$\int^{T}_{0}\int_{K}\boldsymbol{e}_{3}\times\boldsymbol{U}\cdot\boldsymbol{\phi}$.
In the end, passing to the limit for $\varepsilon\rightarrow 0^{+}$ we find

$\mathbb{H}\left(\boldsymbol{e}_{3}\times\boldsymbol{U}\right)\,=\,0\qquad\qquad\text{and
so}\qquad\qquad\boldsymbol{e}_{3}\times\boldsymbol{U}\,=\,\nabla_{x}\Phi\,$

for some potential function $\Phi$. From this relation, which in components
reads

$\begin{pmatrix}-U^{2}\\\ U^{1}\\\
0\end{pmatrix}=\begin{pmatrix}\partial_{1}\Phi\\\ \partial_{2}\Phi\\\
\partial_{3}\Phi\end{pmatrix}\,,$ (2.2.34)

we deduce that $\Phi=\Phi(t,x^{h})$, i.e. $\Phi$ does not depend on $x^{3}$,
and that the same property is inherited by
$\boldsymbol{U}^{h}\,=\,\bigl{(}U^{1},U^{2}\bigr{)}$, i.e.
$\boldsymbol{U}^{h}\,=\,\boldsymbol{U}^{h}(t,x^{h})$. Furthermore, from
(2.2.34), it is also easy to see that the $2$-D flow given by
$\boldsymbol{U}^{h}$ is incompressible, namely ${\rm
div}\,_{\\!h}\,\boldsymbol{U}^{h}\,=\,0$. Combining this fact with (2.2.31),
we infer that $\partial_{3}U^{3}\,=\,0$; on the other hand, thanks to the
boundary condition (2.1.8) we must have
$\bigl{(}\boldsymbol{U}\cdot\boldsymbol{n}\bigr{)}_{|\partial\Omega}\,=\,0$.
Keeping in mind that
$\partial\Omega\,=\,\bigl{(}\mathbb{R}^{2}\times\\{0\\}\bigr{)}\cup\bigl{(}\mathbb{R}^{2}\times\\{1\\}\bigr{)}$,
we finally get $U^{3}\,\equiv\,0$, whence (2.2.28) finally follows.

Next, we observe that we can by now pass to the limit in the weak formulation
(2.1.29) of (NSF${}_{\varepsilon}^{3}$). The argument being analogous to the
one used in [39] (see Paragraph 5.3.2), we only sketch it. First of all,
testing (NSF${}_{\varepsilon}^{3}$) on $\varphi/\varepsilon^{m}$, for some
$\varphi\in C^{\infty}_{c}\bigl{(}[0,T[\,\times\Omega\bigr{)}$, and using
(NSF${}_{\varepsilon}^{1}$), for $\varepsilon>0$ small enough we get

$\displaystyle-\int^{T}_{0}\\!\\!\int_{K}\varrho_{\varepsilon}\left(\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(1,\overline{\vartheta})}{\varepsilon^{m}}\right)\partial_{t}\varphi-\int^{T}_{0}\\!\\!\int_{K}\varrho_{\varepsilon}\left(\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(1,\overline{\vartheta})}{\varepsilon^{m}}\right)\boldsymbol{u}_{\varepsilon}\cdot\nabla_{x}\varphi$
(2.2.35)
$\displaystyle+\int^{T}_{0}\\!\\!\int_{K}\frac{\kappa(\vartheta_{\varepsilon})}{\vartheta_{\varepsilon}}\,\frac{1}{\varepsilon^{m}}\,\nabla_{x}\vartheta_{\varepsilon}\cdot\nabla_{x}\varphi-\frac{1}{\varepsilon^{m}}\,\langle\sigma_{\varepsilon},\varphi\rangle_{[\mathcal{M}^{+},C^{0}]([0,T]\times
K)}=\int_{K}\varrho_{0,\varepsilon}\left(\frac{s(\varrho_{0,\varepsilon},\vartheta_{0,\varepsilon})-s(1,\overline{\vartheta})}{\varepsilon^{m}}\right)\varphi(0)\,.$

