right)&=\rho
b-\sum_{i}\rho_{i}\nabla\mu_{i}-\frac{1}{\theta}(\rho
e+p-\sum_{i}\rho_{i}\mu_{i})\nabla\theta\\\ &\overset{\eqref{intro-gd}}{=}\rho
b-\sum_{i}\rho_{i}\nabla\mu_{i}-\rho\eta\nabla\theta\\\ &=\rho
b+\sum_{i}\mu_{i}\nabla\rho_{i}-\nabla\left(\sum_{i}\rho_{i}\mu_{i}\right)-\rho\eta\nabla\theta\\\
&\overset{\eqref{intro-const3}-\eqref{intro-gd}}{=}\rho b-\nabla p\end{split}$
(2.25)

For more details on the thermodynamic relations used in the calculation above,
we refer to Appendices A and B. Expanding both sides of (2.25), we get:

$\sum_{i}\left(\rho_{i}^{0}b_{i}^{0}-\rho_{i}^{0}\nabla\mu_{i}^{0}-\frac{1}{\theta^{0}}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})\nabla\theta^{0}\right)=\sum_{i}\rho_{i}^{0}b_{i}^{0}-\nabla
p^{0}$ (2.26)

and

$\begin{split}\sum_{i}\Big{(}\rho_{i}^{1}b_{i}^{0}&+\rho_{i}^{0}b_{i}^{1}-\rho_{i}^{1}\nabla\mu_{i}^{0}-\rho_{i}^{0}\nabla\mu_{i}^{1}+\frac{\theta^{1}}{(\theta^{0})^{2}}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})\nabla\theta^{0}-\frac{1}{\theta^{0}}(\rho_{i}^{1}e_{i}^{0}+\rho_{i}^{0}e_{i}^{1}+p_{i}^{1}\\\
&-\rho_{i}^{1}\mu_{i}^{0}-\rho_{i}^{0}\mu_{i}^{1})\nabla\theta^{0}-\frac{1}{\theta^{0}}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})\nabla\theta^{1}\Big{)}=\sum_{i}(\rho_{i}^{1}b_{i}^{0}+\rho_{i}^{0}b_{i}^{1})-\nabla
p^{1}\end{split}$ (2.27)

where $p^{0}=\sum_{i}p_{i}^{0}$ and $p^{1}=\sum_{i}p_{i}^{1}$.

Equation (2.24) along with (2.18) and (2.20) provide a closed system for
determining $(\rho_{1}^{0},\dots,\rho_{n}^{0},v^{0},\theta^{0})$. Now, by
(2.19) follows that $(u_{1}^{1},\dots,u_{n}^{1})$ satisfies the linear system

$-\sum_{j\not=i}b_{ij}\theta^{0}\rho_{i}^{0}\rho_{j}^{0}(u_{i}^{1}-u_{j}^{1})=d_{i}^{0}$
(2.28)

where
$d_{i}^{0}=\partial_{t}(\rho_{i}^{0}v^{0})+\textnormal{div}(\rho_{i}^{0}v^{0}\otimes
v^{0})-\rho_{i}^{0}b_{i}^{0}+\rho_{i}^{0}\nabla\mu_{i}^{0}+\frac{1}{\theta^{0}}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})\nabla\theta^{0}$.
Since $u_{i}^{0}=0$ the first constraint of (2.13) is satisfied trivially and
the second one becomes

$\sum_{i}\rho_{i}^{0}u_{i}^{1}=0.$ (2.29)

Moreover, due to (2.24) and (2.26) we see that $\sum_{i}d_{i}^{0}=0$. This
guarantees the solvability of system (2.28), since assumptions (i)-(iv) are
satisfied according to the analysis of system (2.5). Hence, there exists a
unique solution $(u_{1}^{1},\dots,u_{n}^{1})$ to system (2.28).

Similarly, summing (2.22) over $i$, using the symmetry of $b_{ij}$, the
identity (2.27) and the constraint (2.29) we get

$\partial_{t}(\sum_{i}\rho_{i}^{1}v^{0}+\rho^{0}v^{1})+\textnormal{div}(\sum_{i}\rho_{i}^{1}v^{0}\otimes
v^{0}+\rho^{0}v^{1}\otimes v^{0}+\rho^{0}v^{0}\otimes
v^{1})=\sum_{i}(\rho_{i}^{1}b_{i}^{0}+\rho_{i}^{0}b_{i}^{1})-\nabla p^{1}$
(2.30)

which along with (2.21) and (2.23) provide a closed system for determining
$(\\{\rho_{i}^{1}\\}_{i=1}^{n},v^{1},\theta^{1})$. Note that after using the
constraint (2.29), (2.23) reads

$\begin{split}\partial_{t}\Big{(}\sum_{i}&(\rho_{i}^{0}e_{i}^{1}+\rho_{i}^{1}e_{i}^{0})+\sum_{i}\frac{1}{2}\rho_{i}^{1}(v^{0})^{2}+\rho^{0}v^{0}v^{1}\Big{)}+\textnormal{div}\Big{(}\Big{(}\sum_{i}\rho_{i}^{0}e_{i}^{0}+\sum_{i}\frac{1}{2}\rho_{i}^{0}(v^{0})^{2}\Big{)}v^{1}\\\
&+\Big{(}\sum_{i}(\rho_{i}^{0}e_{i}^{1}+\rho_{i}^{1}e_{i}^{0})+\rho^{0}v^{0}v^{1}+\sum_{i}\frac{1}{2}\rho_{i}^{1}(v^{0})^{2}\Big{)}v^{0}\Big{)}=-\textnormal{div}(p^{0}v^{1}+p^{1}v^{0})\\\
&+\sum_{i}(\rho_{i}^{0}b_{i}^{0}\cdot
v^{1}+(\rho_{i}^{1}b_{i}^{0}+\rho_{i}^{0}b_{i}^{1})\cdot
v^{0})+\sum_{i}\rho_{i}^{0}b_{i}^{0}\cdot
u_{i}^{1}+\sum_{i}(\rho_{i}^{0}r_{i}^{1}+\rho_{i}^{1}r_{i}^{0})\\\
&+\textnormal{div}\left(\kappa^{0}\nabla\theta^{1}+\kappa^{1}\nabla\theta^{0}-\sum_{i}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0})u_{i}^{1}\right)\end{split}$

###  2.2. Reconstruction of the effective equations

Next, we reconstruct the effective equations that are valid asymptotically up
to order $\mathcal{O}(\epsilon^{2})$. We add back (2.18) plus $\epsilon$ times
(2.21), (2.24) plus $\epsilon$ times (2.30) and (2.20) plus $\epsilon$ times
(2.23) to obtain

$\partial_{t}(\rho_{i}^{0}+\epsilon\rho_{i}^{1})+\textnormal{div}(\rho_{i}^{0}v^{0}+\epsilon(\rho_{i}^{0}v^{1}+\rho_{i}^{1}v^{0}))=-\epsilon\textnormal{div}(\rho_{i}^{0}u_{i}^{1})$
(2.31)
$\begin{split}\partial_{t}\Big{(}\rho^{0}v^{0}&+\epsilon(\rho^{1}v^{0}+\rho^{0}v^{1})\Big{)}+\textnormal{div}\left(\rho^{0}v^{0}+\epsilon(\rho^{1}v^{0}\otimes
v^{0}+\rho^{0}v^{1}\otimes v^{0}+\rho^{0}v^{0}\otimes v^{1})\right)\\\
&=\sum_{i}\\{\rho_{i}^{0}b_{i}^{0}+\epsilon(\rho_{i}^{1}b_{i}^{0}+\rho_{i}^{0}b_{i}^{1})\\}-(\nabla
p^{0}+\epsilon\nabla p^{1})\end{split}$ (2.32)
$\begin{split}&\partial_{t}\Big{(}\sum_{i}\rho_{i}^{0}e_{i}^{0}+\epsilon\sum_{i}(\rho_{i}^{0}e_{i}^{1}+\rho_{i}^{1}e_{i}^{0})+\sum_{i}\frac{1}{2}(\rho_{i}^{0}+\epsilon\rho_{i}^{1})(v^{0})^{2})+\epsilon\sum_{i}\frac{1}{2}\rho_{i}^{0}2v^{0}v^{1}\Big{)}\\\
&+\textnormal{div}\Big{(}\sum_{i}\rho_{i}^{0}e_{i}^{0}(v^{0}+\epsilon
v^{1})+\epsilon\sum_{i}(\rho_{i}^{1}e_{i}^{0}+\rho_{i}^{0}e_{i}^{1})v^{0}+\sum_{i}\frac{1}{2}\rho_{i}^{0}(v^{0})^{2}(v^{0}+\epsilon
v^{1})\\\
&+\epsilon\sum_{i}\frac{1}{2}\rho_{i}^{1}(v^{0})^{2}v^{0})+\epsilon\sum_{i}\frac{1}{2}\rho_{i}^{0}2v^{0}(u_{i}^{1}+v^{1})v^{0}\Big{)}=-\textnormal{div}(p^{0}v^{0}+\epsilon(p^{1}v^{0}+p^{0}v^{1})\\\
&+\sum_{i}(\rho_{i}^{0}r_{i}^{0}+\epsilon(\rho_{i}^{0}r_{i}^{1}+\rho_{i}^{1}r_{i}^{0}))+\sum_{i}\Big{(}\rho_{i}^{0}b_{i}^{0}\cdot
v^{0}+\epsilon(\rho_{i}^{0}b_{i}^{0}\cdot
v^{1}+(\rho_{i}^{0}b_{i}^{1}+\rho_{i}^{1}b_{i}^{0})\cdot v^{0})\Big{)}\\\
&+\epsilon\sum_{i}\rho_{i}^{0}b_{i}^{0}\cdot
u_{i}^{1}+\textnormal{div}\left((\kappa^{0}\nabla\theta^{0})+\epsilon(\kappa^{1}\nabla\theta^{0}+\kappa^{1}\nabla\theta^{0})-\epsilon\sum_{i}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0})u_{i}^{1}\right)\end{split}$
(2.33)

Now set

$\rho_{i}^{\epsilon}=\rho_{i}^{0}+\epsilon\rho_{i}^{1}+\mathcal{O}(\epsilon^{2})\,,\qquad\rho^{\epsilon}=\sum_{i}\rho_{i}^{\epsilon}$
(2.34) $u_{i}^{\epsilon}=u_{i}^{0}+\epsilon
u_{i}^{1}+\mathcal{O}(\epsilon^{2})$ (2.35) $v^{\epsilon}=v^{0}+\epsilon
v^{1}+\mathcal{O}(\epsilon^{2})$ (2.36)
$\theta^{\epsilon}=\theta^{0}+\epsilon\theta^{1}+\mathcal{O}(\epsilon^{2})$
(2.37)

We also have the expansions obtained via (2.16) and reading

$\displaystyle e_{i}^{\epsilon}$ $\displaystyle=e_{i}^{0}+\epsilon
e_{i}^{1}+\epsilon^{2}e_{i}^{2}+\mathcal{O}(\epsilon^{3})$ (2.38)
$\displaystyle p_{i}^{\epsilon}$ $\displaystyle=p_{i}^{0}+\epsilon
p_{i}^{1}+\epsilon^{2}p_{i}^{2}+\mathcal{O}(\epsilon^{3})$
$\displaystyle\kappa^{\epsilon}$
$\displaystyle=\kappa^{0}+\epsilon\kappa^{1}+\epsilon^{2}\kappa^{2}+\mathcal{O}(\epsilon^{3})$

Then equations (2.31)-(2.33) read

$\partial_{t}\rho_{i}^{\epsilon}+\textnormal{div}(\rho_{i}^{\epsilon}v^{\epsilon})=-\textnormal{div}(\rho_{i}^{\epsilon}u_{i}^{\epsilon})+\mathcal{O}(\epsilon^{2})$
(2.39)
$\partial_{t}(\rho^{\epsilon}v^{\epsilon})+\textnormal{div}(\rho^{\epsilon}v^{\epsilon}\otimes
v^{\epsilon})=\rho^{\epsilon}b^{\epsilon}-\nabla
p^{\epsilon}+\mathcal{O}(\epsilon^{2})$ (2.40)
$\begin{split}\partial_{t}\left(\rho^{\epsilon}e^{\epsilon}+\frac{1}{2}\rho^{\epsilon}(v^{\epsilon})^{2}\right)&+\textnormal{div}\left(\left(\rho^{\epsilon}e^{\epsilon}+\frac{1}{2}\rho^{\epsilon}(v^{\epsilon})^{2}\right)v^{\epsilon}\right)=-\textnormal{div}(p^{\epsilon}v^{\epsilon})+\rho^{\epsilon}r^{\epsilon}+\rho^{\epsilon}b^{\epsilon}\cdot
v^{\epsilon}\\\ &+\sum_{i}\rho_{i}^{\epsilon}b_{i}^{\epsilon}\cdot
u_{i}^{\epsilon}+\textnormal{div}\left(\kappa^{\epsilon}\nabla\theta^{\epsilon}-\sum_{i}(\rho_{i}^{\epsilon}e_{i}^{\epsilon}+p_{i}^{\epsilon})u_{i}^{\epsilon}\right)+\mathcal{O}(\epsilon^{2})\end{split}$
(2.41)

Finally, we need to construct the formulas determining
$(u_{i}^{\epsilon})_{i}$. Using (2.28) we deduce that

$\begin{split}-\sum_{j\not=i}b_{ij}\theta^{\epsilon}\rho_{i}^{\epsilon}\rho_{j}^{\epsilon}(u_{i}^{\epsilon}-u_{j}^{\epsilon})&=-\epsilon\sum_{j\not=i}b_{ij}\theta^{0}\rho_{i}^{0}\rho_{j}^{0}(u_{i}^{1}-u_{j}^{1})+\mathcal{O}(\epsilon^{2})\\\
&=\epsilon d_{i}^{0}+\mathcal{O}(\epsilon^{2})\end{split}$ (2.42)

where from (2.18), summing (2.18) over $i$ and (2.24):

$\begin{split}d_{i}^{0}&=\partial_{t}(\rho_{i}^{0}v^{0})+\textnormal{div}(\rho_{i}^{0}v^{0}\otimes
v^{0})-\rho_{i}^{0}b_{i}^{0}+\rho_{i}^{0}\nabla\mu_{i}^{0}+\frac{1}{\theta^{0}}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})\nabla\theta^{0}\\\
&=\rho_{i}^{0}\left(\partial_{t}v^{0}+v^{0}\cdot\nabla
v^{0}\right)-\rho_{i}^{0}b_{i}^{0}+\rho_{i}^{0}\nabla\mu_{i}^{0}+\frac{1}{\theta^{0}}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})\nabla\theta^{0}\\\
&=\frac{\rho_{i}^{0}}{\rho^{0}}\left(\partial_{t}(\rho^{0}v^{0})+\textnormal{div}(\rho^{0}v^{0}\otimes
v^{0})\right)-\rho_{i}^{0}b_{i}^{0}+\rho_{i}^{0}\nabla\mu_{i}^{0}+\frac{1}{\theta^{0}}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})\nabla\theta^{0}\\\
&=\frac{\rho_{i}^{0}}{\rho^{0}}\sum_{j}(\rho_{j}^{0}b_{j}^{0}-\nabla
p_{j}^{0})-\rho_{i}^{0}b_{i}^{0}+\rho_{i}^{0}\nabla\mu_{i}^{0}+\frac{1}{\theta^{0}}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})\nabla\theta^{0}\end{split}$
(2.43)

This motivates to define

$d_{i}^{\epsilon}:=\frac{\rho_{i}^{\epsilon}}{\rho^{\epsilon}}\sum_{j}(\rho_{j}^{\epsilon}b_{j}^{\epsilon}-\nabla
p_{j}^{\epsilon})-\rho_{i}^{\epsilon}b_{i}^{\epsilon}+\rho_{i}^{\epsilon}\nabla\mu_{i}^{\epsilon}+\frac{1}{\theta^{\epsilon}}(\rho_{i}^{\epsilon}e_{i}^{\epsilon}+p_{i}^{\epsilon}-\rho_{i}^{\epsilon}\mu_{i}^{\epsilon})\nabla\theta^{\epsilon}$
(2.44)

which by (2.25)-(2.27) sums up to zero and thus the linear system for
determining $\\{u_{i}^{\epsilon}\\}_{i=1}^{n}$ is

$-\sum_{j\not=i}b_{ij}\theta^{\epsilon}\rho_{i}^{\epsilon}\rho_{j}^{\epsilon}(u_{i}^{\epsilon}-u_{j}^{\epsilon})=\epsilon
d_{i}^{\epsilon}+\mathcal{O}(\epsilon^{2})$ (2.45)

and is solvable according to the analysis of (2.5). Moreover, the constraint
becomes

$\sum_{i}\rho_{i}^{\epsilon}u_{i}^{\epsilon}=\mathcal{O}(\epsilon^{2})$ (2.46)

which means that since the variables $\rho_{i}$ and $u_{i}$ are defined up to
order $\mathcal{O}(\epsilon^{2})$ we can set
$\sum_{i}\rho_{i}^{\epsilon}u_{i}^{\epsilon}=0$ up to that order.

