# Low Mach number limit for degenerate Navier-Stokes equations
in presence of strong stratification

Francesco Fanelli The work of F.F. has been partially supported by the LABEX
MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program
“Investissement d’Avenir” (ANR-11-IDEX-0007), and by the projects BORDS
(ANR-16-CE40-0027-01), SingFlows (ANR-18-CE40-0027) and CRISIS
(ANR-20-CE40-0020-01), all operated by the French National Research Agency
(ANR). Ewelina Zatorska The work of E.Z. was partially supported by the EPSRC
Early Career Fellowship no. EP/V000586/1.

(September 3, 2024)

###### Abstract

In this paper, we investigate the low Mach and low Froude numbers limit for
the compressible Navier-Stokes equations with degenerate, density-dependent,
viscosity coefficient, in the strong stratification regime. We consider the
case of a general pressure law with singular component close to vacuum, and
general ill-prepared initial data. We perform our study in the three-
dimensional periodic domain. We rigorously justify the convergence to the
generalised anelastic approximation, which is used extensively to model
atmospheric flows.

${}^{*}\;$Univ. Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208,
Institut Camille Jordan,

43, Boulevard du 11 novembre 1918 – F-69622 Villeurbanne, France

<EMAIL_ADDRESS>
${}^{\dagger}\;$Department of Mathematics, Imperial College London

6M14 Huxley Building, South Kensington Campus – SW7 2AZ, London, UK

<EMAIL_ADDRESS>
Keywords: compressible Navier-Stokes equations; density-dependent viscosity;
low Mach and Froude numbers; strong stratification; cold pressure.

2020 Mathematics Subject Classification: 35Q35 (primary); 35B40, 76M45, 35B25
(secondary).

##  1 Introduction

Flows in the atmosphere are typically characterised by two main features (see
[27]): first of all, they are weakly compressible, moreover they undergo the
combined effect of a strong stratification (due to the action of gravity) and
of a strong Coriolis force (due to the rotation of the Earth, which is very
fast if compared to the space-time scales of the flows).

Neglecting the effects of the Earth rotation, the importance of the other two
factors, i.e. weak compressibility and strong stratification, may be assessed
by introducing two physical a-dimensional parameters, the Mach number and the
Froude number, respectively. The smaller these parameters are, the more
predominant weak compressibility and strong stratification become. Thus, as
usual in Physics, it is natural to look at the regime where both parameters
vanish, to find reduced models for atmospheric flows. They are simpler to deal
with than the corresponding primitive system, both from the analytical and
numerical point of view.

When the Mach number and the Froude number go to zero with the same speed, the
flow becomes incompressible and stratified at the same rate. Formally, this
asymptotic regime was considered already by Ogura and Phillips in [26]. The
limiting system takes the name of anelastic approximation. The physical
importance of the anelastic approximation is discussed, for example in [20] in
the context of various atmospheric flows, and in [1] in the context of
astrophysics models.

###  1.1 The primitive system and the limit dynamics

In this paper we will give a rigorous derivation of what we call the
generalised anelastic approximation, namely an anelastic approximation with
variable viscosity. The starting system (referred to as the primitive system)
is the barotropic Navier-Stokes equations, with bulk viscosity coefficient
equal to $0$ and the shear viscosity coefficient proportional to the density
of the fluid. In particular, the system strongly degenerates close to vacuum.
This choice of the viscosity coefficients is physically relevant, as viscosity
is, in general, hardly expected to be uniform for flows on large scales. Their
specific form allows one to exploit a certain mathematical structure of the
system, called the BD-entropy (see more details in the discussion below).

Assuming that both the Mach and Froude numbers are equal to a small parameter
$\varepsilon>0$, the system of equations reads as follows:

$\begin{split}&\partial_{t}\varrho+\operatorname{div}(\varrho{\bf u})=0,\\\
&\partial_{t}(\varrho{\bf u})+\operatorname{div}(\varrho{\bf u}\otimes{\bf
u})+\frac{1}{\varepsilon^{2}}\nabla
p(\varrho)-\nu\operatorname{div}(\varrho\,{\mathbb{D}}{\bf
u})=\frac{1}{\varepsilon^{2}}\varrho\nabla G.\end{split}$ (1)

The unknowns are the mass density $\varrho\,=\,\varrho(t,x)\,\geq\,0$, and the
velocity vector field ${\bf u}\,=\,{\bf u}(t,x)\in\mathbb{R}^{3}$. The
function $p\,=\,p(\varrho)$ denotes the internal pressure, the constant
$\nu>0$ is the viscosity coefficient, and
${\mathbb{D}}=\frac{1}{2}\left(\nabla+\nabla^{t}\right)$ is the symmetric part
of the gradient. Finally, $G=G(x)$ is a smooth function (say $G\in
C^{3}(\Omega)$) describing a scalar external force acting on the flow. $G$
typically encodes the action of gravity, in which case $G=-gx_{3}$, where $g$
is the gravitational acceleration constant.

Due to the present state-of-the-art of the mathematical theory for system
(LABEL:main), we assume that the fluid occupies the periodic box in
$\mathbb{R}^{3}$, i.e. we consider the equations (LABEL:main) on the space
domain

$\Omega\,:=\,\mathbb{T}^{3}\,.$ (2)

The pressure function $p$ is assumed of the following form:

$\begin{split}p(\varrho)\,=\,p_{e}(\varrho)+p_{c}(\varrho)\,=\,\dfrac{1}{\gamma}\,\varrho^{\gamma}-\frac{1}{\kappa}\varrho^{-\kappa}\,,\qquad\qquad\gamma>1,\quad\kappa\geq\gamma-2,\quad\kappa>3.\end{split}$
(3)

The first part is the standard barotropic pressure, while the second part is
the so-called “cold pressure”, because it is most significant in the region of
temperatures close to zero. The constants $1/\gamma$ and $1/\kappa$ are just
normalisation factors; their presence is useful in some computations. The
restriction on the adiabatic exponent $\gamma>1$ is somehow classical in the
theory of compressible Navier-Stokes equations. The conditions on the exponent
$\kappa$, instead, are of technical nature.

When $\varepsilon\to 0^{+}$ in equations (LABEL:main), we observe a
competition between the large size of the pressure term (low Mach number
effect), which tends to drive the flow to incompressibility, and the large
size of the forcing term (low Froude number effect), which increases the
stratification of the flow. Due to the choice of scaling, those two terms are
in balance in the limit process. Therefore, it is easy to see that, when
$\varepsilon\to 0^{+}$, $\varrho$ will tend, at least formally, to the profile
$b=b(x)$ satisfying

$\nabla p(b)\,=\,b\,\nabla G\,.$ (4)

Smoothness of $G(x)$ and strict monotonicity of $p$ imply that there exists a
smooth function $b\in C^{3}(\Omega)$ satisfying (4). Monotonicity of $p$
implies convexity of the pressure potential $H$ (defined in (9) below) which
provides existence of constants $b_{*},b^{*}\,\in\mathbb{R}$ such that

$\forall\,x\in\Omega\,,\qquad\qquad 0\,<\,b_{*}\,\leq b(x)\,\leq\,b^{*}\,.$
(5)

Note that, for $p(\varrho)=\frac{\varrho^{2}}{2}$, one gets $G=b$ up to the
choice of an irrelevant additive constant, which is the case considered in
[5]. On the other hand, if $G(x)\,=\,-gx_{3}$ is the gravitational potential,
it is easy to see that $b=b(x_{3})$ verifies (5).

Since $\varrho\approx b$ for $\varepsilon\to 0^{+}$, assuming that we can
identify the limits of the non-linear terms appearing in (LABEL:main), it is
not difficult to check that (formally) the limiting system is an anelastic
approximation with variable viscosity coefficient, namely

$\begin{split}&\operatorname{div}(b\,{\bf U})\,=\,0\\\\[4.30554pt]
&\partial_{t}{\bf U}\,+\,({\bf U}\cdot\nabla){\bf
U}\,+\,\nabla\pi\,-\,b^{-1}\,\nu\,\operatorname{div}\big{(}b\,{\mathbb{D}}{\bf
U}\big{)}\,=\,0\,.\end{split}$ (6)

We refer to system (LABEL:eq:limit) as the generalised anelastic
approximation. In the above system, $\pi\,=\,\pi(t,x)$ denotes an unknown
scalar function, and the term $\nabla\pi$ plays the role of a Lagrangian
multiplier associated with the anelastic constraint ${\rm div}\,(b\,{\bf
U})=0$. The limiting system can be also regarded as the viscous counterpart of
the so-called lake equation, whose study was initiated in [21].

The goal of this paper is to rigorously justify the above formal derivation in
the framework of _global in time finite energy weak solutions_ to the
primitive system (LABEL:main)-(3) for general _ill-prepared initial data_.

###  1.2 An overview of related results

Due to the physical importance of the anelastic approximation, its rigorous
derivation has been the object of intensive studies in the past years.

In [24], Masmoudi proved the rigorous derivation of the anelastic
approximation, starting from the classical barotropic Navier-Stokes system. He
considered a bounded domain in $\mathbb{R}^{3}$, supplemented with Dirichlet
boundary conditions, and the limit was performed for ill-prepared initial data
via a compensated compactness argument. Soon after that, Feireisl, Málek,
Novotný and Straškraba proved an analogous result on a periodic box, and for
pressure laws which are small variations of the ideal gas law, see [14].
Finally, we refer to the book by Feireisl and Novotný [15] for a complete
account of the mathematical literature on the low Mach number limit, in the
presence of both low and high stratification effects. They presented the
theory for the full Navier-Stokes-Fourier system and in the framework of
global in time finite energy weak solutions. The literature related to the
incompressible limit of compressible fluid equations is of course much more
extensive (see e.g. the pioneering works [8] by Ebin and [18]-[19] by
Klainerman and Majda) and recalling all relevant results goes far beyond the
scope of this introduction. We thus limit ourselves to quote a couple of
recent works.

In [13], a variant of the anelastic approximation was derived, starting from a
version of the Navier-Stokes-Fourier system with neglected thermal diffusion:
the potential temperature is assumed to be just transported by the velocity
field. The limit system that is identified in [13] reads as a coupling of the
anelastic approximation system (LABEL:eq:limit) with a transport equation for
the limiting temperature. The convergence is proven in the infinite slab
$\mathbb{R}^{2}\times\,]0,1[\,$ through a spectral analysis of the singular
perturbation operator and an application of the celebrated RAGE theorem from
scattering theory. The advantage of that technique, in comparison to the one
used in [24], is that it allows to get the compactness of the sequence of
velocity fields.

On the side of the incompressible limit (with no stratification effects,
though), another interesting result is [7], where the the authors deal with
weakly compressible viscous fluids in a critical regularity functional
framework. In that paper, weak compressibility is obtained by taking a large
bulk viscosity coefficient limit, instead of the classical low Mach number
limit. More recently, in [11], a similar idea was implemented for fast
rotating fluids.

For the degenerate Navier-Stokes system (LABEL:main), as considered in our
paper, the relevant results are much more sparse. The first one to mention is
the incompressible limit in a strong stratification regime considered in [5]
by Bresch, Gisclon and Lin. Their system includes two artificial drag terms in
the momentum equation, in order to improve the available information for the
velocity field close to vacuum. The convergence to the anelastic approximation
is proven using the relative energy method, for a special choice of pressure
law $p(\varrho)=\varrho^{2}/2$ and for well-prepared initial data.

A similar method was used in [3] by Bresch and Desjardins and in [17] by
Jüngel, Lin and Wu, for the low Mach number and low Rossby number limit in a
two-dimensional geometry. The external force in these papers is replaced by
the Coriolis force (whence the low Rossby number regime) and a capillarity
term. The resulting limiting system is a quasi-geostrophic type equation for
the stream function of the limit velocity field. We also refer to [9]-[10] for
a generalisation of these results to the $3$-D setting and to the case of
general ill-prepared initial data.

###  1.3 The content of the paper

In the context depicted above, our work can be seen as a generalisation of the
result from [5], to the case of ill-prepared initial data and of more general
pressure laws (and hence, more general external forces $G$). We work in the
framework of global in time finite energy weak solutions to the primitive
system. Their existence, in presence of a cold part of the pressure (3), has
been established in [30], [25]. The case without this assumption has been
completed much more recently in [29] by Vasseur and Yu. In all these results,
the finite energy condition plays, of course, a major role. However, the
degenerate Navier-Stokes system (LABEL:main) possesses also a second energy
inequality, usually named _BD entropy inequality_ after Bresch and Desjardins,
who investigated this second energy conservation law in [3].

The BD entropy estimate provides a control on the gradient of a certain
function of the density, whose exact form depends on the form of the viscosity
coefficient. For system (LABEL:main), this function is $\nabla\sqrt{\varrho}$.
The BD entropy also allows to control the skew-symmetric part
$\mathbb{A}=\frac{1}{2}\left(\nabla-\nabla^{t}\right)$ of the gradient of the
velocity. This, when combined with the classical energy, provides the
corresponding bound for the full gradient of the velocity.

The classical energy inequality, the BD entropy inequality, and all the bounds
that follow, are essential also in the present paper. As a matter of fact, for
any value of parameter $\varepsilon\in\,]0,1]$, we consider a finite energy
weak solution $\big{(}\varrho_{\varepsilon},{\bf u}_{\varepsilon}\big{)}$ to
system (LABEL:main), which satisfy both those energy inequalities. However,
proving that the BD entropy estimate is satisfied uniformly with respect to
$\varepsilon$ requires some effort, especially when a general pressure law is
considered: this is one of the first problems solved in our paper.

Having all these estimates satisfied uniformly for a sequence of finite energy
weak solutions $\big{(}\varrho_{\varepsilon},{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}$, the rest of the proof of the
derivation of the generalised anelastic approximation (LABEL:eq:limit) boils
down to showing that the weak limit $(b,{\bf U})$ is indeed a solution to
(LABEL:eq:limit). It is well known that passing to the limit in the weak
formulation of equations (LABEL:main), especially in its nonlinear parts, is
problematic. This is because the singular terms, along with the ill-prepared
initial data, are responsible for fast time oscillations of the solutions (the
so-called _acoustic waves_), which may prevent, in the end, the convergence of
the nonlinear terms to the expected limit. Showing that this does not happen
is the core of the whole proof.

The main concern is the convergence of the convective term in the momentum
equation. To that purpose, we use a different technique than the one from [5].
Our approach is inspired by the previous works [23] and [24] on the
incompressible limit for the classical barotropic Navier-Stokes system, and is
based on a compensated compactness argument. More precisely, we first
regularise the primitive equations, which we recast in the form of a wave
system. After that, we exploit two pieces of information coming from the wave
system. First of all, we may deduce the compactness of the rotational part of
the velocity fields. On the other hand, by direct but elaborated algebraic
manipulations, we may infer that the interaction of the potential part of the
velocity fields in the convective term gives rise to small quantities, which
tend to vanish when $\varepsilon\rightarrow 0^{+}$. It is worth to point that
this argument is robust enough to deal with other variants of the system
(LABEL:main). For instance, we could trade the cold component of the pressure
function, which basically provides us with some integrability properties for
$\nabla{\bf u}$, for a turbulent drag term $\varrho|{\bf u}|{\bf u}$, which
would give a better integrability of the momentum ${\bf V}\,:=\,\varrho{\bf
u}$. In that case, most of the steps are the same, although the derivation of
essential estimates becomes significantly more laborious. The only problem,
and the breaking point, arises when one wants to pass to the limit in this
artificial drag term. This term turns out to be even more non-linear than the
convective term, because of the presence of the norm $|{\bf u}|$ of the vector
${\bf u}$. It is not clear how to bypass this difficulty in our framework, and
so, the problem remains open.

We conclude with a short outline of the paper. In the next section, we collect
our main hypotheses on the initial data, we give the definition of finite
energy weak solutions, and we state our main result. In Section 3, we deduce,
from the energy inequality and the BD entropy inequality, a long list of
uniform bounds for the family of solutions $\big{(}\varrho_{\varepsilon},{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}$ we consider. That part of the study is
rather delicate, due to the degeneracy of the system close to vacuum. In
Section 4, we use the previous uniform bounds to extract a weakly convergent
subsequence, and to derive first basic properties on its weak limit point. At
this stage we reformulate the primitive equations into the wave system, which
describes the propagation of the acoustic waves. In Section 5 we rigorously
perform the convergence in the weak formulation of equations (LABEL:main), and
conclude the derivation of the anelastic approximation (LABEL:eq:limit). For
the convenience of the reader, we collect some tools from Fourier analysis
which we need in our study in the Appendix at the end of the paper.

##  2 Statement of the main result

In this section, we first introduce our assumptions on the initial data, then
we define the notion of finite energy weak solutions to system
(LABEL:main)-(3), and finally we formulate our main theorem.

##### Initial data.

Problem (LABEL:main)- (3) is supplemented by general _ill-prepared_ initial
data. Namely, for any small parameter $\varepsilon\in\,]0,1]$ fixed, we pick
initial data

$\begin{split}(\varrho,{\bf u})|_{t=0}=\big{(}\varrho_{0,\varepsilon},{\bf
u}_{0,\varepsilon}\big{)}\end{split}$ (7)

satisfying the following conditions:

  * (i)

the initial densities $\varrho_{0,\varepsilon}\geq 0$ are assumed to be small
perturbations of the static state $b$, defined by (4): more precisely, we
assume111Here and throughout this paper, we make use of the following
notation: given a normed space $X$ and a sequence of functions
$\big{(}f_{\varepsilon}\big{)}_{\varepsilon}$ all belonging to $X$, we write
$\big{(}f_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,X$ implicitly meaning
that the sequence is also _bounded_ in $X$. that

$\varrho_{0,\varepsilon}\,=\,b\,+\,\varepsilon\,\phi_{0,\varepsilon}\,,\qquad\qquad\mbox{
with }\qquad\big{(}\phi_{0,\varepsilon}\big{)}_{\varepsilon}\subset
L^{\infty}(\Omega)\quad\mbox{ and
}\quad\left(\nabla\ln\frac{\varrho_{0,\varepsilon}}{b}\right)_{\varepsilon}\subset
L^{\infty}(\Omega)\,;$

  * (ii)

the initial velocity fields ${\bf u}_{0,\varepsilon}$ are such that
$\big{(}{\bf u}_{0,\varepsilon}\big{)}_{\varepsilon}\subset
L^{\infty}(\Omega)$.

Thus, up to extraction of a subsequence, not relabeled here, we may suppose
that

$\begin{split}\phi_{0,\varepsilon}\,\to\,\phi_{0}\qquad\mbox{ and
}\qquad\mbox{{\bf u}}_{0,\varepsilon}\to{\bf u}_{0}\qquad\qquad\mbox{
weakly-$*$ \ in }\ L^{\infty}(\Omega)\,.\end{split}$ (8)

##### Energy functionals.

Next, we need to introduce various energy functionals. The internal energy
function (sometimes called _pressure potential_) is defined by the ODE

$\varrho\,H^{\prime}(\varrho)\,-\,H(\varrho)\,=\,p(\varrho)\,,$

which implies in particular that

$H^{\prime\prime}(\varrho)\,=\,\frac{p^{\prime}(\varrho)}{\varrho}\,.$

Notice that $H$ is defined up to the sum of an affine function. Here, we fix
the classical choice

$\begin{split}H(\varrho)\,=\,\varrho\int_{1}^{\varrho}\frac{p(z)}{z^{2}}\,{\rm
d}z\,=\,\frac{1}{\gamma\,(\gamma-1)}\,\varrho\,\big{(}\varrho^{\gamma-1}\,-\,1\big{)}\,+\,\frac{1}{\kappa\,(\kappa+1)}\,\varrho\,\big{(}\varrho^{-\kappa-1}\,-\,1\big{)}.\end{split}$
(9)

We now denote

$\displaystyle\mathcal{E}\Big{(}\varrho,{\bf u}\,\Big{|}\,b\Big{)}\,$
$\displaystyle:=\,\frac{1}{2}\,\int_{\Omega}\varrho\,|{\bf u}|^{2}\ {\rm
d}{x}\,+\,\frac{1}{\varepsilon^{2}}\,\int_{\Omega}\Big{(}H(\varrho)\,-\,H(b)\,-\,H^{\prime}(b)\,(\varrho-b)\Big{)}\
{\rm d}{x}$ (10) $\displaystyle\mathcal{F}\Big{(}\varrho,{\bf
u}\,\Big{|}\,b\Big{)}\,$ $\displaystyle:=\,\int_{\Omega}\varrho\,\left|{\bf
u}\,+\,\nu\nabla\ln\frac{\varrho}{b}\right|^{2}\ {\rm d}{x}$ (11)

to be the _classical energy_ and the _BD entropy_ functions. We also set
${\cal E}\Big{(}\varrho,{\bf u}\,\Big{|}\,b\Big{)}(T)\,:=\,{\cal
E}\Big{(}\varrho(T),{\bf u}(T)\,\Big{|}\,b\Big{)}$ and ${\cal
E}\Big{(}\varrho,{\bf u}\,\Big{|}\,b\Big{)}(0)\,:=\,{\cal
E}\Big{(}\varrho_{0},{\bf u}_{0}\,\Big{|}\,b\Big{)}$, and similarly for the
function $\mathcal{F}$.