To begin with, let us decompose

$\displaystyle\varrho_{\varepsilon}\left(\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(1,\overline{\vartheta})}{\varepsilon^{m}}\right)=$
(2.2.36)
$\displaystyle=[\varrho_{\varepsilon}]_{{\rm{ess}}}\left(\frac{[s(\varrho_{\varepsilon},\vartheta_{\varepsilon})]_{{\rm{ess}}}-s(1,\overline{\vartheta})}{\varepsilon^{m}}\right)+\left[\frac{\varrho_{\varepsilon}}{\varepsilon^{m}}\right]_{{\rm{res}}}\left([s(\varrho_{\varepsilon},\vartheta_{\varepsilon})]_{{\rm{ess}}}-s(1,\overline{\vartheta})\right)+\left[\frac{\varrho_{\varepsilon}\,s(\varrho_{\varepsilon},\vartheta_{\varepsilon})}{\varepsilon^{m}}\right]_{{\rm{res}}}\,.$

Thanks to (2.2.8), we discover that the second term in the right-hand side
strongly converges to $0$ in $L_{T}^{\infty}(L^{5/3}_{\rm loc})$. Also the
third term converges to $0$ in the space $L_{T}^{2}(L^{30/23}_{\rm loc})$, as
a consequence of (2.2.5) and (2.2.13). Notice that these terms converge to $0$
even when multiplied by $\boldsymbol{u}_{\varepsilon}$: to see this, it is
enough to put (2.2.5), (2.2.13), (2.2.12) and the previous properties
together.

As for the first term in the right-hand side of (2.2.36), Propositions 2.2.5
and 2.1.7 and estimate (2.2.21) imply that it weakly converges to
$\partial_{\varrho}s(1,\overline{\vartheta})\,R\,+\,\partial_{\vartheta}s(1,\overline{\vartheta})\,\Theta$,
where $R$ and $\Theta$ are defined respectively in (2.2.25) and (2.2.26). On
the other hand, an application of the Div-Curl Lemma (see Theorem B.14) gives

$[\varrho_{\varepsilon}]_{{\rm{ess}}}\left(\frac{[s(\varrho_{\varepsilon},\vartheta_{\varepsilon})]_{{\rm{ess}}}-s(1,\overline{\vartheta})}{\varepsilon^{m}}\right)\,\boldsymbol{u}_{\varepsilon}\,\rightharpoonup\,\Bigl{(}\partial_{\varrho}s(1,\overline{\vartheta})\,R\,+\,\partial_{\vartheta}s(1,\overline{\vartheta})\,\Theta\Bigr{)}\,\boldsymbol{U}$

in the space $L_{T}^{2}(L^{3/2}_{\rm loc})$. In addition, from (2.2.3) we
deduce that

$\frac{1}{\varepsilon^{m}}\,\langle\sigma_{\varepsilon},\varphi\rangle_{[\mathcal{M}^{+},C^{0}]([0,T]\times
K)}\,\longrightarrow\,0$

when $\varepsilon\rightarrow 0^{+}$. Finally, a separation into essential and
residual part of the coefficient
$\kappa(\vartheta_{\varepsilon})/\vartheta_{\varepsilon}$, together with
(2.1.16), (2.2.4), (2.2.8), (2.2.11) and (2.2.14) gives

$\frac{\kappa(\vartheta_{\varepsilon})}{\vartheta_{\varepsilon}}\,\frac{1}{\varepsilon^{m}}\,\nabla_{x}\vartheta_{\varepsilon}\,\rightharpoonup\,\frac{\kappa(\overline{\vartheta})}{\overline{\vartheta}}\,\nabla_{x}\Theta\qquad\qquad\mbox{
in }\qquad L^{2}\bigl{(}[0,T];L^{1}_{\rm loc}(\Omega)\bigr{)}\,.$

In the end, we have proved that equation (2.2.35) converges, for
$\varepsilon\rightarrow 0^{+}$, to equation

$\displaystyle-\int^{T}_{0}\int_{\Omega}\Bigl{(}\partial_{\varrho}s(1,\overline{\vartheta})R+\partial_{\vartheta}s(1,\overline{\vartheta})\,\Theta\Bigr{)}\left(\partial_{t}\varphi+\boldsymbol{U}\cdot\nabla_{x}\varphi\right)\,{\rm
d}x{\rm d}t\,+$ (2.2.37)
$\displaystyle\qquad\qquad+\int^{T}_{0}\int_{\Omega}\frac{\kappa(\overline{\vartheta})}{\overline{\vartheta}}\nabla_{x}\Theta\cdot\nabla_{x}\varphi\,{\rm
d}x{\rm
d}t=\int_{\Omega}\Bigl{(}\partial_{\varrho}s(1,\overline{\vartheta})\,R_{0}\,+\,\partial_{\vartheta}s(1,\overline{\vartheta})\,\Theta_{0}\Bigr{)}\,\varphi(0)\,{\rm
d}x\,,$

for all $\varphi\in C_{c}^{\infty}([0,T[\,\times\Omega)$, with $T>0$ any
arbitrary time. Relation (2.2.37) means that the quantity $\Upsilon$, defined
in (2.2.30), is a weak solution of that equation, related to the initial datum
$\Upsilon_{0}:=\partial_{\varrho}s(1,\overline{\vartheta})\,R_{0}\,+\,\partial_{\vartheta}s(1,\overline{\vartheta})\,\Theta_{0}$.
Equation (2.2.30) is in fact an equation for $\Theta$ only, keep in mind
Remark 2.1.11.