Therefore, omitting the $\epsilon$-notation and the higher-order terms, we
conclude that system (1.1)-(1.8) is approximated within
$\mathcal{O}(\epsilon^{2})$ by the system:

$\partial_{t}\rho_{i}+\textnormal{div}(\rho_{i}v)=-\textnormal{div}(\rho_{i}u_{i})$
(2.47) $\partial_{t}(\rho v)+\textnormal{div}(\rho v\otimes v)=\rho b-\nabla
p$ (2.48) $\begin{split}\partial_{t}\left(\rho e+\frac{1}{2}\rho
v^{2}\right)+\textnormal{div}\left((\rho e+\frac{1}{2}\rho
v^{2})v\right)&=-\textnormal{div}(pv)+\rho r+\rho b\cdot
v+\sum_{i}\rho_{i}b_{i}\cdot u_{i}\\\
&+\textnormal{div}\left(\kappa\nabla\theta-\sum_{i}(\rho_{i}e_{i}+p_{i})u_{i}\right)\end{split}$
(2.49)

where $u_{i}$ are determined by solving the constrained linear system:

$\displaystyle-\sum_{j\not=i}b_{ij}\theta\rho_{i}\rho_{j}(u_{i}-u_{j})$
$\displaystyle=\epsilon\left(\frac{\rho_{i}}{\rho}(-\nabla p+\rho
b)+\rho_{i}\theta\nabla\frac{\mu_{i}}{\theta}-\theta(\rho_{i}e_{i}+p_{i})\nabla\frac{1}{\theta}-\rho_{i}b_{i}\right)$
(2.50) $\displaystyle\sum_{i}\rho_{i}u_{i}$ $\displaystyle=0$

Our system (2.47)-(2.50) agrees with the Type-I model obtained in [2] using
the entropy invariant method, in the case of an inviscid, non-reactive mixture
with zero thermal diffusivities [2, Section 8]. We note here that the present
work concerns only the derivation of the corresponding Type-I model in the
formal level. For a validation of the expansion and a proof of the
convergence, however valid in the isothermal case, we refer to [4, 16].

###  2.3. Asymptotic expansion of the entropy equation

As explained in Appendix C or in [2, Section 5], given the Type-II system
(1.1)-(1.3), one derives the entropy equation:

$\begin{split}\partial_{t}(\rho\eta)+\textnormal{div}(\rho\eta
v)&=\textnormal{div}\left(\frac{\kappa\nabla\theta-\sum_{i}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})u_{i}}{\theta}\right)+\frac{1}{\theta^{2}}\kappa|\nabla\theta|^{2}\\\
&+\frac{1}{2\epsilon}\sum_{i}\sum_{j}b_{ij}\rho_{i}\rho_{j}(u_{i}-u_{j})^{2}+\frac{\rho
r}{\theta}\end{split}$ (2.51)

There are two approaches for deriving the entropy equation of the asymptotic
limit system (2.47)-(2.49): (i) to derive it by using the system and the
constitutive relations as was done in Appendix C to obtain (2.51) and (ii) to
expand the entropy equation of the Type-II model and obtain its
$\epsilon^{2}-$approximation the same way we obtained system (2.47)-(2.49).
Here we present the latter way, but it is easy to verify that the two results
coincide.

As in subsection 2.1, we introduce the same Hilbert expansion, insert it into
(2.51) and identify terms of the same order:

(i) Terms of order $\mathcal{O}(1/\epsilon)$:

$\frac{1}{2}\sum_{i,j}b_{ij}\rho_{i}^{0}\rho_{j}^{0}(u_{i}^{0}-u_{j}^{0})^{2}=0$
(2.52)

which is consistent with $u_{i}^{0}=0$ in subsection 2.1.

Next, using $u_{i}^{0}=0$, the expansion (2.15) for $\frac{1}{\theta}$ and

$\frac{1}{\theta^{2}}=\frac{1}{(\theta^{0})^{2}}-\epsilon\frac{2\theta^{1}}{(\theta^{0})^{3}}+\epsilon^{2}\frac{3(\theta^{1})^{2}-\theta^{2}\theta^{0}}{(\theta^{0})^{4}}+\mathcal{O}(\epsilon^{3})\,,$

as well as the expansions (2.17), (2.38) we obtain:

(ii) at order $\mathcal{O}(1)$:

$\begin{split}\partial_{t}(\rho^{0}\eta^{0})&+\textnormal{div}(\rho^{0}\eta^{0}v^{0})=\textnormal{div}\left(\frac{1}{\theta^{0}}\kappa^{0}\nabla\theta^{0}\right)+\frac{1}{(\theta^{0})^{2}}\kappa^{0}(\nabla\theta^{0})^{2}+\frac{\rho^{0}r^{0}}{\theta^{0}}\end{split}$
(2.53)

(iii) at order $\mathcal{O}(\epsilon)$:

$\begin{split}&\partial_{t}(\rho^{1}\eta^{0}+\rho^{0}\eta^{1})+\textnormal{div}((\rho^{1}\eta^{0}+\rho^{0}\eta^{1})v^{0}+\rho^{0}\eta^{0}v^{1})=\textnormal{div}\Big{(}\frac{1}{\theta^{0}}(\kappa^{1}\nabla\theta^{0}+\kappa^{0}\nabla\theta^{1})\\\
&-\frac{\theta^{1}}{(\theta^{0})^{2}}\kappa^{0}\nabla\theta^{0}-\frac{1}{\theta^{0}}\sum_{i}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})u_{i}^{1}\Big{)}-\frac{2\theta^{1}}{(\theta^{0})^{3}}\kappa^{0}(\nabla\theta^{0})^{2}+\frac{1}{(\theta^{0})^{2}}\kappa^{1}(\nabla\theta^{0})^{2}\\\
&+\frac{1}{(\theta^{0})^{2}}\kappa^{0}2\nabla\theta^{0}\nabla\theta^{1}+\frac{1}{2}\sum_{i,j}b_{ij}\rho_{i}^{0}\rho_{j}^{0}(u_{i}^{1}-u_{j}^{1})^{2}+\frac{\rho^{1}r^{0}+\rho^{0}r^{1}}{\theta^{0}}-\frac{\rho^{0}r^{0}\theta^{1}}{(\theta^{0})^{2}}\end{split}$
(2.54)

Moreover, using (2.28) in the entropy expansion, the third last term of (2.54)
reads

$\begin{split}\frac{1}{2}\sum_{i,j}b_{ij}\rho_{i}^{0}\rho_{j}^{0}(u_{i}^{1}-u_{j}^{1})^{2}&=\frac{1}{2}\sum_{i}u_{i}^{1}\sum_{j}b_{ij}\rho_{i}^{0}\rho_{j}^{0}(u_{i}^{1}-u_{j}^{1})-\frac{1}{2}\sum_{j}u_{j}^{1}\sum_{i}b_{ij}\rho_{i}^{0}\rho_{j}^{0}(u_{i}^{1}-u_{j}^{1})\\\
&=-\frac{1}{2\theta^{0}}\sum_{i}u_{i}^{1}d_{i}^{0}-\frac{1}{2\theta^{0}}\sum_{j}u_{j}^{1}d_{j}^{0}\\\
&=-\frac{1}{\theta^{0}}\sum_{i}u_{i}^{1}\cdot d_{i}^{0}\end{split}$

Thus, in the reconstruction, we add (2.53) plus $\epsilon$ times (2.54), to
obtain

$\begin{split}&\partial_{t}(\rho^{0}\eta^{0}+\epsilon(\rho^{1}\eta^{0}+\rho^{0}\eta^{1}))+\textnormal{div}(\rho^{0}\eta^{0}v^{0}+\epsilon((\rho^{1}\eta^{0}+\rho^{0}\eta^{1})v^{0}+\rho^{0}\eta^{0}v^{1}))\\\
&=\textnormal{div}\left[\frac{1}{\theta^{0}}\kappa^{0}\nabla\theta^{0}+\epsilon\left(\frac{1}{\theta^{0}}(\kappa^{1}\nabla\theta^{0}+\kappa^{0}\nabla\theta^{1})-\frac{\theta^{1}}{(\theta^{0})^{2}}\kappa^{0}\nabla\theta^{0}-\frac{1}{\theta^{0}}\sum_{i}(\rho_{i}^{0}e_{i}^{0}+p_{i}^{0}-\rho_{i}^{0}\mu_{i}^{0})u_{i}^{1}\right)\right]\\\
&+\frac{1}{(\theta^{0})^{2}}\kappa^{0}(\nabla\theta^{0})^{2}+\epsilon\left(-\frac{2\theta^{1}}{(\theta^{0})^{3}}\kappa^{0}(\nabla\theta^{0})^{2}+\frac{1}{(\theta^{0})^{2}}\kappa^{1}(\nabla\theta^{0})^{2}+\frac{1}{(\theta^{0})^{2}}\kappa^{0}2\nabla\theta^{0}\nabla\theta^{1}\right)\\\
&-\frac{\epsilon}{\theta^{0}}\sum_{i}u_{i}^{1}\cdot
d_{i}^{0}+\frac{\rho^{0}r^{0}}{\theta^{0}}+\epsilon\left(\frac{\rho^{1}r^{0}+\rho^{0}r^{1}}{\theta^{0}}-\frac{\rho^{0}r^{0}\theta^{1}}{(\theta^{0})^{2}}\right)\end{split}$
(2.55)

which in turn gives

$\begin{split}\partial_{t}(\rho^{\epsilon}\eta^{\epsilon})+\textnormal{div}(\rho^{\epsilon}\eta^{\epsilon}v^{\epsilon})&=\textnormal{div}\left(\kappa^{\epsilon}\frac{\nabla\theta^{\epsilon}}{\theta^{\epsilon}}-\sum_{i}\frac{\rho_{i}^{\epsilon}e_{i}^{\epsilon}+p_{i}^{\epsilon}-\rho_{i}^{\epsilon}\mu_{i}^{\epsilon}}{\theta^{\epsilon}}u_{i}^{\epsilon}\right)+\kappa^{\epsilon}\frac{(\nabla\theta^{\epsilon})^{2}}{(\theta^{\epsilon})^{2}}\\\
&+\frac{\rho^{\epsilon}r^{\epsilon}}{\theta^{\epsilon}}-\frac{1}{\theta^{\epsilon}}\sum_{i}u_{i}^{\epsilon}\cdot
d_{i}^{\epsilon}+\mathcal{O}(\epsilon^{2})\end{split}$ (2.56)

Omitting the $O(\epsilon^{2})$ correction, the entropy equation for the Type-I
system (2.47)-(2.49) reads

$\begin{split}\partial_{t}(\rho\eta)+\textnormal{div}(\rho\eta
v)&=\textnormal{div}\left(\frac{1}{\theta}\kappa\nabla\theta-\frac{1}{\theta}\sum_{i}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})u_{i}\right)\\\
&+\frac{\rho
r}{\theta}+\frac{1}{\theta^{2}}\kappa|\nabla\theta|^{2}-\frac{1}{\theta}\sum_{i}u_{i}\cdot
d_{i}\,.\end{split}$ (2.57)

We conclude that the order $O(\epsilon^{2})$ expansion of the entropy
dissipation structure (2.51) for the Type-II model yields the entropy
dissipation structure (2.57) for the emerging Type-I model.

##  3\. Dissipative structure of the limiting hyperbolic-parabolic system

For simplicity we select the external fields $r_{i}$ and $b_{i}$ to be zero
and proceed to study the mathematical structure of the Type-I model, which can
be written as follows:

$\displaystyle\partial_{t}\rho_{i}+\textnormal{div}(\rho_{i}v)$
$\displaystyle=\textnormal{div}(-\rho_{i}u_{i})$ (3.1)
$\displaystyle\partial_{t}(\rho v)+\textnormal{div}(\rho v\otimes
v+p\mathbb{I})$ $\displaystyle=0$ (3.2) $\displaystyle\partial_{t}\left(\rho
e+\frac{1}{2}\rho v^{2}\right)+\textnormal{div}\left((\rho e+\frac{1}{2}\rho
v^{2})v+pv\right)$
$\displaystyle=\textnormal{div}\left(\kappa\nabla\theta-\sum_{i}(\rho_{i}e_{i}+p_{i})u_{i}\right)$
(3.3)

where $u_{i}$ are determined by inverting the constrained linear system

$\displaystyle-\sum_{j\not=i}b_{ij}\theta\rho_{i}\rho_{j}(u_{i}-u_{j})$
$\displaystyle=\epsilon\left(-\frac{\rho_{i}}{\rho}\nabla
p+\rho_{i}\theta\nabla\frac{\mu_{i}}{\theta}-\theta(\rho_{i}e_{i}+p_{i})\nabla\frac{1}{\theta}\right)$
(3.4) $\displaystyle\sum_{i}\rho_{i}u_{i}$ $\displaystyle=0\,.$ (3.5)

Given the constitutive choices outlined in section 2 (see also Appendix B)
smooth solutions of (3.1)-(3.5) satisfy the entropy identity

$\partial_{t}(-\rho\eta)+\textnormal{div}(-\rho\eta
v)=\textnormal{div}\left(\frac{1}{\theta}\sum_{i}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})u_{i}-\frac{1}{\theta}\kappa\nabla\theta\right)-\frac{1}{\theta^{2}}\kappa|\nabla\theta|^{2}+\frac{1}{\theta}\sum_{i}u_{i}\cdot
d_{i}$ (3.6)

Notice we multiplied the entropy identity by minus one, because we want to
study the mathematical entropy, which is defined as the negative thermodynamic
entropy.

Next, setting $U=[\rho_{1},\dots,\rho_{n},v,\theta]^{\top}$ with
$v=(v_{1},v_{2},v_{3})$, system (3.1)-(3.3) can be written in the form:

$\partial_{t}(A(U))+\textnormal{div}(F(U))=\textnormal{div}(\varepsilon
B(U)\nabla U)$ (3.7)

where

$A(U)=\begin{bmatrix}\rho_{1}\\\ \vdots\\\ \rho_{n}\\\ \rho v_{1}\\\ \rho
v_{2}\\\ \rho v_{3}\\\ \rho e+\frac{1}{2}\rho
v^{2}\end{bmatrix},~{}~{}F(U)=\begin{bmatrix}\rho_{1}v_{1}&\rho_{1}v_{2}&\rho_{1}v_{3}\\\
\vdots&\vdots&\vdots\\\ \rho_{n}v_{1}&\rho_{n}v_{2}&\rho_{n}v_{3}\\\ \rho
v_{1}^{2}+p&\rho v_{1}v_{2}&\rho v_{1}v_{3}\\\ \rho v_{1}v_{2}&\rho
v_{2}^{2}+p&\rho v_{2}v_{3}\\\ \rho v_{1}v_{3}&\rho v_{2}v_{3}&\rho
v_{3}^{2}+p\\\ (\rho e+\frac{1}{2}\rho v^{2}+p)v_{1}&(\rho e+\frac{1}{2}\rho
v^{2}+p)v_{2}&(\rho e+\frac{1}{2}\rho v^{2}+p)v_{3}\end{bmatrix}$ (3.8)

$B^{\kappa}(U)=\begin{bmatrix}0&\cdots&0&0&0&0&0\\\
\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\\ 0&\cdots&0&0&0&0&0\\\
0&\cdots&0&0&0&0&0\\\ 0&\cdots&0&0&0&0&0\\\ 0&\cdots&0&0&0&0&0\\\
0&\cdots&0&0&0&0&1\\\ \end{bmatrix},~{}~{}\nabla
U=\begin{bmatrix}\nabla\rho_{1}\\\ \vdots\\\ \nabla\rho_{n}\\\ \nabla v_{1}\\\
\nabla v_{2}\\\ \nabla v_{3}\\\ \nabla\theta\end{bmatrix}$

$B^{\epsilon}(U)=\begin{bmatrix}A_{11}&\cdots&A_{1n}&0&0&0&A_{1,n+4}\\\
\vdots&\ddots&\vdots&0&0&0&\vdots\\\ A_{n1}&\cdots&A_{nn}&0&0&0&A_{n,n+4}\\\
0&\cdots&0&0&0&0&0\\\ 0&\cdots&0&0&0&0&0\\\ 0&\cdots&0&0&0&0&0\\\
\sum_{i=1}^{n}\frac{h_{i}}{\rho_{i}}A_{i1}&\cdots&\sum_{i=1}^{n}\frac{h_{i}}{\rho_{i}}A_{in}&0&0&0&\sum_{i=1}^{n}\frac{h_{i}}{\rho_{i}}A_{i,n+4}\end{bmatrix}$

where we denote $h_{i}:=\rho_{i}e_{i}+p_{i}$ and write

$-\rho_{i}u_{i}=\sum_{j=1}^{n}A_{ij}\nabla\rho_{j}+A_{i,n+4}\nabla\theta,~{}\textnormal{for
$i=1,\dots,n$}$

and in (3.7) we set $\varepsilon=(\kappa,\epsilon)$ and
$B(U)=(B^{\kappa}(U),B^{\epsilon}(U))$. We note that the coefficients $A_{ij}$
and $A_{i,n+4}$ are determined by solving system (3.4)-(3.5), as explained
later in this section.

According to [7] system (3.7) fits into the class of hyperbolic-parabolic
systems (see also [20]) provided the following conditions hold:

(i) $A$ is a $C^{2}$ and bijective map from its domain onto its range, with
$\nabla A(U)$ non-singular for any $U$ in the domain of $A$

(ii) there is an entropy-entropy flux pair $(H(U),Q_{i}(U))$, $i=1,2,3$,
generated by a smooth (vector valued) multiplier $G(U)$, such that

$\nabla H=G\cdot\nabla A\quad\textnormal{and}\quad\nabla Q_{i}=G\cdot\nabla
F_{i}\quad\textnormal{for}~{}i=1,2,3\,.$

(iii) the matrix $\nabla^{2}H-G\cdot\nabla^{2}A$ is symmetric and positive
definite

(iv) smooth solutions of (3.7) satisfy an additional conservation law (the so
called entropy identity)

$\partial_{t}(H(U))+\text{div}(Q(U))=\text{div}(G(U)\varepsilon B(U)\nabla
U)-\nabla U^{\top}\nabla G(U)^{\top}\varepsilon B(U)\nabla U$ (3.9)

with the matrices $\nabla G^{\top}B$ inducing entropy dissipation, i.e.