##### Weak solutions to the primitive system.

After this preparation, we are ready to give the definition of weak solutions
to system (LABEL:main)-(3) which are relevant for us.

###### Definition 2.1

Let $\big{(}\varrho_{0},{\bf u}_{0}\big{)}$ be such that $\,{\cal
E}\Big{(}\varrho_{0},{\bf u}_{0}\,\Big{|}\,b\Big{)}+{\cal
F}\Big{(}\varrho_{0},{\bf u}_{0}\,\Big{|}\,b\Big{)}<+\infty$.
We say that the couple $(\varrho,{\bf u})$ is a _finite energy weak solution_
of (LABEL:main)-(3) in $[0,T[\,\times\Omega$, with the initial datum
$\big{(}\varrho_{0},{\bf u}_{0}\big{)}$, provided the following conditions are
satisfied:

  1. (1)

$\varrho\geq 0$ almost everywhere, with $\varrho\in
L^{\infty}\big{(}[0,T[\,;L^{\gamma}(\Omega)\big{)}$ and $\varrho^{-1}\in
L^{\infty}\big{(}[0,T[\,;L^{\kappa}(\Omega)\big{)}$, $\nabla\sqrt{\varrho}\in
L^{\infty}\big{(}[0,T[\,;L^{2}(\Omega)\big{)}$ and
$\sqrt{\frac{p^{\prime}(\varrho)}{\varrho}}\nabla\varrho\in
L^{2}\big{(}[0,T[\,;L^{2}(\Omega)\big{)}$;

  2. (2)

$\sqrt{\varrho}{\bf u}\in L^{\infty}\big{(}[0,T[\,;L^{2}(\Omega)\big{)}$ and
$\sqrt{\varrho}\nabla{\bf u}\in L^{2}\big{(}[0,T[\,;L^{2}(\Omega)\big{)}$;

  3. (3)

the equations of system (LABEL:main) are satisfied in the sense of
distributions: more precisely, we have

$\begin{split}\int_{\Omega}\varrho_{0}\xi(0)\ {\rm
d}{x}+\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\Big{(}\varrho\partial_{t}\xi\,+\,\varrho{\bf
u}\cdot\nabla\xi\Big{)}\ {\rm d}{x}\,{\rm d}t\,=\,0\end{split}$ (12)

for any test function $\xi\in\mathcal{D}\big{(}[0,T[\,\times\Omega\big{)}$,
and

$\begin{split}&\int_{\Omega}\varrho_{0}{\bf u}_{0}\cdot\psi(0)\ {\rm
d}{x}+\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\Big{(}\varrho{\bf
u}\cdot\partial_{t}\psi+\varrho{\bf u}\otimes{\bf u}:\nabla\psi\Big{)}\ {\rm
d}{x}\,{\rm d}t\\\
&\quad+\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}p(\varrho)\operatorname{div}\psi\
{\rm d}{x}\,{\rm
d}t+\frac{1}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\varrho\nabla
G\cdot\psi\ {\rm d}{x}\,{\rm
d}t-\nu\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\varrho\,{\mathbb{D}}{\bf
u}:\nabla\psi\ {\rm d}{x}\,{\rm d}t\,=\,0\end{split}$ (13)

for any test function
$\psi\in\mathcal{D}\big{(}[0,T[\,\times\Omega;\mathbb{R}^{3}\big{)}$;

  4. (4)

for almost every $t\in[0,T[\,$, the following energy inequalities hold true:

$\begin{split}&\mathcal{E}\Big{(}\varrho,{\bf
u}\,\Big{|}\,b\Big{)}(t)+\nu\int_{0}^{t}\\!\\!\int_{\Omega}\varrho|{\mathbb{D}}{\bf
u}|^{2}\ {\rm d}{x}\,{\rm d}s\leq\mathcal{E}\Big{(}\varrho_{0},{\bf
u}_{0}\,\Big{|}\,b\Big{)}\,,\\\ &\mathcal{F}\Big{(}\varrho,{\bf
u}\,\Big{|}\,b\Big{)}(t)+\,\frac{\nu}{\varepsilon^{2}}\int_{0}^{t}\\!\\!\int_{\Omega}b^{2}\,\frac{p^{\prime}(\varrho)}{\varrho}\,\left|\nabla\left(\frac{\varrho}{b}\right)\right|^{2}\
{\rm d}{x}\,{\rm d}s+\nu\int_{0}^{t}\\!\\!\int_{\Omega}\varrho|\mathbb{A}{\bf
u}|^{2}\ {\rm d}{x}\,{\rm d}s\leq C_{0}\,e^{C_{0}(1+T)}\,,\end{split}$ (14)

where the constant $C_{0}>0$ may depend on $\mathcal{E}\Big{(}\varrho_{0},{\bf
u}_{0}\,\Big{|}\,b\Big{)}$ and $\mathcal{F}\Big{(}\varrho_{0},{\bf
u}_{0}\,\Big{|}\,b\Big{)}$ but is independent of $\varepsilon$.

The solution $(\varrho,{\bf u})$ is said _global in time_ if the previous
properties hold true for any $T>0$.

For any $\varepsilon\in\,]0,1]$ fixed, the existence of global in time finite
energy weak solutions to system (LABEL:main) in the sense of previous
definition was proven in [30] and [25], in the case $G=0$ (corresponding to
$b=const.$). The argument of those papers apply in a fairly direct way also to
the case considered in this paper, where $G\neq 0$ and $b$ is non-constant: we
explain in the next section how to modify the estimates of [30]-[25] in order
to include the force.

##### Main result.

Before stating the main result of this paper, we need some additional tools
and notation. Following [24]-[14] (see also [22]), we introduce the twisted
Leray-Helmholtz projector $\mathbb{P}_{b}$, related to the smooth function $b$
satisfying (5), as follows: for any smooth vector field ${\bf v}$ on $\Omega$,
we write

${\bf v}\,=\,\mathbb{P}_{b}[{\bf v}]\,+\,b\,\nabla\Psi\,,$

where $\Psi$ is the unique solution to the Neumann problem

$\operatorname{div}\big{(}b\,\nabla\Psi\big{)}\,=\,\operatorname{div}{\bf
v}\quad\mbox{ in }\ \Omega\,,\qquad\qquad\int_{\Omega}\Psi\ {\rm
d}{x}\,=\,0\,.$

Remark that $\mathbb{P}_{b}[{\bf v}]$ and $\mathbb{Q}_{b}[{\bf
v}]\,:=\,b\,\nabla\Psi$ are orthogonal in the weighted Hilbert space
$L^{2}_{b}(\Omega;\mathbb{R}^{3})$, which is defined as the space of functions
$f:\Omega\longrightarrow\mathbb{R}^{3}$ which are $L^{2}$-summable with
respect to the measure $\frac{1}{b}\,{\rm d}{x}$.

Similarly to the case of the classical Leray-Helmholtz projector
$\mathbb{P}\,=\,\mathbb{P}_{1}$ and its $L^{2}$-orthognal projector
$\mathbb{Q}\,=\,\mathbb{Q}_{1}$, it is possible to prove that both
$\mathbb{P}_{b}$ and $\mathbb{Q}_{b}$ are bounded continuous functionals on
$L^{p}(\Omega;\mathbb{R}^{3})$, for any $1<p<+\infty$.

We can now state the main result of this paper, which is contained in the
following theorem.

###### Theorem 2.2

Let $\gamma>1$ and $\kappa\geq\gamma-2,\;\kappa>3$ in (3).
Let $\big{(}\varrho_{0,\varepsilon},{\bf
u}_{0,\varepsilon}\big{)}_{\varepsilon}$ be a family of initial data
satisfying hypotheses (i)-(ii) fixed above, so in particular the condition

$\sup_{\varepsilon\in\,]0,1]}\mathcal{E}\Big{(}\varrho_{0,\varepsilon},{\bf{\bf
u}}_{0,\varepsilon}\,\Big{|}\,b\Big{)}\,+\,\sup_{\varepsilon\in\,]0,1]}\mathcal{F}\Big{(}\varrho_{0,\varepsilon},{\bf{\bf
u}}_{0,\varepsilon}\,\Big{|}\,b\Big{)}\,<\,+\infty$ (15)

holds. Define the couple $(\phi_{0},{\bf u}_{0})$ as in (8).
Let $\big{(}\varrho_{\varepsilon},{\bf u}_{\varepsilon}\big{)}_{\varepsilon}$
be a family of global in time weak solutions to system (LABEL:main)-(3), in
the sense of Definition 2.1, corresponding to the previous initial data.
Define the scalar quantity
$\phi_{\varepsilon}:=\frac{\varrho_{\varepsilon}-b}{\varepsilon}$.

Then, there exists a couple of functions $\big{(}\phi,{\bf U}\big{)}$ such
that, passing to a suitable subsequence as the case may be, in the limit
$\varepsilon\to 0$ one has

$\displaystyle\varrho_{\varepsilon}\,\to\,b\qquad\mbox{ strongly in }\
L^{\infty}_{\rm loc}\big{(}\mathbb{R}_{+};L^{p}(\Omega)\big{)}\,,\quad\mbox{
for any }\ p\in[1,3[\,,$
$\displaystyle\phi_{\varepsilon}\,\to\,\phi\qquad\mbox{ weakly in}\ L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};W^{1,\min\\{\gamma,2\\}}(\Omega)\big{)}\,,$
$\displaystyle{\bf u}_{\varepsilon}\,\to\,{\bf U}\qquad\mbox{ weakly-$*$ in }\
L^{\infty}\big{(}\mathbb{R}_{+};L^{p_{1}}(\Omega)\big{)}\,\cap\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};W^{1,p_{1}}(\Omega)\big{)},\quad\mbox{ where }\
p_{1}\,:=\,\frac{2\kappa}{\kappa+1}\,.$

In addition, $\phi\,=\,\phi(b)$ is a function of the static profile $b$, while
${\bf U}$ is a solution of the target system (LABEL:eq:limit) in the weak
sense, related to the initial datum ${\bf U}|_{t=0}\,=\,{\bf
U}_{0}\,:=\,\frac{1}{b}\,\mathbb{P}_{b}[b\,{\bf u}_{0}]$, i.e. one has
$\operatorname{div}(b{\bf U})=0$ almost everywhere in
$\mathbb{R}_{+}\times\Omega$ and

$\begin{split}\int_{\Omega}b\,{\bf u}_{0}\cdot\psi(0)\ {\rm
d}{x}+\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\Big{(}b\,{\bf
U}\cdot\partial_{t}\psi\,+\,b\,{\bf U}\otimes{\bf U}:\nabla\psi\Big{)}\ {\rm
d}{x}\,{\rm d}t-\nu\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}b\,\nabla{\bf
U}:\nabla\psi\ {\rm d}{x}\,{\rm d}t\,=\,0\end{split}$ (16)

for any $T>0$ and any test function
$\psi\in\mathcal{D}\big{(}[0,T[\,\times\Omega;\mathbb{R}^{3}\big{)}$ such that
$\operatorname{div}(b\psi)=0$.

###### Remark 2.3

Note that the initial condition equals

$\begin{split}\int_{\Omega}b\,{\bf u}_{0}\cdot\zeta\ {\rm
d}{x}=\int_{\Omega}\mathbb{P}_{b}[b\,{\bf u}_{0}]\cdot\zeta\ {\rm
d}{x}=\int_{\Omega}b\,{\bf U}_{0}\cdot\zeta\ {\rm d}{x}\end{split}$

for any test function $\zeta\in\mathcal{D}\big{(}\Omega;\mathbb{R}^{3}\big{)}$
such that $\operatorname{div}(b\zeta)=0$.

##  3 A priori estimates

In this section, we derive _uniform bounds_ for the family of weak solutions
$\big{(}\varrho_{\varepsilon},{\bf u}_{\varepsilon}\big{)}_{\varepsilon}$ to
the original Navier-Stokes system (LABEL:main). The main tools for this are
the classical _energy inequality_ and the so-called _BD entropy estimate_.

Here and everywhere in the text, we adopt the following notation: given a
Banach space $X$ and any $p\in[1,+\infty]$, we set
$L^{p}_{T}(X)\,:=\,L^{p}([0,T];X)$; in the case $T=+\infty$, instead, we
explicitly write $L^{p}(\mathbb{R}_{+};X)$. When convenient, we will use also
the notation $L^{p}_{\rm
loc}(\mathbb{R}_{+};X)\,:=\,\bigcap_{T>0}L^{p}_{T}(X)$.

###  3.1 Bounds coming from the energy inequality

The energy inequality for $\big{(}\varrho_{\varepsilon},{\bf
u}_{\varepsilon}\big{)}$, which is satisfied by assumption, reads as follows:
for almost any time $T>0$, we have

$\displaystyle{\cal E}\Big{(}\varrho_{\varepsilon},{\bf
u}_{\varepsilon}\,\Big{|}\,b\Big{)}(T)+\nu\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\varrho_{\varepsilon}\,|{\mathbb{D}}{\bf
u}_{\varepsilon}|^{2}\ {\rm d}{x}\,{\rm d}t\,\leq\,{\cal
E}\Big{(}\varrho_{0,\varepsilon},{\bf
m}_{0,\varepsilon}\,\Big{|}\,b\Big{)}\,,$ (17)

where the function $\mathcal{E}\Big{(}\varrho,{\bf u}\,\Big{|}\,b\Big{)}$ has
been defined in (10) and we recall that we have set ${\cal
E}\Big{(}\varrho_{\varepsilon},{\bf
u}_{\varepsilon}\,\Big{|}\,b\Big{)}(T)\,:=\,{\cal
E}\Big{(}\varrho_{\varepsilon}(T),{\bf
u}_{\varepsilon}(T)\,\Big{|}\,b\Big{)}$. Notice that, due to the cold
pressure, at any value of $\varepsilon\in\,]0,1]$ fixed, the velocity field
${\bf u}_{\varepsilon}$ is well-defined , thus the previous notation makes
sense.

From the energy inequality (17), we now derive first uniform bounds for the
family $\big{(}\varrho_{\varepsilon},{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}$. In fact, owing to our assumptions on
the initial data, and in particular to (15), the right-hand side of (17) is
_uniformly bounded_ : specifically, we have

$\sup_{\varepsilon\in\,]0,1]}\mathcal{E}\Big{(}\varrho_{0,\varepsilon},{\bf{\bf
u}}_{0,\varepsilon}\,\Big{|}\,b\Big{)}\,<\,+\infty\,.$

Then, it is easy to deduce the following uniform bounds:

$\displaystyle\big{(}\sqrt{\varrho_{\varepsilon}}\,{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}\,$
$\displaystyle\subset\,L^{\infty}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)}\,,$
(18) $\displaystyle\big{(}\sqrt{\varrho_{\varepsilon}}\,{\mathbb{D}}{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}\,$
$\displaystyle\subset\,L^{2}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)}\,.$
(19)

Let us now focus on the density functions. To begin with, following the
approach of [15], it is convenient to decompose any function $h$ into its
_essential_ and _residual_ parts. Thus, for almost every time $t>0$ and all
$\varepsilon\in\,]0,1]$, we introduce the sets

$\Omega_{\rm
ess}^{\varepsilon}(t)\,:=\,\left\\{x\in\Omega\;\Big{|}\quad\frac{b_{*}}{2}\,\leq\,\varrho_{\varepsilon}(t,x)\,\leq\,2\,b^{*}\right\\}\,,\qquad\Omega^{\varepsilon}_{\rm
res}(t)\,:=\,\Omega\setminus\Omega^{\varepsilon}_{\rm ess}(t)\,,$

where the constants $b_{*}$ and $b^{*}$ have been defined in (5) Then, given a
function $h$, we can write

$h\,=\,\left[h\right]_{\rm ess}\,+\,\left[h\right]_{\rm
res}\,,\qquad\qquad\mbox{ where }\qquad\left[h\right]_{\rm
ess}\,:=\,h\,\mathds{1}_{\Omega_{\rm ess}^{\varepsilon}(t)}\,.$

Here above, $\mathds{1}_{A}$ denotes the characteristic function of a set
$A\subset\Omega$.

For later use, it is convenient to divide the residual set
$\Omega^{\varepsilon}_{\rm res}(t)$ further: we define

$\Omega^{\varepsilon}_{{\rm
res},B}(t)\,:=\,\left\\{x\in\Omega^{\varepsilon}_{\rm res}(t)\;\Big{|}\quad
0\,\leq\,\varrho_{\varepsilon}(t,x)\,<\,\frac{b_{*}}{2}\right\\}\,,\qquad\Omega^{\varepsilon}_{{\rm
res},UB}(t)\,:=\,\left\\{x\in\Omega^{\varepsilon}_{\rm
res}(t)\;\Big{|}\quad\varrho_{\varepsilon}(t,x)\,>\,2\,b^{*}\right\\}$

as the regions where $\varrho_{\varepsilon}$, respectively, stays bounded and
may become unbounded.

After this preparation, let us come back to (17) and derive uniform bounds for
the family $\big{(}\varrho_{\varepsilon}\big{)}_{\varepsilon}$. In the
essential set, we can perform Taylor’s expansion of the function $H$; we thus
get

$\Big{[}H(\varrho)\,-\,H(b)\,-\,H^{\prime}(b)\,(\varrho-b)\Big{]}_{\rm
ess}\,\geq\,c\,\big{|}\varrho_{\varepsilon}\,-\,b\big{|}^{2}\,\mathds{1}_{\Omega^{\varepsilon}_{\rm
ess}(t)}\,,$

which implies that

$\sup_{t\in\mathbb{R}_{+}}\left\|\frac{1}{\varepsilon}\;\left[\varrho_{\varepsilon}-b\right]_{\rm
ess}\right\|_{L^{2}(\Omega)}\,\leq\,C\,.$ (20)

On the other hand, using the convexity of the function $H$, the fact that
$\left|\big{[}\varrho_{\varepsilon}-b\big{]}_{\rm res}\right|\geq b_{*}/2$ and
equation (9), we discover (see e.g. [16] for details) the following bounds on
the residual set:

$\displaystyle\sup_{t\in\mathbb{R}_{+}}\left\|\left[\varrho_{\varepsilon}\right]_{\rm
res}\right\|^{\gamma}_{L^{\gamma}(\Omega)}\,+\sup_{t\in\mathbb{R}_{+}}\left\|\left[\varrho_{\varepsilon}^{-1}\right]_{\rm
res}\right\|^{\kappa}_{L^{\kappa}(\Omega)}\,+\,\sup_{t\in\mathbb{R}_{+}}\left\|\left[1\right]_{\rm
res}\right\|_{L^{1}(\Omega)}\,\leq\,C\,\varepsilon^{2}\,.$ (21)

The previous estimate immediately implies that

$\sup_{t\in\mathbb{R}_{+}}\mathcal{L}\big{(}\Omega^{\varepsilon}_{\rm
res}(t)\big{)}\,\leq\,C\,\varepsilon^{2}\,,$ (22)

where we have denoted by $\mathcal{L}(A)$ the Lebesgue measure of a set
$A\subset\Omega$.

At this point, let us define the quantity

$\phi_{\varepsilon}\,=\,\frac{1}{\varepsilon}\,\big{(}\varrho_{\varepsilon}\,-\,b\big{)}\,.$

From the uniform bound (20), we may deduce that

$\left\|\big{[}\phi_{\varepsilon}\big{]}_{\rm
ess}\right\|_{L^{\infty}(\mathbb{R}_{+};L^{2})}\,\leq\,C\,.$ (23)

On the other hand, using (21) we can compute that for any $p\leq\gamma$, we
have

$\int_{\Omega}\left|\big{[}\phi_{\varepsilon}\big{]}_{\rm
res}\right|^{p}\,dx\,\leq\,\frac{C}{\varepsilon^{p}}\left(\int_{\Omega}\left|\big{[}\varrho_{\varepsilon}\big{]}_{\rm
res}\right|^{p}\,dx\,+\,\int_{\Omega}[1]_{\rm
res}\,dx\right)\,\leq\,C\,\varepsilon^{2-p}\,.$

Thus, we finally infer that

$\forall\;1\leq
p\leq\gamma\,,\qquad\qquad\left\|\big{[}\phi_{\varepsilon}\big{]}_{\rm
res}\right\|_{L^{\infty}(\mathbb{R}_{+};L^{p})}\,\leq\,C\,\varepsilon^{(2-p)/p}\,.$
(24)

Of course, this estimate will be useful only in the case when $p$ satisfies
the additional restriction $p\leq 2$.