######  2.2.2.3 The case of isotropic scaling

We focus now on the case of isotropic scaling, namely $m=1$. Recall that, in
this instance, we also set $F=0$. In this case, the fast rotation and weak
compressibility effects are of the same order; in turn, this allows to reach
the so-called _quasi-geostrophic balance_ in the limit (see equation (2.2.38)
below).

###### Proposition 2.2.9

Take $m=1$ and $F=0$ in system (NSF${}_{\varepsilon}^{1}$) to
(NSF${}_{\varepsilon}^{4}$). Let
$\left(\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon},\vartheta_{\varepsilon}\right)_{\varepsilon}$
be a family of weak solutions to (NSF${}_{\varepsilon}^{1}$) to
(NSF${}_{\varepsilon}^{4}$), associated with initial data
$\left(\varrho_{0,\varepsilon},\boldsymbol{u}_{0,\varepsilon},\vartheta_{0,\varepsilon}\right)$
verifying the hypotheses fixed in Paragraph 2.1.1.4. Let
$(R,\boldsymbol{U},\Theta)$ be a limit point of the sequence
$\left(R_{\varepsilon},\boldsymbol{u}_{\varepsilon},\Theta_{\varepsilon}\right)_{\varepsilon}$,
as identified in Subsection 2.2.2.1. Then,

$\displaystyle\boldsymbol{U}\,=\,\,\Big{(}\boldsymbol{U}^{h}\,,\,0\Big{)}\,,\qquad\qquad\mbox{
with }\qquad\boldsymbol{U}^{h}\,=\,\boldsymbol{U}^{h}(t,x^{h})\quad\mbox{ and
}\quad{\rm div}\,_{\\!h}\,\boldsymbol{U}^{h}\,=\,0$
$\displaystyle\boldsymbol{U}^{h}\,=\,\nabla^{\perp}_{h}q\;\mbox{ a.e. in
}\;\,]0,T[\,\times\Omega\,,\quad\mbox{ with }$ (2.2.38) $\displaystyle
q\,=\,q(t,x^{h})\,:=\,\partial_{\varrho}p(1,\overline{\vartheta})R+\partial_{\vartheta}p(1,\overline{\vartheta})\Theta-G-1/2$
(2.2.39) $\displaystyle\partial_{t}\Upsilon+{\rm
div}_{h}\left(\Upsilon\boldsymbol{U}^{h}\right)-\frac{\kappa(\overline{\vartheta})}{\overline{\vartheta}}\Delta\Theta=0\,,\qquad\quad\mbox{
with }\qquad\Upsilon_{|t=0}\,=\,\Upsilon_{0}\,,$

where $\Upsilon$ and $\Upsilon_{0}$ are the same quantities defined in
Proposition 2.2.7.

Proof:  Arguing as in the proof of Proposition 2.2.7, it is easy to pass to
the limit in the continuity equation and in the entropy balance. In
particular, we obtain again equations (2.2.31) and (2.2.37).

The only changes concern the analysis of the momentum equation, written in its
weak formulation (2.1.28). We start by testing it on
$\varepsilon\,\boldsymbol{\phi}$, for a smooth compactly supported
$\boldsymbol{\phi}$. Similarly to what done above, the uniform bounds of
Subsection 2.2.1 allow us to say that the only quantity which does not vanish
in the limit is the sum of the terms involving the Coriolis force, the
pressure and the gravitational force:

$\boldsymbol{e}_{3}\times\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\,+\frac{\nabla_{x}\left(p(\varrho_{\varepsilon},\vartheta_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon},\vartheta_{\varepsilon})\right)}{\varepsilon}\,-\,\left(\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}\right)\nabla_{x}G\,=\,\mathcal{O}(\varepsilon)\,.$

From this relation, following the same computations performed in the proof of
Proposition 2.2.7, in the limit $\varepsilon\rightarrow 0^{+}$ we obtain that

$\boldsymbol{e}_{3}\times\boldsymbol{U}+\nabla_{x}\left(\partial_{\varrho}p(1,\overline{\vartheta})\,\varrho^{(1)}\,+\,\partial_{\vartheta}p(1,\overline{\vartheta})\,\Theta\right)\,=\,0\qquad\qquad\mbox{
a.e. in }\;\mathbb{R}_{+}\times\Omega\,.$

After defining $q$ as in (2.2.39), i.e.