$\nabla U^{\top}\nabla G(U)^{\top}\varepsilon B(U)\nabla U\geq 0\,.$

Let us check whether these conditions are satisfied in the case of system
(3.1)-(3.5). Indeed, setting

$H(U):=-\rho\eta(\rho_{1},\dots,\rho_{n},\theta)\,,\quad
Q_{i}(U):=-\rho\eta(\rho_{1},\dots,\rho_{n},\theta)v_{i}\,,$ (3.10)

we see that (3.9) is precisely equation (3.6). Concerning (i), if
$U=[\rho_{1},\dots,\rho_{n},v,\theta]^{\top}$ and
$\tilde{U}=[\tilde{\rho}_{1},\dots,\tilde{\rho}_{n},\tilde{v},\tilde{\theta}]^{\top}$
we have

$A(U)=A(\tilde{U})\Leftrightarrow\left\\{\begin{array}[]{c}\rho_{i}=\tilde{\rho}_{i}\\\
v=\tilde{v}\\\
e(\rho_{1},\dots,\rho_{n},\theta)=e(\tilde{\rho}_{1},\dots,\tilde{\rho}_{n},\tilde{\theta})\end{array}\right.$

where the last equality implies that $\theta=\tilde{\theta}$, provided the
internal energy is an increasing function of the temperature thus one-to-one.
This follows from assuming that $e_{\theta}\equiv c_{v}>0$, a property
connected to the stability of equilibrium states (see [6] for more). Hence, if
$\psi\in C^{3}$ and $\rho>0$, $A$ is $C^{2}$ and bijective in its domain, with

$\begin{split}\textnormal{det}(\nabla
A)&=\textnormal{det}\begin{bmatrix}1&0&\dots&0&0&0&0&0\\\
0&1&\dots&0&0&0&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\dots&1&0&0&0&0\\\ v_{1}&v_{1}&\dots&v_{1}&\rho&0&0&0\\\
v_{2}&v_{2}&\dots&v_{2}&0&\rho&0&0\\\ v_{3}&v_{3}&\dots&v_{3}&0&0&\rho&0\\\
(\rho e)_{\rho_{1}}+\frac{1}{2}v^{2}&(\rho
e)_{\rho_{2}}+\frac{1}{2}v^{2}&\dots&(\rho e)_{\rho_{n}}+\frac{1}{2}v^{2}&\rho
v_{1}&\rho v_{2}&\rho v_{3}&\rho c_{v}\end{bmatrix}\\\
&=\rho^{4}c_{v}>0\end{split}$ (3.11)

For (ii) using the notation
$\rho\hat{\eta}=\rho\eta(\rho_{1},\dots,\rho_{n},\theta)$ and
$\rho\tilde{\eta}=\rho\eta(\rho_{1},\dots,\rho_{n},\rho e)$ and the
thermodynamic relations from Appendix A and B we compute the partial
derivatives of the entropy

$(\rho\hat{\eta})_{\rho_{i}}=(\rho\tilde{\eta})_{\rho
e}(\rho\hat{e})_{\rho_{i}}+(\rho\tilde{\eta})_{\rho_{i}}=\frac{(\rho
e)_{\rho_{i}}-\mu_{i}}{\theta}$ $(\rho\hat{\eta})_{\theta}=\left(\frac{\rho
e-\rho\psi}{\theta}\right)_{\theta}=\frac{[(\rho
e)_{\theta}-(\rho\psi)_{\theta}]\theta-(\rho
e-\rho\psi)}{\theta^{2}}=\frac{\rho e_{\theta}}{\theta}=\frac{\rho
c_{v}}{\theta}.$

By virtue of the relation $\nabla(-\rho\eta)=G\cdot\nabla A$ this determines
the multiplier $G$ to be

$G(U)=\bigg{(}G_{j}(U)\bigg{)}_{j=1}^{n+4}=\frac{1}{\theta}\begin{bmatrix}\mu_{1}-\frac{1}{2}v^{2}\\\
\vdots\\\ \mu_{n}-\frac{1}{2}v^{2}\\\ v_{1}\\\ v_{2}\\\ v_{3}\\\
-1\end{bmatrix}$ (3.12)

while relations $\nabla(-\rho\eta v_{i})=G\cdot\nabla F_{i}$ serve as a way of
verifying that our calculations are correct; indeed, they hold for the $G$ we
found, using the properties

$(\mu_{i})_{\theta}=((\rho\psi)_{\rho_{i}})_{\theta}=((\rho\psi)_{\theta})_{\rho_{i}}=(-\rho\hat{\eta})_{\rho_{i}}=\frac{\mu_{i}-(\rho
e)_{\rho_{i}}}{\theta}$ (3.13)

Now, for (iii), note that
$\displaystyle{G\cdot\nabla^{2}A:=\sum_{j=1}^{n+4}G_{j}(U)\nabla^{2}A_{j}(U)}$.
One easily sees that $\nabla^{2}A_{j}=0$ for $j=1,\dots,n$. Then a tedious but
straightforward calculation, using the thermodynamic relations (1.4)-(1.7) and

$\displaystyle\frac{1}{\theta}(\rho e)_{\theta}-(\rho\eta)_{\theta}$
$\displaystyle=\frac{1}{\theta}\big{(}(\rho\psi)_{\theta}+\rho\eta\big{)}=0$
$\displaystyle\frac{1}{\theta}(\rho
e)_{\theta\theta}-(\rho\eta)_{\theta\theta}$
$\displaystyle=\frac{1}{\theta}(\rho\psi)_{\theta\theta}+\frac{2}{\theta}(\rho\eta)_{\theta}=\frac{\rho}{\theta}\eta_{\theta}=\frac{\rho}{\theta^{2}}e_{\theta}\,,$

implies

$\nabla^{2}(-\rho\eta)-G\cdot\nabla^{2}A=\frac{1}{\theta}\begin{bmatrix}(\rho\psi)_{\rho_{1}\rho_{1}}&\cdots&(\rho\psi)_{\rho_{1}\rho_{n}}&0&0&0&0\\\
\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
(\rho\psi)_{\rho_{n}\rho_{1}}&\cdots&(\rho\psi)_{\rho_{n}\rho_{n}}&0&0&0&0\\\
0&\cdots&0&\rho&0&0&0\\\ 0&\cdots&0&0&\rho&0&0\\\ 0&\cdots&0&0&0&\rho&0\\\
0&\cdots&0&0&0&0&\frac{1}{\theta}\rho c_{v}\\\ \end{bmatrix}$ (3.14)

Clearly, for $\psi\in C^{3}$, $\rho,\theta>0$ and $c_{v}>0$, the latter will
be positive definite provided the matrix $(\rho\psi)_{\rho_{i}\rho_{j}}$ is
positive definite. Indeed, for any
$\xi\in\mathbb{R}^{n+4}\setminus\\{(0,\dots,0)\\}$ we see that

$\xi^{\top}(\nabla^{2}(-\rho\eta)-G\cdot\nabla^{2}A)\xi=\frac{1}{\theta}\sum_{i=1}^{n}\sum_{j=1}^{n}(\rho\psi)_{\rho_{i}\rho_{j}}\xi_{i}\xi_{j}+\frac{\rho}{\theta}(\xi_{n+1}^{2}+\xi_{n+2}^{2}+\xi_{n+3}^{2})+\frac{\rho
c_{v}}{\theta^{2}}\xi_{n+4}^{2}>0$

The assumption that $(\rho\psi)_{\rho_{i}\rho_{j}}$ is positive definite is a
natural assumption in thermodynamics that is related to the convexity of the
entropy (again we refer to [6] regarding the stability of equilibrium states,
or to [2]).

For the last condition, we need to show that $\nabla U^{\top}\nabla
G(U)^{\top}\varepsilon B(U)\nabla U\geq 0$, in other words, the entropy
production is non-negative:

$\zeta=\frac{1}{\theta^{2}}\kappa|\nabla\theta|^{2}-\frac{1}{\theta}\sum_{i}u_{i}\cdot
d_{i}\geq 0.$ (3.15)

Clearly the first term is non-negative, since $\kappa\geq 0$. For the second
term we need to invert system (3.4), in order to determine $\rho_{i}u_{i}$.

Various methods for the inversion of the Maxwell-Stefan system are available
in the literature, see for example [2, 16] and [12, Sec. 7.7]. In the present
work, we invert (3.4) using the Bott-Duffin inverse, following the analysis of
[17]. The advantage of this method is that it provides an explicit formula for
the solution of the linear system (3.4), (3.5), which is helpful to estimate
the dissipation.

The need to introduce the Bott-Duffin inverse arises from the fact that the
desired inversion has to respect the constraint $\sum_{i}\rho_{i}u_{i}=0$,
i.e. we invert within the appropriate subspace. More precisely, we consider
the solution of the generic system

$Mx+y=w,~{}~{}x\in L,~{}y\in L^{\perp}$ (3.16)

where $M\in\mathbb{R}^{m\times m}$, for some $m\in\mathbb{N}$, is a matrix and
$L\subset\mathbb{R}^{m}$ a subspace. Let $\mathbb{P}_{L}$ and
$\mathbb{P}_{L^{\perp}}$ be the projection operators onto the subspaces $L$
and $L^{\perp}$, respectively. Then, the set of solutions of system (3.16) is
the same as the set of solutions of the system

$(M\mathbb{P}_{L}+\mathbb{P}_{L^{\perp}})z=w$ (3.17)

and $[x,y]^{\top}$ solves (3.16) if and only if $x=\mathbb{P}_{L}z$ and
$y=\mathbb{P}_{L^{\perp}}z=w-M\mathbb{P}_{L}z$. Now, if the matrix
$M\mathbb{P}_{L}+\mathbb{P}_{L^{\perp}}$ is invertible, we define the Bott-
Duffin inverse of $M$ with respect to $L$ by

$M^{BD}=\mathbb{P}_{L}(M\mathbb{P}_{L}+\mathbb{P}_{L^{\perp}})^{-1}$ (3.18)

so that the solution of (3.16) is given by

$x=M^{BD}d,~{}~{}y=w-Mx$ (3.19)

In our context, if we introduce the molar fractions

$c_{i}=\frac{\rho_{i}}{\rho}$ (3.20)

the left-hand side of (3.4) reads:

$\begin{split}-\sum_{j\not=i}b_{ij}\theta\rho_{i}\rho_{j}(u_{i}-u_{j})&=-\rho\theta\left(\sqrt{\rho_{i}}\sum_{j\not=i}b_{ij}c_{j}(\sqrt{\rho_{i}}u_{i})-\sqrt{\rho_{i}}\sum_{j\not=i}b_{ij}\sqrt{c_{i}}\sqrt{c_{j}}(\sqrt{\rho_{j}}u_{j})\right)\\\
&=-\rho\theta\sqrt{\rho_{i}}\sum_{j=1}^{n}\left(\sum_{k\not=i}c_{k}b_{ik}\delta_{ij}-\sqrt{c_{i}c_{j}}b_{ij}\right)\sqrt{\rho_{j}}u_{j}\\\
&=-\rho\theta\sqrt{\rho_{i}}\sum_{j=1}^{n}M_{ij}\sqrt{\rho_{j}}u_{j}\end{split}$

where we introduce the matrix $M=(M_{ij})$ given by

$M_{ij}=\left\\{\begin{array}[]{cc}\sum_{k\not=i}c_{k}b_{ik}&i=j\\\
-\sqrt{c_{i}c_{j}}b_{ij}&i\not=j\end{array}\right.$ (3.21)

and we are interested in the constrained inversion $Mx=w$, $x\in L$, where
$w_{i}=-\frac{\epsilon d_{i}}{\rho\theta\sqrt{\rho_{i}}}$, using $d_{i}$ from
section 2, $x_{i}=\sqrt{\rho_{i}}u_{i}$ and
$L=\\{(y_{1},\dots,y_{n})\in\mathbb{R}^{n}:\sum_{i=1}^{n}\sqrt{\rho_{i}}y_{i}=0\\}$.
Moreover, the projection matrix $\mathbb{P}_{L}$ on $L$ is given by

$(\mathbb{P}_{L})_{ij}=\delta_{ij}-\frac{\sqrt{\rho_{i}\rho_{j}}}{\rho}$
(3.22)

In [17] it was proven that the matrix $M$ from (3.21) satisfies the relation

$z^{\top}Mz\geq\mu|\mathbb{P}_{L}z|^{2}~{}~{}\forall
z\in\mathbb{R}^{n}~{}~{}\textnormal{where $\mu=\min_{i\not=j}b_{ij}$}$ (3.23)

which in turn implies that the Bott-Duffin inverse of $M$, namely $M^{BD}$, is
well-defined, symmetric and satisfies

$z^{\top}M^{BD}z\geq\lambda|\mathbb{P}_{L}z|^{2}~{}~{}\forall
z\in\mathbb{R}^{n}~{}~{}\textnormal{where
$\lambda=\left(2\sum_{i\not=j}(b_{ij}+1)\right)^{-1}$}$ (3.24)

Therefore, system (3.4) is written

$\sum_{j=1}^{n}M_{ij}\sqrt{\rho_{j}}u_{j}=-\frac{\epsilon
d_{i}}{\rho\theta\sqrt{\rho_{i}}}$ (3.25)

and can be inverted

$\sqrt{\rho_{i}}u_{i}=-\sum_{j=1}^{n}M_{ij}^{BD}\frac{\epsilon
d_{j}}{\rho\theta\sqrt{\rho_{j}}}$ (3.26)

Plugging (3.26) into the last term of (3.15) we get

$\begin{split}-\frac{1}{\theta}\sum_{i}u_{i}\cdot
d_{i}&=-\frac{1}{\theta}\sum_{i}\sqrt{\rho_{i}}u_{i}\cdot\frac{d_{i}}{\sqrt{\rho_{i}}}\\\
&=\frac{\epsilon}{\rho\theta^{2}}\sum_{i}\sum_{j}M_{ij}^{BD}\frac{d_{i}}{\sqrt{\rho_{i}}}\frac{d_{j}}{\sqrt{\rho_{j}}}\\\
&\geq 0\end{split}$

by (3.24). Hence, (3.15) is satisfied, which means that the matrices $\nabla
G^{\top}B$ induce entropy dissipation as we wanted, concluding that system
(3.1)-(3.3) along with the linear system (3.4) subject to the constraint (3.5)
is of hyperbolic-parabolic type. In fact, condition (3.15) is the minimum
framework inducing entropy dissipation along the evolution and at the same
time allowing for degenerate diffusion matrices (see [7]). In section 4 we
shall show that our problem enjoys a stronger dissipative structure, which
allows us to establish some convergence results.

A particular case of system (3.1)-(3.5) is the system without mass-diffusion
and heat-conduction, obtained by setting $\kappa=\epsilon=0$:

$\partial_{t}\rho_{i}+\textnormal{div}(\rho_{i}v)=0$ (3.27) $\partial_{t}(\rho
v)+\textnormal{div}(\rho v\otimes v+p\mathbb{I})=0$ (3.28)
$\partial_{t}\left(\rho e+\frac{1}{2}\rho
v^{2}\right)+\textnormal{div}\left((\rho e+\frac{1}{2}\rho
v^{2})v+pv\right)=0$ (3.29)

equipped with the entropy identity

$\partial_{t}(-\rho\eta)+\textnormal{div}(-\rho\eta v)=0$ (3.30)

We would like to show that (3.27)-(3.29) is hyperbolic under the assumptions
$(\rho\psi)_{\rho_{i}\rho_{j}}$ is positive definite and $e_{\theta}>0$, as
long as the total mass $\rho$ remains away from zero. To do so, one needs to
rewrite system (3.27)-(3.29) in the form

$\nabla A(U)\partial_{t}U+\sum_{\alpha=1}^{3}\nabla
F_{\alpha}(U)\partial_{x_{\alpha}}U=0$

where $A,F_{\alpha},U$ are as in (3.7). Since $\nabla A$ is non-singular, we
proceed to find the characteristic speeds of the system by solving, for any
$N=(N_{1},N_{2},N_{3})$ on the sphere, the eigenvalue problem

$\left[\sum_{\alpha=1}^{3}\nabla F_{\alpha}(U)N_{\alpha}-\lambda(U,N)\nabla
A(U)\right]r(U,N)=0$

where $\lambda,r$ are the eigenvalues and eigenvectors respectively and

$\begin{split}&\sum_{\alpha=1}^{3}\nabla
F_{\alpha}(U)N_{\alpha}-\lambda(U,N)\nabla A(U)\\\ &=\begin{bmatrix}v\cdot
N-\lambda&\cdots&0&\rho_{1}N_{1}&\rho_{1}N_{2}&\rho_{1}N_{3}&0\\\
\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\\ 0&\cdots&v\cdot
N-\lambda&\rho_{n}N_{1}&\rho_{n}N_{2}&\rho_{n}N_{3}&0\\\
p_{\rho_{1}}N_{1}&\cdots&p_{\rho_{n}}N_{1}&\rho(v\cdot
N-\lambda)&0&0&N_{1}p_{\theta}\\\
p_{\rho_{1}}N_{2}&\cdots&p_{\rho_{n}}N_{2}&0&\rho(v\cdot
N-\lambda)&0&N_{2}p_{\theta}\\\
p_{\rho_{1}}N_{3}&\cdots&p_{\rho_{n}}N_{3}&0&0&\rho(v\cdot
N-\lambda)&N_{3}p_{\theta}\\\ 0&\cdots&0&N_{1}\theta p_{\theta}&N_{2}\theta
p_{\theta}&N_{3}\theta p_{\theta}&\rho c_{v}(v\cdot
N-\lambda)\end{bmatrix}\end{split}$

By using the property

$\textnormal{det}\begin{bmatrix}A&B\\\
C&D\end{bmatrix}=\textnormal{det}(A)\textnormal{det}(D-CA^{-1}B)$

from [29, Sec. 5], for any block of matrices $A,B,C,D$, we determine the
characteristic equation:

$\rho^{4}c_{v}(v\cdot N-\lambda)^{n+2}\left((v\cdot
N-\lambda)^{2}-\frac{1}{\rho}\sum_{i}\rho_{i}p_{\rho_{i}}-\frac{\theta
p_{\theta}^{2}}{c_{v}\rho^{2}}\right)=0$

which yields the wave speeds

$\lambda_{1}=\cdots=\lambda_{n+2}=v\cdot N,~{}\lambda_{n+3,n+4}=v\cdot
N\pm\sqrt{\frac{1}{\rho}\sum_{i}\rho_{i}p_{\rho_{i}}+\frac{\theta
p_{\theta}^{2}}{c_{v}\rho^{2}}}$

Therefore, system (3.27)-(3.29) is hyperbolic if all eigenvalues are real,
which holds under the hypotheses $\rho,\theta>0$, $c_{v}>0$ and
$(\rho\psi)_{i,j}>0$. The last hypothesis ensures that the term
$\sum_{i}\rho_{i}p_{\rho_{i}}$ is positive, since by (D.3)

$\sum_{i}\rho_{i}p_{\rho_{i}}=\sum_{i,j}\rho_{i}\rho_{j}(\mu_{i})_{\rho_{j}}>0$

In fact, the eigenvectors corresponding to the repeated eigenvalues are given
by the formula

$\xi=(\xi_{1},\xi_{2},\dots,\xi_{n},\xi_{n+1},\xi_{n+2},-\frac{N_{1}}{N_{3}}\xi_{n+1}-\frac{N_{2}}{N_{3}}\xi_{n+2},-\frac{1}{p_{\theta}}\sum_{i=1}^{n}\xi_{i}p_{\rho_{i}}),~{}\textnormal{for}~{}(\xi_{1},\dots,\xi_{n+2})\in\mathbb{R}^{n+2}$

and thus the dimension of the eigenspace is $n+2$.