###  3.2 The Bresch-Desjardins estimate

As it is apparent from the bounds of the previous subsection, the difficulty
with system (LABEL:main) is that we lose any control on the velocity fields
${\bf u}_{\varepsilon}$ and their gradient $\nabla{\bf u}_{\varepsilon}$ close
to vacuum, specifically in the region $\Omega^{\varepsilon}_{{\rm res},B}$.
The cold pressure term $p_{c}$ is of great help in order to bypass that
difficulty.

However, the cold pressure term alone is not strong enough to give us all the
pieces of information we need to pass to the limit. On the other hand, system
(LABEL:main) possesses a nice underlying structure, as evidenced for the first
time by Bresch and Desjardins (see e.g. [4], [3]). By taking advantage of that
structure, it is possible to derive, via the so-called _BD entropy estimates_
, some uniform estimates on the gradient of the density functions
$\varrho_{\varepsilon}$. This is the goal of the next lemma.

A bound coming from BD entropy estimates has been required in the definition
of weak solutions, see point (4) in Definition 2.1. Here we show that such a
bound holds _uniformly_ with respect to the small parameter
$\varepsilon\in\,]0,1]$. Note that the result in Lemma 3.1 is stated for
smooth solutions to the Navier-Stokes system (LABEL:main). This is solely to
justify the manipulations required to derive the inequality. Once the
inequality is proven for the smooth solutions, it is possible to deduce that
it is inherited also by the finite energy weak solutions considered in this
paper (see [30] and[25] for details).

In the next statement, we resort to the notation introduced in (11), and we
recall that we denote by $\mathbb{A}{\bf u}\,=\,\big{(}\nabla{\bf
u}\,-\,\nabla^{t}{\bf u}\big{)}/2$ the skew-symmetric part of the Jacobian
matrix of the vector field ${\bf u}$.

###### Lemma 3.1

Let $(\varrho_{\varepsilon},{\bf u}_{\varepsilon})$ be the smooth solution to
(LABEL:main)-(3). Then we have the inequality

$\begin{split}&\sup_{t\in\,]0,T[}\left(\mathcal{F}\Big{(}\varrho_{\varepsilon},{\bf
u}_{\varepsilon}\,\Big{|}\,b\Big{)}(t)\,+\,\frac{1}{\varepsilon^{2}}\int_{\Omega}\left(H(\varrho_{\varepsilon})\,-\,H(b)\,-\,H^{\prime}(b)(\varrho_{\varepsilon}-b)\right)\
{\rm d}{x}\right)\\\
&\qquad\qquad\qquad+\,\frac{\nu}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}b^{2}\,\frac{p^{\prime}(\varrho_{\varepsilon})}{\varrho_{\varepsilon}}\,\left|\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)\right|^{2}\
{\rm d}{x}\,{\rm
d}t\,+\,\nu\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\varrho_{\varepsilon}\,|\mathbb{A}{\bf
u}_{\varepsilon}|^{2}\ {\rm d}{x}\,{\rm d}t\,\leq\,C\,,\end{split}$ (25)

where the constant $C>0$ depends only on the initial data and on $T$. In
particular, the previous bound is uniform with respect to
$\varepsilon\in\,]0,1]$.

Proof. The proof of this estimate follows closely [5], with the only
modifications associated with more general forms of the pressure and of the
force. We proceed in several steps, assuming that $\varrho_{\varepsilon},{\bf
u}_{\varepsilon}$ are smooth enough to justify all the computations. For
notational simplicity, in what follows we write $(\varrho,{\bf u})$ instead of
$\big{(}\varrho_{\varepsilon},{\bf u}_{\varepsilon}\big{)}$.

Step 1. From Lemma 5.1 in [5] it follows that for sufficiently smooth
solutions of the continuity equation in (LABEL:main), the following equality
holds true

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}\varrho\left|\nabla\ln\frac{\varrho}{b}\right|^{2}\
{\rm d}{x}+\int_{\Omega}\varrho\nabla{\bf
u}\cdot\nabla\ln\frac{\varrho}{b}\nabla\ln\frac{\varrho}{b}\ {\rm d}{x}$
$\displaystyle\qquad\qquad+\int_{\Omega}\varrho\nabla{\bf u}\cdot\nabla\ln
b\nabla\ln\frac{\varrho}{b}\ {\rm d}{x}+\int_{\Omega}\varrho{\bf
u}\cdot\nabla\nabla\ln b\nabla\ln\frac{\varrho}{b}\ {\rm
d}{x}+\int_{\Omega}\varrho\nabla\operatorname{div}{\bf
u}\nabla\ln\frac{\varrho}{b}\ {\rm d}{x}=0.$

Step 2. In this step, one multiplies the momentum equation by
$\nu\nabla\ln\frac{\varrho}{b}$, we get

$\nu\int_{\Omega}\varrho\left(\partial_{t}{\bf u}+{\bf u}\cdot\nabla{\bf
u}\right)\cdot\nabla\ln\frac{\varrho}{b}\ {\rm
d}{x}+\nu^{2}\int_{\Omega}\varrho{\mathbb{D}}{\bf
u}:\nabla^{2}\ln\frac{\varrho}{b}\ {\rm
d}{x}+\frac{\nu}{\varepsilon^{2}}\int_{\Omega}\left(\nabla
p(\varrho)-\varrho\nabla G\right)\cdot\nabla\ln\frac{\varrho}{b}\ {\rm
d}{x}=0.$

Let us now rewrite each term from the above expression. First, note that the
transport term gives

$\nu\int_{\Omega}\varrho\left(\partial_{t}{\bf u}+{\bf u}\cdot\nabla{\bf
u}\right)\cdot\nabla\ln\frac{\varrho}{b}\ {\rm
d}{x}=\nu\frac{d}{dt}\int_{\Omega}b{\bf u}\cdot\nabla\frac{\varrho}{b}\ {\rm
d}{x}-\nu\int_{\Omega}\varrho\nabla{\bf u}:\nabla^{t}{\bf u}\ {\rm
d}{x}+\nu\int_{\Omega}\varrho{\bf u}\cdot\nabla(\nabla\ln b){\bf u}\ {\rm
d}{x}.$

For the diffusion term, after noticing that $\nabla^{2}$ is always symmetric,
we have

$\displaystyle\nu^{2}\int_{\Omega}\varrho{\mathbb{D}}{\bf
u}:\nabla^{2}\ln\frac{\varrho}{b}\ {\rm d}{x}$
$\displaystyle=\nu^{2}\int_{\Omega}\varrho\nabla{\bf
u}:\nabla^{2}\ln\frac{\varrho}{b}\ {\rm d}{x}$
$\displaystyle=\nu^{2}\int_{\Omega}b\nabla{\bf u}:\nabla^{2}\frac{\varrho}{b}\
{\rm d}{x}-\nu^{2}\int_{\Omega}\varrho\nabla{\bf
u}\cdot\nabla\ln\frac{\varrho}{b}\nabla\ln\frac{\varrho}{b}\ {\rm d}{x}$
$\displaystyle=-\nu^{2}\int_{\Omega}\varrho\nabla{\bf u}\cdot\nabla\ln
b\nabla\ln\frac{\varrho}{b}\ {\rm
d}{x}-\nu^{2}\int_{\Omega}\varrho\nabla\operatorname{div}{\bf
u}\nabla\ln\frac{\varrho}{b}\ {\rm d}{x}$
$\displaystyle\qquad\qquad\qquad-\nu^{2}\int_{\Omega}\varrho\nabla{\bf
u}\cdot\nabla\ln\frac{\varrho}{b}\nabla\ln\frac{\varrho}{b}\ {\rm d}{x}.$

Finally, for the pressure and force term, using (4), we obtain

$\nu\int_{\Omega}\frac{\nabla p(\varrho)-\varrho\nabla
G}{\varepsilon^{2}}\cdot\nabla\ln\frac{\varrho}{b}\ {\rm
d}{x}=\frac{\nu}{\varepsilon^{2}}\int_{\Omega}b^{2}\frac{p^{\prime}(\varrho)}{\varrho}\left|\nabla\frac{\varrho}{b}\right|^{2}\
{\rm
d}{x}+\frac{\nu}{\varepsilon^{2}}\int_{\Omega}\big{(}p^{\prime}(\varrho)-p^{\prime}(b)\big{)}\nabla
b\cdot\nabla\frac{\varrho}{b}\ {\rm d}{x}.$

Step 3. Now, we sum up the equalities from the previous steps along with the
energy estimate. In our case, after setting
$H(\varrho;b)\,:=\,H(\varrho)-H^{\prime}(b)(\varrho-b)-H(b)$, the statement of
Lemma 5.2 from [5] gives

$\begin{split}&\frac{d}{dt}\int_{\Omega}\left(\mathcal{F}\Big{(}\varrho,{\bf
u}\,\Big{|}\,b\Big{)}+\frac{1}{\varepsilon^{2}}\,H(\varrho;b)\right)\ {\rm
d}{x}+\frac{\nu}{\varepsilon^{2}}\int_{\Omega}b^{2}\frac{p^{\prime}(\varrho)}{\varrho}\left|\nabla\left(\frac{\varrho}{b}\right)\right|^{2}\
{\rm d}{x}+\nu\int_{\Omega}\varrho|{\mathbb{D}}{\bf u}|^{2}\ {\rm d}{x}\\\
&=\nu\int_{\Omega}\varrho\nabla{\bf u}:\nabla^{t}{\bf u}\ {\rm
d}{x}-\nu\int_{\Omega}\varrho{\bf u}\cdot(\nabla\nabla\ln b){\bf u}\ {\rm
d}{x}-\nu^{2}\int_{\Omega}\varrho{\bf u}\nabla\nabla\ln
b\nabla\ln\frac{\varrho}{b}\ {\rm d}{x}\\\
&\quad-\frac{\nu}{\varepsilon^{2}}\int_{\Omega}\big{(}p^{\prime}(\varrho)-p^{\prime}(b)\big{)}\nabla
b\cdot\nabla\frac{\varrho}{b}\ {\rm d}{x}\,.\end{split}$ (26)

Now notice that

$\nu\int_{\Omega}\varrho|{\mathbb{D}}{\bf u}|^{2}\ {\rm
d}{x}-\nu\int_{\Omega}\varrho\nabla{\bf u}:\nabla^{t}{\bf u}\ {\rm
d}{x}=\nu\int_{\Omega}\varrho|\mathbb{A}{\bf u}|^{2}\ {\rm d}{x},$

and so, we finally get

$\begin{split}&\frac{d}{dt}\int_{\Omega}\left(\mathcal{F}\Big{(}\varrho,{\bf
u}\,\Big{|}\,b\Big{)}+\frac{1}{\varepsilon^{2}}\,H(\varrho;b)\right)\ {\rm
d}{x}+\frac{\nu}{\varepsilon^{2}}\int_{\Omega}b^{2}\frac{p^{\prime}(\varrho)}{\varrho}\left|\nabla\left(\frac{\varrho}{b}\right)\right|^{2}\
{\rm d}{x}+\nu\int_{\Omega}\varrho|\mathbb{A}{\bf u}|^{2}\ {\rm d}{x}\\\
&=-\nu\int_{\Omega}\varrho{\bf u}\cdot(\nabla\nabla\ln b){\bf u}\ {\rm
d}{x}-\nu^{2}\int_{\Omega}\varrho{\bf u}\nabla\nabla\ln
b\nabla\ln\frac{\varrho}{b}\ {\rm
d}{x}-\frac{\nu}{\varepsilon^{2}}\int_{\Omega}\big{(}p^{\prime}(\varrho)-p^{\prime}(b)\big{)}\nabla
b\cdot\nabla\frac{\varrho}{b}\ {\rm d}{x}\\\
&=\sum_{i=1}^{3}J_{i}.\end{split}$ (27)

Step 4. We have to control the terms $J_{1},J_{2},J_{3}$ appearing in the
right-hand side of (LABEL:BD). To begin with, we can easily estimate

$\displaystyle\int_{0}^{T}J_{1}\ {\rm d}t$
$\displaystyle=-\nu\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\varrho{\bf
u}\cdot(\nabla\nabla\ln b){\bf u}\ {\rm d}{x}\,{\rm d}t\leq\|\nabla^{2}\ln
b\|_{L^{\infty}_{T}(L^{\infty})}\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\varrho|{\bf
u}|^{2}\ {\rm d}{x}\,{\rm d}t\leq C\,.$

For $J_{2}$, instead, we get

$\displaystyle\int_{0}^{T}J_{2}\ {\rm d}t$
$\displaystyle=-\nu^{2}\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\varrho{\bf
u}\nabla\nabla\ln b\nabla\ln\frac{h}{b}\ {\rm d}{x}\,{\rm
d}t\,\leq\,\nu^{2}\|\nabla^{2}\ln
b\|_{L^{\infty}_{T}(L^{\infty})}\int^{T}_{0}\mathcal{F}\Big{(}\varrho,{\bf
u}\,\Big{|}\,b\Big{)}\,{\rm d}t\,,$

hence $J_{2}$ can be controlled by means of a Grönwall argument.
Finally, we have to deal with $J_{3}$. This estimate is a bit more involved
than the previous ones, as this term depends on the pressure. We have to
distinguish some different cases.

Case 1: integral over $\Omega^{\varepsilon}_{\rm ess}$. We start by bounding
the part of $J_{3}$ which is restricted to the essential set. For this, we use
the Taylor expansion
$p^{\prime}(\varrho)=p^{\prime}(b)+p^{\prime\prime}(z)(\varrho-b)$, for some
$z$ between $\varrho$ and $b$, and the fact that $\varrho$ is bounded in
$\Omega_{\rm ess}$ to write

$\displaystyle\left|\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{\rm
ess}^{\varepsilon}}{\big{(}p^{\prime}(\varrho)-p^{\prime}(b)\big{)}\nabla
b\cdot\nabla\frac{\varrho}{b}}\,{\rm d}{x}\right|\,$
$\displaystyle\lesssim\,\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{\rm
ess}^{\varepsilon}}|\varrho-b|\,|\nabla
b|\left|\sqrt{\frac{p^{\prime}(\varrho)}{\varrho}}\,b\,\nabla\left(\frac{\varrho}{b}\right)\right|\,{\rm
d}{x}$ $\displaystyle\lesssim\,C_{\delta}\,\left\|\big{[}\phi\big{]}_{\rm
ess}\right\|^{2}_{L^{2}}\,+\,\delta\,\frac{\nu}{\varepsilon^{2}}\int_{\Omega}b^{2}\,\frac{p^{\prime}(\varrho)}{\varrho}\,\left|\nabla\left(\frac{\varrho}{b}\right)\right|^{2}\
{\rm d}{x}\,,$

where we have also applied the Young inequality in the last step. Here,
$\delta>0$ can be taken arbitrarily small, to the price of increasing the
value of $C_{\delta}$. Thus, taking $\delta$ small enough, after integration
in time, we can absorbe the last term of the previous estimate in the left-
hand side of (LABEL:BD), while the other term is bounded by a uniform constant
$C$ times $T$, in view of (23).

Case 2: integral over $\Omega_{\rm res}^{\varepsilon}$. In the residual set,
it is convenient to split $J_{3}$ into two pieces, as follows:

$\displaystyle\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{\rm
res}^{\varepsilon}}{\big{(}p^{\prime}(\varrho)-p^{\prime}(b)\big{)}\nabla
b\cdot\nabla\frac{\varrho}{b}}\,{\rm d}{x}\,$
$\displaystyle=\,\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{\rm
res}^{\varepsilon}}{p^{\prime}(\varrho)\nabla
b\cdot\nabla\frac{\varrho}{b}}\,{\rm
d}{x}\,-\,\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{\rm
res}^{\varepsilon}}{p^{\prime}(b)\nabla b\cdot\nabla\frac{\varrho}{b}}\,{\rm
d}{x}$ (28)

The bound of the first term in the last equality is easy. We start by writing

$\begin{split}\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{\rm
res}^{\varepsilon}}{p^{\prime}(\varrho)\nabla
b\cdot\nabla\frac{\varrho}{b}}\,{\rm
d}{x}\,=\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{\rm
res}^{\varepsilon}}{\sqrt{p^{\prime}(\varrho)\,\varrho}\;\nabla\ln
b\cdot\left(\sqrt{\frac{p^{\prime}(\varrho)}{\varrho}}\,b\,\nabla\frac{\varrho}{b}\right)}\,{\rm
d}{x}.\end{split}$

Then, we observe that

$p^{\prime}(\varrho)\,\varrho\,=\,\varrho^{\gamma}+\varrho^{-\kappa}\qquad\qquad\Longrightarrow\qquad\qquad\left[\sqrt{p^{\prime}(\varrho)\,\varrho}\right]_{\rm
res}\,\in\,L^{\infty}\big{(}\mathbb{R}_{+};L^{2}\big{)}$

uniformly in $\varepsilon>0$, owing to (21). Indeed, one has
$\left\|\left[\sqrt{p^{\prime}(\varrho)\,\varrho}\right]_{\rm
res}\right\|_{L^{2}}\,\leq\,\left\|\varrho\right\|_{L^{\gamma}}^{\gamma/2}+\,\left\|\varrho^{-1}\right\|_{L^{\kappa}}^{\kappa/2}$.
Thus, using (21) quantitatively and arguing as in Case 1, we deduce that

$\displaystyle\left|\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{\rm
res}^{\varepsilon}}{\sqrt{p^{\prime}(\varrho)\,\varrho}\;\nabla\ln
b\cdot\left(\sqrt{\frac{p^{\prime}(\varrho)}{\varrho}}\,b\,\nabla\frac{\varrho}{b}\right)}\,{\rm
d}{x}\right|\,$
$\displaystyle\lesssim\,\frac{C_{\delta}}{\varepsilon^{2}}\left\|\left[\sqrt{p^{\prime}(\varrho)\,\varrho}\right]_{\rm
res}\right\|^{2}_{L^{2}}\,+\,\delta\,\frac{\nu}{\varepsilon^{2}}\int_{\Omega}b^{2}\,\frac{p^{\prime}(\varrho)}{\varrho}\,\left|\nabla\left(\frac{\varrho}{b}\right)\right|^{2}\
{\rm d}{x}$
$\displaystyle\lesssim\,C_{\delta}\,+\,\delta\,\frac{\nu}{\varepsilon^{2}}\int_{\Omega}b^{2}\,\frac{p^{\prime}(\varrho)}{\varrho}\,\left|\nabla\left(\frac{\varrho}{b}\right)\right|^{2}\
{\rm d}{x}\,.$

Once again, after integration in time, the last term on the right can be
absorbed in the left-hand side of (LABEL:BD), for $\delta>0$ small enough.