$q\,:=\,\partial_{\varrho}p(1,\overline{\vartheta})R+\partial_{\vartheta}p(1,\overline{\vartheta})\Theta-G-1/2$

and keeping Remark 2.2.8 in mind, this equality can be equivalently written as

$\boldsymbol{e}_{3}\times\boldsymbol{U}+\nabla_{x}q\,=\,0\qquad\qquad\mbox{
a.e. in }\;\mathbb{R}_{+}\times\Omega\,.$

As done in the proof to Proposition 2.2.7, from this relation we immediately
deduce that $q=q(t,x^{h})$ and
$\boldsymbol{U}^{h}=\boldsymbol{U}^{h}(t,x^{h})$. In addition, we get
$\boldsymbol{U}^{h}=\nabla^{\perp}_{h}q$, whence we gather that $q$ can be
viewed as a stream function for $\boldsymbol{U}^{h}$. Using (2.2.31), we infer
that $\partial_{3}U^{3}=0$, which in turn implies that $U^{3}\equiv 0$, thanks
to (2.1.8). The proposition is thus proved.

###### Remark 2.2.10

Notice that $q$ is defined up to an additive constant. We fix it to be $-1/2$,
in order to compensate the vertical mean of $G$ and have a cleaner expression
for $\langle q\rangle$ (see Theorem 2.1.12). As a matter of fact, it is
$\langle q\rangle$ the natural quantity to look at, see also Subsection 2.4.3
in this respect.

####  2.3 Convergence in presence of the centrifugal force

In this section we complete the proof of Theorem 2.1.10, in the case when
$m\geq 2$ and $F\neq 0$. In the case $m>1$ and $F=0$, some arguments of the
proof slightly change, due to the absence of the (unbounded) centrifugal
force: we refer to Section 2.4 below for more details.

The uniform bounds of Subsection 2.2.1 allow us to pass to the limit in the
mass and entropy equations, but they are not enough for proving convergence in
the weak formulation of the momentum equation: the main problem relies on
identifying the weak limit of the convective term
$\varrho_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}$.
For this, we need to control the strong oscillations in time of the solutions:
this is the aim of Subsection 2.3.1. In Subsection 2.3.2, by using a
compensated compactness argument together with Aubin-Lions Theorem (see
Theorem B.13), we establish strong convergence of suitable quantities related
to the velocity fields. This property, which deeply relies on the structure of
the wave system, allows us to pass to the limit in our equations (see
Subsection 2.3.3).

#####  2.3.1 Analysis of the acoustic waves

The goal of the present subsection is to describe oscillations of solutions.
First of all, we recast our equations into a wave system; there we also
implement a localisation procedure, due to the presence of the centrifugal
force. Then, we establish uniform bounds for the quantities appearing in the
wave system. Finally, we apply a regularisation in space for all the
quantities, which is preparatory in view of the computations of Subsection
2.3.2.

######  2.3.1.1 Formulation of the acoustic equation

Let us define

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

We start by writing the continuity equation in the form

$\varepsilon^{m}\,\partial_{t}\varrho^{(1)}_{\varepsilon}\,+\,{\rm
div}\,\boldsymbol{V}_{\varepsilon}\,=\,0\,.$ (2.3.1)

Of course, this relation, as well as the other ones which will follow, has to
be read in the weak form.

Using continuity equation and resorting to the time lifting (2.1.32) of the
measure $\sigma_{\varepsilon}$, straightforward computations lead us to the
following form of the entropy balance:

$\varepsilon^{m}\partial_{t}\\!\left(\varrho_{\varepsilon}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}-\frac{1}{\varepsilon^{m}}\Sigma_{\varepsilon}\right)\,=\,\varepsilon^{m}\,{\rm
div}\,\\!\\!\left(\frac{\kappa(\vartheta_{\varepsilon})}{\vartheta_{\varepsilon}}\frac{\nabla_{x}\vartheta_{\varepsilon}}{\varepsilon^{m}}\right)+s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta}){\rm
div}\,\\!\\!\left(\varrho_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\right)-{\rm
div}\,\\!\\!\left(\varrho_{\varepsilon}s(\varrho_{\varepsilon},\vartheta_{\varepsilon})\boldsymbol{u}_{\varepsilon}\right),$

where, with a little abuse of notation, we use the identification
$\int_{\Omega_{\varepsilon}}\Sigma_{\varepsilon}\,\varphi\,dx\,=\,\langle\Sigma_{\varepsilon},\varphi\rangle_{[\mathcal{M}^{+},C^{0}]}$.
Next, since $\widetilde{\varrho}_{\varepsilon}$ is smooth (recall relation
(2.1.18) above), the previous equation can be finally written as