The reader should notice that hyperbolicity is not valid at $\rho=0$ (as $\det
A$ vanishes) but strict hyperbolicity still holds when some of the
$\rho_{i}$’s vanish provided $\rho\neq 0$.

##  4\. The zero-diffusion limit to multicomponent non-isothermal flows

As in section 3, we consider the hyperbolic-parabolic system

$\partial_{t}\rho_{i}+\textnormal{div}(\rho_{i}v)=\textnormal{div}(-\rho_{i}u_{i})$
(4.1) $\partial_{t}(\rho v)+\textnormal{div}(\rho v\otimes v+p\mathbb{I})=0$
(4.2) $\partial_{t}\left(\rho e+\frac{1}{2}\rho
v^{2}\right)+\textnormal{div}\left((\rho e+\frac{1}{2}\rho
v^{2})v+pv\right)=\textnormal{div}\left(\kappa\nabla\theta-\sum_{i}(\rho_{i}e_{i}+p_{i})u_{i}\right)\,,$
(4.3)

where $u_{i}$ are determined by solving

$\displaystyle-\sum_{j\not=i}b_{ij}\theta\rho_{i}\rho_{j}(u_{i}-u_{j})$
$\displaystyle=\epsilon\left(-\frac{\rho_{i}}{\rho}\nabla
p+\rho_{i}\theta\nabla\frac{\mu_{i}}{\theta}-\theta(\rho_{i}e_{i}+p_{i})\nabla\frac{1}{\theta}\right)$
(4.4) $\displaystyle\sum_{i}\rho_{i}u_{i}$ $\displaystyle=0\,,$ (4.5)

and which is endowed with the dissipation structure

$\displaystyle\partial_{t}(-\rho\eta)+\textnormal{div}(-\rho\eta v)$
$\displaystyle+\frac{1}{\theta^{2}}\kappa|\nabla\theta|^{2}-\frac{1}{\theta}\sum_{i}u_{i}\cdot
d_{i}$ (4.6)
$\displaystyle=\textnormal{div}\left(\frac{1}{\theta}\sum_{i}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})u_{i}-\frac{1}{\theta}\kappa\nabla\theta\right)\,.$

Moreover, consider the system obtained when neglecting the mass diffusive
effects ($\epsilon=0$) but including heat conduction:

$\partial_{t}\rho_{i}+\textnormal{div}(\rho_{i}v)=0$ (4.7) $\partial_{t}(\rho
v)+\textnormal{div}(\rho v\otimes v+p\mathbb{I})=0$ (4.8)
$\partial_{t}\left(\rho e+\frac{1}{2}\rho
v^{2}\right)+\textnormal{div}\left((\rho e+\frac{1}{2}\rho
v^{2})v+pv\right)=\textnormal{div}\left(\kappa\nabla\theta\right)$ (4.9)

endowed with the limiting dissipation structure

$\partial_{t}(-\rho\eta)+\textnormal{div}(-\rho\eta
v)=\textnormal{div}\left(-\frac{1}{\theta}\kappa\nabla\theta\right)-\frac{1}{\theta^{2}}\kappa|\nabla\theta|^{2}\,.$
(4.10)

Local existence results for smooth solutions are available for multicomponent
systems (see [13] and [12, Ch. 8]). For hyperbolic-parabolic systems like the
ones above with general initial data one can expect existence of a unique
smooth solution, which however can break down at finite time and the time of
existence in general depends on the diffusion constants $\varepsilon$ and
$\kappa$. More precisely, under sufficient conditions on the initial data, the
diffusion coefficients and the free energy function, if
$Q_{T}=\Omega\times(0,T)$, where $\Omega$ is bounded and $T>0$, there exists a
$\tau\in(0,T]$ such that the above problem possesses a unique solution

$(\rho_{1},\dots,\rho_{n})\in W^{1}_{p}(Q_{\tau};\mathbb{R}_{+}^{n}),~{}v\in
W^{2,1}_{p}(Q_{\tau};\mathbb{R}^{3}),~{}\theta\in
W^{2,1}(Q_{\tau};\mathbb{R}_{+})$

where the spaces above are defined as follows:

$W^{2,1}_{p}(Q_{T})=\\{u\in
L^{p}(Q_{T}):\partial_{t}^{\beta}\partial_{x}^{\alpha}u\in
L^{p}(Q_{T})~{}\textnormal{for all}~{}0<2\beta+|\alpha|\leq 2\\}$
$W^{1,0}_{p}(Q_{T})=\\{u\in L^{p}(Q_{T}):\partial_{x}^{\alpha}u\in
L^{p}(Q_{T})~{}\textnormal{for all}~{}|\alpha|=1\\}$

For more details we refer to [11].

In this section, we show that smooth solutions of (4.1)-(4.5) converge to
solutions of (4.7)-(4.10) as $\epsilon$ tends to zero, so long as the
solutions of the latter remain in the smooth regime and the theory developed
should be understood as indicating conditions for convergence of
thermomechanical theories in the smooth regime.

In order to show the convergence from (3.7) as $\varepsilon\to 0$ we use the
following ingredients:

(a) The method of relative entropy introduced in [8], here employed in the
form proposed in [7] :

$H(U|\bar{U})=H(U)-H(\bar{U})-G(\bar{U})\cdot(A(U)-A(\bar{U}))\,.$ (4.11)

By [7, Appendix A], whenever conditions (i)-(iv) for (3.9) hold, the relative
entropy can be written as $H(U|\bar{U})=\hat{H}(A(U)|A(\bar{U}))$, where
$\hat{H}(V)$ is a strictly convex function. Hence, $H(U|\bar{U})$ vanishes if
and only if $A(U)=A(\bar{U})$ (and by (i) if $U=\bar{U}$) and it can serve to
measure the distance between two solutions.

(b) A second ingredient is the control of diffusion by dissipation (see
[7],[9, Sec 4.6]), that is a hypothesis that there exist constants $\nu_{1}>0$
and $\nu_{2}>0$ such that

$\sum_{\alpha,\beta}\nabla
G(U)\partial_{\alpha}UB_{\alpha\beta}^{\kappa}(U)\partial_{\beta}U\geq\nu_{1}\sum_{\alpha}\left|\sum_{\beta}B_{\alpha\beta}^{\kappa}(U)\partial_{\beta}U\right|^{2}$
(4.12) $\sum_{\alpha,\beta}\nabla
G(U)\partial_{\alpha}UB_{\alpha\beta}^{\epsilon}(U)\partial_{\beta}U\geq\nu_{2}\sum_{\alpha}\left|\sum_{\beta}B_{\alpha\beta}^{\epsilon}(U)\partial_{\beta}U\right|^{2}$
(4.13)

Next, we list hypotheses used on the thermodynamic functions. For the internal
energy, when expressed in the form $\rho
e=\rho\tilde{e}(\rho_{1},\dots,\rho_{n},\rho\eta)$, we require

$\displaystyle\frac{\partial(\rho\tilde{e})}{\partial(\rho\eta)}>0\,,\quad\frac{\partial^{2}(\rho\tilde{e})}{\partial(\rho\eta)^{2}}>0$
(H1)
$\displaystyle\nabla^{2}_{(\rho_{1},\dots,\rho_{n},\rho\eta)}(\rho\tilde{e})>0$
(H2)

Hypothesis (H1) is natural in thermodynamics stating that the temperature
$\theta>0$ and ensuring convexity of the energy as a function of entropy.
Hypothesis (H2) implies that the system (3.27)-(3.29) is hyperbolic and
excludes various interesting models related to pressure laws of Van-der-Waals
type.

Using (H1) one may invert the equation
$\theta=\frac{\partial(\rho\tilde{e})}{\partial(\rho\eta)}$ and define the
inverse function $\rho\eta=(\rho\eta)^{*}(\rho_{1},...,\rho_{n},\theta)$. This
yields the Legendre transform [6, Sec 5]

$\rho\psi=\rho\tilde{e}(\rho_{1},\dots,\rho_{n},(\rho\eta)^{*})-\theta(\rho\eta)^{*}\quad\mbox{where}\quad\frac{\partial(\rho\tilde{e})}{\partial(\rho\eta)}(\rho_{1},\dots,\rho_{n},(\rho\eta)^{*})=\theta$
(4.14)

A computation shows that (H1), (H2) imply

$\nabla^{2}_{(\rho_{1},\dots,\rho_{n})}(\rho\psi)>0\,,\quad(\rho\psi)_{\theta\theta}<0\,.$
(4.15)

The latter should be compared to (3.14) and property (iii).

An alternative is to define the Legendre transform through the direct formula

$(\rho\psi)(\rho_{1},\dots,\rho_{n},\theta)=\inf_{0<\rho\eta<\infty}\big{\\{}\rho\tilde{e}(\rho_{1},\dots,\rho_{n},\rho\eta)-\theta\rho\eta\big{\\}}$
(4.16)

Under (H1) and

$\lim_{\rho\eta\to
0}\frac{\partial(\rho\tilde{e})}{\partial(\rho\eta)}=0\,,\quad\lim_{\rho\eta\to\infty}\frac{\rho\tilde{e}}{\rho\eta}=\infty\,$
(4.17)

this problem has at most one solution computed via (4.16). An advantage of
this approach is that the convexity conditions (4.15) follow directly from the
minimization formula (4.16). On the other hand to solve (4.16) requires the
assumption (4.17). This formulation and (4.17) is consistent with the third
law of thermodynamics (that the entropy vanishes at the state of zero-
temperature, see [6, Sec 1.10]) but the popular model of the ideal gas (with
constant heat capacity) violates the third law and presents negative
entropies. Nevertheless, the relation (4.14) between internal energy $\rho e$
and Helmholtz free energy $\rho\psi$ is still valid.

The adaptation of the general framework to the system (4.1)-(4.5) requires
some computations, and it is remarkable that such a complicated system hides a
simple structure. Let $U=(\rho_{1},\dots,\rho_{n},v,\theta)^{\top}$ be a
solution of (4.1)-(4.6) and
$\bar{U}=(\bar{\rho}_{1},\dots,\bar{\rho}_{n},\bar{v},\bar{\theta})^{\top}$ of
(4.7)-(4.10). Using (3.10), (3.12) and (1.4)-(1.7) we arrive at

$\displaystyle H(U|\bar{U})$
$\displaystyle=-\rho\eta+\bar{\rho}\bar{\eta}-\frac{1}{\bar{\theta}}\sum_{j=1}^{n}(\bar{\mu}_{j}-\tfrac{1}{2}\bar{v}^{2})(\rho_{j}-\bar{\rho}_{j})-\frac{\bar{v}}{\bar{\theta}}\cdot(\rho
v-\bar{\rho}\bar{v})$ (4.18)
$\displaystyle\qquad+\frac{1}{\bar{\theta}}\Big{(}\rho e+\tfrac{1}{2}\rho
v^{2}-\bar{\rho}\bar{e}-\tfrac{1}{2}\bar{\rho}\bar{v}^{2}\Big{)}$
$\displaystyle=\frac{1}{2\bar{\theta}}\rho|v-\bar{v}|^{2}+\frac{1}{\bar{\theta}}\Big{[}\Big{(}\rho\psi-\bar{\rho}\bar{\psi}-\sum_{j=1}^{n}\bar{\mu}_{j}(\rho_{j}-\bar{\rho}_{j})+\bar{\rho}\bar{\eta}(\theta-\bar{\theta})\Big{)}+(\theta-\bar{\theta})(\rho\eta-\bar{\rho}\bar{\eta})\Big{]}$
$\displaystyle=\frac{1}{\bar{\theta}}\left(\frac{1}{2}\rho|v-\bar{v}|^{2}+J(\rho_{1},\dots,\rho_{n},\theta|\bar{\rho}_{1},\dots,\bar{\rho}_{n},\bar{\theta})\right)$

where we set $\omega=(\rho_{1},\dots,\rho_{n},\theta)$,
$\bar{\omega}=(\bar{\rho}_{1},\dots,\bar{\rho}_{n},\bar{\theta})$ and use
(1.5) to write

$\displaystyle J(\omega|\bar{\omega})$
$\displaystyle:=\rho\psi-\bar{\rho}\bar{\psi}-\sum_{i}\bar{\mu}_{i}(\rho_{i}-\bar{\rho}_{i})+\bar{\rho}\bar{\eta}(\theta-\bar{\theta})+(\theta-\bar{\theta})(\rho\eta-\bar{\rho}\bar{\eta})$
(4.19) $\displaystyle=\rho
e-\bar{\rho}\bar{e}-\sum_{i}\bar{\mu}_{i}(\rho_{i}-\bar{\rho}_{i})-\bar{\theta}(\rho\eta-\bar{\rho}\bar{\eta})$
$\displaystyle I(U|\bar{U})$
$\displaystyle:=\bar{\theta}H(U|\bar{U})=\frac{1}{2}\rho|v-\bar{v}|^{2}+J(\omega|\bar{\omega})\,.$
(4.20)

Due to hypothesis (H2) and (A.11), the quantity $J(\omega|\bar{\omega})$ will
serve as a measure of the distance between the states $\omega$ and
$\bar{\omega}$, in analogy to the situation in single component fluids [8].
This suggests to calculate the evolution of the quantity (4.20).

Subtracting the entropy identities (4.10) from (4.6) and multiplying by
$\bar{\theta}$, we obtain

$\begin{split}&\partial_{t}(-\bar{\theta}\rho\eta+\bar{\theta}\bar{\rho}\bar{\eta})+\textnormal{div}(-\rho\eta
v\bar{\theta}+\bar{\rho}\bar{\eta}\bar{v}\bar{\theta})=\textnormal{div}\left(\frac{\bar{\theta}}{\theta}\sum_{j}(h_{j}-\rho_{j}\mu_{j})u_{j}-\frac{\bar{\theta}}{\theta}\kappa\nabla\theta\right)\\\
&-\partial_{t}\bar{\theta}(\rho\eta-\bar{\rho}\bar{\eta})+\nabla\bar{\theta}\cdot(-\rho\eta
v+\bar{\rho}\bar{\eta}\bar{v})-\frac{1}{\theta}\sum_{j}(h_{j}-\rho_{j}\mu_{j})u_{j}\cdot\nabla\bar{\theta}+\frac{1}{\theta}\kappa\nabla\theta\cdot\nabla\bar{\theta}\\\
&-\frac{\bar{\theta}}{\theta^{2}}\kappa|\nabla\theta|^{2}+\frac{\bar{\theta}}{\theta}\sum_{j}u_{j}\cdot
d_{j}+\textnormal{div}\left(\frac{\bar{\theta}}{\bar{\theta}}\bar{\kappa}\nabla\bar{\theta}\right)-\frac{1}{\bar{\theta}}\bar{\kappa}\nabla\bar{\theta}\cdot\nabla\bar{\theta}+\frac{\bar{\theta}}{\bar{\theta}^{2}}\bar{\kappa}|\nabla\bar{\theta}|^{2}\end{split}$
(4.21)

Likewise, subtracting system (4.7)-(4.9) from system (4.1)-(4.3) and
multiplying the result by $-\bar{\theta}G(\bar{U})$, where $G$ is the
multiplier from (3.12), we obtain:

$\begin{split}&\partial_{t}\left(\sum_{i}\left(\frac{1}{2}\bar{v}^{2}-\bar{\mu}_{i}\right)(\rho_{i}-\bar{\rho}_{i})-\bar{v}(\rho
v-\bar{\rho}\bar{v})+\left(\rho e+\frac{1}{2}\rho
v^{2}-\bar{\rho}\bar{e}-\frac{1}{2}\bar{\rho}\bar{v}^{2}\right)\right)\\\
&+\textnormal{div}\Big{(}\sum_{i}\left(\frac{1}{2}\bar{v}^{2}-\bar{\mu}_{i}\right)(\rho_{i}v-\bar{\rho}_{i}\bar{v})-\bar{v}(\rho
v\otimes v-\bar{\rho}\bar{v}\otimes\bar{v}+(p-\bar{p})\mathbb{I})\\\
&+\left(\rho e+\frac{1}{2}\rho
v^{2}+p\right)v-\left(\bar{\rho}\bar{e}+\frac{1}{2}\bar{\rho}\bar{v}^{2}+\bar{p}\right)\bar{v}\Big{)}=\sum_{i}\partial_{t}\left(\frac{1}{2}\bar{v}^{2}-\bar{\mu}_{i}\right)(\rho_{i}-\bar{\rho_{i}})\\\
&+\sum_{i}\nabla\left(\frac{1}{2}\bar{v}^{2}-\bar{\mu}_{i}\right)\cdot(\rho_{i}v-\bar{\rho}_{i}\bar{v})-\sum_{i}\left(\frac{1}{2}\bar{v}^{2}-\bar{\mu}_{i}\right)\textnormal{div}(\rho_{i}u_{i})-\partial_{t}\bar{v}\cdot(\rho
v-\bar{\rho}\bar{v})\\\ &-\rho v\nabla\bar{v}\cdot
v+\bar{\rho}\bar{v}\nabla\bar{v}\cdot\bar{v}-(p-\bar{p})\textnormal{div}\bar{v}+\textnormal{div}\left(\kappa\nabla\theta-\sum_{j}h_{j}u_{j}\right)-\textnormal{div}(\bar{\kappa}\nabla\bar{\theta})\end{split}$
(4.22)

where we used the abbreviation $h_{j}:=\rho_{j}e_{j}+p_{j}$.