It remains to deal with the last term appearing in (28). We start by writing

$\begin{split}\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{\rm
res}^{\varepsilon}}{p^{\prime}(b)\nabla b\cdot\nabla\frac{\varrho}{b}}\,{\rm
d}{x}=\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{\rm
res}^{\varepsilon}}{\sqrt{\frac{\varrho}{p^{\prime}(\varrho)}}\,\frac{p^{\prime}(b)}{b}\nabla
b\cdot\left(\sqrt{\frac{p^{\prime}(\varrho)}{\varrho}}\,b\,\nabla\frac{\varrho}{b}\right)}\,{\rm
d}{x}\,.\end{split}$

First note that

$\frac{\varrho}{p^{\prime}(\varrho)}\,=\,\frac{\varrho}{p_{E}^{\prime}(\varrho)+p^{\prime}_{c}(\varrho)}\,=\,\frac{\varrho^{\kappa+2}}{\varrho^{\gamma+\kappa}+1}\,.$

Thus, on the one hand we get

$\left|\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{{\rm
res},B}^{\varepsilon}}{\sqrt{\frac{\varrho}{p^{\prime}(\varrho)}}\,\frac{p^{\prime}(b)}{b}\nabla
b\cdot\left(\sqrt{\frac{p^{\prime}(\varrho)}{\varrho}}\,b\,\nabla\frac{\varrho}{b}\right)}\,{\rm
d}{x}\right|\,\lesssim\,\frac{\nu}{\varepsilon^{2}}\,\Big{(}\mathcal{L}(\Omega^{\varepsilon}_{\rm
res})\Big{)}^{1/2}\,\left\|\sqrt{\frac{p^{\prime}(\varrho)}{\varrho}}\,b\,\nabla\frac{\varrho}{b}\right\|_{L^{2}}\,.$

On the other hand, we also have

$\mathds{1}_{\Omega_{{\rm
res},UB}^{\varepsilon}}\,\sqrt{\frac{\varrho}{p^{\prime}(\varrho)}}\,\lesssim\,\mathds{1}_{\Omega_{{\rm
res},UB}^{\varepsilon}}\,\varrho_{\varepsilon}^{(2-\gamma)/2}\,.$

Now, in view of (21), for $2-\gamma\,\geq\,0\,$ we have that

$\left[\varrho\right]_{\rm
res}^{(2-\gamma)/2}\,\in\,L^{\infty}(\mathbb{R}_{+};L^{q})\,,\quad
q\,:=\,\frac{2\gamma}{2-\gamma}\,>\,2\,,\qquad\qquad\mbox{ with
}\qquad\left\|\left[\varrho\right]_{\rm
res}^{(2-\gamma)/2}\right\|_{L^{q}}\,=\,\left\|\left[\varrho\right]_{\rm
res}\right\|_{L^{\gamma}}^{\gamma/q}\,\lesssim\,\varepsilon^{2/q}\,.$

If, on the other hand $2-\gamma\,<\,0$, then we have

$\left[\varrho\right]_{\rm
res}^{(2-\gamma)/2}\,\in\,L^{\infty}(\mathbb{R}_{+};L^{q})\,,\quad
q\,:=\,\frac{2\kappa}{\gamma-2}\,,\qquad\qquad\mbox{ with
}\qquad\left\|\left[\varrho\right]_{\rm
res}^{(2-\gamma)/2}\right\|_{L^{q}}\,=\,\left\|\left[\varrho^{-1}\right]_{\rm
res}\right\|_{L^{\kappa}}^{\kappa/q}\,\lesssim\,\varepsilon^{2/q}\,.$

In the latter case, in order to have $q\geq 2$, we need the condition

$\kappa\,\geq\,\gamma\,-\,2\,.$

In any case, and under the assumption that the previous requirement is
satisfied in the case $\gamma>2$, we can thus bound

$\left|\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{{\rm
res},UB}^{\varepsilon}}{\sqrt{\frac{\varrho}{p^{\prime}(\varrho)}}\,\frac{p^{\prime}(b)}{b}\nabla
b\cdot\left(\sqrt{\frac{p^{\prime}(\varrho)}{\varrho}}\,b\,\nabla\frac{\varrho}{b}\right)}\,{\rm
d}{x}\right|\,\lesssim\,\frac{1}{\varepsilon^{2}}\,\left\|\left[\varrho\right]_{\rm
res}^{(2-\gamma)/2}\right\|_{L^{q}}\,\left\|\sqrt{\frac{p^{\prime}(\varrho)}{\varrho}}\,b\,\nabla\frac{\varrho}{b}\right\|_{L^{2}}\,\Big{(}\mathcal{L}(\Omega^{\varepsilon}_{\rm
res})\Big{)}^{1/m}\,,$

where $1/q\,+\,1/m\,=\,1/2$. In the end, using (21), (22) and the definition
of $m$, after an application of Young’s inequality we arrive at

$\left|\frac{\nu}{\varepsilon^{2}}\int_{\Omega_{\rm
res}^{\varepsilon}}{\sqrt{\frac{\varrho}{p^{\prime}(\varrho)}}\,\frac{p^{\prime}(b)}{b}\nabla
b\cdot\left(\sqrt{\frac{p^{\prime}(\varrho)}{\varrho}}\,b\,\nabla\frac{\varrho}{b}\right)}\,{\rm
d}{x}\right|\,\lesssim\,C_{\delta}\,+\,\delta\,\frac{\nu}{\varepsilon^{2}}\,\left\|\sqrt{\frac{p^{\prime}(\varrho)}{\varrho}}\,b\,\nabla\frac{\varrho}{b}\right\|^{2}_{L^{2}}\,.$

Finally, putting all those inequalities together, we see that

$\int^{T}_{0}\big{|}J_{3}\big{|}\,{\rm
d}t\,\lesssim\,C_{\delta}\,+\,3\,\delta\,\frac{\nu}{\varepsilon^{2}}\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}b^{2}\,\frac{p^{\prime}(\varrho)}{\varrho}\,\left|\nabla\left(\frac{\varrho}{b}\right)\right|^{2}\
{\rm d}{x}\,{\rm d}t\,.$

Hence, for $\delta>0$ small enough, we can absorbe the last term of the
previous inequality into the left-hand side of (LABEL:BD). The proof of the
lemma is thus completed. $\Box$

Before moving on, let us observe that the constant in Lemma 3.1 depends on
time. Therefore, all the bounds which we are going to derive from the BD
entropy estimate will be only _local in time_ , on any arbitrarily large but
compact interval $[0,T]$. This contrasts with the bounds coming from the
classical energy inequality, which instead are global in time.

###  3.3 Consequences of the BD entropy estimate

Lemma 3.1 provides us with at least three additional pieces of information
with respect to the classical energy inequality (17). Those additional bounds
will be fundamental in order to prove convergence in our setting.

First of all, estimate (LABEL:BD_estimate) can be used to improve (19):
indeed, as $\nabla{\bf u}\,=\,{\mathbb{D}}{\bf u}\,+\,\mathbb{A}{\bf u}$, we
have

$\begin{split}\big{(}\sqrt{\varrho_{\varepsilon}}\,\nabla{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}\,&\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)}\,.\end{split}$ (29)

Next, combining (LABEL:BD_estimate) with (17), we gather that

$\sqrt{\varrho_{\varepsilon}}\,\nabla\ln\left(\frac{\varrho_{\varepsilon}}{b}\right)\,=\,\frac{b}{\sqrt{\varrho_{\varepsilon}}}\,\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)\;\in\,L^{\infty}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)}\,,$ (30)

with _uniform_ inclusion (of course, uniform with respect to
$\varepsilon\in\,]0,1]$). At this point, an easy computation shows that

$\frac{b}{\sqrt{\varrho_{\varepsilon}}}\,\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)\,=\,2\,\nabla\sqrt{\varrho_{\varepsilon}}\,-\,\sqrt{\varrho_{\varepsilon}}\,\nabla\ln
b\,=\,2\,\nabla\sqrt{\varrho_{\varepsilon}}\,-\,\left(\sqrt{[\varrho_{\varepsilon}]_{\rm
ess}}\,+\,\sqrt{[\varrho_{\varepsilon}]_{\rm res}}\right)\,\nabla\ln b\,.$

Observe that $\big{(}[\varrho_{\varepsilon}]_{\rm ess}\big{)}_{\varepsilon}$
is uniformly bounded in $L^{\infty}(\mathbb{R}_{+}\times\Omega)$, whereas
$\left(\sqrt{[\varrho_{\varepsilon}]_{\rm res}}\right)_{\varepsilon}$ is
uniformly bounded in
$L^{\infty}(\mathbb{R}_{+};L^{2\gamma})\,\hookrightarrow\,L^{\infty}(\mathbb{R}_{+};L^{2})$,
in view of (21). Putting this information together with (30), by Sobolev
embeddings we finally deduce that

$\left(\nabla\sqrt{\varrho_{\varepsilon}}\right)_{\varepsilon}\,\subset\,L^{\infty}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)}\qquad\Longrightarrow\qquad\big{(}\varrho_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{\infty}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{3}(\Omega)\big{)}\,.$ (31)

The previous property provides higher integrability for
$\varrho_{\varepsilon}$, _globally_ (i.e. in the whole $\Omega$, without
having to distinguish between essential and residual parts) and _uniformly_
with respect to $\varepsilon\in\,]0,1]$.

Next, we consider the quantity

$\left(\frac{1}{\varepsilon^{2}}\,{\frac{p^{\prime}(\varrho_{\varepsilon})}{\varrho_{\varepsilon}}}\,b^{2}\,\left|\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)\right|^{2}\right)_{\varepsilon}\,\subset\,L^{1}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{1}(\Omega)\big{)}\,.$ (32)

The term inside the parentheses can be written as

$\displaystyle{\frac{p^{\prime}(\varrho_{\varepsilon})}{\varrho_{\varepsilon}}}\,b^{2}\,\left|\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)\right|^{2}\,$
$\displaystyle=\,\left(\varrho_{\varepsilon}^{\gamma-2}+\varrho_{\varepsilon}^{-\kappa-2}\right)\,b^{2}\,\left|\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)\right|^{2}\,$
$\displaystyle=\,\left|b^{\gamma/2}\,\left(\frac{\varrho_{\varepsilon}}{b}\right)^{(\gamma-2)/2}\,\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)\right|^{2}\,+\left|b^{-\kappa/2}\,\left(\frac{\varrho_{\varepsilon}}{b}\right)^{(-\kappa-2)/2}\,\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)\right|^{2}\,$
$\displaystyle=\,\left|\frac{2}{\gamma}\,b^{\gamma/2}\,\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)^{\gamma/2}\right|^{2}\,+\left|-\frac{2}{\kappa}\,b^{-\kappa/2}\,\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)^{-\kappa/2}\right|^{2}\,.$

Thus, we get the following uniform embeddings:

$\left(\frac{1}{\varepsilon}\,\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)^{\gamma/2}\right)_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)},\qquad\left(\frac{1}{\varepsilon}\,\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)^{-\kappa/2}\right)_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)}\,.$ (33)

This is another fundamental piece of information, since it gives, roughly
speaking, a uniform control on the gradient of $\varrho_{\varepsilon}/b$, with
a quantitative bound $O(\varepsilon)$ in a suitable norm. However, in order to
be able to fully exploit this information, some preparatory work is needed.

To begin with, we write

$\frac{1}{\varepsilon}\,\nabla\left(\frac{\varrho_{\varepsilon}}{b}\right)^{\gamma/2}\,=\,\frac{1}{\varepsilon}\nabla\left(\frac{\varrho_{\varepsilon}^{\gamma/2}\,-\,b^{\gamma/2}}{b^{\gamma/2}}\right)\,.$

Therefore, the uniform bound (33) implies that

$\left(\frac{1}{\varepsilon}\,\left(\varrho_{\varepsilon}^{\gamma/2}\,-\,b^{\gamma/2}\right)\right)_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{6}(\Omega)\big{)}\,.$ (34)

Let us derive a couple of properties from (34). We notice that we can write

$\varrho_{\varepsilon}^{\gamma/2}\,-\,b^{\gamma/2}\,=\,\frac{\varrho_{\varepsilon}^{\gamma}\,-\,b^{\gamma}}{\varrho_{\varepsilon}^{\gamma/2}\,+\,b^{\gamma/2}}\,.$

On the one hand, from the previous relation we immediately deduce that

$\left(\frac{1}{\varepsilon}\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},B}\cup\Omega^{\varepsilon}_{\rm
ess}}\,\left(\varrho_{\varepsilon}^{\gamma}\,-\,b^{\gamma}\right)\right)_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{6}(\Omega)\big{)}\,.$

We use a Taylor expansion of the function $f(s)\,=\,s^{\gamma}$ to get, for a
suitable point $z_{\varepsilon}\,=\,z_{\varepsilon}(t,x)$ between
$\varrho_{\varepsilon}(t,x)$ and $b(x)$, the following fact:

$\varrho_{\varepsilon}^{\gamma}\,-\,b^{\gamma}\,=\,\gamma\,b^{\gamma-1}\,\big{(}\varrho_{\varepsilon}\,-\,b\big{)}\,+\,\gamma\,(\gamma-1)\,z_{\varepsilon}^{\gamma-2}\,\big{(}\varrho_{\varepsilon}\,-\,b\big{)}^{2}\,\geq\,\gamma\,b^{\gamma-1}\,\big{(}\varrho_{\varepsilon}\,-\,b\big{)}\,.$

This inequality, together with (34) above, implies that

$\left(\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},B}\cup\Omega^{\varepsilon}_{\rm
ess}}\,\phi_{\varepsilon}\right)_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{6}(\Omega)\big{)}\,.$ (35)

On the other hand, on the set $\Omega^{\varepsilon}_{{\rm res},UB}$ we have
that $\varrho_{\varepsilon}\,\geq\,2\,b^{*}$. Hence, on that set we can write

$\frac{\varrho_{\varepsilon}^{\gamma}\,-\,b^{\gamma}}{\varrho_{\varepsilon}^{\gamma/2}\,+\,b^{\gamma/2}}\,\geq\,\left(\frac{\varrho_{\varepsilon}}{2}\right)^{\gamma}\,\frac{2^{\gamma}-1}{2\,\varrho_{\varepsilon}^{\gamma/2}}\,\geq\,C\,\varrho_{\varepsilon}^{\gamma/2}\,.$

Using (34) again, we gather that

$\left(\frac{1}{\varepsilon}\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\varrho^{\gamma/2}_{\varepsilon}\right)_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{6}(\Omega)\big{)}\qquad\quad\Longrightarrow\qquad\quad\left(\frac{1}{\varepsilon}\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\varrho_{\varepsilon}\right)_{\varepsilon}\,\subset\,L^{\gamma}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{3\gamma}(\Omega)\big{)}\,.$ (36)

The main point of the last computations is that (33) does not really provide a
uniform control on the gradient of the functions $\phi_{\varepsilon}$ in
$L^{2}_{T}(L^{2})$, whenever $\gamma\neq 2$. Such a control, which is true
when $\gamma=2$, would give higher integrability of the $\phi_{\varepsilon}$’s
in $L^{2}_{T}(L^{6})$. Estimate (35) shows that we are not too far from
getting that property, but this fact is true only when $\varrho_{\varepsilon}$
stays bounded. Since we do not have $L^{\infty}$ bounds on
$\varrho_{\varepsilon}$, the density functions may grow in some part of the
domain $\Omega$ (which has to be small, recall (21) above). Anyway, inequality
(36) provides us with a useful control in that region.

Actually, in $\Omega^{\varepsilon}_{{\rm res},B}$ we can derive an even better
control on the $\varrho_{\varepsilon}$’s than (35). Indeed, arguing in a
similar way as for getting (36), from the second piece of information in (33)
we deduce

$\left(\frac{1}{\varepsilon}\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},B}}\,\varrho_{\varepsilon}^{-\kappa/2}\right)_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{6}(\Omega)\big{)}\qquad\quad\Longrightarrow\qquad\quad\left(\frac{1}{\varepsilon}\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},B}}\,\varrho_{\varepsilon}^{-1}\right)_{\varepsilon}\,\subset\,L^{\kappa}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{3\kappa}(\Omega)\big{)}\,.$ (37)

We conclude this part by showing a third consequence of the BD entropy
estimate (LABEL:BD_estimate), that is a control on the gradient functions
$\nabla\phi_{\varepsilon}$ in suitable Lebesgue norms. Since this is a key
property, we put it in the form of a proposition.

###### Proposition 3.2

Let $\gamma>1$ and $\kappa>0$ such that $\kappa\geq\gamma-2$. Let
$\big{(}\varrho_{\varepsilon},{\bf u}_{\varepsilon}\big{)}_{\varepsilon}$ be a
family of global in time finite energy weak solutions to system (LABEL:main),
in the sense of Definition 2.1, related to initial data
$\big{(}\varrho_{0,\varepsilon},{\bf u}_{0,\varepsilon}\big{)}_{\varepsilon}$
such that (15) holds.

Then:

  * •

if $\gamma\geq 2$, one has the uniform embedding
$\;\big{(}\phi_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};H^{1}(\Omega)\big{)}$;

  * •

when $\gamma<2$, one deduces instead
$\;\big{(}\phi_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};W^{1,\gamma}(\Omega)\big{)}$.

Proof. We start by observing that, as a consequence of (32), we get in
particular

$\left({\varrho_{\varepsilon}}^{-(1-\gamma/2)}\,b\,\nabla\left(\frac{\phi_{\varepsilon}}{b}\right)\right)_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)},\quad\left({\varrho_{\varepsilon}}^{-(1+\kappa/2)}\,b\,\nabla\left(\frac{\phi_{\varepsilon}}{b}\right)\right)_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)}\,.$ (38)

Since on the set $\Omega^{\varepsilon}_{\rm
ess}\,\cup\,\Omega^{\varepsilon}_{{\rm res},B}$ we have
$0\leq\varrho_{\varepsilon}\leq 2b^{*}$, we easily get

$\left|\mathds{1}_{\Omega^{\varepsilon}_{\rm
ess}\,\cup\,\Omega^{\varepsilon}_{{\rm
res},B}}\;b\,\nabla\left(\frac{\phi_{\varepsilon}}{b}\right)\right|\,\lesssim\,\left|\mathds{1}_{\Omega^{\varepsilon}_{\rm
ess}\,\cup\,\Omega^{\varepsilon}_{{\rm
res},B}}\;{\varrho_{\varepsilon}}^{-(1+\kappa/2)}\,b\,\nabla\left(\frac{\phi_{\varepsilon}}{b}\right)\right|\,,$

which implies that, for all $T>0$ fixed, one has

$\begin{split}\left(\mathds{1}_{\Omega^{\varepsilon}_{\rm
ess}\,\cup\,\Omega^{\varepsilon}_{{\rm
res},B}}\;b\,\nabla\left(\frac{\phi_{\varepsilon}}{b}\right)\right)_{\varepsilon}\,\subset\,L^{2}_{T}(L^{2})\,.\end{split}$
(39)

On the subset $\Omega^{\varepsilon}_{{\rm res},UB}$, we need to consider two
cases.

_Case 1._ For $1-\gamma/2<0$ we can employ the first part of (38) and an
argument similar to the one used above to deduce that

$\forall\;T\,>\,0\,,\qquad\qquad\left(\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\;b\,\nabla\left(\frac{\phi_{\varepsilon}}{b}\right)\right)_{\varepsilon}\,\subset\,L^{2}_{T}(L^{2})\,.$

_Case 2._ For $1-\gamma/2\geq 0$ instead, we can write

$\displaystyle\left|\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\;b\,\nabla\left(\frac{\phi_{\varepsilon}}{b}\right)\right|\,$
$\displaystyle=\,\left|\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\;{\varrho_{\varepsilon}}^{-(1-\gamma/2)}\,b\,\nabla\left(\frac{\phi_{\varepsilon}}{b}\right)\right|\;\big{[}\varrho_{\varepsilon}\big{]}_{{\rm
res},UB}^{1-\gamma/2}\,.$

Since, for any $T>0$ fixed, $\big{(}\big{[}\varrho_{\varepsilon}\big{]}_{\rm
res}\big{)}_{\varepsilon}$ is uniformly bounded in
$L^{\infty}_{T}(L^{\gamma})$, we have that
$\left(\big{[}\varrho_{\varepsilon}\big{]}_{{\rm
res},UB}^{1-\gamma/2}\right)_{\varepsilon}$ is uniformly bounded in
$L^{\infty}_{T}(L^{2\gamma/(2-\gamma)})$. In turn, combining this information
with (38) implies that

$\forall\;T\,>\,0\,,\qquad\qquad\left(\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\;b\,\nabla\left(\frac{\phi_{\varepsilon}}{b}\right)\right)_{\varepsilon}\,\subset\,L^{2}_{T}(L^{\gamma})\,.$

We have thus discovered that the family of
$\nabla\big{(}\phi_{\varepsilon}/b\big{)}$’s is uniformly bounded in the space
$L^{2}_{T}(L^{\gamma})$, for any time $T>0$.