$\displaystyle\varepsilon^{m}\,\partial_{t}\left(\varrho_{\varepsilon}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,-\,\frac{1}{\varepsilon^{m}}\Sigma_{\varepsilon}\right)\,=$
(2.3.2) $\displaystyle\qquad\quad=\,\varepsilon^{m}\,\biggl{(}{\rm
div}\,\\!\left(\frac{\kappa(\vartheta_{\varepsilon})}{\vartheta_{\varepsilon}}\,\frac{\nabla_{x}\vartheta_{\varepsilon}}{\varepsilon^{m}}\right)\,-\,\varrho_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\,\cdot\,\frac{1}{\varepsilon^{m}}\,\nabla_{x}s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})\,-\,{\rm
div}\,\\!\left(\varrho_{\varepsilon}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,\boldsymbol{u}_{\varepsilon}\right)\biggr{)}\,.$

Now, we turn our attention to the momentum equation. By (2.1.17) we find

$\displaystyle\varepsilon^{m}\,\partial_{t}\boldsymbol{V}_{\varepsilon}\,+\,\nabla_{x}\left(\frac{p(\varrho_{\varepsilon},\vartheta_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\right)\,+\,\varepsilon^{m-1}\,\boldsymbol{e}_{3}\times\boldsymbol{V}_{\varepsilon}\,=\,\varepsilon^{2(m-1)}\frac{\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}}{\varepsilon^{m}}\nabla_{x}F\,+$
(2.3.3)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\,\varepsilon^{m}\left({\rm
div}\,\mathbb{S}\\!\left(\vartheta_{\varepsilon},\nabla_{x}\boldsymbol{u}_{\varepsilon}\right)\,-\,{\rm
div}\,\\!\left(\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}\right)\,+\,\frac{\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}}{\varepsilon^{m}}\nabla_{x}G\right)\,.$

At this point, let us introduce two real numbers $\mathcal{A}$ and
$\mathcal{B}$, such that the following relations are satisfied:

$\mathcal{A}\,+\,\mathcal{B}\,\partial_{\varrho}s(1,\overline{\vartheta})\,=\,\partial_{\varrho}p(1,\overline{\vartheta})\qquad\mbox{
and
}\qquad\mathcal{B}\,\partial_{\vartheta}s(1,\overline{\vartheta})\,=\,\partial_{\vartheta}p(1,\overline{\vartheta})\,.$
(2.3.4)

Due to Gibbs’ law (2.1.3) and the structural hypotheses of Paragraph 2.1.1.2
(see also Chapter 8 of [39] and [44]), we notice that $\mathcal{A}$ is given
by formula (2.1.38), and $\mathcal{A}>0$.

Taking a linear combination of (2.3.1) and (2.3.2), with coefficients
respectively $\mathcal{A}$ and $\mathcal{B}$, and keeping in mind equation
(2.3.3), we finally get the wave system

$\left\\{\begin{array}[]{l}\varepsilon^{m}\,\partial_{t}Z_{\varepsilon}\,+\,\mathcal{A}\,{\rm
div}\,\boldsymbol{V}_{\varepsilon}\,=\,\varepsilon^{m}\,\left({\rm
div}\,\boldsymbol{X}^{1}_{\varepsilon}\,+\,X^{2}_{\varepsilon}\right)\\\\[4.30554pt]
\varepsilon^{m}\,\partial_{t}\boldsymbol{V}_{\varepsilon}\,+\,\nabla_{x}Z_{\varepsilon}\,+\,\varepsilon^{m-1}\,\boldsymbol{e}_{3}\times\boldsymbol{V}_{\varepsilon}\,=\,\varepsilon^{m}\,\left({\rm
div}\,\mathbb{Y}^{1}_{\varepsilon}\,+\,\boldsymbol{Y}^{2}_{\varepsilon}\,+\,\nabla_{x}Y^{3}_{\varepsilon}\right)\,,\qquad\big{(}\boldsymbol{V}_{\varepsilon}\cdot\boldsymbol{n}\big{)}_{|\partial\Omega_{\varepsilon}}\,=\,0\,,\end{array}\right.$
(2.3.5)