Next, we add equations (4.21) and (4.22) and perform a series of calculations
detailed in Appendix D to re-organize the terms in the right hand side. The
resulting relative entropy identity reads:

$\begin{split}&\partial_{t}I(U|\bar{U})+\textnormal{div}\Big{[}vI(U|\bar{U})+(p-\bar{p})(v-\bar{v})+\sum_{j}\rho_{j}u_{j}(\mu_{j}-\bar{\mu}_{j})\\\
&-(\theta-\bar{\theta})\left(\frac{1}{\theta}\kappa\nabla\theta-\frac{1}{\bar{\theta}}\bar{\kappa}\nabla\bar{\theta}\right)+\frac{1}{\theta}(\theta-\bar{\theta})\sum_{j}(h_{j}-\rho_{j}\mu_{j})u_{j}\Big{]}+\bar{\theta}\kappa\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)^{2}\\\
&-\frac{\bar{\theta}}{\theta}\sum_{j}u_{j}\cdot
d_{j}=(\partial_{t}\bar{\theta}+\bar{v}\cdot\nabla\bar{\theta})(-\rho\eta)(\omega|\bar{\omega})-p(\omega|\bar{\omega})\textnormal{div}\bar{v}-(\eta-\bar{\eta})\rho(v-\bar{v})\cdot\nabla\bar{\theta}\\\
&-\sum_{j}\nabla\bar{\mu}_{j}\left(\frac{\rho_{j}}{\rho}-\frac{\bar{\rho}_{j}}{\bar{\rho}}\right)\rho(v-\bar{v})-\rho(v-\bar{v})\nabla\bar{v}\cdot(v-\bar{v})-\sum_{j}\nabla\bar{\mu}_{j}\cdot\rho_{j}u_{j}\\\
&-\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)\frac{\nabla\bar{\theta}}{\bar{\theta}}(\bar{\theta}\kappa-\theta\bar{\kappa})-\frac{1}{\bar{\theta}}\nabla\bar{\theta}\sum_{j}(h_{j}-\rho_{j}\mu_{j})u_{j}\end{split}$
(4.23)

where

$\displaystyle p(\omega|\bar{\omega})$
$\displaystyle=p-\bar{p}-\sum_{j}\bar{p}_{\rho_{j}}(\rho_{j}-\bar{\rho}_{j})-\bar{p}_{\theta}(\theta-\bar{\theta})$
$\displaystyle(-\rho\eta)(\omega|\bar{\omega})$
$\displaystyle=-\rho\eta+\bar{\rho}\bar{\eta}+\sum_{j}(\bar{\rho}\bar{\eta})_{\rho_{j}}(\rho_{j}-\bar{\rho}_{j})+(\bar{\rho}\bar{\eta})_{\theta}(\theta-\bar{\theta})$

In the sequel, let $\mathcal{U}\subset(\mathbb{R}^{+})^{n+1}$ be a set in the
positive cone $(\mathbb{R}^{+})^{n+1}$ with $\overline{\mathcal{U}}$ compact,
and suppose that states $\omega,\bar{\omega}\in\mathcal{U}$ satisfy

$0<\rho_{j},\bar{\rho}_{j}\leq M$ (4.24) $0<\delta\leq\rho,\bar{\rho}\leq M$
(4.25) $0<\delta\leq\theta,\bar{\theta}\leq M$ (4.26)

for some $\delta,M>0$.

######  Lemma 1.

Let $\omega,\bar{\omega}\in\mathcal{U}$ satisfy (4.24)-(4.26) and suppose that
$\psi(\rho_{1},\dots,\rho_{n},\theta)\in C^{3}(\overline{\mathcal{U}})$
satisfies the thermodynamic relations (1.4)-(1.8) and the hypotheses (H1),(H2)
(and thus (4.15)). There exist constants $c_{1},c_{2},c_{3}>0$ depending on
$\delta,M$, such that for $\omega,\bar{\omega}\in\mathcal{U}$ we have

$c_{1}|\omega-\bar{\omega}|^{2}\leq J(\omega|\bar{\omega})$ (4.27)
$|p(\omega|\bar{\omega})|\leq c_{2}J(\omega|\bar{\omega})$ (4.28)
$|(-\rho\eta)(\omega|\bar{\omega})|\leq c_{3}J(\omega|\bar{\omega})\,.$ (4.29)

###### Proof.

Consider the form (4.19) and use (A.11) and the convexity of $\rho
e=\rho\tilde{e}(\rho_{1},\dots,\rho_{n},\rho\eta)$ in the variables
$(\rho_{1},\dots,\rho_{n},\rho\eta)$ to obtain

$\begin{split}J(\omega|\bar{\omega})&=\rho\tilde{e}-\overline{\rho\tilde{e}}-\sum_{i}\overline{\left(\frac{\partial\rho\tilde{e}}{\partial\rho_{i}}\right)}(\rho_{i}-\bar{\rho}_{i})-\overline{\left(\frac{\partial\rho\tilde{e}}{\partial\rho\eta}\right)}(\rho\eta-\bar{\rho}\bar{\eta})\\\
&\geq
c\left(\sum_{i}|\rho_{i}-\bar{\rho}_{i}|^{2}+|\rho\eta-\bar{\rho}\bar{\eta}|^{2}\right)\end{split}$
(4.30)

where
$c=\inf_{(\rho_{1},\dots,\rho_{n},\rho\eta)\in\overline{\mathcal{U}}}\nabla^{2}_{(\rho_{1},\dots,\rho_{n},\rho\eta)}\rho\tilde{e}>0$,
for $0<\delta\leq\rho<M$. Next, the map
$(\rho_{1},\dots,\rho_{n},\theta)\mapsto(\rho_{1},\dots,\rho_{n},\rho\eta)$
defined by $\eta(\rho_{1},\dots,\rho_{n},\theta)=-\psi_{\theta}$ can be
inverted on the set $\overline{\mathcal{U}}$ and since
$\frac{\partial(\rho\eta)}{\partial\theta}=\frac{1}{\theta}\rho c_{v}>0$, for
$\rho,\theta>0$, the inverse map is Lipschitz and

$|\theta-\bar{\theta}|^{2}+\sum_{i}|\rho_{i}-\bar{\rho}_{i}|^{2}\leq
C\left(|\rho\eta-\bar{\rho}\bar{\eta}|^{2}+\sum_{i}|\rho_{i}-\bar{\rho}_{i}|^{2}\right)$
(4.31)

where $C$ depends on
$\min_{\overline{\mathcal{U}}}\frac{\partial(\rho\eta)}{\partial\theta}$.
Combining (4.30) with (4.31) gives (4.27).

The bounds (4.28)-(4.29) follow from the Taylor theorem, which provides

$p(\omega|\bar{\omega})=(\omega-\bar{\omega})^{\top}\left(\int_{0}^{1}\int_{0}^{s}\nabla^{2}p(\tau\omega+(1-\tau)\bar{\omega})d\tau
ds\right)(\omega-\bar{\omega})$ (4.32)
$(-\rho\eta)(\omega|\bar{\omega})=(\omega-\bar{\omega})^{\top}\left(\int_{0}^{1}\int_{0}^{s}\nabla^{2}(-\rho\eta)(\tau\omega+(1-\tau)\bar{\omega})d\tau
ds\right)(\omega-\bar{\omega})$ (4.33)

and the regularity of $\psi$ implies that $p,\rho\eta\in
C^{2}(\overline{\mathcal{U}})$. ∎

Let $\mathbb{T}^{3}=(\mathbb{R}/2\pi\mathbb{Z})^{3}$ be the three-dimensional
torus. We have the following convergence result:

######  Theorem 2.

Let $\bar{U}^{\kappa}$ be a classical solution of (4.7)-(4.9) and let
$U^{\epsilon,\kappa}$ be a family of classical solutions of (4.1)-(4.3)
defined on $\mathbb{T}^{3}\times[0,T]$ for some $T<T^{*}$, which emanate from
smooth data $\bar{U}_{0}^{\kappa}$, $U_{0}^{\epsilon,\kappa}$, respectively,
and satisfy the uniform bounds (4.24)-(4.26) for $\delta,M>0$. Moreover,
assume that $\psi\in C^{3}(\bar{\mathcal{U}})$ satisfies (H1), (H2), that

$\sum_{j}\left|\frac{h_{j}-\rho_{j}\mu_{j}}{\sqrt{\rho_{j}}}\right|^{2}\leq\alpha(\rho
e+1)$ (4.34)

for some $\alpha>0$ and that $0\leq\kappa(\rho_{1},\dots,\rho_{n},\theta)\leq
M$. Then, there exist constants $c,C>0$ depending on $\delta,M,\alpha$ but
otherwise independent of $\epsilon$ such that

$\begin{split}\int_{\mathbb{T}^{3}}I(U^{\epsilon,\kappa}|\bar{U}^{\kappa})dx&\leq
c\int_{\mathbb{T}^{3}}I(U^{\epsilon,\kappa}_{0}|\bar{U}^{\kappa}_{0})dx+\epsilon
C\end{split}$ (4.35)

In particular, if
$\int_{\mathbb{T}^{3}}I(U^{\epsilon,\kappa}_{0}|\bar{U}^{\kappa}_{0})dx\to 0$
as $\epsilon\to 0$, then

$\sup_{t\in(0,T)}\int_{\mathbb{T}^{3}}I(U^{\epsilon,\kappa}|\bar{U}^{\kappa})dx\to
0~{}~{}\textnormal{as $\epsilon\to 0$}.$ (4.36)

###### Proof.

We first integrate the relative entropy identity (4.23) to obtain

$\begin{split}&\frac{d}{dt}\int_{\mathbb{T}^{3}}I(U^{\varepsilon}|\bar{U})dx+\int_{\mathbb{T}^{3}}\bar{\theta}\kappa\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)^{2}dx-\int_{\mathbb{T}^{3}}\frac{\bar{\theta}}{\theta}\sum_{j}u_{j}\cdot
d_{j}dx\\\
&=\int_{\mathbb{T}^{3}}(\partial_{t}\bar{\theta}+\bar{v}\cdot\nabla\bar{\theta})(-\rho\eta)(U^{\varepsilon}|\bar{U})dx-\int_{\mathbb{T}^{3}}p(U^{\varepsilon}|\bar{U})\textnormal{div}\bar{v}dx\\\
&+\int_{\mathbb{T}^{3}}(\eta-\bar{\eta})\rho(v-\bar{v})\cdot\nabla\bar{\theta}dx-\int_{\mathbb{T}^{3}}\sum_{j}\nabla\bar{\mu}_{j}\left(\frac{\rho_{j}}{\rho}-\frac{\bar{\rho}_{j}}{\bar{\rho}}\right)\rho(v-\bar{v})dx\\\
&-\int_{\mathbb{T}^{3}}\rho(v-\bar{v})\nabla\bar{v}\cdot(v-\bar{v})dx-\int_{\mathbb{T}^{3}}\sum_{j}\nabla\bar{\mu}_{j}\cdot\rho_{j}u_{j}dx\\\
&-\int_{\mathbb{T}^{3}}\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)\frac{\nabla\bar{\theta}}{\bar{\theta}}\kappa(\bar{\theta}-\theta)dx-\int_{\mathbb{T}^{3}}\frac{1}{\bar{\theta}}\nabla\bar{\theta}\sum_{j}(h_{j}-\rho_{j}\mu_{j})u_{j}dx\\\
&=:I_{1}+\cdots+I_{8}\,.\end{split}$ (4.37)

Our strategy is to control terms $I_{1}$ to $I_{5}$ by the integral of
$I(U|\bar{U})$ and terms $I_{6}$ to $I_{8}$ by the dissipation on the left-
hand side. In particular, to control $I_{1}$ and $I_{2}$, we use (4.29) and
(4.28) respectively. For $I_{3}$, using the regularity of $\psi$, and thus
$\eta$, we have

$\begin{split}(\eta-\bar{\eta})\rho(v-\bar{v})\cdot\nabla\bar{\theta}&\leq
c\left(\rho|\eta-\bar{\eta}|^{2}+\rho|v-\bar{v}|^{2}\right)\\\ &\leq
c\left(\sum_{j}|\rho_{j}-\bar{\rho}_{j}|^{2}+|\theta-\bar{\theta}|^{2}+\rho|v-\bar{v}|^{2}\right)\end{split}$

Regarding $I_{4}$, we have

$\begin{split}\sum_{j}\nabla\bar{\mu}_{j}\left(\frac{\rho_{j}}{\rho}-\frac{\bar{\rho}_{j}}{\bar{\rho}}\right)\rho(v-\bar{v})&\leq
c\left(\rho\sum_{j}\left(\frac{\rho_{j}}{\rho}-\frac{\bar{\rho}_{j}}{\bar{\rho}}\right)^{2}+\rho(v-\bar{v})^{2}\right)\\\
&\leq
c\left(\sum_{j}(\rho_{j}-\bar{\rho}_{j})^{2}+\rho(v-\bar{v})^{2}\right)\end{split}$

since the map
$f_{j}:(\rho_{1},\dots,\rho_{n})\mapsto\frac{\rho_{j}}{\rho_{1}+\cdots+\rho_{n}}$,
for $j\in\\{1,\dots,n\\}$ is Lipschitz under the assumption
$0<\delta\leq\rho$. Indeed

$\frac{\partial
f_{j}}{\partial\rho_{i}}=\begin{cases}-\frac{\rho_{j}}{\rho^{2}},&i\not=j\\\
\frac{\rho-\rho_{j}}{\rho^{2}},&i=j\end{cases}$

hence

$\left|\frac{\partial
f_{j}}{\partial\rho_{i}}\right|\leq\frac{c}{\rho}\leq\frac{c}{\delta}\,.$

As for $I_{5}$

$\rho(v-\bar{v})\nabla\bar{v}\cdot(v-\bar{v})\leq c\rho|v-\bar{v}|^{2}$

Now, if we set $\rho_{j}u_{j}=\epsilon\rho_{j}\tilde{u}_{j}$ we have, by
Young’s inequality and for $\mu=\min_{i\not=j}b_{ij}>0$:

$\begin{split}I_{6}&\leq\frac{\epsilon}{\mu\delta}\int_{\mathbb{T}^{3}}\sum_{j}\frac{\rho_{j}}{\rho}|\nabla\bar{\mu}_{j}|^{2}dx+\frac{\epsilon\mu\delta}{4}\int_{\mathbb{T}^{3}}\rho\sum_{j}|\sqrt{\rho_{j}}\tilde{u}_{j}|^{2}dx\\\
&\leq\epsilon
C+\frac{\epsilon\mu\delta}{4}\int_{\mathbb{T}^{3}}\rho\sum_{j}|\sqrt{\rho_{j}}\tilde{u}_{j}|^{2}dx=:I_{61}+I_{62}\end{split}$

Likewise, using assumption (4.34), we have that

$\begin{split}I_{8}&\leq\frac{\epsilon}{\mu\delta}\int_{\mathbb{T}^{3}}\frac{1}{\rho}\sum_{j}\left|\frac{h_{j}-\rho_{j}\mu_{j}}{\sqrt{\rho_{j}}}\right|^{2}\left|\frac{\nabla\bar{\theta}}{\bar{\theta}}\right|^{2}dx+\frac{\epsilon\mu\delta}{4}\int_{\mathbb{T}^{3}}\rho\sum_{j}|\sqrt{\rho_{j}}\tilde{u}_{j}|^{2}dx\\\
&\leq\epsilon
C+\frac{\epsilon\mu\delta}{4}\int_{\mathbb{T}^{3}}\rho\sum_{j}|\sqrt{\rho_{j}}\tilde{u}_{j}|^{2}dx=:I_{81}+I_{82}\end{split}$

Finally

$\begin{split}I_{7}&\leq\frac{1}{2}\int_{\mathbb{T}^{3}}\bar{\theta}\kappa\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)^{2}dx+\frac{1}{2}\int_{\mathbb{T}^{3}}\kappa\frac{|\nabla\bar{\theta}|^{2}}{\bar{\theta}^{3}}(\theta-\bar{\theta})^{2}dx\\\
&\leq\frac{1}{2}\int_{\mathbb{T}^{3}}\bar{\theta}\kappa\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)^{2}dx+c\int_{\mathbb{T}^{3}}(\theta-\bar{\theta})^{2}dx=:I_{71}+I_{72}\end{split}$

Using (4.27) we conclude that

$I_{1}+\cdots+I_{5}+I_{72}\leq
c\int_{\mathbb{T}^{3}}I(U^{\varepsilon}|\bar{U})dx\,.$

The error terms $I_{62}$, $I_{71}$ and $I_{82}$ are controlled by the
dissipation on the left-hand side of (4.37). This is due to the assumption
that $\delta\leq\bar{\theta}$ and the following estimate, in which we use
(3.25) and (3.23):