In order to conclude, we simply write

$\nabla\left(\frac{\phi_{\varepsilon}}{b}\right)\,=\,\frac{1}{b}\,\nabla\phi_{\varepsilon}\,-\,\frac{1}{b^{2}}\,\phi_{\varepsilon}\,\nabla
b\,,$

and, when $\gamma\leq 2$, we bound
$\big{(}\phi_{\varepsilon}\big{)}_{\varepsilon}$ uniformly in
$L^{\infty}_{T}(L^{\gamma})$, thanks to (23) and (24). When $\gamma>2$,
instead, we use (23) again, together with the fact that, by definition of
$\phi_{\varepsilon}$, we have

$\displaystyle\int_{\Omega_{{\rm
res},B}}\left|\phi_{\varepsilon}\right|^{2}\,{\rm d}{x}\,$
$\displaystyle=\,\dfrac{1}{\varepsilon^{2}}\int_{\Omega_{{\rm
res},B}}\left|\varrho_{\varepsilon}\,-\,b\right|^{2}\,{\rm
d}{x}\;\lesssim\;\dfrac{1}{\varepsilon^{2}}\,\mathcal{L}\big{(}\Omega_{\rm
res}\big{)}\;\lesssim\;1$ $\displaystyle\int_{\Omega_{{\rm
res},UB}}\left|\phi_{\varepsilon}\right|^{2}\,{\rm d}{x}\,$
$\displaystyle=\,\dfrac{1}{\varepsilon^{2}}\int_{\Omega_{{\rm
res},UB}}\left|\varrho_{\varepsilon}\,-\,b\right|^{2}\,{\rm
d}{x}\;\lesssim\;\dfrac{1}{\varepsilon^{2}}\int_{\Omega_{{\rm
res},UB}}\varrho_{\varepsilon}^{2}\,{\rm d}{x}$
$\displaystyle\lesssim\,\dfrac{1}{\varepsilon^{2}}\,\left\|\big{[}\varrho_{\varepsilon}\big{]}_{\rm
res}\right\|^{2}_{L^{\gamma}}\,\Big{(}\mathcal{L}\big{(}\Omega_{\rm
res}\big{)}\Big{)}^{(\gamma-2)/\gamma}\;\lesssim\;1\,,$

where we have used also the bounds provided by (21). We thus conclude that,
when $\gamma>2$, we have
$\big{(}\phi_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{\infty}_{T}(L^{2})$
for any $T>0$ fixed.

In the end, the proposition is proved. $\Box$

We also notice that, as a consequence of Proposition 3.2 and Sobolev
embedding, we get that

$\mbox{if
}1<\gamma<2\,,\quad\big{(}\phi_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{3\gamma/(3-\gamma)}(\Omega)\big{)}\,,\qquad\quad\mbox{and
if }\gamma\geq
2\,,\quad\big{(}\phi_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{6}(\Omega)\big{)}\,.$ (40)

##  4 The singular perturbation operator and the wave system

In this section, we study in detail the singular part of the primitive
equations (LABEL:main). To begin with, in Subsection 4.1 we establish first
convergence properties for the sequence of solutions
$\big{(}\varrho_{\varepsilon},{\bf u}_{\varepsilon}\big{)}_{\varepsilon}$
towards some targe profile $\big{(}\overline{\varrho},\overline{\bf
u}\big{)}$. Then, in Subsection 4.2 we derive some constraints that the limit
point $\big{(}\overline{\varrho},\overline{\bf u}\big{)}$ has to satisfy.
Finally, in Subsection 4.3 we introduce the system of waves, which encodes the
propagation of fast time oscillations.

###  4.1 Preliminary convergence properties

From the uniform bounds exhibited in Subsections 3.1, 3.2 and 3.3, we can
derive first convergence properties for the family of solutions
$\big{(}\varrho_{\varepsilon},{\bf u}_{\varepsilon}\big{)}_{\varepsilon}$ of
our primitive system (LABEL:main). Of course, the convergence we are going to
establish is only in weak topologies, therefore it will not be enough for
deriving the limit system (LABEL:eq:limit).

Here below, it is convenient to work on time intervals $[0,T]$, for arbitrary
large but fixed $T>0$. Also, in the notation we imply that all the
convergences are taken in the limit $\varepsilon\rightarrow 0^{+}$.

##### The density functions.

We start by considering the sequence of the density functions
$\varrho_{\varepsilon}$. First of all, from (20) and (21), we see that we can
decompose

$\varrho_{\varepsilon}\,=\,b\,+\,\varrho_{\varepsilon}^{(1)}\,+\,\varrho_{\varepsilon}^{(2)}\,,\qquad\mbox{
with }\qquad\varrho_{\varepsilon}^{(1)}\,\longrightarrow\,0\ \mbox{ in }\
L^{\infty}_{T}(L^{2})\,,\quad\varrho_{\varepsilon}^{(2)}\,\longrightarrow\,0\
\mbox{ in }\ L^{\infty}_{T}(L^{\gamma})\,.$ (41)

Since $\Omega$ is of finite measure, we can interpolate those convergence
properties with (31) to deduce that

$\varrho_{\varepsilon}\,-\,b\,=\,r_{\varepsilon}\,,\qquad\qquad\mbox{ with
}\qquad\qquad r_{\varepsilon}\,\longrightarrow\,0\qquad\mbox{ in }\quad
L^{\infty}_{T}(L^{p})\qquad\forall\;p<3\,.$

The uniform bounds of Section 3 allow us to find more quantitative convergence
properties. Indeed, from (23) and (35) it follows that there exists a function
$\phi\in L^{\infty}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)}\cap L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{6}(\Omega)\big{)}$ such that, up to the
extraction of a suitable subsequence, one has

$\big{[}\phi_{\varepsilon}\big{]}_{\rm
ess}\,\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\,\phi\qquad\qquad\mbox{ in
}\qquad L^{\infty}_{T}(L^{2})\,\cap\,L^{2}_{T}(L^{6})\,,$ (42)

where the symbol $\stackrel{{\scriptstyle*}}{{\rightharpoonup}}$ stands for
the weak-$*$ convergence in the respective functional space. On the other
hand, owing to (24), we know that

$\big{[}\phi_{\varepsilon}\big{]}_{\rm
res}\,\longrightarrow\,0\qquad\qquad\mbox{ in }\qquad
L^{\infty}_{T}(L^{p})\,,\qquad\forall\,p\in[1,\min\\{\gamma,2\\}[\,.$ (43)

Of course, if $\gamma\geq 2$, we have the previous strong convergence only for
all $1\leq p<2$; however, interpolating with the bounds of (40), in that case
we get

$\big{[}\phi_{\varepsilon}\big{]}_{\rm
res}\,\longrightarrow\,0\qquad\qquad\mbox{ in }\qquad
L^{2}_{T}(L^{p})\,,\qquad\forall\,p\in[1,6[\,.$

##### The velocity fields.

As it is apparent from equations (LABEL:main), any information on the velocity
fields ${\bf u}_{\varepsilon}$ and their gradients is lost in regions close to
vacuum. This is one of the main difficulties arising in the analysis of system
(LABEL:main). On the other hand, at least at a first sight, it is not so clear
which is the right quantity to look at; for instance, keep in mind
inequalities (18), (29). Here, we decide to work with the momentum

${\bf V}_{\varepsilon}\,:=\,\varrho_{\varepsilon}\,{\bf u}_{\varepsilon}\,.$

However, the first step is to get some uniform bounds on the velocity fields
${\bf u}_{\varepsilon}$. This is the goal of the next proposition.

###### Proposition 4.1

Let $\big{(}\varrho_{\varepsilon},{\bf u}_{\varepsilon}\big{)}_{\varepsilon}$
be a family of global in time finite energy weak solutions to system
(LABEL:main), in the sense of Definition 2.1, related to initial data
$\big{(}\varrho_{0,\varepsilon},{\bf u}_{0,\varepsilon}\big{)}_{\varepsilon}$
such that (15) holds.

Then we have the uniform estimates

$\big{(}{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{\infty}\big{(}\mathbb{R}_{+};L^{p_{1}}(\Omega)\big{)}\qquad\mbox{
and }\qquad\big{(}\nabla{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{p_{1}}(\Omega)\big{)}\,,\qquad\qquad
p_{1}\,:=\,\frac{2\kappa}{\kappa+1}\,.$

In particular, we also have that

$\big{(}{\bf u}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{p_{2}}(\Omega)\big{)}\,,\qquad\qquad\qquad\mbox{
where }\qquad p_{2}\,:=\,\frac{6\kappa}{\kappa+3}\,.$

###### Remark 4.2

Note that, due to our assumption $\kappa>3$, one has $p_{1}>3/2$ and $p_{2}>3$
in the above proposition.

Proof. We start by writing ${\bf
u}_{\varepsilon}\,=\,\varrho_{\varepsilon}^{-1/2}\,\sqrt{\varrho_{\varepsilon}}\,{\bf
u}_{\varepsilon}$, from which, by use of (18) and (21), we immediately deduce
that $\big{(}{\bf u}_{\varepsilon}\big{)}_{\varepsilon}$ is bounded in
$L^{\infty}_{T}(L^{p_{1}})$.

For the gradient terms, the argument is analogous, since we can write
$\nabla{\bf
u}_{\varepsilon}\,=\,\varrho_{\varepsilon}^{-1/2}\,\sqrt{\varrho_{\varepsilon}}\,\nabla{\bf
u}_{\varepsilon}$ and use, this time, (29) and (21). This implies the claimed
bound on the sequence $\big{(}\nabla{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}$.

Finally, the last uniform bound for the family of ${\bf u}_{\varepsilon}$’s
follows from the previous property and Sobolev embeddings. $\Box$

We are now ready to derive some important estimates for the velocity fields
${\bf V}_{\varepsilon}$.

###### Proposition 4.3

Let $\big{(}\varrho_{\varepsilon},{\bf u}_{\varepsilon}\big{)}_{\varepsilon}$
be a family of global in time finite energy weak solutions to system
(LABEL:main), in the sense of Definition 2.1, related to initial data
$\big{(}\varrho_{0,\varepsilon},{\bf m}_{0,\varepsilon}\big{)}_{\varepsilon}$
such that (15) holds. For all $\varepsilon\in\,]0,1]$, define ${\bf
V}_{\varepsilon}\,:=\,\varrho_{\varepsilon}\,{\bf u}_{\varepsilon}$.

Then, the following facts hold:

  1. (i)

the sequence $\big{(}{\bf V}_{\varepsilon}\big{)}_{\varepsilon}$ is uniformly
bounded in the space $L^{\infty}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{3/2}(\Omega)\big{)}\,\cap\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};W^{1,p_{3}}(\Omega)\big{)}$, where
$p_{3}:=6\kappa/(5\kappa+3)$;

  2. (ii)

there exist sequences $\big{(}\mathcal{V}_{\varepsilon}\big{)}_{\varepsilon}$
and $\big{(}{\bf W}_{\varepsilon}\big{)}_{\varepsilon}$ of vector fields such
that

$\forall\,\varepsilon\in\,]0,1]\,,\qquad{\bf
V}_{\varepsilon}\,=\,\mathcal{V}_{\varepsilon}\,+\,\varepsilon\,{\bf
W}_{\varepsilon,M}\,,$

with the uniform embedding properties

$\big{(}\mathcal{V}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{\infty}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)}\qquad\mbox{ and
}\qquad\big{(}{\bf W}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{3/2}(\Omega)\big{)}\,;$

  3. (iii)

after setting $\widetilde{\mathcal{V}}_{\varepsilon}\,:=\,b\,{\bf
u}_{\varepsilon}$, we can also write

${\bf
V}_{\varepsilon}\,=\,\widetilde{\mathcal{V}}_{\varepsilon}\,+\,\varepsilon\,\widetilde{{\bf
W}}_{\varepsilon}\,,$

with the uniform embedding properties

$\big{(}\widetilde{\mathcal{V}}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};W^{1,p_{1}}(\Omega)\big{)}\qquad\mbox{ and
}\qquad\big{(}\widetilde{{\bf
W}}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{1}(\Omega)\big{)}\,.$

Proof. Let $T>0$ be arbitrary, but fixed throughout the following
computations.

The proof of the $L^{\infty}_{T}(L^{3/2})$ bound of item (i) is easy to get:
it is enough to write ${\bf
V}_{\varepsilon}\,=\,\sqrt{\varrho_{\varepsilon}}\;\sqrt{\varrho_{\varepsilon}}\,{\bf
u}_{\varepsilon}$ and use the uniform bounds given in (18) and (31). Next, let
us focus on the bounds for the gradient. For any $1\leq j\leq 3$, we compute

$\partial_{j}{\bf
V}_{\varepsilon}\,=\,\sqrt{\varrho_{\varepsilon}}\,\sqrt{\varrho_{\varepsilon}}\,\partial_{j}{\bf
u}_{\varepsilon}\,+\,{\bf
u}_{\varepsilon}\,\partial_{j}\varrho_{\varepsilon}\,=\,\sqrt{\varrho_{\varepsilon}}\,\sqrt{\varrho_{\varepsilon}}\,\partial_{j}{\bf
u}_{\varepsilon}\,+\,2\,\sqrt{\varrho_{\varepsilon}}\,{\bf
u}_{\varepsilon}\,\partial_{j}\sqrt{\varrho_{\varepsilon}}\,:=\,A_{\varepsilon}\,+\,B_{\varepsilon}\,.$

Repeating the previous argument, using this time (29) and (31), it is easy to
see that
$\big{(}A_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{3/2})$. As
for $B_{\varepsilon}$, we start by observing that, owing to Proposition 4.1,
we have $\big{(}{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{p})$ for $p\leq
p_{2}:=\frac{6\kappa}{\kappa+3}$. On the other hand, we have
$\big{(}\nabla\sqrt{\varrho_{\varepsilon}}\big{)}_{\varepsilon}\,\subset\,L^{\infty}_{T}(L^{2})$
and
$\big{(}\sqrt{\varrho_{\varepsilon}}\big{)}_{\varepsilon}\,\subset\,L^{\infty}_{T}(L^{6})$.
Hence, we get
$\big{(}B_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{p_{3}})$
where $p_{3}=\frac{6\kappa}{5\kappa+3}$. Notice that $p_{3}>1$ if and only if
$\kappa>3$: this is precisely the place where the strongest assumption on
$\kappa$ appears. Item (i) is thus proven.

For showing the decomposition in item (ii), we notice that

${\bf V}_{\varepsilon}\,=\,\sqrt{b}\,\sqrt{\varrho_{\varepsilon}}\,{\bf
u}_{\varepsilon}\,+\,\varepsilon\,\frac{\sqrt{\varrho_{\varepsilon}}\,-\,\sqrt{b}}{\varepsilon}\,\sqrt{\varrho_{\varepsilon}}\,{\bf
u}_{\varepsilon}\,.$

Thus, if we define
$\mathcal{V}_{\varepsilon}\,:=\,\sqrt{b}\,\sqrt{\varrho_{\varepsilon}}\,{\bf
u}_{\varepsilon}$, from (18) we immediately infer that
$\big{(}\mathcal{V}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{\infty}_{T}(L^{2})$.
Next, we define

${\bf
W}_{\varepsilon}\,:=\,\frac{\sqrt{\varrho_{\varepsilon}}\,-\,\sqrt{b}}{\varepsilon}\,\sqrt{\varrho_{\varepsilon}}\,{\bf
u}_{\varepsilon}\,.$

At this point, we use that
$\left|\sqrt{\varrho_{\varepsilon}}-\sqrt{b}\right|\,\leq\,b_{*}^{-1/2}\,\left|\varrho_{\varepsilon}-b\right|$.
Hence, on the one hand, by using (35), we see that

$\left(\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},B}\cup\Omega^{\varepsilon}_{\rm
ess}}\,\frac{\sqrt{\varrho_{\varepsilon}}\,-\,\sqrt{b}}{\varepsilon}\right)_{\varepsilon}\,\subset\,L^{2}_{T}(L^{6})\,;$

on the other hand, since

$0\,\leq\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\left(\sqrt{\varrho_{\varepsilon}}-\sqrt{b}\right)\,\leq\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\sqrt{\varrho_{\varepsilon}}\,\lesssim\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\varrho_{\varepsilon}^{\gamma/2}\,,$

in view of (36) we gather that also
$\left(\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\frac{\sqrt{\varrho_{\varepsilon}}\,-\,\sqrt{b}}{\varepsilon}\right)_{\varepsilon}$
is uniformly bounded in $L^{2}_{T}(L^{6})$. Therefore, we get that

$\left(\frac{\sqrt{\varrho_{\varepsilon}}\,-\,\sqrt{b}}{\varepsilon}\right)_{\varepsilon}\,\subset\,L^{2}_{T}(L^{6})\qquad\quad\Longrightarrow\qquad\quad\big{(}{\bf
W}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{3/2})\,.$

The proof of item (iii) is similar. This time, we write
$\varrho_{\varepsilon}-b\,=\,\big{(}\sqrt{\varrho}_{\varepsilon}-\sqrt{b}\big{)}\big{(}\sqrt{\varrho}_{\varepsilon}+\sqrt{b}\big{)}$
and get

${\bf V}_{\varepsilon}\>=\,b\,{\bf
u}_{\varepsilon}\,+\,\big{(}\sqrt{\varrho}_{\varepsilon}-\sqrt{b}\big{)}\,\sqrt{\varrho_{\varepsilon}}\,{\bf
u}_{\varepsilon}\,+\,\big{(}\sqrt{\varrho}_{\varepsilon}-\sqrt{b}\big{)}\,\sqrt{b}\,{\bf
u}_{\varepsilon}\,.$

If we set $\widetilde{\mathcal{V}}_{\varepsilon}\,:=\,b\,{\bf
u}_{\varepsilon}$, Proposition 4.1 ensures us that the claimed bounds for this
quantity are satisfied. Next, we claim that the sequence of

$\widetilde{{\bf
W}}_{\varepsilon}\,:=\,\frac{\sqrt{\varrho}_{\varepsilon}-\sqrt{b}}{\varepsilon}\,\sqrt{\varrho_{\varepsilon}}\,{\bf
u}_{\varepsilon}\,+\,\frac{\sqrt{\varrho}_{\varepsilon}-\sqrt{b}}{\varepsilon}\,\sqrt{b}\,{\bf
u}_{\varepsilon}$

is uniformly bounded in $L^{2}_{T}(L^{1})$, for all $T>0$ fixed. For this,
owing to (18), the last part of Proposition 4.1 and Remark 4.2, it is enough
to show that

$\sqrt{\varrho_{\varepsilon}}\,-\,\sqrt{b}\quad\mbox{ is of order
}\;\;O(\varepsilon)\qquad\qquad\mbox{ in }\qquad L^{\infty}_{T}(L^{2})\,.$
(44)

As a matter of fact, since
$\left|\sqrt{\varrho_{\varepsilon}}-\sqrt{b}\right|\,\leq\,b_{*}^{-1/2}\,\left|\varrho_{\varepsilon}-b\right|$,
from (20) and (21) we immediately get that
$\left(\left(\mathds{1}_{\Omega^{\varepsilon}_{\rm
ess}}\,+\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},B}}\right)\frac{\sqrt{\varrho_{\varepsilon}}-\sqrt{b}}{\varepsilon}\right)_{\varepsilon}$
is uniformly bounded in $L^{\infty}_{T}(L^{2})$. Finally, as done above, we
have

$0\,\leq\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\left(\sqrt{\varrho_{\varepsilon}}-\sqrt{b}\right)\,\leq\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\sqrt{\varrho_{\varepsilon}}\,,$

which, in view of (21) again, implies that

$\displaystyle\left\|\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\left(\sqrt{\varrho_{\varepsilon}}-\sqrt{b}\right)\right\|_{L^{2}}\,$
$\displaystyle\lesssim\,\left\|\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\sqrt{\varrho_{\varepsilon}}\right\|_{L^{2\gamma}}\,\Big{(}\mathcal{L}(\Omega^{\varepsilon}_{\rm
res})\Big{)}^{1/2-1/2\gamma}\,\lesssim\,\varepsilon\,.$

In the end, we have shown (44), which in turn implies the sought bound for the
vector fields $\widetilde{{\bf W}}_{\varepsilon}$. $\Box$

From the previous proposition, we immediately deduce the following corollary.
The proof is rather straightforward, hence omitted.

###### Corollary 4.4

Under the assumptions of Proposition 4.3, there exists a vector field ${\bf
V}$, belonging to $L^{\infty}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{3/2}(\Omega)\big{)}\,\cap\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};W^{1,p_{3}}(\Omega)\big{)}$ such that, up to the
extraction of a suitable subsequence, one has ${\bf
V}_{\varepsilon}\,\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\,{\bf V}$ in
that space.

In addition, ${\bf V}$ also belongs to $L^{\infty}_{\rm
loc}\big{(}\mathbb{R}_{+};L^{2}(\Omega)\big{)}\,\cap\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};W^{1,p_{1}}(\Omega)\big{)}$, and, up to further
extractions, for all $T>0$ one has that
$\mathcal{V}_{\varepsilon}\,\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\,{\bf
V}$ in $L^{\infty}_{T}(L^{2})$ and
$\widetilde{\mathcal{V}}_{\varepsilon}\,\rightharpoonup\,{\bf V}$ in
$L^{2}_{T}(W^{1,p_{1}})$.

To conclude this part, we define the target velocity field ${\bf U}$ as

${\bf U}\,:=\,\frac{1}{b}\,{\bf V}\,,$ (45)

where ${\bf V}$ is the vector field identified in Corollary 4.4.