where we have defined the quantities

$\displaystyle Z_{\varepsilon}$ $\displaystyle:=$
$\displaystyle\mathcal{A}\,\varrho^{(1)}_{\varepsilon}\,+\,\mathcal{B}\,\left(\varrho_{\varepsilon}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,-\,\frac{1}{\varepsilon^{m}}\Sigma_{\varepsilon}\right)$
$\displaystyle\boldsymbol{X}^{1}_{\varepsilon}$ $\displaystyle:=$
$\displaystyle\mathcal{B}\left(\frac{\kappa(\vartheta_{\varepsilon})}{\vartheta_{\varepsilon}}\,\frac{\nabla_{x}\vartheta_{\varepsilon}}{\varepsilon^{m}}\,-\,\varrho_{\varepsilon}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,\boldsymbol{u}_{\varepsilon}\right)$
$\displaystyle X^{2}_{\varepsilon}$ $\displaystyle:=$
$\displaystyle-\,\mathcal{B}\,\varrho_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\,\cdot\,\frac{1}{\varepsilon^{m}}\,\nabla_{x}s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})$
$\displaystyle\mathbb{Y}^{1}_{\varepsilon}$ $\displaystyle:=$
$\displaystyle\mathbb{S}\\!\left(\vartheta_{\varepsilon},\nabla\boldsymbol{u}_{\varepsilon}\right)\,-\,\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}$
$\displaystyle\boldsymbol{Y}^{2}_{\varepsilon}$ $\displaystyle:=$
$\displaystyle\frac{\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}}{\varepsilon^{m}}\nabla_{x}G\,+\,\varepsilon^{m-2}\,\frac{\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}}{\varepsilon^{m}}\nabla_{x}F$
$\displaystyle Y^{3}_{\varepsilon}$ $\displaystyle:=$
$\displaystyle\frac{1}{\varepsilon^{m}}\left(\mathcal{A}\,\frac{\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}}{\varepsilon^{m}}\,+\mathcal{B}\,\varrho_{\varepsilon}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,-\,\mathcal{B}\,\frac{1}{\varepsilon^{m}}\Sigma_{\varepsilon}\,-\,\frac{p(\varrho_{\varepsilon},\vartheta_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\right)\,.$

We remark that system (2.3.5) has to be read in the weak sense: for any
$\varphi\in
C_{c}^{\infty}\bigl{(}[0,T[\,\times\overline{\Omega}_{\varepsilon}\bigr{)}$,
one has

$-\,\varepsilon^{m}\,\int^{T}_{0}\int_{\Omega_{\varepsilon}}Z_{\varepsilon}\,\partial_{t}\varphi\,-\,\mathcal{A}\,\int^{T}_{0}\int_{\Omega_{\varepsilon}}\boldsymbol{V}_{\varepsilon}\cdot\nabla_{x}\varphi\,=\,\varepsilon^{m}\int_{\Omega_{\varepsilon}}Z_{0,\varepsilon}\,\varphi(0)\,+\,\varepsilon^{m}\,\int^{T}_{0}\int_{\Omega_{\varepsilon}}\left(-\,\boldsymbol{X}^{1}_{\varepsilon}\cdot\nabla_{x}\varphi\,+\,X^{2}_{\varepsilon}\,\varphi\right)\,,$

and also, for any $\boldsymbol{\psi}\in
C_{c}^{\infty}\bigl{(}[0,T[\,\times\overline{\Omega}_{\varepsilon};\mathbb{R}^{3}\bigr{)}$
such that
$\big{(}\boldsymbol{\psi}\cdot\boldsymbol{n}_{\varepsilon}\big{)}_{|\partial{\Omega_{\varepsilon}}}=0$,
one has