$\begin{split}-\frac{1}{\theta}\sum_{j}u_{j}\cdot
d_{j}&=\epsilon\rho\sum_{i,j}M_{ij}\sqrt{\rho_{i}}\tilde{u}_{i}\cdot\sqrt{\rho_{j}}\tilde{u}_{j}\\\
&\geq\epsilon\rho\mu|\mathbb{P}_{L}\sqrt{\rho}\tilde{u}|^{2}\end{split}$

where $\sqrt{\rho}\tilde{u}$ is the vector with components
$\sqrt{\rho_{i}}\tilde{u}_{i}$, whose projection is computed as

$\begin{split}\mathbb{P}_{L}\sqrt{\rho}\tilde{u}&=\sum_{k}(\mathbb{P}_{L})_{ik}\sqrt{\rho_{k}}\tilde{u}_{k}\\\
&=\sqrt{\rho_{i}}\tilde{u}_{i}-\frac{\sqrt{\rho_{i}}}{\rho}\sum_{k}\rho_{k}\tilde{u}_{k}\\\
&=\sqrt{\rho_{i}}\tilde{u}_{i}\end{split}$

Putting everything together we obtain:

$\begin{split}\frac{d}{dt}\int_{\mathbb{T}^{3}}I(U^{\varepsilon}|\bar{U})dx&+\frac{\delta}{2}\int_{\mathbb{T}^{3}}\kappa\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)^{2}dx+\frac{\delta\epsilon\mu}{2}\int_{\mathbb{T}^{3}}\rho\sum_{j}|\sqrt{\rho_{j}}\tilde{u}_{j}|^{2}dx\\\
&\leq c\int_{\mathbb{T}^{3}}I(U^{\varepsilon}|\bar{U})dx+\epsilon
C\end{split}$ (4.38)

for appropriate constants $c,C>0$ independent of $\epsilon$. The dissipation
terms are neglected and we obtain the differential inequality

$\frac{d\varphi^{\varepsilon}(t)}{dt}\leq c\varphi^{\varepsilon}(t)+\epsilon
C$ (4.39)

for

$\varphi^{\varepsilon}(t)=\int_{\mathbb{T}^{3}}I(U^{\varepsilon}(x,t)|\bar{U}(x,t))dx$

Then, (4.35) follows by Grönwall’s Lemma. ∎

A particular case of the above analysis is the convergence to the adiabatic
theory, i.e. when also $\kappa=0$. Consider the hyperbolic-parabolic system
(4.1)-(4.6) with the linear system (4.4) and the constraint (4.5) and its
hyperbolic counterpart (3.27)-(3.30). Then, following the same process as
before, we can obtain that the hyperbolic-parabolic system (4.1)-(4.6)
converges as $\epsilon,\kappa\to 0$ to the hyperbolic system without diffusion
and heat conduction (3.27)-(3.30):

######  Theorem 3.

Let $\bar{U}$ be a classical solution of (3.27)-(3.29) defined on a maximal
interval of existence $\mathbb{T}^{3}\times[0,T^{*})$ and let
$U^{\epsilon,\kappa}$ be a family of classical solutions of (4.1)-(4.3)
defined on $\mathbb{T}^{3}\times[0,T]$, for $T<T^{*}$, emanating from smooth
data $\bar{U}_{0},U_{0}^{\epsilon,\kappa}$ respectively. Under the hypotheses
of theorem 2, there exist constants $c,C_{1},C_{2}>0$ independent of
$\epsilon$ and $\kappa$ such that

$\begin{split}\int_{\mathbb{T}^{3}}I(U^{\epsilon,\kappa}|\bar{U})dx&\leq
c\int_{\mathbb{T}^{3}}I(U^{\epsilon,\kappa}_{0}|\bar{U}_{0})dx+\|\kappa\|_{\infty}C_{1}+\epsilon
C_{2}\end{split}$ (4.40)

In particular, if
$\int_{\mathbb{T}^{3}}I(U^{\epsilon,\kappa}_{0}|\bar{U}_{0})dx\to 0$ as
$\epsilon,\kappa\to 0$, then

$\sup_{t\in(0,T)}\int_{\mathbb{T}^{3}}I(U^{\epsilon,\kappa}|\bar{U})dx\to
0~{}~{}\textnormal{as $\epsilon,\kappa\to 0$}.$ (4.41)

###### Proof.

The proof is identical to the proof of Theorem 2, with the only difference
that $\bar{\kappa}=0$ (whereas in the previous case $\kappa=\bar{\kappa}$) and
thus $I_{7}$ is controlled by:

$\begin{split}I_{7}&=-\int_{\mathbb{T}^{3}}\kappa\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)\cdot\nabla\bar{\theta}\\\
&\leq\frac{1}{2}\int_{\mathbb{T}^{3}}\bar{\theta}\kappa\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)^{2}dx+\frac{1}{2}\int_{\mathbb{T}^{3}}\kappa\frac{|\nabla\bar{\theta}|^{2}}{\bar{\theta}}dx\\\
&\leq\frac{1}{2}\int_{\mathbb{T}^{3}}\bar{\theta}\kappa\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)^{2}dx+\|\kappa\|_{\infty}C_{1}\end{split}$

∎

Next, we present a commentary on the hypotheses for Theorems 2 and 3. The goal
here is to discuss aspects of the theory of smooth solutions; the situation
for weak solutions presents serious challenges. The hypotheses on the bounds
(4.25), (4.26) reflect the loss of strict hyperbolicity (and even
hyperbolicity) of the model at $\rho=0$. Also, we would expect that such
continuum models are not valid for very large temperatures or temperature near
zero.

The hypothesis $\psi\in C^{3}(\bar{U})$ is a drawback as it does not hold at
$\rho_{i}=0$ for realistic models and at the same time one would insist on the
range (4.24) that guarantees some of the components may disappear as an
outcome of interactions. The main problematic term is $I_{3}$. Realistic
models are discussed in section 5 dealing with the multicomponent ideal gas.
The reader will notice that $\psi\in C^{3}(\bar{U})$ holds on the restricted
range

$0<\delta\leq\rho_{j},\bar{\rho}_{j}\leq M$ (4.42)

and gives convergence for solutions taking values in that range (4.42),
(4.25), (4.26).

The need concerning (4.34) originates from a deficiency of the models
discussed here. For the general model, the partial energies $e_{i}$, pressures
$p_{i}$ and enthalpies $h_{i}$ are not determined from the total free energy
$\rho\psi$ via (A.19), yet they enter the balance equations (1.9)-(1.12) and
thus extra constitutive relations have to be supplied. Hypothesis (4.34)
concerns these extra constitutive relations. For the case of simple mixtures,
$\rho\psi=\sum_{i}\rho_{i}\psi_{i}(\rho_{i},\theta)$, the need for extra
constitutive relations does not arise; one computes using (B.11)-(B.14) that
$\frac{1}{\sqrt{\rho_{j}}}(h_{j}-\rho_{j}\mu_{j})=-\theta\sqrt{\rho_{j}}\frac{\partial{\psi_{j}}}{\partial\theta}$
and the terms are bounded from (4.24)-(4.26).

##  5\. Multicomponent ideal gases

We present the constitutive model of an ideal multicomponent gas and compute
the relative constitutive functions that appear in the relative entropy
formula (4.23). We refer to Callen [6, Sec 13] and Giovangigli [12, Sec 6] for
details on the multicomponent ideal gas laws. Here, we outline a constitutive
model defined in terms of densities $\rho_{i}$ and temperature $\theta$. The
model is a simple mixture of ideal gases where the free energy of each
component is given by
$\rho_{i}\psi_{i}=R_{i}\theta\rho_{i}\log\rho_{i}-c_{i}\rho_{i}\theta\log\theta$,
where $R_{i}>0$ is the engineering gas constant and $c_{i}>0$ the constant
heat capacity of the $i$-th component, and the mixture free energy is

$\rho\psi=\sum_{i}\rho_{i}\psi_{i}=\sum_{i}R_{i}\theta\rho_{i}\log\rho_{i}-\sum_{i}c_{i}\rho_{i}\theta\log\theta\,.$
(5.1)

We then have

$\displaystyle\mu_{j}$
$\displaystyle=\frac{\partial(\rho\psi)}{\partial\rho_{j}}=R_{j}(1+\log\rho_{j})\theta-
c_{j}\theta\log\theta$ (5.2) $\displaystyle\rho\eta$
$\displaystyle=-\frac{\partial(\rho\psi)}{\partial\theta}=-\sum_{i}R_{i}\rho_{i}\log\rho_{i}+\sum_{i}c_{i}\rho_{i}(1+\log\theta)$
$\displaystyle\rho e$ $\displaystyle=\sum_{i}c_{i}\rho_{i}\theta$
$\displaystyle p$
$\displaystyle=-\rho\psi+\sum_{j}\rho_{j}\mu_{j}=\sum_{j}R_{j}\rho_{j}\theta$

The relative quantities are computed as follows: Using (4.19), we have

$\displaystyle J(\omega|\bar{\omega})$
$\displaystyle=\bar{\theta}\sum_{i}R_{i}\Big{(}\rho_{i}\log\rho_{i}-\bar{\rho}_{i}\log\bar{\rho}_{i}-(1+\log\bar{\rho}_{i})(\rho_{i}-\bar{\rho}_{i})\Big{)}$
(5.3)
$\displaystyle\qquad+\bar{\theta}\sum_{i}c_{i}\rho_{i}\Big{(}-\log\theta+\log\bar{\theta}+\frac{1}{\bar{\theta}}(\theta-\bar{\theta})\Big{)}$
$\displaystyle=\bar{\theta}\sum_{i}R_{i}(x\log
x)\big{(}\rho_{i}|\bar{\rho}_{i}\big{)}+\sum_{i}c_{i}\rho_{i}\bar{\theta}(-\log
y)(\theta|\bar{\theta})$

where $(x\log x)\big{(}\rho_{i}|\bar{\rho}_{i}\big{)}$ is the quadratic part
of the Taylor expansion of $x\log x$ and ditto for $(-\log
y)(\theta|\bar{\theta})$. Due to the convexity of $(x\log x)$ and $(-\log y)$
both terms are positive.

Similarly, we compute

$p(\omega|\bar{\omega})=\sum_{j}R_{j}(\rho_{j}-\bar{\rho}_{j})(\theta-\bar{\theta})$
(5.4)

and

$\displaystyle(-\rho\eta)(\omega|\bar{\omega})$
$\displaystyle=-\rho\eta+\bar{\rho}\bar{\eta}+\sum_{j}\overline{\frac{\partial(\rho\eta)}{\partial\rho_{j}}}(\rho_{j}-\bar{\rho}_{j})+\overline{\frac{\partial(\rho\eta)}{\partial\theta}}(\theta-\bar{\theta})$
(5.5) $\displaystyle=\sum_{i}R_{i}\big{(}x\log
x\big{)}\big{(}\rho_{i}|\bar{\rho}_{i}\big{)}+\sum_{i}c_{i}\rho_{i}(-\log
y)(\theta|\bar{\theta})-\sum_{i}\frac{c_{i}}{\bar{\theta}}(\rho_{i}-\bar{\rho}_{i})(\theta-\bar{\theta})$

Finally observe that the relative entropy (4.20) takes the form

$I(U|\bar{U})=\tfrac{1}{2}\rho|v-\bar{v}|^{2}+\bar{\theta}\sum_{i}R_{i}\big{(}x\log
x\big{)}\big{(}\rho_{i}|\bar{\rho}_{i}\big{)}+\bar{\theta}\sum_{i}c_{i}\rho_{i}\big{(}-\log
y\big{)}\big{(}\theta|\bar{\theta}\big{)}$ (5.6)

Using the explicit formulas (5.3), (5.4) and (5.5) the facts $(x\log
x)^{\prime\prime}=\frac{1}{x}$ and $(-\log x)^{\prime\prime}=\frac{1}{x^{2}}$
we obtain an analog of Lemma 1 for the ideal gas:

######  Lemma 4.

Let $\omega,\bar{\omega}\in\mathcal{U}$ satisfy (4.24)-(4.26) and let
$\rho\psi$ be given in (5.1). There exist constants $c_{1},c_{2}>0$ depending
on $\delta,M$, such that

$c_{1}|\omega-\bar{\omega}|^{2}\leq J(\omega|\bar{\omega})$ (5.7)
$|p(\omega|\bar{\omega})|+|(-\rho\eta)(\omega|\bar{\omega})|\leq
c_{2}J(\omega|\bar{\omega})\,.$ (5.8)

The missing element to conclude the proof of an analog of Theorem 2 for the
multicomponent ideal gas is an estimate of the type

$\begin{split}|(\eta-\bar{\eta})\rho(v-\bar{v})\cdot\nabla\bar{\theta}|\leq
C\Big{(}\frac{1}{2}\rho|v-\bar{v}|^{2}+J(\omega|\bar{\omega})\Big{)}\end{split}$

However, such an estimate is not valid for solutions that take value on the
range (4.24)-(4.26).

##  Appendix A Equilibrium Thermodynamics

Following [6, sec 2,3,5] we describe some elements of the thermodynamic theory
in equilibrium, known as thermostatics, which seeks to describe the
equilibrium states to which systems eventually evolve. It is assumed that
there exist equilibrium states of simple systems and that they are
characterized by the internal energy of the system $E$, the volume $V$ and the
mole number of the components of the system $M_{1},\dots,M_{n}$. These
quantities are called extensive parameters. It is postulated that there exists
a function $H$ of the extensive parameters, called entropy, defined for all
equilibrium states, so that the values assumed by the extensive parameters in
the absence of internal constraints, are those that maximize the entropy over
the manifold of constrained equilibria. We refer to [6, sec 2] for the precise
statements and details. Note that the existence of the entropy is postulated
only for equilibrium states, that the entropy of the system is postulated to
be provided by an extremum principle and to be known as a function of the
extensive parameters,

$H=H(E,V,M_{1},\dots,M_{n})\,.$ (A.1)

This function, known as the fundamental relation, contains all thermodynamic
information about the system. As the entropy of a composite system is additive
over the constituent subsystem, this function has to be homogeneous of first
order,

$H(\lambda E,\lambda V,\lambda M_{1},\dots,\lambda M_{n})=\lambda
H(E,V,M_{1},\dots,M_{n}),~{}\forall\lambda>0,$ (A.2)

it is assumed to be differentiable, and is postulated to be an increasing
function of the internal energy. As a result, the fundamental relation can be
inverted with respect to the energy leading to a differentiable function of
$H,V,M_{1},\dots,M_{n}$ and an alternative form of the fundamental relation

$E=E(H,V,M_{1},\dots,M_{n})\,,$ (A.3)

for which homogeneity of first order still holds.

With the aforementioned postulates we are ready to proceed to the
thermodynamic analysis. Since the energy is a quantity that is easier to
understand intuitively and also to measure in experiments, we prefer to use
the energy representation of the fundamental relation, instead of the entropy
one. In this case, the extremum principle is no longer the maximization of the
entropy in equilibrium, but rather the minimization of the energy. The
condition that guarantees the equivalence between the two principles is the
fact that the entropy is an increasing function of the energy.

Now, the differential of the internal energy $E$ is:

$dE=\frac{\partial E}{\partial H}dH+\frac{\partial E}{\partial
V}dV+\sum_{j=1}^{n}\frac{\partial E}{\partial M_{j}}dM_{j}$ (A.4)

The partial derivatives in (A.4) are called intensive parameters and occur so
frequently that we introduce special symbols for them:

$\frac{\partial E}{\partial H}=\theta,~{}~{}\textnormal{the temperature}$
(A.5) $\frac{\partial E}{\partial V}=-p,~{}~{}\textnormal{the negative (total)
pressure}$ (A.6) $\frac{\partial E}{\partial
M_{i}}=\mu_{i},~{}~{}\textnormal{the chemical potential of the $i$-th
component}$ (A.7)

With this notation, (A.4) reads:

$dE=\theta dH-pdV+\sum_{j}\mu_{j}dM_{j}$ (A.8)

We briefly mention here that in the (equivalent) entropy representation the
intensive parameters are

$\frac{\partial H}{\partial E}=\frac{1}{\theta}~{},~{}\frac{\partial
H}{\partial V}=\frac{p}{\theta}~{},~{}\frac{\partial H}{\partial
M_{i}}=-\frac{\mu_{i}}{\theta}$

The first-order homogeneity of the fundamental equation (A.3) provides an
essential relation that will be used later in Appendix B. Differentiating

$E(\lambda H,\lambda V,\lambda M_{1},\dots,\lambda M_{n})=\lambda
E(H,V,M_{1},\dots,M_{n})$

with respect to $\lambda$ and then taking $\lambda=1$ gives

$E=\theta H-pV+\sum_{j}\mu_{j}M_{j}$ (A.9)

Equation (A.9) is known as Euler equation, some authors refer to it as Gibbs-
Duhem relation and we will retain this name here.