We notice that ${\bf U}$ is (up to a further extraction) the weak-limit point
of the sequence $\big{(}{\bf u}_{\varepsilon}\big{)}_{\varepsilon}$ in the
functional spaces identified in Proposition 4.1. We point out also that ${\bf
U}$ belongs to the same functional spaces to which ${\bf V}$ belongs.

###  4.2 Constraints on the limit

In the previous subsection, we have identified the limit points $b$, $\phi$
and ${\bf V}$ of the families
$\big{(}\varrho_{\varepsilon}\big{)}_{\varepsilon}$,
$\big{(}\phi_{\varepsilon}\big{)}_{\varepsilon}$ and $\big{(}{\bf
V}_{\varepsilon}\big{)}_{\varepsilon}$, respectively. Our next goal is to find
some properties those limit points have to satisfy. We point out that these
conditions do not fully characterise the limit dynamics, which will be deduced
by passing to the limit in the momentum equation.

###### Proposition 4.5

Let $\big{(}\varrho_{\varepsilon},{\bf u}_{\varepsilon}\big{)}_{\varepsilon}$
be a family of global in time finite energy weak solutions to system
(LABEL:main), in the sense of Definition 2.1, related to initial data
$\big{(}\varrho_{0,\varepsilon},{\bf u}_{0,\varepsilon}\big{)}_{\varepsilon}$
such that (15) holds. For all $\varepsilon\in\,]0,1]$, define ${\bf
V}_{\varepsilon}\,:=\,\varrho_{\varepsilon}\,{\bf u}_{\varepsilon}$. Let
$\phi$ be the scalar function identified in (42), ${\bf V}$ the vector field
identified in Corollary 4.4 and ${\bf U}$ the vector field defined in (45).

Then ${\bf U}$ has to satisfy the _anelastic constraint_

${\rm div}\,\big{(}b\,{\bf U}\big{)}\,=\,0\qquad\qquad\mbox{ in
}\qquad\mathcal{D}^{\prime}\big{(}\mathbb{R}_{+}\times\Omega\big{)}\,.$

On the other hand, $\phi\,=\,\phi(b)$ is determined as a function of $b$ only.

Before proving the previous proposition, let us state a simple lemma. It will
be useful to understand how to deal with the singularity of the pressure and
gravitational terms, and to put in evidence the right singularity in the
momentum equation.

###### Lemma 4.6

Let the assumptions of Proposition 4.5 be in force. Then, after defining

$\Pi(\varrho_{\varepsilon};b)\,:=\,p(\varrho_{\varepsilon})\,-\,p(b)\,-\,p^{\prime}(b)\,\big{(}\varrho_{\varepsilon}\,-\,b\big{)}\,,$

we have the equality

$\frac{1}{\varepsilon^{2}}\,\Big{(}\nabla
p(\varrho_{\varepsilon})\,-\,\varrho_{\varepsilon}\nabla
G\Big{)}\,=\,\frac{1}{\varepsilon}\,b\,\nabla\big{(}H^{\prime\prime}(b)\,\phi_{\varepsilon}\big{)}\,+\,\frac{1}{\varepsilon^{2}}\nabla\Pi(\varrho_{\varepsilon};b)\,.$

In addition, one has that
$\big{(}\frac{1}{\varepsilon^{2}}\,\Pi(\varrho_{\varepsilon};b)\big{)}_{\varepsilon}$
is uniformly bounded in
$L^{\infty}\big{(}\mathbb{R}_{+};L^{1}(\Omega)\big{)}$.

Proof. The claimed identity follows from a simple algebraic computation:

$\displaystyle\frac{1}{\varepsilon^{2}}\,\Big{(}\nabla
p(\varrho_{\varepsilon})\,-\,\varrho_{\varepsilon}\nabla G\Big{)}\,$
$\displaystyle=\,\frac{1}{\varepsilon^{2}}\,\Big{(}\nabla\big{(}p(\varrho_{\varepsilon})\,-\,p(b)\big{)}\,-\,\big{(}\varrho_{\varepsilon}\,-\,b\big{)}\,\nabla
G\Big{)}$
$\displaystyle=\,\frac{1}{\varepsilon^{2}}\nabla\Pi(\varrho_{\varepsilon};b)\,+\,\frac{1}{\varepsilon^{2}}\,\bigg{(}\nabla\Big{(}p^{\prime}(b)\,\big{(}\varrho_{\varepsilon}\,-\,b\big{)}\Big{)}\,-\,\big{(}\varrho_{\varepsilon}\,-\,b\big{)}\,\nabla
G\bigg{)}$
$\displaystyle=\,\frac{1}{\varepsilon}\,b\,\nabla\big{(}H^{\prime\prime}(b)\,\phi_{\varepsilon}\big{)}\,+\,\frac{1}{\varepsilon^{2}}\nabla\Pi(\varrho_{\varepsilon};b)\,,$

where, in the last step, we have used the equality $\nabla
G\,=\,H^{\prime\prime}(b)\,\nabla b$, which holds owing to (4).

It remains us to show the uniform bounds for the family
$\big{(}\frac{1}{\varepsilon^{2}}\,\Pi(\varrho_{\varepsilon};b)\big{)}_{\varepsilon}$.
The argument is similar to the proof of Lemma 4.1 in [11]; however, since that
paper did not deal with non-constant limit density profiles $b$ nor with the
presence of a cold component $p_{c}$ of the pressure, we report here most of
the details.

First of all, using Taylor formula at the second order for the pressure
function $p$ together with (23), we easily see that

$\frac{1}{\varepsilon^{2}}\left[\Pi(\varrho_{\varepsilon};b)\right]_{\rm
ess}\,\approx\,\phi_{\varepsilon}^{2}\qquad\qquad\Longrightarrow\qquad\qquad\left(\frac{1}{\varepsilon^{2}}\left[\Pi(\varrho_{\varepsilon};b)\right]_{\rm
ess}\right)_{\varepsilon}\,\subset\,L^{\infty}(\mathbb{R}_{+};L^{1})\,.$

Next, let us denote by $\Pi_{E}(\varrho_{\varepsilon};b)$ and
$\Pi_{c}(\varrho_{\varepsilon};b)$ the functions defined as
$\Pi(\varrho_{\varepsilon};b)$, but using respectively $p_{E}$ and $p_{c}$
instead of the full pressure function $p$. Then, a Taylor expansion again and
(22) imply that

$\frac{1}{\varepsilon^{2}}\left(\big{[}\Pi_{E}(\varrho_{\varepsilon};b)\big{]}_{{\rm
res},B}\,+\,\big{[}\Pi_{c}(\varrho_{\varepsilon};b)\big{]}_{{\rm
res},UB}\right)\,\subset\,L^{\infty}(\mathbb{R}_{+};L^{1})\,.$

Finally, we notice that

$\left|\big{[}\Pi_{E}(\varrho_{\varepsilon};b)\big{]}_{{\rm
res},UB}\right|\,\lesssim\,\big{[}\varrho_{\varepsilon}\big{]}^{\gamma}_{{\rm
res}}\qquad\qquad\mbox{ and
}\qquad\qquad\left|\big{[}\Pi_{c}(\varrho_{\varepsilon};b)\big{]}_{{\rm
res},B}\right|\,\lesssim\,\big{[}\varrho_{\varepsilon}\big{]}^{-\kappa}_{{\rm
res}}\,,$

for which one can deduce the sought bounds by using the controls in (21).

In the end, the lemma is completely proved. $\Box$

We can now turn our attention to the proof of Proposition 4.5.

Proof of Proposition 4.5. We start by considering the weak form of the
momentum equation: given a test function
$\varphi\in\mathcal{D}\big{(}\mathbb{R}_{+}\times\Omega\big{)}$, with ${\rm
Supp}\,\varphi\subset[0,T[\,\times\Omega$ for some positive time $T>0$, we
have

$-\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\varrho_{\varepsilon}\,\partial_{t}\varphi\
{\rm d}{x}\,{\rm
d}t\,-\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\varrho_{\varepsilon}\,{\bf
u}_{\varepsilon}\cdot\nabla\varphi\ {\rm d}{x}\,{\rm
d}t\,=\,\int_{\Omega}\varrho_{0,\varepsilon}\,\varphi(0)\ {\rm d}{x}\,.$

By assumption on the initial data, we know that
$\varrho_{0,\varepsilon}\,-\,b\,\longrightarrow\,0$ in $L^{2}\cap L^{\infty}$,
so it is easy to pass to the limit in the right-hand side of the previous
relation. Moreover, in view of (41) and Corollary 4.4, we know that
$\varrho_{\varepsilon}\,\longrightarrow\,b$ and $\varrho_{\varepsilon}\,{\bf
u}_{\varepsilon}\,=\,{\bf V}_{\varepsilon}\,\longrightarrow\,{\bf V}$ in
$\mathcal{D}^{\prime}([0,T[\,\times\Omega)$. Thus, passing to the limit in the
previous equality yields the constraint

$\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}{\bf V}\cdot\nabla\varphi\ {\rm
d}{x}\,{\rm
d}t\,=\,0\qquad\forall\,\varphi\in\mathcal{D}\qquad\qquad\Longrightarrow\qquad\qquad{\rm
div}\,{\bf V}\,=\,0\qquad\mbox{ in }\quad\mathcal{D}^{\prime}\,.$

Now, using the definition of ${\bf U}$ given in (45) immediately gives the
anelastic constraint.

Let us turn our attention to the momentum equation. As appears in the
statement, the momentum equation does not give any relevant information on the
limit points: we will discover that $\phi$ is a quantity which play no role in
the limit dynamics.

Indeed, we can test the momentum equation by $\varepsilon\,\psi$, where
$\psi\in\mathcal{D}(\mathbb{R}_{+}\times\Omega;\mathbb{R}^{3})$ is a test
function with support (say) included in $[0,T[\,\times\Omega$, for some $T>0$.
By the uniform bounds established in Section 3, and using the identity in
Lemma 4.6 above, it is possible to see that all the terms in the momentum
equation tend to $0$, except for the term
$b\,\nabla\big{(}H^{\prime\prime}(b)\,\phi_{\varepsilon}\big{)}$. Thus,
passing to the limit for $\varepsilon\rightarrow 0^{+}$, we find that

$\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}H^{\prime\prime}(b)\,\phi\,{\rm
div}\,\big{(}b\,\psi\big{)}\ {\rm d}{x}\,{\rm
d}t\,=\,0\qquad\qquad\forall\,\psi\in\mathcal{D}\,.$

Since $b$ never vanishes, this implies that
$\nabla\big{(}H^{\prime\prime}(b)\,\phi\big{)}\,=\,0$, and then
$H^{\prime\prime}\big{(}b(x)\big{)}\,\phi(t,x)\,=\,c(t)$ a.e. on
$\mathbb{R}_{+}\times\Omega$, for a suitable function $c(t)$ only depending on
the time variable. We claim that $c(t)\,\equiv\,c(0)$ is in fact constant, and
does not depend on time neither. Indeed, let us denote by $\langle f\rangle$
the space average over $\Omega$ of a function $f=f(t,x)$, namely

$\langle f\rangle\,=\,\langle
f\rangle(t)\,:=\,\frac{1}{\mathcal{L}(\Omega)}\,\int_{\Omega}f(t,x)\ {\rm
d}{x}\,.$

Using the definition of $\phi_{\varepsilon}$, we can recast the continuity
equation as

$\partial_{t}\phi_{\varepsilon}\,+\,\frac{1}{\varepsilon}\,{\rm div}\,{\bf
V}_{\varepsilon}\,=\,0\,.$ (46)

Taking the mean value with respect to the space variable, we discover that
$\partial_{t}\langle\phi\rangle\,=\,0$, thus the mean value of $\phi$ is
preserved in time: $\langle\phi\rangle(t)\,=\,\langle\phi_{0}\rangle$ for
almost all times $t\geq 0$, where $\phi_{0}$ is the weak limit point of the
initial data $\big{(}\phi_{0,\varepsilon}\big{)}_{\varepsilon}$ specified in
(8). On the other hand, coming back to the equality
$H^{\prime\prime}(b)\,\phi\,=\,c(t)$ and computing the space average, we have
that $\langle\phi\rangle(t)\,=\,\langle 1/H^{\prime\prime}(b)\rangle\,c(t)$,
which in turn implies that $c(t)\equiv c_{0}$ has to be a constant function.

The proposition is now proved. $\Box$

As already pointed out, the previous proposition does not allow us to identify
the limit dynamics yet. The main problem consists in passing to the limit in
the momentum equation, which reveals a not so easy task, owing to the non-
linear terms appearing in it. In order to succeed, we first need to understand
the propagation of fast oscillations in time: this is the goal of the next
subsection.

###  4.3 Acoustic equation

Lemma 4.6 and Proposition 4.5 allow us to identify the singular part of the
equations of motion. Because of the ill-preparation of the initial data, this
singular part is responsible for fast oscillations in time of the solution,
the so called _acoustic waves_ , which may eventually prevent the convergence
of the non-linear terms. In order to study those oscillations and be able to
pass to the limit in the equations, we reformulate our system (LABEL:main) as
a wave system.

Let us recall that we have denoted

$\phi_{\varepsilon}\,=\,\frac{\varrho_{\varepsilon}\,-\,b}{\varepsilon}\qquad\qquad\mbox{
and }\qquad\qquad{\bf V}_{\varepsilon}\,=\,\varrho_{\varepsilon}\,{\bf
u}_{\varepsilon}\,.$

As already remarked in the proof of Proposition 4.5, in terms of those
quantities the continuity equation can be rewritten as in (46). Concerning the
momentum equation, instead, we can take advantage of the identity of Lemma 4.6
to combine the pressure and gravitation terms together.

In the end, we discover that system (LABEL:main) can be recasted in the
following form:

$\left\\{\begin{array}[]{l}\varepsilon\,\partial_{t}\phi_{\varepsilon}\,+\,\operatorname{div}{\bf
V}_{\varepsilon}\,=\,0\\\\[4.30554pt] \varepsilon\,\partial_{t}{\bf
V}_{\varepsilon}\,+\,b\,\nabla\big{(}H^{\prime\prime}(b)\,\phi_{\varepsilon}\big{)}\,=\,\varepsilon\,{\bf
f}_{\varepsilon}\,,\end{array}\right.$ (47)

where we have defined

${\bf
f}_{\varepsilon}\,=\,\nu\,\operatorname{div}(\varrho_{\varepsilon}\,{\mathbb{D}}{\bf
u}_{\varepsilon})\,-\,\operatorname{div}(\varrho_{\varepsilon}{\bf
u}_{\varepsilon}\otimes{\bf
u}_{\varepsilon})\,-\,\frac{1}{\varepsilon^{2}}\nabla\Pi(\varrho_{\varepsilon};b)\,.$
(48)

By the uniform bounds (18), we get that the sequence
$\big{(}\varrho_{\varepsilon}{\bf u}_{\varepsilon}\otimes{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}$ is uniformly bounded in
$L^{\infty}_{T}(L^{1})$, for all times $T>0$. On the other hand, by arguing as
in the proof of Proposition 4.3, it is easy to see that
$\big{(}\varrho_{\varepsilon}\,{\mathbb{D}}{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}$ is uniformly bounded in
$L^{2}_{T}(L^{3/2})$. Finally, the term $\Pi(\varrho_{\varepsilon};b)$ has
been dealt with in Lemma 4.6. Therefore, owing to the Sobolev embedding
$H^{s}(\Omega)\subset L^{\infty}(\Omega)$ for any $s>3/2$, we get that

$\forall\,T>0\,,\quad\forall\,s>5/2\,,\qquad\qquad\big{(}{\bf
f}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(H^{-s})\,.$ (49)

For later use, it is convenient to introduce a regularised version of the wave
system (47). For this, we employ the low frequency cut-off operator $S_{M}$,
with $M\in\mathbb{N}$, introduced in relation (67) in the Appendix.

Since $S_{M}$ commutes with all partial derivatives, applying operator $S_{M}$
to (47) yields

$\left\\{\begin{array}[]{l}\varepsilon\,\partial_{t}\phi_{\varepsilon,M}\,+\,\operatorname{div}{\bf
V}_{\varepsilon,M}\,=\,0\\\\[4.30554pt] \varepsilon\,\partial_{t}{\bf
V}_{\varepsilon,M}\,+\,b\,\nabla\big{(}H^{\prime\prime}(b)\,\phi_{\varepsilon,M}\big{)}\,=\,\varepsilon\,{\bf
f}_{\varepsilon,M}\,+\,{\bf h}_{\varepsilon,M}\,,\end{array}\right.$ (50)

where we have denoted $\phi_{\varepsilon,M}\,=\,S_{M}(\phi_{\varepsilon})$,
${\bf V}_{\varepsilon,M}\,=\,S_{M}(V_{\varepsilon})$ and ${\bf
f}_{\varepsilon,M}\,=\,S_{M}({\bf f}_{\varepsilon})$, and we have set

${\bf
h}_{\varepsilon,M}\,:=\,\big{[}b,S_{M}\big{]}\nabla\big{(}H^{\prime\prime}(b)\phi_{\varepsilon}\big{)}\,+\,b\,\nabla\big{(}\big{[}H^{\prime\prime}(b),S_{M}\big{]}\phi_{\varepsilon}\big{)}\,.$

Thanks to the uniform bounds for the sequence
$\big{(}\phi_{\varepsilon}\big{)}_{\varepsilon}$ in $L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};W^{1,\gamma}(\Omega)\big{)}$ if $\gamma<2$, in
$L^{2}_{\rm loc}\big{(}\mathbb{R}_{+};H^{1}(\Omega)\big{)}$ when $\gamma\geq
2$ (keep in mind Proposition 3.2 above), standard commutator estimates (see
e.g. Lemma 2.97 of [2]) imply that

$\forall\;T\,>\,0\,,\qquad\qquad\left\|{\bf
h}_{\varepsilon,M}\right\|_{L^{2}_{T}(L^{\gamma})}\,+\,\left\|{\rm curl}\,{\bf
h}_{\varepsilon,M}\right\|_{L^{2}_{T}(L^{\gamma})}\,\lesssim\,2^{-M}\,,$ (51)

where we agree that the Lebesgue exponent $\gamma$ is changed into $2$
whenever $\gamma\geq 2$.

We explicitly point out that, in the above estimate (51), the multiplicative
constant is uniform with respect to _both_ $M\in\mathbb{N}$ and
$\varepsilon\in\,]0,1]$, but it may depend on the fixed time $T>0$. Notice
that, for the uniform bound on ${\rm curl}\,{\bf h}_{\varepsilon,M}$, the
gradient structure of the commutator terms play a key role. We remark also
that the commutator term ${\bf h}_{\varepsilon,M}$ is much better controlled
than the corresponding term in [24], thanks to the uniform bounds provided by
the BD entropy estimates.

In addition, as an immediate consequence of (49), we get that

$\forall\;T\,>\,0\,,\quad\forall\;s\,\geq\,0\,,\qquad\qquad\left\|{\bf
f}_{\varepsilon,M}\right\|_{L^{2}_{T}(H^{s})}\,\leq\,C(s,M)\,,$ (52)

where the constants $C(s,M)>0$ depend only on the quantities inside the
parentheses. Notice that these constants blow up when $M\rightarrow+\infty$,
but they have finite value for any $M\in\mathbb{N}$ fixed.

We conclude this part by showing uniform bounds on the two sequences
$\big{(}\phi_{\varepsilon,M}\big{)}_{\varepsilon,M}$ and
$\big{(}V_{\varepsilon,M}\big{)}_{\varepsilon,M}$. As a matter of fact, in
view of our computations in Section 5 below, it is important to introduce a
fine decomposition of those terms.

###### Lemma 4.7

Let $T>0$ be arbitrarily large, but fixed. For any $M\in\mathbb{N}$, define
the quantities $\phi_{\varepsilon,M}\,=\,S_{M}(\phi_{\varepsilon})$ and ${\bf
V}_{\varepsilon,M}\,=\,S_{M}({\bf V}_{\varepsilon})$ as above.

  1. (i)

Both sequences $\big{(}\phi_{\varepsilon,M}\big{)}_{\varepsilon}$ and
$\big{(}{\bf V}_{\varepsilon,M}\big{)}_{\varepsilon}$ are sequences of smooth
functions in space, which are uniformly bounded (with respect to
$\varepsilon$, but _not_ with respect to $M$) in the space $L^{2}_{T}(H^{s})$,
for any $s\geq 0$ arbitrarily large.