$\displaystyle-\,\varepsilon^{m}\,\int^{T}_{0}\int_{\Omega_{\varepsilon}}\boldsymbol{V}_{\varepsilon}\cdot\partial_{t}\boldsymbol{\psi}\,-\,\int^{T}_{0}\int_{\Omega_{\varepsilon}}Z_{\varepsilon}\,{\rm
div}\,\boldsymbol{\psi}\,+\,\varepsilon^{m-1}\int^{T}_{0}\int_{\Omega_{\varepsilon}}\boldsymbol{e}_{3}\times\boldsymbol{V}_{\varepsilon}\cdot\boldsymbol{\psi}$
$\displaystyle\qquad\qquad\qquad\qquad=\,\varepsilon^{m}\int_{\Omega_{\varepsilon}}\boldsymbol{V}_{0,\varepsilon}\cdot\boldsymbol{\psi}(0)\,+\,\varepsilon^{m}\,\int^{T}_{0}\int_{\Omega_{\varepsilon}}\left(-\,\mathbb{Y}^{1}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}\,+\,\boldsymbol{Y}^{2}_{\varepsilon}\cdot\boldsymbol{\psi}\,-\,Y^{3}_{\varepsilon}\,{\rm
div}\,\boldsymbol{\psi}\right)\,,$

where we have set

$Z_{0,\varepsilon}\,=\,\mathcal{A}\,\varrho^{(1)}_{0,\varepsilon}\,+\,\mathcal{B}\,\left(\varrho_{0,\varepsilon}\,\frac{s(\varrho_{0,\varepsilon},\vartheta_{0,\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\right)\qquad\mbox{
and
}\qquad\boldsymbol{V}_{0,\varepsilon}\,=\,\varrho_{0,\varepsilon}\,\boldsymbol{u}_{0,\varepsilon}\,.$
(2.3.6)

At this point, analogously to [34], for any fixed $l>0$, let us introduce a
smooth cut-off

$\displaystyle\chi_{l}\in C^{\infty}_{c}(\mathbb{R}^{2})\quad\mbox{ radially
decreasing}\,,\qquad\mbox{ with }\quad 0\leq\chi_{l}\leq 1\,,$ (2.3.7)
$\displaystyle\mbox{such that }\quad\chi_{l}\equiv 1\ \mbox{ on }\
\mathbb{B}_{l}\,,\quad\chi_{l}\equiv 0\ \mbox{ out of }\
\mathbb{B}_{2l}\,,\quad\left|\nabla_{h}\chi_{l}(x^{h})\right|\,\leq\,C(l)\;\;\text{for
any}\;x^{h}\in\mathbb{R}^{2}\,.$

Then we define

$\Lambda_{\varepsilon,l}\,:=\,\chi_{l}\,Z_{\varepsilon}\,=\,\chi_{l}\,\mathcal{A}\,\varrho^{(1)}_{\varepsilon}\,+\,\chi_{l}\,\mathcal{B}\,\left(\varrho_{\varepsilon}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,-\,\frac{1}{\varepsilon^{m}}\Sigma_{\varepsilon}\right)\quad\mbox{
and
}\quad\boldsymbol{W}_{\varepsilon,l}\,:=\,\chi_{l}\,\boldsymbol{V}_{\varepsilon}\,.$
(2.3.8)

For notational convenience, in what follows we keep using the notation
$\Lambda_{\varepsilon}$ and $\boldsymbol{W}_{\varepsilon}$ instead of
$\Lambda_{\varepsilon,l}$ and $\boldsymbol{W}_{\varepsilon,l}\,$, tacitly
meaning the dependence on $l$. So system (2.3.5) becomes

$\left\\{\begin{array}[]{l}\varepsilon^{m}\,\partial_{t}\Lambda_{\varepsilon}\,+\,\mathcal{A}\,{\rm
div}\,\boldsymbol{W}_{\varepsilon}\,=\,\varepsilon^{m}f_{\varepsilon}\\\\[4.30554pt]
\varepsilon^{m}\,\partial_{t}\boldsymbol{W}_{\varepsilon}\,+\,\nabla_{x}\Lambda_{\varepsilon}\,+\,\varepsilon^{m-1}\,\boldsymbol{e}_{3}\times\boldsymbol{W}_{\varepsilon}\,=\,\varepsilon^{m}\boldsymbol{G}_{\varepsilon}\,,\qquad\big{(}\boldsymbol{W}_{\varepsilon}\cdot\boldsymbol{n}\big{)}_{|\partial\Omega_{\varepsilon}}\,=\,0\,,\end{array}\right.$
(2.3.9)

where we have defined $f_{\varepsilon}\,:=\,{\rm
div}\,\boldsymbol{F}^{1}_{\varepsilon}\,+\,F^{2}_{\varepsilon}\;$ and
$\;\boldsymbol{G}_{\varepsilon}\,:=\,{\rm
div}\,\mathbb{G}^{1}_{\varepsilon}\,+\,\boldsymbol{G}^{2}_{\varepsilon}\,+\,\nabla_{x}G^{3}_{\varepsilon}$,
with