Back to the homogeneity of the fundamental relation, we can choose
$\lambda=\frac{1}{V}$ and by introducing the partial mass densities
$\rho_{i}=\frac{M_{i}}{V}$ and the total mass density $\rho=\frac{1}{V}$
(normalizing the total mass $\sum M_{i}=1$), the fundamental relation reads

$\rho e=\rho e(\rho\eta,\rho_{1},\dots,\rho_{n})$ (A.10)

where $\rho e$ is the specific internal energy and $\rho\eta$ the specific
entropy. Then (A.5)-(A.7) read

$\frac{\partial(\rho e)}{\partial(\rho\eta)}=\theta~{},~{}\frac{\partial(\rho
e)}{\partial\rho_{i}}=\mu_{i}$ (A.11)

and the Gibbs-Duhem relation takes the form

$\rho e=\rho\eta\theta-p+\sum_{j}\rho_{j}\mu_{j}$ (A.12)

In thermodynamics, we often need to pass from a set of variables that contains
only extensive parameters to another that contains some intensive parameters
as well. For instance, throughout this paper, we select the temperature is one
of the prime variables and we need to express the equations of state through
this variable. To this end, we use partial Legendre transforms of the internal
energy replacing some extensive variables by the corresponding intensive ones.
These partial Legendre transforms are called thermodynamic potentials. A well-
known example is the enthalpy: the enthalpy is the partial Legendre transform
of the internal energy $E$ that replaces the volume $V$ by the pressure $p$ as
an independent variable. Thus, the enthalpy $h$ is a function of
$H,p,M_{1},\dots,M_{n}$, defined by

$h=E+pV$ (A.13)

where the replacement is achieved by first solving the relation

$\frac{\partial E}{\partial V}=-p$

with respect to $V$ and then eliminating $V$ from (A.13). In our case, we are
interested in passing from the set $\\{H,V,M_{1},\dots,M_{n}\\}$ to
$\\{\theta,V,M_{1},\dots,M_{n}\\}$, i.e. to replace the entropy by the
temperature. The thermodynamic potential that allows us to do so is the
Helmholtz free energy defined by

$\Psi=E-H\theta$ (A.14)

The replacement is achieved by solving

$\frac{\partial E}{\partial H}=\theta$

with respect to $H$ and eliminating $H$ from (A.14). Using the notation of
(A.10), we have

$\rho\psi=\rho\psi(\theta,\rho_{1},\dots,\rho_{n})$ (A.15) $\rho\psi=\rho
e-\rho\eta\theta$ (A.16)
$e(\theta,\rho_{1},\dots,\rho_{n})=-\theta^{2}\left(\frac{\psi}{\theta}\right)_{\theta}$
(A.17)

and the Gibbs-Duhem relation reads

$\rho\psi+p=\sum_{j}\rho_{j}\mu_{j}\,.$ (A.18)

Furthermore, one can use the relations (A.11) to obtain

$\frac{\partial(\rho\psi)}{\partial\rho_{i}}=\mu_{i}~{},~{}\frac{\partial(\rho\psi)}{\partial\theta}=-\rho\eta$
(A.19)

Finally, we note that the first postulate states that the entropy attains its
maximum in equilibrium. This is translated as follows: the first variation of
the entropy functional must vanish and the second one must be negative. The
second condition in particular determines the stability of the predicted
equilibrium states and suggests that the entropy is a concave function. A
byproduct of this, is that the specific heat at constant volume
$c_{v}:=e_{\theta}(\theta,\rho_{1},\dots,\rho_{2})$ is positive. Should we
reformulate the stability criteria in the energy representation, we see that
the energy attains its minimum in equilibrium, hence it is a convex function.
These conditions can be extended to the thermodynamic potentials, as well. In
particular, the Helmholtz free energy is a concave function of the
temperature.

##  Appendix B Consistency with the Clausius-Duhem inequality

In this appendix we outline the derivation of the Type-II model (1.1)-(1.3)
for multicomponent flows, employed in this work, and derived in [2]. The
framework is that of continuum mechanics, in which bodies are modeled as
continuous media and their behavior is governed by (i) balance laws and (ii)
constitutive relations that connect the various quantities and characterize
the material response. For more details on modeling multicomponent fluids we
refer to [10, 12].

The emphasis of this presentation and the development in [2] is on the
thermodynamic structure of the model so that consistency is guaranteed with
the Clausius-Duhem inequality. The model is using as primitive variables the
mass densities, the velocities of the constituents and the temperature of the
mixture. (The model in [2] uses the specific entropy instead of the
temperature but this minor deviation does not cause major differences.) A
Type-II model is considered employing the following list of equations for the
partial mass, partial momentum and (total) energy balance respectively:

$\partial_{t}\rho_{i}+\textnormal{div}(\rho_{i}v_{i})=m_{i}$ (B.1)
$\partial_{t}(\rho_{i}v_{i})+\textnormal{div}(\rho_{i}v_{i}\otimes
v_{i})=\textnormal{div}(S_{i})+\rho_{i}b_{i}+f_{i}$ (B.2)
$\begin{split}\partial_{t}(\rho
e+\sum_{i}\frac{1}{2}\rho_{i}v_{i}^{2})+\textnormal{div}((\rho
e&+\sum_{i}\frac{1}{2}\rho_{i}v_{i}^{2})v)=\textnormal{div}(q-\sum_{i}(\rho_{i}e_{i}+p_{i}+\frac{1}{2}\rho_{i}v_{i}^{2})u_{i})\\\
&-\textnormal{div}(pv)+\textnormal{div}(v_{i}\cdot\sigma_{i})+\rho b\cdot
v+\rho r+\sum_{i}\rho_{i}b_{i}\cdot u_{i}\end{split}$ (B.3)

where $\rho_{i}$ is the mass density of the $i$-th component, $m_{i}$ the
production of mass, $v_{i}$ the velocity, $S_{i}$ the partial stress tensor,
$b_{i}$ the body force on the $i$-th component, $f_{i}$ the momentum
production, $e_{i}$ the specific internal energy, $q_{i}$ the heat flux,
$r_{i}$ the radiative heat supply and $\ell_{i}$ the internal energy
production. Moreover, we define the total mass $\rho:=\sum_{i}\rho_{i}$, the
barycentric velocity of the mixture $v$ such that $\rho
v=\sum_{i}\rho_{i}v_{i}$. In (B.3) we have also introduced the diffusional
velocities $u_{i}:=v_{i}-v$, the thermal energy $\rho
e:=\sum_{i}\rho_{i}e_{i}$, the total pressure $p:=\sum_{i}p_{i}$, total heat
flux $q:=\sum_{i}q_{i}$, the net force applied to the mixture $\rho
b:=\sum_{i}\rho_{i}b_{i}$ and the total radiative heat supply $\rho
r:=\sum_{i}\rho_{i}r_{i}$.

Some remarks on the model:

(a) There is a single temperature $\theta$ common to all components that is
posiive. For the thermodynamic quantities internal energy $e$, entropy $\eta$,
and pressures $p_{i}$ we employ the relations (A.15), (A.16), (A.18) and
(A.19).

(b) The production of mass is often due to chemical reactions; here, we
consider non-reactive fluids, i.e. we take $m_{i}=0$. For a treatment of the
same model with chemical reactions we refer to [2, 12].

(c) The momentum productions $f_{i}$ model binary interactions between the
species, e.g. due to friction. Likewise, the energy production $\ell_{i}$ is
neglected due to (d), as we consider only the total energy equation (B.3),
which is obtained by summing up all partial energy equations.

(d) Even though we allow for mass, momentum and energy production, we require
that the total mass, total momentum and total energy be conserved, thus having
the constraints

$\sum_{i=1}^{n}m_{i}=\sum_{i=1}^{n}f_{i}=\sum_{i=1}^{n}\ell_{i}=0$ (B.4)

where $n\in\mathbb{N}$ is the number of constituents.

(e) The consideration of the terms $b_{i}$ and $r_{i}$ is not essential, since
both are external factors and can be modified accordingly (e.g. they can be
taken equal to zero).

(f) The partial stresses are decomposed as
$S_{i}=-p_{i}\mathbb{I}+\sigma_{i}$, where $-p_{i}\mathbb{I}$ is the elastic
part of the stress tensor, with $p_{i}$ the partial pressures and $\mathbb{I}$
the identity tensor and $\sigma_{i}$ the viscous part. Here, for simplicity,
we take $\sigma_{i}=0$ and thus $\textnormal{div}(S_{i})$ reduces to $-\nabla
p_{i}$. For the treatment of the theory with viscosity we refer to [2].

In order for our system to be thermodynamically complete, it needs to be
consistent with the second law of thermodynamics expressed in the form of the
Clausius-Duhem inequality:

$\partial_{t}(\rho\eta)+\textnormal{div}(\rho\eta
v)\geq\textnormal{div}\Phi+\frac{\rho r}{\theta}$ (B.5)

where $\rho\eta$ stands for the specific entropy, $\Phi$ for the entropy flux
and $\theta$ for the temperature. The fact that we have an inequality
signifies that there is an entropy production $\zeta$ that balances the two
sides in (B.5). The entropy production is not a-priori specified. Instead, the
only information available is that according to the second law of
thermodynamics, the entropy production must be non-negative. As opposed to the
balance laws of mass, momentum and energy, which determine the thermodynamic
process from the assigned body force $b$, heat supply $r$, boundary and
initial conditions, the Clausius-Duhem inequality plays the role of an
admissibility criterion for thermodynamic processes that already comply with
the balance laws. In the case of one species the entropy flux is the heat flux
divided by the temperature, however there is no available information for the
multicomponent case, and $\Phi$ is determined along with the entropy
production, by the process of checking consistency with the Clausius-Duhem
inequality.

For the thermodynamic reduction, one introduces the Helmholtz free energy
$\rho\psi$, which inherits the relations (A.15)-(A.19) as if we were in
equilibrium and using the entropy inequality (B.5) and the balance laws
(B.1)-(B.3), one constructs an inequality known as dissipation inequality.
Then, by controlling the body forces $b_{i}$ and heat supplies $r_{i}$ we can
construct smooth processes that satisfy the mass, momentum and energy balances
but attain at some point $(x,t)$ arbitrarily prescribed values for the
primitive variables and their derivatives. Hence, the dissipation inequality
is violated unless certain relations hold (see [2]). From this process, one
determines the entropy flux and entropy production

$\Phi=\frac{q}{\theta}-\frac{1}{\theta}\sum_{i}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})u_{i}$
(B.6)
$\zeta=\frac{1}{\theta^{2}}q\cdot\nabla\theta-\frac{1}{\theta}\sum_{i}u_{i}\cdot\left(f_{i}-\nabla
p_{i}+\rho_{i}\nabla\mu_{i}+\frac{1}{\theta}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})\nabla\theta\right)$
(B.7)

Equation (B.7) is too complicated to allow for investigating necessary and
sufficient conditions so that $\zeta\geq 0$ holds everywhere. However,
following Onsager’s reciprocal relations, one decouples the entropy production
terms that correspond to different dissipative mechanisms and asks for linear
dependence with a symmetric and positive semi-definite matrix of
phenomenological coefficients, According to this, a sufficient condition for
the heat conduction term to be nonnegative is

$q=\kappa\nabla\theta$ (B.8)

with $\kappa=\kappa(\rho_{1},\dots,\rho_{n},\theta)\geq 0$, which is the
standard Fourier’s law. Following a more sophisticated argument (originally
due to Truesdell), one obtains the form for the momentum production

$f_{i}=\nabla
p_{i}-\rho_{i}\nabla\mu_{i}-\frac{1}{\theta}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})\nabla\theta-\theta\sum_{j\not=i}b_{ij}\rho_{i}\rho_{j}(v_{i}-v_{j})\,,$
(B.9)

where $b_{ij}=b_{ij}(\rho_{i},\rho_{j},\theta)$ are symmetric, nonnegative
binary-type interactions. For the full derivation we refer to [2] section 7.
After plugging the two closures in (B.7) and using the symmetry of $b_{ij}$
the entropy production reads:

$\zeta=\frac{1}{\theta^{2}}\kappa|\nabla\theta|^{2}+\frac{1}{2}\sum_{i=1}^{n}\sum_{j\not=i}b_{ij}\rho_{i}\rho_{j}|u_{i}-u_{j}|^{2}$
(B.10)

An interesting comment is that even after determining the constitutive
relations via the thermodynamic reduction, our model is not fully closed.
Indeed, although we can compute the thermal energy from (A.17) and the total
pressure from the Gibbs-Duhem relation (A.18), it turns out that if we are
just given the free energy density $\psi$ there is no way to determine the
partial pressures $p_{i}$ and partial internal energies $e_{i}$. In other
words, for a Type-II model it is not sufficient to be given solely the
Helmholtz free energy, as it is for a Type-I model [2]. Therefore, we require
some more information: a first approach is to set up constitutive functions at
the level of the components for $(e_{i})_{i}$ and $(p_{i})_{i}$. Since the
total pressure and energy can be computed only $n-1$ constitutive functions
need to be specified. Another approach that is often used is the so-called
simple mixture theory. In this simplification each component is assumed to
have its own thermodynamical structure (as if it were on its own), that is, it
is described by its own free energy density
$\psi_{i}=\psi_{i}(\rho_{i},\theta)$ and the thermodynamic functions of the
component are determined via the relations:

$e_{i}=e_{i}(\rho_{i},\theta)=-\theta^{2}\left(\frac{\psi_{i}}{\theta}\right)_{\theta}$
(B.11) $\mu_{i}=\mu_{i}(\rho_{i},\theta)=(\rho_{i}\psi_{i})_{\rho_{i}}$ (B.12)
$-\eta_{i}=(\psi_{i})_{\theta}$ (B.13)
$p_{i}=p_{i}(\rho_{i},\theta)=-\rho_{i}\psi_{i}+\rho_{i}\mu_{i}$ (B.14)

This also has the implication that $\rho\psi=\sum_{i}\rho_{i}\psi_{i}$ and
$\rho\eta=\sum_{i}\rho_{i}\eta_{i}$. For an extensive discussion on simple
mixtures we refer to [23], while for special cases of simple mixtures to [2,
sec 15].

##  Appendix C Derivation of the Entropy Equation for the Type-II system

For the derivation of equation (2.51), one multiplies the mass equations (1.1)
by $\frac{1}{2}v_{i}^{2}-\mu_{i}$, the momentum equations (1.2) by $-v_{i}$
and the energy equation (1.3) by $1$ and sums them up to obtain, by making use
of the definition $u_{i}=v_{i}-v$ and noticing that $u_{i}-u_{j}=v_{i}-v_{j}$
the following:

$\begin{split}\partial_{t}&(\rho\eta\theta)-\partial_{t}p+\operatorname{div}(\rho\eta\theta
v)-\operatorname{div}\left(\sum_{i}\rho_{i}\mu_{i}u_{i}\right)-\operatorname{div}\left(\frac{1}{2}\sum_{i}\rho_{i}v_{i}^{2}u_{i}\right)-\operatorname{div}(pv)\\\
&=-\sum_{i}\rho_{i}\partial_{t}\mu_{i}-\sum_{i}\rho_{i}v_{i}\cdot\nabla\mu_{i}-\sum_{i}\rho_{i}b_{i}\cdot
v_{i}+\sum_{i}\rho_{i}\nabla\mu_{i}\cdot
v_{i}+\frac{\nabla\theta}{\theta}\sum_{i}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})v_{i}\\\
&+\frac{\theta}{\epsilon}\sum_{i,j}b_{ij}\rho_{i}\rho_{j}(v_{i}-v_{j})v_{i}+\operatorname{div}(\kappa\nabla\theta)-\operatorname{div}\left(\sum_{i}(\rho_{i}e_{i}+p_{i})u_{i}\right)-\operatorname{div}\left(\frac{1}{2}\sum_{i}\rho_{i}v_{i}^{2}u_{i}\right)\\\
&-\operatorname{div}(pv)+\rho b\cdot v+\rho r+\sum_{i}\rho_{i}b_{i}\cdot
u_{i}\end{split}$

Now doing the necessary simplifications and using the Gibbs-Duhem relation,
one obtains

$\begin{split}\partial_{t}&(\rho\eta\theta)+\operatorname{div}(\rho\eta\theta
v)=\operatorname{div}\left(\kappa\nabla\theta-\sum_{i}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})u_{i}\right)+\frac{\nabla\theta}{\theta}\sum_{i}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})u_{i}\\\
&+\partial_{t}p-\sum_{i}\rho_{i}\partial_{t}\mu_{i}+\rho\eta
v\cdot\nabla\theta+\frac{\theta}{\epsilon}\sum_{i,j}b_{ij}\rho_{i}\rho_{j}(v_{i}-v_{j})v_{i}+\rho
r\end{split}$

If we differentiate the Gibbs-Duhem relation with respect to time we get

$\partial_{t}p=\rho\eta\partial_{t}\theta+\sum_{i}\rho_{i}\partial_{t}\mu_{i}$

thus, we divide by $\theta$ to obtain

$\begin{split}\partial_{t}(\rho\eta)+\operatorname{div}(\rho\eta
v)&=\operatorname{div}\left(\frac{1}{\theta}\kappa\nabla\theta-\frac{1}{\theta}\sum_{i}(\rho_{i}e_{i}+p_{i}-\rho_{i}\mu_{i})u_{i}\right)\\\
&+\frac{1}{\theta^{2}}\kappa|\nabla\theta|^{2}+\frac{1}{\epsilon}\sum_{i,j}b_{ij}\rho_{i}\rho_{j}(v_{i}-v_{j})v_{i}+\frac{\rho
r}{\theta}\end{split}$

Finally, note that due to the symmetry of $b_{ij}$ and the fact that
$u_{i}-u_{j}=v_{i}-v_{j}$ we find that

$\frac{1}{\epsilon}\sum_{i,j}b_{ij}\rho_{i}\rho_{j}(v_{i}-v_{j})v_{i}=\frac{1}{2\epsilon}\sum_{i,j}b_{ij}\rho_{i}\rho_{j}(v_{i}-v_{j})^{2}=\frac{1}{2\epsilon}\sum_{i,j}b_{ij}\rho_{i}\rho_{j}(u_{i}-u_{j})^{2}$

which concludes the computation.