  2. (ii)

For any $M\in\mathbb{N}$ and any $\varepsilon\in\,]0,1]$, we can write

$\phi_{\varepsilon,M}\,=\,\varphi_{\varepsilon,M}\,+\,\varepsilon^{(2-\gamma)/\gamma}\,\pi_{\varepsilon,M}\,,$
(53)

where we have the uniform estimates

$\sup_{M\in\mathbb{N}}\,\sup_{\varepsilon\in\,]0,1]}\left\|\varphi_{\varepsilon,M}\right\|_{L^{\infty}_{T}(L^{2})\cap
L^{2}_{T}(L^{6})}\,\leq\,C\qquad\mbox{ and }\qquad\forall\,s\geq
0\,,\quad\sup_{\varepsilon\in\,]0,1]}\left\|\pi_{\varepsilon,M}\right\|_{L^{\infty}_{T}(H^{s})}\,\leq\,C(s,M)\,,$

for suitable positive constants $C$ and $C(s,M)$, which depend only on $T>0$
and on the quantities in the brackets. In the case $\gamma\geq 2$, one can
simply take $\pi_{\varepsilon,M}\equiv 0$.

  3. (iii)

For any $M\in\mathbb{N}$ and $\varepsilon\in\,]0,1]$, one has

${\bf V}_{\varepsilon,M}\,=\,\mathcal{V}_{\varepsilon.M}\,+\,\varepsilon\,{\bf
W}_{\varepsilon,M}\,,$ (54)

with the uniform bounds

$\sup_{M\in\mathbb{N}}\,\sup_{\varepsilon\in\,]0,1]}\left\|\mathcal{V}_{\varepsilon,M}\right\|_{L^{\infty}_{T}(L^{2})}\,\leq\,C\qquad\quad\mbox{
and }\qquad\quad\forall\,s\geq
0\,,\quad\sup_{\varepsilon\in\,]0,1]}\left\|{\bf
W}_{\varepsilon,M}\right\|_{L^{2}_{T}(H^{s})}\,\leq\,C(s,M)\,.$

  4. (iv)

We can also write

${\bf
V}_{\varepsilon,M}\,=\,\widetilde{\mathcal{V}}_{\varepsilon.M}\,+\,\varepsilon\,\widetilde{{\bf
W}}_{\varepsilon,M}\,,$ (55)

with the uniform estimates

$\sup_{M\in\mathbb{N}}\,\sup_{\varepsilon\in\,]0,1]}\left\|\widetilde{\mathcal{V}}_{\varepsilon,M}\right\|_{L^{2}_{T}(W^{1,p_{1}})}\,\leq\,C\qquad\quad\mbox{
and }\qquad\quad\forall\,s\geq
0\,,\quad\sup_{\varepsilon\in\,]0,1]}\left\|\widetilde{{\bf
W}}_{\varepsilon,M}\right\|_{L^{2}_{T}(H^{s})}\,\leq\,C(s,M)\,.$

Proof. The properties claimed in item (i) are an immediate consequence of the
bounds for the families $\big{(}\phi_{\varepsilon}\big{)}_{\varepsilon}$ and
$\big{(}{\bf V}_{\varepsilon}\big{)}_{\varepsilon}$, combined with the
smoothing effect of the low frequency cut-off operators $S_{M}$. Keep in mind
(23), (24) and Propositions 3.2 and 4.3.

The decompositions of items (iii) and (iv), together with the corresponding
uniform estimates, also follow from Proposition 4.3, simply setting
$\mathcal{V}_{\varepsilon,M}\,:=\,S_{M}(\mathcal{V}_{\varepsilon})$ and ${\bf
W}_{\varepsilon,M}\>=\,S_{M}({\bf W}_{\varepsilon})$, and similarly for
$\widetilde{\mathcal{V}}_{\varepsilon,M}$ and $\widetilde{{\bf
W}}_{\varepsilon,M}$.

Finally, let us prove item (ii). We start by considering the case
$1<\gamma<2$. In this case, we decompose

$\phi_{\varepsilon}\,=\,\big{[}\phi_{\varepsilon}\big{]}_{\rm
ess}\,+\,\big{[}\phi_{\varepsilon}\big{]}_{\rm res}\,.$

Thus, if we set
$\varphi_{\varepsilon,M}\,:=\,S_{M}\left(\big{[}\phi_{\varepsilon}\big{]}_{\rm
ess}\right)$, estimates (23) and (35) easily imply the uniform boundedness
(_both_ with respect to $M\in\mathbb{N}$ and $\varepsilon\in\,]0,1]$) of the
sequence $\big{(}\varphi_{\varepsilon,M}\big{)}_{\varepsilon,M}$. Next, we
define

$\pi_{\varepsilon}\,:=\,\frac{1}{\varepsilon^{(2-\gamma)/\gamma}}\,\big{[}\phi_{\varepsilon}\big{]}_{\rm
res}\qquad\qquad\mbox{ and
}\qquad\qquad\pi_{\varepsilon,M}\,:=\,S_{M}(\pi_{\varepsilon})\,,$

and we conclude with the help of (24). When $\gamma\geq 2$, instead, one has
that $\big{(}\phi_{\varepsilon}\big{)}_{\varepsilon}$ is uniformly bounded in
$L^{\infty}_{T}(L^{2})\cap L^{2}_{T}(H^{1})$, thanks to Proposition 3.2.
Hence, in this case one can simply define
$\varphi_{\varepsilon,M}\,=\,S_{M}(\phi_{\varepsilon})$ and
$\pi_{\varepsilon,M}\,=\,0$. This completes the proof of the lemma. $\Box$

##  5 Passage to the limit

In this section, we finish the proof of our main theorem, showing the
convergence (up to an extraction) of weak solutions to (LABEL:main) to weak
solutions to the target system (LABEL:eq:limit).

The main problem is to pass to the limit in the most non-linear term, namely
the convective term in the momentum equation. This convergence will be proved
in the following two subsections, by use of a _compensated compactness_
argument. Finally, in Subsection 5.3 we will take care of the other terms, and
complete the proof of the convergence.

Before going into the details, let us recall that convergence will be proved
for any test function lying in the kernel of the singular perturbation
operator, namely (in view of Proposition 4.5) for any test function

$\psi\,\in\,\mathcal{D}\big{(}\mathbb{R}_{+}\times\Omega;\mathbb{R}^{3}\big{)}\,,\qquad\qquad\mbox{
such that }\qquad{\rm div}\,\big{(}b\,\psi\big{)}\,=\,0\,.$ (56)

Also, it is useful to introduce the following notation: we denote by
$R_{\varepsilon,M}$ any remainder term, that is any term such that

$\lim_{M\rightarrow+\infty}\,\limsup_{\varepsilon\rightarrow
0^{+}}\left|\int_{0}^{T}\langle
R_{\varepsilon,M}\,,\,\psi\rangle_{\mathcal{D}^{\prime}\times\mathcal{D}}\,{\rm
d}t\right|\,=\,0\,,$ (57)

for some given time $T>0$ and test function
$\psi\in\mathcal{D}\big{(}[0,T[\,\times\Omega;\mathbb{R}^{3}\big{)}$ taken as
above. Similarly, we will use the notation $\mathcal{R}_{\varepsilon,M}$ for
any scalar term such that

$\lim_{M\rightarrow+\infty}\,\limsup_{\varepsilon\rightarrow
0^{+}}\;\mathcal{R}_{\varepsilon,M}\,=\,0\,.$ (58)

Typically, we will have

$\mathcal{R}_{\varepsilon,M}\,=\,\int_{0}^{T}\langle
R_{\varepsilon,M}\,,\,\psi\rangle_{\mathcal{D}^{\prime}\times\mathcal{D}}\,{\rm
d}t\,.$

In the next computations, the precise values of $R_{\varepsilon,M}$ and
$\mathcal{R}_{\varepsilon,M}$ may change from one line to another.

###  5.1 Approximation of the convective term

Passing to the limit in the convective term is based on a compensated
compactness argument, following work [24] of Masmoudi. This argument relies on
using _algebraically_ the structure of the wave system (47) and performing
direct computations on it. Of course, for that, we need first of all to smooth
out all the quantities with respect to the space variables: this is the scope
of the next lemma.

Notice that the approximation argument here is delicate, due to the degeneracy
of the system close to vacuum. The cold pressure term $p_{c}$ will be of great
help.

###### Lemma 5.1

For any $T>0$ fixed and any smooth test function $\psi$ as in (56), such that
${\rm Supp}\,\psi\subset[0,T[\,\times\Omega$, we have

$\lim_{M\rightarrow+\infty}\,\limsup_{\varepsilon\rightarrow
0^{+}}\left|\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\varrho_{\varepsilon}{\bf
u}_{\varepsilon}\otimes{\bf u}_{\varepsilon}:\nabla\psi\ {\rm d}{x}\,{\rm
d}t\,-\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\frac{1}{b}\,{\bf
V}_{\varepsilon,M}\otimes{\bf V}_{\varepsilon,M}:\nabla\psi\ {\rm d}{x}\,{\rm
d}t\,\right|\,=\,0\,.$

Proof. We start by observing that, in view of Propositions 4.1 and 4.3, we
have $\big{(}{\bf
V}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{3/2})$ and
$\big{(}{\bf u}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{p})$
for all $p\in[1,p_{2}]$, where $p_{2}\,=\,6\kappa/(\kappa+3)$. We also notice
that, under our assumption $\kappa>3$, we have $2/3\,+\,1/p_{2}<1$, i.e.
$p_{2}>3$. Thus, given any $M\in\mathbb{N}$, we can write

$\displaystyle I\,$
$\displaystyle:=\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\varrho_{\varepsilon}{\bf
u}_{\varepsilon}\otimes{\bf u}_{\varepsilon}:\nabla\psi\ {\rm d}{x}\,{\rm
d}t\,$ $\displaystyle=\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\frac{1}{b}\,{\bf
V}_{\varepsilon}\otimes S_{M}(b\,{\bf u}_{\varepsilon}):\nabla\psi\ {\rm
d}{x}\,{\rm d}t\,+\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\frac{1}{b}\,{\bf
V}_{\varepsilon}\otimes({\rm Id}\,-S_{M})(b\,{\bf
u}_{\varepsilon}):\nabla\psi\ {\rm d}{x}\,{\rm d}t\,,$

where all terms are well-defined, and the last term on the right can be
bounded as follows:

$\left|\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\frac{1}{b}\,{\bf
V}_{\varepsilon}\otimes({\rm Id}\,-S_{M})(b\,{\bf
u}_{\varepsilon}):\nabla\psi\ {\rm d}{x}\,{\rm
d}t\,\right|\,\lesssim\,\left\|{\bf
V}_{\varepsilon}\right\|_{L^{2}_{T}(L^{3/2})}\,\left\|({\rm
Id}\,-S_{M})(b\,{\bf u}_{\varepsilon})\right\|_{L^{2}_{T}(L^{3})}\,.$

At this point, taking advantage of Bernstein’s inequalities of Lemma A.1 in
the Appendix, we can bound

$\displaystyle\left\|({\rm Id}\,-S_{M})(b\,{\bf
u}_{\varepsilon})\right\|_{L^{3}}\,$ $\displaystyle\leq\,\sum_{j\geq
M}\left\|\Delta_{j}(b\,{\bf
u}_{\varepsilon})\right\|_{L^{3}}\,\lesssim\,\sum_{j\geq
M}2^{-j}\left\|\Delta_{j}\nabla(b\,{\bf u}_{\varepsilon})\right\|_{L^{3}}$
(59) $\displaystyle\lesssim\,\sum_{j\geq
M}2^{j\left[-1\,+\,3\left(\frac{1}{p_{1}}-\frac{1}{3}\right)\right]}\left\|\Delta_{j}(b\,{\bf
u}_{\varepsilon})\right\|_{L^{p_{1}}}\,\leq\,2^{-\alpha
M}\,\left\|\nabla(b\,{\bf
u}_{\varepsilon})\right\|_{B^{0}_{p_{1},\infty}}\,\sum_{j\geq 0}2^{-\alpha
j}\,,$

where $p_{1}=2\kappa/(\kappa+1)$ has been defined in Proposition 4.1 and we
have set $\alpha\,=\,(\kappa-3)/2\kappa\,>\,0$. Owing to the embedding
$L^{p}\,\hookrightarrow\,B^{0}_{p,\infty}$ for any $p\in[1,+\infty]$, we
finally get

$\left\|({\rm Id}\,-S_{M})(b\,{\bf
u}_{\varepsilon})\right\|_{L^{3}}\,\lesssim\,2^{-\alpha
M}\,\left\|\nabla(b\,{\bf
u}_{\varepsilon})\right\|_{L^{p_{1}}}\,\lesssim\,2^{-\alpha M}\,.$

As a result, the previous computations show that, in the sense of (58), one
has

$I\,=\,\int^{T}_{0}\int_{\Omega}\frac{1}{b}\,{\bf V}_{\varepsilon}\otimes
S_{M}(b\,{\bf u}_{\varepsilon}):\nabla\psi\ {\rm d}{x}\,{\rm
d}t\,+\,\mathcal{R}_{\varepsilon,M}\,.$

Next, we use again the uniform bound $\big{(}{\bf
u}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{p_{2}})$,
together with the fact that $\big{(}S_{M}(b\,{\bf
u}_{\varepsilon})\big{)}_{\varepsilon}$ is uniformly bounded (with respect to
$\varepsilon$, but _not_ to $M$) in the space $L^{2}_{T}(L^{\infty})$, to get

$\displaystyle\int^{T}_{0}\\!\\!\\!\int_{\Omega}\frac{1}{b}\,{\bf
V}_{\varepsilon}\otimes S_{M}(b\,{\bf u}_{\varepsilon}):\nabla\psi\ {\rm
d}{x}\,{\rm d}t$ (60)
$\displaystyle\qquad\qquad\qquad=\,\int^{T}_{0}\\!\\!\\!\int_{\Omega}{\bf
u}_{\varepsilon}\otimes S_{M}(b\,{\bf u}_{\varepsilon}):\nabla\psi\ {\rm
d}{x}\,{\rm
d}t\,+\,\varepsilon\int^{T}_{0}\\!\\!\\!\int_{\Omega}\frac{1}{b}\,\phi_{\varepsilon}\,{\bf
u}_{\varepsilon}\otimes S_{M}(b\,{\bf u}_{\varepsilon}):\nabla\psi\ {\rm
d}{x}\,{\rm d}t$
$\displaystyle\qquad\qquad\qquad=\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\frac{1}{b}\,S_{M}\big{(}b\,{\bf
u}_{\varepsilon}\big{)}\otimes S_{M}(b\,{\bf u}_{\varepsilon}):\nabla\psi\
{\rm d}{x}\,{\rm d}t\,+\,\mathcal{R}_{\varepsilon,M}\,.$

Indeed, first note that, due to (40), we can write, for $\gamma<2$, the
following estimate:

$\left|\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\frac{1}{b}\,\phi_{\varepsilon}\,{\bf
u}_{\varepsilon}\otimes S_{M}(b\,{\bf u}_{\varepsilon}):\nabla\psi\ {\rm
d}{x}\,{\rm d}t\,{\rm
d}t\right|\,\lesssim\,\left\|\phi_{\varepsilon}\right\|_{L^{\infty}_{T}(L^{3\gamma/(3-\gamma)})}\,\left\|{\bf
u}_{\varepsilon}\right\|_{L^{2}_{T}(L^{p_{2}})}\,\left\|S_{M}(b\,{\bf
u}_{\varepsilon})\right\|_{L^{2}_{T}(L^{\infty})}\,\lesssim\,1\,.$

We observe that this makes sense whenever
$(3-\gamma)/3\gamma\,+\,(\kappa+3)/6\kappa\,\leq\,1$, hence for
$\kappa\,\geq\,3\gamma/(7\gamma-6)$. But, for $\gamma>1$, one always has
$3\gamma/(7\gamma-6)<3$, therefore the previous estimate is satisfied for all
$\kappa>3$. Notice that, since $p_{2}>3$, we can repeat the exact same
computations also when $\gamma\geq 2$, up to use the right bound from (40).

On the other hand, after noticing that $1/p_{1}\,+\,1/p_{2}\,<\,1$ for
$\kappa>3$ and arguing in a similar way as above, we can estimate

$\displaystyle\left|\int^{T}_{0}\\!\\!\\!\int_{\Omega}\frac{1}{b}\,({\rm
Id}\,-S_{M})(b\,{\bf u}_{\varepsilon})\otimes S_{M}(b\,{\bf
u}_{\varepsilon}):\nabla\psi\ {\rm d}{x}\,{\rm d}t\right|\,$
$\displaystyle\lesssim\,\left\|({\rm Id}\,-S_{M})(b\,{\bf
u}_{\varepsilon})\right\|_{L^{2}_{T}(L^{p_{1}})}\,\left\|S_{M}(b\,{\bf
u}_{\varepsilon})\right\|_{L^{2}_{T}(L^{p_{2}})}$
$\displaystyle\lesssim\,2^{-M}\,\left\|\nabla(b\,{\bf
u}_{\varepsilon})\right\|^{2}_{L^{p_{1}}}\,\lesssim\,2^{-M}\,.$

Thus, we have proven that the last equality in (60) holds true.

At this point, to conclude the argument we may use the decomposition of Lemma
4.7. Indeed, keeping in mind the definition of the vector fields
$\widetilde{\mathcal{V}}_{\varepsilon}$ given in Proposition 4.3 above, we
notice that $S_{M}(b\,{\bf
u}_{\varepsilon})\,=\,\widetilde{\mathcal{V}}_{\varepsilon,M}\,=\,{\bf
V}_{\varepsilon,M}\,-\,\varepsilon\,\widetilde{{\bf W}}_{\varepsilon,M}$.
Thus, using the bounds collected in item (iv) of Lemma 4.7 we easily see that

$\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\frac{1}{b}\,S_{M}\big{(}b\,{\bf
u}_{\varepsilon}\big{)}\otimes S_{M}(b\,{\bf u}_{\varepsilon}):\nabla\psi\
{\rm d}{x}\,{\rm
d}t\,=\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\frac{1}{b}\,{\bf
V}_{\varepsilon,M}\otimes{\bf V}_{\varepsilon,M}:\nabla\psi\ {\rm d}{x}\,{\rm
d}t\,+\,\mathcal{R}_{\varepsilon,M}\,.$

This last equality finally ends the proof of the lemma. $\Box$

###  5.2 Compensated compactness

Owing to Lemma 5.1, it is enough to compute the limit of the approximate
convective term

$-\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\frac{1}{b}\,{\bf
V}_{\varepsilon,M}\otimes{\bf V}_{\varepsilon,M}:\nabla\psi\ {\rm d}{x}\,{\rm
d}t\,=\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}{\rm
div}\,\left(\frac{1}{b}\,{\bf V}_{\varepsilon,M}\otimes{\bf
V}_{\varepsilon,M}\right)\cdot\psi\ {\rm d}{x}\,{\rm d}t\,,$

for any test function
$\psi\in\mathcal{D}\big{(}[0,T[\,\times\Omega;\mathbb{R}^{3}\big{)}$ belonging
to the kernel of the singular perturbation operator, namely such that ${\rm
div}\,(b\psi)\,=\,0$. Here above, we have performed an integration by parts,
because the vector fields ${\bf V}_{\varepsilon,M}$ are smooth in the space
variable.

####  5.2.1 Preliminary reductions

Now, we compute

$\displaystyle{\rm div}\,\left(\frac{1}{b}\,{\bf V}_{\varepsilon,M}\otimes{\bf
V}_{\varepsilon,M}\right)\,$
$\displaystyle=\,\frac{1}{b}\,\operatorname{div}({\bf
V}_{\varepsilon,M})\,{\bf V}_{\varepsilon,M}\,+\,{\bf
V}_{\varepsilon,M}\cdot\nabla\left(\frac{1}{b}\,{\bf
V}_{\varepsilon,M}\right)$ (61)
$\displaystyle=\,\frac{1}{b}\,\operatorname{div}({\bf
V}_{\varepsilon,M})\,{\bf
V}_{\varepsilon,M}\,+\,\frac{b}{2}\,\nabla\left|\frac{1}{b}\,{\bf
V}_{\varepsilon,M}\right|^{2}\,-\,{\rm curl}\,\left(\frac{1}{b}\,{\bf
V}_{\varepsilon,M}\right)\times{\bf V}_{\varepsilon,M}\,,$

where the symbol $\times$ denotes the ususal external product of vectors in
$\mathbb{R}^{3}$ and, for a $3$-D vector field ${\bf U}$, we have ${\rm
curl}\,{\bf U}:=\nabla\times{\bf U}$. Notice that the second term in the last
line identically vanishes, whenever tested against a test function $\psi$ as
in (56). Thus, this term contributes as a remainder $R_{\varepsilon,M}$ to the
limit, in the sense of relation (57).