##  Appendix D Relative Entropy Identity

In this appendix we present the computations leading to the derivation of
(4.23). We first add (4.21) and (4.22) and use the formula (4.18) and

$\begin{split}Q(U|\bar{U})&=-\rho\eta
v\bar{\theta}+\bar{\rho}\bar{\eta}\bar{v}\bar{\theta}+\sum_{i}\left(\frac{1}{2}\bar{v}^{2}-\bar{\mu}_{i}\right)(\rho_{i}v-\bar{\rho}_{i}\bar{v})-\bar{v}\rho
v^{2}\\\ &+\bar{v}\bar{\rho}\bar{v}^{2}-\bar{v}(p-\bar{p})+\left(\rho
e+\frac{1}{2}\rho
v^{2}+p\right)v-\left(\bar{\rho}\bar{e}+\frac{1}{2}\bar{\rho}\bar{v}^{2}+\bar{p}\right)\bar{v}\\\
&=vI(U|\bar{U})+(p-\bar{p})(v-\bar{v})\end{split}$ (D.1)

to arrive at

$\partial_{t}I(U|\bar{U})+\textnormal{div}Q(U|\bar{U})=T_{1}+T_{2}+T_{3}+T_{4}$
(D.2)

where

$T_{1}:=\sum_{i}\partial_{t}\left(\frac{1}{2}\bar{v}^{2}-\bar{\mu}_{i}\right)(\rho_{i}-\bar{\rho_{i}})-\partial_{t}\bar{v}\cdot(\rho
v-\bar{\rho}\bar{v})-\partial_{t}\bar{\theta}(\rho\eta-\bar{\rho}\bar{\eta})$
$T_{2}:=\sum_{i}\nabla\left(\frac{1}{2}\bar{v}^{2}-\bar{\mu}_{i}\right)\cdot(\rho_{i}v-\bar{\rho}_{i}\bar{v})-\rho
v\nabla\bar{v}\cdot
v+\bar{\rho}\bar{v}\nabla\bar{v}\cdot\bar{v}-(p-\bar{p})\textnormal{div}\bar{v}+\nabla\bar{\theta}\cdot(-\rho\eta
v+\bar{\rho}\bar{\eta}\bar{v})$
$\begin{split}T_{3}&:=\textnormal{div}\left(-\frac{\bar{\theta}}{\theta}\kappa\nabla\theta\right)+\frac{1}{\theta}\kappa\nabla\theta\cdot\nabla\bar{\theta}-\frac{\bar{\theta}}{\theta^{2}}\kappa|\nabla\theta|^{2}+\textnormal{div}\left(\frac{\bar{\theta}}{\bar{\theta}}\bar{\kappa}\nabla\bar{\theta}\right)-\frac{1}{\bar{\theta}}\bar{\kappa}\nabla\bar{\theta}\cdot\nabla\bar{\theta}\\\
&+\frac{\bar{\theta}}{\bar{\theta}^{2}}\bar{\kappa}|\nabla\bar{\theta}|^{2}+\textnormal{div}\left(\kappa\nabla\theta\right)-\textnormal{div}(\bar{\kappa}\nabla\bar{\theta})\end{split}$

and

$\begin{split}T_{4}&:=\textnormal{div}\left(\frac{\bar{\theta}}{\theta}\sum_{j}(h_{j}-\rho_{j}\mu_{j})u_{j}\right)-\frac{1}{\theta}\sum_{j}(h_{j}-\rho_{j}\mu_{j})u_{j}\cdot\nabla\bar{\theta}+\frac{\bar{\theta}}{\theta}\sum_{j}u_{j}\cdot
d_{j}\\\
&-\sum_{i}\left(\frac{1}{2}\bar{v}^{2}-\bar{\mu}_{i}\right)\textnormal{div}(\rho_{i}u_{i})+\textnormal{div}\left(-\sum_{j}h_{j}u_{j}\right)\end{split}$

Now

$\begin{split}T_{1}&=-\rho(v-\bar{v})\partial_{t}\bar{v}+\partial_{t}\bar{\theta}(-\rho\eta)(U|\bar{U})-(\bar{\rho}\bar{\eta})_{\theta}(\theta-\bar{\theta})\partial_{t}\bar{\theta}-\sum_{j}\sum_{i}(\bar{\mu}_{j})_{\rho_{i}}(\rho_{i}-\bar{\rho}_{i})\partial_{t}\bar{\rho}_{j}\\\
&=:T_{11}+T_{12}+T_{13}+T_{14}\end{split}$

where we have used the fact that $(\mu_{i})_{\theta}=-(\rho\eta)_{\rho_{i}}$
and introduced the relative quantity

$(-\rho\eta)(U|\bar{U})=-\rho\eta+\bar{\rho}\bar{\eta}+\sum_{j}(\bar{\rho}\bar{\eta})_{\rho_{j}}(\rho_{j}-\bar{\rho}_{j})+(\bar{\rho}\bar{\eta})_{\theta}(\theta-\bar{\theta})$

Using the entropy balance (4.10) we get

$\begin{split}T_{13}&=\partial_{t}(-\bar{\rho}\bar{\eta})(\theta-\bar{\theta})+\sum_{j}(\bar{\rho}\bar{\eta})_{\rho_{j}}\partial_{t}\bar{\rho}_{j}(\theta-\bar{\theta})\\\
&=\textnormal{div}(\bar{\rho}\bar{\eta}\bar{v})(\theta-\bar{\theta})+\textnormal{div}\left(-\frac{1}{\bar{\theta}}\bar{\kappa}\nabla\bar{\theta}\right)(\theta-\bar{\theta})-\frac{1}{\bar{\theta}^{2}}\bar{\kappa}|\nabla\bar{\theta}|^{2}(\theta-\bar{\theta})\\\
&-\sum_{j}(\bar{\rho}\bar{\eta})_{\rho_{j}}\nabla\bar{\rho}_{j}\cdot\bar{v}(\theta-\bar{\theta})-\sum_{j}(\bar{\rho}\bar{\eta})_{\rho_{j}}\bar{\rho}_{j}\textnormal{div}\bar{v}(\theta-\bar{\theta})\\\
&=:T_{131}+T_{132}+T_{133}+T_{134}+T_{135}\end{split}$

where

$\begin{split}T_{131}&=\nabla(\bar{\rho}\bar{\eta})\cdot\bar{v}(\theta-\bar{\theta})+\bar{\rho}\bar{\eta}\textnormal{div}\bar{v}(\theta-\bar{\theta})\\\
&=:T_{1311}+T_{1312}\end{split}$

Furthermore

$\begin{split}T_{14}&=\sum_{j}\sum_{i}(\bar{\mu}_{i})_{\rho_{j}}(\rho_{i}-\bar{\rho}_{i})\nabla\bar{\rho}_{j}\cdot\bar{v}+\sum_{j}\sum_{i}(\bar{\mu}_{i})_{\rho_{j}}(\rho_{i}-\bar{\rho}_{i})\bar{\rho_{j}}\textnormal{div}\bar{v}\\\
&=\sum_{i}\nabla\bar{\mu}_{i}\cdot\bar{v}(\rho_{i}-\bar{\rho}_{i})-\sum_{i}(\bar{\mu}_{i})_{\theta}\nabla\bar{\theta}\cdot\bar{v}(\rho_{i}-\bar{\rho}_{i})+\sum_{j}\sum_{i}(\bar{\mu}_{i})_{\rho_{j}}(\rho_{i}-\bar{\rho}_{i})\bar{\rho_{j}}\textnormal{div}\bar{v}\\\
&=:T_{141}+T_{142}+T_{143}\end{split}$

Introducing the relative pressure

$p(U|\bar{U})=p-\bar{p}-\sum_{j}\bar{p}_{\rho_{j}}(\rho_{j}-\bar{\rho}_{j})-\bar{p}_{\theta}(\theta-\bar{\theta})$

we get

$\begin{split}T_{2}&=-\rho(v-\bar{v})\nabla\bar{v}\cdot(v-\bar{v})-\sum_{i}\nabla\bar{\mu}_{i}\cdot(\rho_{i}v-\bar{\rho}_{i}\bar{v})-p(U|\bar{U})\textnormal{div}\bar{v}-\sum_{j}\bar{p}_{\rho_{j}}(\rho_{j}-\bar{\rho}_{j})\textnormal{div}\bar{v}\\\
&-\bar{p}_{\theta}(\theta-\bar{\theta})\textnormal{div}\bar{v}+\bar{v}\cdot\nabla\bar{\theta}(-\rho\eta)(U|\bar{U})-\bar{v}\cdot\nabla\bar{\theta}\sum_{j}(\bar{\rho}\bar{\eta})_{\rho_{j}}(\rho_{j}-\bar{\rho}_{j})-\bar{v}\cdot\nabla\bar{\theta}(\bar{\rho}\bar{\eta})_{\theta}(\theta-\bar{\theta})\\\
&+\nabla\bar{\theta}(-\rho\eta)\cdot(v-\bar{v})-\rho\bar{v}\nabla\bar{v}\cdot(v-\bar{v})\\\
&=:T_{21}+\cdots+T_{210}\end{split}$

where

$\begin{split}T_{22}&=-\sum_{i}\nabla\bar{\mu}_{i}\cdot(\rho_{i}-\bar{\rho}_{i})v-\sum_{i}\nabla\bar{\mu}_{i}\cdot\bar{\rho}_{i}(v-\bar{v})\\\
&=-\sum_{i}\nabla\bar{\mu}_{i}\cdot(\rho_{i}-\bar{\rho}_{i})(v-\bar{v})-\sum_{i}\nabla\bar{\mu}_{i}\cdot(\rho_{i}-\bar{\rho}_{i})\bar{v}+(v-\bar{v})\bar{\rho}\bar{\eta}\cdot\nabla\bar{\theta}-(v-\bar{v})\cdot\nabla\bar{p}\\\
&=:T_{221}+T_{222}+T_{223}+T_{224}\end{split}$

Now, $T_{28}$ and $T_{134}$ cancel with $T_{1311}$ and the same holds for
$T_{141}$ with $T_{222}$ and $T_{142}$ with $T_{27}$. Moreover

$T_{29}+T_{223}=-(\rho\eta-\bar{\rho}\bar{\eta})(v-\bar{v})\cdot\nabla\bar{\theta}$

and by the momentum balance (4.8)

$\begin{split}T_{224}&=\frac{1}{\bar{\rho}}(\rho-\bar{\rho})(v-\bar{v})\cdot\nabla\bar{p}-\frac{\rho}{\bar{\rho}}(v-\bar{v})\cdot\nabla\bar{p}\\\
&=\frac{1}{\bar{\rho}}(\rho-\bar{\rho})(v-\bar{v})\cdot\nabla\bar{p}-\rho(v-\bar{v})\cdot\partial_{t}\bar{v}+\rho\bar{v}\nabla\bar{v}\cdot(v-\bar{v})\\\
&=:T_{2241}+T_{2242}+T_{2243}\end{split}$

where $T_{2242}$ cancels with $T_{11}$ and $T_{2243}$ with $T_{210}$.

Differentiation of the Gibbs-Duhem relation (1.8) with respect to mass density
and temperature, respectively, gives

$p_{\rho_{i}}=\sum_{j}\rho_{j}(\mu_{j})_{\rho_{i}}\,,\qquad
p_{\theta}=\rho\eta+\sum_{j}\rho_{j}(\mu_{j})_{\theta}\,.$ (D.3)

Hence $T_{25},T_{1312}$ cancel with $T_{135}$ and $T_{24}$ with $T_{143}$.
Finally, taking the gradient of the Gibbs-Duhem relation and using (D.3), we
obtain the following identity

$\nabla p=\rho\eta\nabla\theta+\sum_{j}\rho_{j}\nabla\mu_{j}$

which allows us to write

$\begin{split}&-(\rho\eta-\bar{\rho}\bar{\eta})(v-\bar{v})\cdot\nabla\bar{\theta}-\sum_{j}\nabla\bar{\mu}_{j}\cdot\left(\rho_{j}-\bar{\rho}_{j}\right)(v-\bar{v})+\frac{1}{\bar{\rho}}(\rho-\bar{\rho})(v-\bar{v})\cdot\nabla\bar{p}\\\
&=-(\eta-\bar{\eta})\rho(v-\bar{v})\cdot\nabla\bar{\theta}-\sum_{j}\nabla\bar{\mu}_{j}\left(\frac{\rho_{j}}{\rho}-\frac{\bar{\rho}_{j}}{\bar{\rho}}\right)\cdot\rho(v-\bar{v})\end{split}$

Therefore

$\begin{split}T_{1}+T_{2}&=(\partial_{t}\bar{\theta}+\bar{v}\cdot\nabla\bar{\theta})(-\rho\eta)(\omega|\bar{\omega})-p(\omega|\bar{\omega})\textnormal{div}\bar{v}-(\eta-\bar{\eta})\rho(v-\bar{v})\cdot\nabla\bar{\theta}\\\
&\;\;-\sum_{j}\nabla\bar{\mu}_{j}\left(\frac{\rho_{j}}{\rho}-\frac{\bar{\rho}_{j}}{\bar{\rho}}\right)\cdot\rho(v-\bar{v})-\rho(v-\bar{v})\nabla\bar{v}\cdot(v-\bar{v})+T_{132}+T_{133}\end{split}$
(D.4)

Regarding $T_{3}$, we have

$\begin{split}T_{3}&=-\textnormal{div}\left(\bar{\theta}\frac{1}{\theta}\kappa\nabla\theta-\bar{\theta}\frac{1}{\bar{\theta}}\bar{\kappa}\nabla\bar{\theta}\right)+\textnormal{div}\left(\kappa\nabla\theta-\bar{\kappa}\nabla\bar{\theta}\right)\\\
&+\nabla\bar{\theta}\left(\frac{1}{\theta}\kappa\nabla\theta-\frac{1}{\bar{\theta}}\bar{\kappa}\nabla\bar{\theta}\right)-\bar{\theta}\left(\frac{1}{\theta^{2}}\kappa|\nabla\theta|^{2}-\frac{1}{\bar{\theta}^{2}}\bar{\kappa}|\nabla\bar{\theta}|^{2}\right)\end{split}$

and thus

$\begin{split}T_{3}+T_{132}+T_{133}&=\textnormal{div}\left[(\theta-\bar{\theta})\left(\frac{1}{\theta}\kappa\nabla\theta-\frac{1}{\bar{\theta}}\bar{\kappa}\nabla\bar{\theta}\right)\right]-\bar{\theta}\kappa\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)^{2}\\\
&-\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)\frac{\nabla\bar{\theta}}{\bar{\theta}}(\bar{\theta}\kappa-\theta\bar{\kappa})\end{split}$
(D.5)

As for $T_{4}$, we have

$\begin{split}T_{4}&=-\textnormal{div}\Big{[}\sum_{j}\rho_{j}u_{j}(\mu_{j}-\bar{\mu}_{j})+\frac{1}{\theta}(\theta-\bar{\theta})\sum_{j}(h_{j}-\rho_{j}\mu_{j})u_{j}\Big{]}+\frac{\bar{\theta}}{\theta}\sum_{j}u_{j}\cdot
d_{j}\\\
&-\sum_{j}\nabla\bar{\mu}_{j}\cdot\rho_{j}u_{j}-\frac{1}{\bar{\theta}}\nabla\bar{\theta}\sum_{j}(h_{j}-\rho_{j}\mu_{j})u_{j}\end{split}$
(D.6)

Then, by (D.1)-(D.2) and (D.4)-(D.6) we obtain the relative entropy identity
(4.23).

A more elegant, however not more advantageous way of writing (4.23) is by
noting that

$(-\rho\eta)(\omega|\bar{\omega})\bar{v}\cdot\nabla\bar{\theta}-(\eta-\bar{\eta})\rho(v-\bar{v})\cdot\nabla\bar{\theta}=(-\rho\eta
v)(U|\bar{U})\cdot\nabla\bar{\theta}+\bar{\eta}(\rho-\bar{\rho})(v-\bar{v})\cdot\nabla\bar{\theta}$
(D.7)

Indeed,

$\begin{split}(-\rho\eta
v)(U|\bar{U})+\bar{\eta}(\rho-\bar{\rho})(v-\bar{v})&=-\rho\eta
v+\bar{\rho}\bar{\eta}\bar{v}+\sum_{i}(\bar{\rho}\bar{\eta})_{\rho_{i}}\bar{v}(\rho_{i}-\bar{\rho}_{i})+\bar{\rho}\bar{\eta}(v-\bar{v})\\\
&+(\bar{\rho}\bar{\eta})_{\theta}\bar{v}(\theta-\bar{\theta})+\bar{\eta}v\rho-\bar{\eta}v\bar{\rho}-\bar{\eta}\bar{v}\rho+\bar{\eta}\bar{v}\bar{\rho}\\\
&=(-\rho\eta)(\omega|\bar{\omega})\bar{v}+\rho\eta\bar{v}-\rho\eta
v+\rho\bar{\eta}v-\rho\bar{\eta}\bar{v}\\\
&=(-\rho\eta)(\omega|\bar{\omega})\bar{v}-(\eta-\bar{\eta})\rho(v-\bar{v})\end{split}$

in which case the right-hand side RHS of (4.23) reads

$\begin{split}\textnormal{RHS}=&\;\partial_{t}\bar{\theta}(-\rho\eta)(\omega|\bar{\omega})-\rho(v-\bar{v})\nabla\bar{v}(v-\bar{v})-p(\omega|\bar{\omega})\textnormal{div}\bar{v}+(-\rho\eta
v)(U|\bar{U})\cdot\nabla\bar{\theta}\\\
&+\bar{\eta}(\rho-\bar{\rho})(v-\bar{v})\cdot\nabla\bar{\theta}-\sum_{i}\nabla\bar{\mu}_{i}\left(\frac{\rho_{i}}{\rho}-\frac{\bar{\rho}_{i}}{\bar{\rho}}\right)\rho(v-\bar{v})-\sum_{i}\nabla\bar{\mu}_{i}\cdot\rho_{i}u_{i}\\\
&-\left(\frac{\nabla\theta}{\theta}-\frac{\nabla\bar{\theta}}{\bar{\theta}}\right)\frac{\nabla\bar{\theta}}{\bar{\theta}}(\bar{\theta}\kappa-\theta\bar{\kappa})-\frac{1}{\bar{\theta}}\nabla\bar{\theta}\sum_{i}(h_{i}-\rho_{i}\mu_{i})u_{i}.\end{split}$

Acknowledgement We would like to thank the anonymous referees for their very
helpful comments as well as bringing certain references to our attention.

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