Therefore, resorting to the first equation in (50) for dealing with the first
term, we can write

$\displaystyle{\rm div}\,\left(\frac{1}{b}\,{\bf V}_{\varepsilon,M}\otimes{\bf
V}_{\varepsilon,M}\right)\,$
$\displaystyle=\,-\,\frac{\varepsilon}{b}\,\partial_{t}\phi_{\varepsilon,M}\,{\bf
V}_{\varepsilon,M}\,-\,{\rm curl}\,\left(\frac{1}{b}\,{\bf
V}_{\varepsilon,M}\right)\times{\bf V}_{\varepsilon,M}\,+\,R_{\varepsilon,M}$
$\displaystyle=\,\frac{\varepsilon}{b}\,\phi_{\varepsilon,M}\,\partial_{t}{\bf
V}_{\varepsilon,M}\,-\,{\rm curl}\,\left(\frac{1}{b}\,{\bf
V}_{\varepsilon,M}\right)\times{\bf
V}_{\varepsilon,M}\,+\,R_{\varepsilon,M}\,,$

where, in the second step, we have included the total time derivative
$\varepsilon\,\partial_{t}\big{(}\phi_{\varepsilon,M}\,{\bf
V}_{\varepsilon,M}\big{)}$ into the remainder $R_{\varepsilon,M}$. Indeed, the
time derivative can be put on the test function and the family
$\big{(}\phi_{\varepsilon,M}\,{\bf V}_{\varepsilon,M}\big{)}_{\varepsilon}$ is
uniformly bounded in e.g. $L^{2}_{T}(L^{2})$, owing to item (i) of Lemma 4.7.

At this point, we use the second equation in (50) to deal with the term
presenting the time derivative, and we get

$\displaystyle{\rm div}\,\left(\frac{1}{b}\,{\bf V}_{\varepsilon,M}\otimes{\bf
V}_{\varepsilon,M}\right)$
$\displaystyle\qquad\qquad=\,-\,\phi_{\varepsilon,M}\,\nabla\big{(}H^{\prime\prime}(b)\,\phi_{\varepsilon,M}\big{)}\,+\,\frac{\varepsilon}{b}\,\phi_{\varepsilon,M}\,{\bf
f}_{\varepsilon,M}\,+\,\frac{1}{b}\,\phi_{\varepsilon,M}\,{\bf
h}_{\varepsilon,M}\,-\,{\rm curl}\,\left(\frac{1}{b}\,{\bf
V}_{\varepsilon,M}\right)\times{\bf
V}_{\varepsilon,M}\,+\,R_{\varepsilon,M}\,.$

Owing to (52) and item (i) of Lemma 4.7, it is clear that the second term in
the right-hand side is a remainder, in the sense of (57). Next, we claim that
also the third term, i.e. $\big{(}\phi_{\varepsilon,M}\,{\bf
h}_{\varepsilon,M}/b\big{)}_{\varepsilon,M}$, is a remainder. For proving this
claim, in the case $\gamma\geq 2$ it is enough to employ Proposition 3.2 and
recall that estimate (51) holds true with the Lebesgue exponent $\gamma$
replaced by $2$. When $\gamma\in\,]1,2[\,$, instead, we notice that the piece
of information coming from (40) is not good enough to cover all possible
values of $\gamma$ in that interval. Our approach is instead based on the use
of item (ii) of Lemma 4.7, which allows us to write

$\frac{1}{b}\,\phi_{\varepsilon,M}\,{\bf
h}_{\varepsilon,M}\,=\,\frac{1}{b}\,\varphi_{\varepsilon,M}\,{\bf
h}_{\varepsilon,M}\,+\,\varepsilon^{(2-\gamma)/\gamma}\,\frac{1}{b}\,\pi_{\varepsilon,M}\,{\bf
h}_{\varepsilon,M}\,.$

Thanks to (51) and item (ii) of Lemma 4.7, we can estimate

$\displaystyle\left\|\frac{1}{b}\,\varphi_{\varepsilon,M}\,{\bf
h}_{\varepsilon,M}\right\|_{L^{1}_{T}(L^{6\gamma/(6-\gamma)})}\,\lesssim\,\left\|\varphi_{\varepsilon,M}\right\|_{L^{2}_{T}(L^{6})}\,\left\|{\bf
h}_{\varepsilon,M}\right\|_{L^{2}_{T}(L^{\gamma})}\,\lesssim\,2^{-M}\,,$

which implies that this term satisfies (57). In addition, it is easy to see
that also $\big{(}\varepsilon^{(2-\gamma)/\gamma}\,\pi_{\varepsilon,M}\,{\bf
h}_{\varepsilon,M}/b\big{)}_{\varepsilon}$ verifies (57). Indeed, recall that
in the case $\gamma\geq 2$, one can simply take $\pi_{\varepsilon,M}\equiv 0$.
For $1<\gamma<2$ one has only to notice that

$\displaystyle\left\|\varepsilon^{(2-\gamma)/\gamma}\,\frac{1}{b}\,\pi_{\varepsilon,M}\,{\bf
h}_{\varepsilon,M}\right\|_{L^{2}_{T}(L^{\gamma})}\,\lesssim\,\varepsilon^{(2-\gamma)/\gamma}\,\left\|\pi_{\varepsilon,M}\right\|_{L^{\infty}_{T}(L^{\infty})}\,\left\|{\bf
h}_{\varepsilon,M}\right\|_{L^{\infty}_{T}(L^{\gamma})}\,\lesssim\,C(M)\,\varepsilon^{(2-\gamma)/\gamma}\,.$

As a result of the previous computations, we infer that

${\rm div}\,\left(\frac{1}{b}\,{\bf V}_{\varepsilon,M}\otimes{\bf
V}_{\varepsilon,M}\right)\,=\,-\,\phi_{\varepsilon,M}\,\nabla\big{(}H^{\prime\prime}(b)\,\phi_{\varepsilon,M}\big{)}\,-\,{\rm
curl}\,\left(\frac{1}{b}\,{\bf V}_{\varepsilon,M}\right)\times{\bf
V}_{\varepsilon,M}\,+\,R_{\varepsilon,M}\,.$ (62)

####  5.2.2 Compactness of the rotational part

The next lemma takes care of the convergence of the ${\rm curl}\,$ term in
(62).

###### Lemma 5.2

Denote by ${\bf V}$ the weak-limit of $\big{(}{\bf
V}_{\varepsilon}\big{)}_{\varepsilon}$ identified in Corollary 4.4, so that
${\bf V}_{M}\,=\,S_{M}{\bf V}$ is the weak-limit of $\big{(}{\bf
V}_{\varepsilon,M}\big{)}_{\varepsilon}$ when $\varepsilon\rightarrow 0^{+}$.
Then, for any
$\psi\in\mathcal{D}\big{(}\mathbb{R}_{+}\times\Omega;\mathbb{R}^{3}\big{)}$,
one has

$\lim_{M\rightarrow+\infty}\limsup_{\varepsilon\rightarrow
0^{+}}\left(\int^{T}_{0}\\!\\!\\!\int_{\Omega}{\rm
curl}\,\left(\frac{1}{b}\,{\bf V}_{\varepsilon,M}\right)\times{\bf
V}_{\varepsilon,M}\cdot\psi\ {\rm d}{x},{\rm
d}t\,-\,\int^{T}_{0}\\!\\!\\!\int_{\Omega}S_{M}{\rm
curl}\,\left(\frac{1}{b}\,{\bf V}\right)\times{\bf V}_{M}\cdot\psi\ {\rm
d}{x}\,{\rm d}t\right)\,=\,0\,.$

Proof. We start the proof by recalling that, owing to Proposition 4.3, we have
$\big{(}{\bf
V}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(W^{1,p_{3}})$, hence
$\big{(}{\rm curl}\,{\bf
V}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{p_{3}})$. Observe
that, by dual Sobolev embeddings (see e.g. Theorem 0.5 of [15]), we have that,
for any compact $K\subset\Omega$, the space $L^{p_{3}}(K)$ is compactly
embedded into $H^{-2}(K)$, for instance.

Next, consider the (not regularised) wave system (47). Dividing the momentum
equation by $b$ and then taking the ${\rm curl}\,$ (which simply consists in
an adequate choice of the test function in the weak formulation), we deduce

$\partial_{t}{\rm curl}\,\left(\frac{1}{b}{\bf V}_{\varepsilon}\right)={\rm
curl}\,\left(\frac{{\bf f}_{\varepsilon}}{b}\right)\,.$

In turn, this relation, together with (49) and the fact that $b\in
C^{3}(\Omega)$, implies that $\big{(}\partial_{t}{\rm
curl}\,\left(\frac{1}{b}{\bf V}_{\varepsilon}\right)\big{)}_{\varepsilon}$ is
uniformly bounded in the space $L^{2}_{T}(H^{-s})$, for any $s>7/2$.

Putting together these pieces of information and applying the Aubin-Lions
lemma (see e.g. Lemma 3.7 of [28]), we gather that

$\bigg{(}{\rm curl}\,\left(\frac{1}{b}{\bf
V}_{\varepsilon}\right)\bigg{)}_{\varepsilon}\qquad\qquad\mbox{ is compact in
}\qquad\qquad L^{2}_{T}\big{(}H^{-2}(K)\big{)}\,,$

for any compact set $K\subset\Omega$. This implies that, for any
$M\in\mathbb{N}$ and any compact $K\subset\Omega$ fixed, the sequence

$\bigg{(}S_{M}{\rm curl}\,\left(\frac{1}{b}{\bf
V}_{\varepsilon}\right)\bigg{)}_{\varepsilon}\qquad\qquad\mbox{ is compact
(with respect to $\varepsilon$) in }\qquad\qquad
L^{2}_{T}\big{(}L^{2}(K)\big{)}\,.$ (63)

Next, we write

$\displaystyle\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}{\rm
curl}\,\left(\frac{1}{b}\,{\bf V}_{\varepsilon,M}\right)\times{\bf
V}_{\varepsilon,M}\cdot\psi\ {\rm d}{x}\,{\rm d}t$
$\displaystyle\qquad=\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}S_{M}{\rm
curl}\,\left(\frac{1}{b}\,{\bf V}_{\varepsilon}\right)\times{\bf
V}_{\varepsilon,M}\cdot\psi\ {\rm d}{x}\,{\rm
d}t\,+\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}{\rm
curl}\,\left(\left[\frac{1}{b},S_{M}\right]\,{\bf
V}_{\varepsilon}\right)\times{\bf V}_{\varepsilon,M}\cdot\psi\ {\rm
d}{x}\,{\rm d}t\,.$

By what we have just said, we have that

$\lim_{\varepsilon\rightarrow
0^{+}}\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}S_{M}{\rm
curl}\,\left(\frac{1}{b}\,{\bf V}_{\varepsilon}\right)\times{\bf
V}_{\varepsilon,M}\cdot\psi\ {\rm d}{x}\,{\rm
d}t\,=\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}S_{M}{\rm
curl}\,\left(\frac{1}{b}\,{\bf V}\right)\times{\bf V}_{M}\cdot\psi\ {\rm
d}{x}\,{\rm d}t\,.$

On the other hand, using the embedding $\big{(}{\bf
V}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(W^{1,p_{3}})$ again,
we have

$\left\|{\rm curl}\,\left(\left[\frac{1}{b},S_{M}\right]\,{\bf
V}_{\varepsilon}\right)\right\|_{L^{2}_{T}(L^{p_{3}})}\,\lesssim\,2^{-M}\,,$

whereas item (iv) of Lemma 4.7 and Bernstein’s inequalities (see Lemma A.1 in
the Appendix) imply

$\left\|{\bf
V}_{\varepsilon,M}\right\|_{L^{2}_{T}(L^{p_{3}^{\prime}})}\,\lesssim\,2^{3M(\frac{1}{p_{2}}\,-\,\frac{1}{p_{3}^{\prime}})}\,+\,\varepsilon\,C_{M}\,,$

for a constant $C_{M}$ which blows up when $M\rightarrow+\infty$, but which is
uniform in $\varepsilon>0$. Observe that

$3\left(\frac{1}{p_{2}}\,-\,\frac{1}{p_{3}^{\prime}}\right)\,=\,3\left(\frac{\kappa+3}{6\kappa}\,-\,1\,+\,\frac{5\kappa+3}{6\kappa}\right)\,=\frac{3}{\kappa}\,,\qquad\qquad\mbox{
with }\qquad\frac{3}{\kappa}\,<\,1\,.$

Thus, we deduce that

$\lim_{M\rightarrow+\infty}\limsup_{\varepsilon\rightarrow
0^{+}}\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}{\rm
curl}\,\left(\left[\frac{1}{b},S_{M}\right]\,{\bf
V}_{\varepsilon}\right)\times{\bf V}_{\varepsilon,M}\cdot\psi\ {\rm
d}{x}\,{\rm d}t\,=\,0\,.$

The proof of the lemma is now completed. $\Box$

####  5.2.3 Handling the pressure term

Before computing the limit with respect to $M\rightarrow+\infty$ in the
previous lemma, let us treat the first term in the right-hand side of (62). As
a matter of fact, we need to couple it with a term coming from the pressure
function, namely

$-\,\frac{1}{\varepsilon^{2}}\,\int_{\Omega}\Pi(\varrho_{\varepsilon};b)\,\operatorname{div}\psi\
{\rm d}{x}\,,$

where $\Pi(\varrho_{\varepsilon};b)$ has been defined in Lemma 4.6. We will
use a fundamental cancellation (appearing after regularisation), which is
already present in [24]. For this, we need the following preparatory lemma.

###### Lemma 5.3

For any $T>0$ fixed and any test function $\psi$ as in (56), such that ${\rm
Supp}\,\psi\subset[0,T[\,\times\Omega$, the following relation holds true, in
the sense of (58):

$-\frac{1}{\varepsilon^{2}}\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\Pi(\varrho_{\varepsilon};b)\,\operatorname{div}\psi\
{\rm d}{x}\,{\rm
d}t\,=\,-\,\frac{1}{2}\int^{T}_{0}\\!\\!\\!\int_{\Omega^{\varepsilon}_{\rm
ess}}{p^{\prime\prime}(b)\,\phi_{\varepsilon}^{2}\,\operatorname{div}\psi}\,{\rm
d}{x}\,{\rm d}t\,+\,\mathcal{R}_{\varepsilon,M}\,.$

Proof. Let us start by decomposing the term on the left-hand side of the
claimed equality into two integrals, one on the essential set and the other on
the residual set:

$\frac{1}{\varepsilon^{2}}\,\int_{0}^{T}\\!\\!\\!\\!\int_{\Omega}\Pi(\varrho_{\varepsilon};b)\,\operatorname{div}\psi\
{\rm d}{x}\,{\rm
d}t\,=\,\frac{1}{\varepsilon^{2}}\,\int^{T}_{0}\\!\\!\\!\int_{\Omega^{\varepsilon}_{\rm
ess}}{\Pi(\varrho_{\varepsilon};b)\,\operatorname{div}\psi}\,{\rm d}{x}\,{\rm
d}t\,+\,\frac{1}{\varepsilon^{2}}\,\int^{T}_{0}\\!\\!\\!\int_{\Omega^{\varepsilon}_{\rm
res}}{\Pi(\varrho_{\varepsilon};b)\,\operatorname{div}\psi}\,{\rm d}{x}\,{\rm
d}t\,.$

Notice that, in order to treat the integral over $\Omega^{\varepsilon}_{\rm
res}$, the bounds of Lemma 4.6 are not enough. Instead, in order to take
advantage of the smallness bounds in (21), we need to introduce a finer
decomposition and split that term further into two pieces.

To begin with, let us consider
$\Pi_{E}(\varrho_{\varepsilon};b)\,=\,p_{E}(\varrho_{\varepsilon})-p_{E}(b)-p_{E}^{\prime}(b)(\varrho_{\varepsilon}-b)$
only. On the one hand, thanks the uniform bounds of (35), we have, for some
$z_{\varepsilon}=z_{\varepsilon}(t,x)$ belonging to the interval
$\,]\varrho_{\varepsilon}(t,x),b(x)[\,$, the estimate

$\displaystyle\left|\frac{1}{\varepsilon^{2}}\,\int_{\Omega^{\varepsilon}_{{\rm
res},B}}{\Pi_{E}(\varrho_{\varepsilon};b)\,\operatorname{div}\psi}\,{\rm
d}{x}\right|\,$
$\displaystyle\lesssim\,\frac{1}{\varepsilon}\,\int_{\Omega^{\varepsilon}_{{\rm
res},B}}{\Big{(}p^{\prime}_{E}(z_{\varepsilon})+p^{\prime}_{E}(b)\Big{)}\,\big{|}\phi_{\varepsilon}\big{|}\,\big{|}\operatorname{div}\psi\big{|}}\,{\rm
d}{x}$
$\displaystyle\lesssim\,\frac{1}{\varepsilon}\,\left\|\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},B}}\,\phi_{\varepsilon}\right\|_{L^{2}_{T}(L^{6})}\,\Big{(}\mathcal{L}(\Omega^{\varepsilon}_{\rm
res})\Big{)}^{5/6}\,\lesssim\,\varepsilon^{-1+5/3}\,=\,\varepsilon^{2/3}\,,$

which tells us that the contribution coming from this term is a remainder, in
the sense of (57). On the other hand, proceeding similarly and employing the
assumption $p_{E}^{\prime}(z)\approx z^{\gamma-1}$, we get

$\displaystyle\left|\frac{1}{\varepsilon^{2}}\,\int_{\Omega^{\varepsilon}_{{\rm
res},UB}}{\Pi_{E}(\varrho_{\varepsilon};b)\,\operatorname{div}\psi}\,{\rm
d}{x}\right|\,$
$\displaystyle\lesssim\,\frac{1}{\varepsilon}\,\int_{\Omega^{\varepsilon}_{{\rm
res},UB}}{\varrho_{\varepsilon}^{\gamma-1}\,\big{|}\phi_{\varepsilon}\big{|}\,\big{|}\operatorname{div}\psi\big{|}}\,{\rm
d}{x}\,.$

Now, owing to (36), we gather the uniform bound

$\left(\frac{1}{\varepsilon}\,\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\varrho_{\varepsilon}^{\gamma-1}\right)_{\varepsilon}\,\subset\,L^{r}_{T}(L^{p})\,,\qquad\qquad
r\,:=\,\frac{\gamma}{\gamma-1}\quad\mbox{ and }\quad
p\,:=\,\frac{3\,\gamma}{\gamma-1}\,,$ (64)

whereas we can use (40) to control the $\phi_{\varepsilon}$ term. After
checking that

$\forall\;1\,<\,\gamma\,\leq\,2\,,\qquad\qquad\frac{\gamma-1}{\gamma}\,+\,\frac{1}{2}\,\leq\,1\qquad\mbox{
and
}\qquad\frac{1}{q}\,:=\,\frac{3-\gamma}{3\,\gamma}\,+\,\frac{\gamma-1}{3\,\gamma}\,<\,1\,,$

we can thus bound

$\displaystyle\left|\frac{1}{\varepsilon^{2}}\,\int^{T}_{0}\\!\\!\\!\int_{\Omega^{\varepsilon}_{{\rm
res},UB}}{\Pi_{E}(\varrho_{\varepsilon};b)\,\operatorname{div}\psi}\,{\rm
d}{x}\,{\rm d}t\right|\,$
$\displaystyle\lesssim\,\left(\frac{1}{\varepsilon}\,\left\|\mathds{1}_{\Omega^{\varepsilon}_{{\rm
res},UB}}\,\varrho_{\varepsilon}^{\gamma-1}\right\|_{L^{r}_{T}(L^{p})}\right)\,\left\|\phi_{\varepsilon}\right\|_{L^{2}_{T}(L^{3\gamma/(3-\gamma)})}\,\Big{(}\mathcal{L}(\Omega_{\rm
res})\Big{)}^{1-1/q}$ $\displaystyle\lesssim\,\varepsilon^{2(1-1/q)}\,.$

A direct computation shows that $1-1/q\,>\,0$. Therefore, for $1<\gamma\leq
2$, we have proved that the contribution coming from the integral over the
residual set is indeed a remainder, in the sense of (57).

In the case $\gamma>2$, instead, we interpolate between the
$L^{\infty}_{T}(L^{\gamma/(\gamma-1)})$ bound on
$\big{(}\mathds{1}_{\Omega^{\varepsilon}_{{\rm