$\displaystyle\boldsymbol{F}^{1}_{\varepsilon}\,=\,\chi_{l}\,\boldsymbol{X}^{1}_{\varepsilon}\qquad\qquad\mbox{
and }\qquad\qquad
F^{2}_{\varepsilon}\,=\,\chi_{l}X^{2}_{\varepsilon}\,-\,\boldsymbol{X}^{1}_{\varepsilon}\cdot\nabla_{x}\chi_{l}\,+\,\mathcal{A}\,\boldsymbol{V}_{\varepsilon}\cdot\nabla_{x}\chi_{l}\,$
$\displaystyle\mathbb{G}^{1}_{\varepsilon}\,=\,\chi_{l}\,\mathbb{Y}^{1}_{\varepsilon}\;,\qquad\boldsymbol{G}^{2}_{\varepsilon}\,=\,\chi_{l}\,\boldsymbol{Y}^{2}_{\varepsilon}\,+\,\left(\frac{Z_{\varepsilon}}{\varepsilon^{m}}-Y^{3}_{\varepsilon}\right)\,\nabla_{x}\chi_{l}\,-\,^{t}\mathbb{Y}^{1}_{\varepsilon}\cdot\nabla_{x}\chi_{l}\qquad\mbox{
and }\qquad G^{3}_{\varepsilon}\,=\,\chi_{l}\,Y^{3}_{\varepsilon}\,.$

######  2.3.1.2 Uniform bounds

Here we use estimates of Subsection 2.2.1 in order to show uniform bounds for
the solutions and the data in the wave equation (2.3.9). We start by dealing
with the “unknowns” $\Lambda_{\varepsilon}$ and
$\boldsymbol{W}_{\varepsilon}$.

###### Lemma 2.3.1

Let $\bigl{(}\Lambda_{\varepsilon}\bigr{)}_{\varepsilon}$ and
$\bigl{(}\boldsymbol{W}_{\varepsilon}\bigr{)}_{\varepsilon}$ be defined as
above. Then, for any $T>0$ and all $\varepsilon\in\,]0,1]$, one has

$\|\Lambda_{\varepsilon}\|_{L^{\infty}_{T}(L^{2}+L^{5/3}+L^{1}+\mathcal{M}^{+})}\leq
c(l)\,,\quad\quad\|\boldsymbol{W}_{\varepsilon}\|_{L^{2}_{T}(L^{2}+L^{30/23})}\leq
c(l)\,.$

Proof:  We start by writing
$\boldsymbol{W}_{\varepsilon}\,=\,\boldsymbol{W}^{1}_{\varepsilon}\,+\,\boldsymbol{W}_{\varepsilon}^{2}$,
where

$\boldsymbol{W}^{1}_{\varepsilon}\,:=\,\chi_{l}\,[\varrho_{\varepsilon}]_{{\rm{ess}}}\,\boldsymbol{u}_{\varepsilon}\qquad\qquad\mbox{
and
}\qquad\qquad\boldsymbol{W}^{2}_{\varepsilon}\,:=\,\chi_{l}\,[\varrho_{\varepsilon}]_{{\rm{res}}}\,\boldsymbol{u}_{\varepsilon}\,.$

Since the density and temperature are uniformly bounded on the essential set,
by (2.2.12) we infer that $\boldsymbol{W}_{\varepsilon}^{1}$ is uniformly
bounded in $L_{T}^{2}(L^{2})$. On the other hand, by (2.2.8) and (2.2.12)
again, we easily deduce that $\boldsymbol{W}_{\varepsilon}^{2}$ is uniformly
bounded in $L_{T}^{2}(L^{p})$, where $3/5+1/6\,=\,1/p$. The claim about
$\boldsymbol{W}_{\varepsilon}$ is hence proved.

Let us now consider $\Lambda_{\varepsilon}$, defined in (2.3.8), i.e.

$\Lambda_{\varepsilon}=\Lambda_{\varepsilon,l}\,:=\,\chi_{l}\,Z_{\varepsilon}\,=\,\chi_{l}\,\mathcal{A}\,\varrho^{(1)}_{\varepsilon}\,+\,\chi_{l}\,\mathcal{B}\,\left(\varrho_{\varepsilon}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,-\,\frac{1}{\varepsilon^{m}}\Sigma_{\varepsilon}\right)\,.$

First of all, owing to the bounds
$\left\|\Sigma_{\varepsilon}\right\|_{L^{\infty}_{T}(\mathcal{M}^{+})}\,\leq\,C\,\|\sigma_{\varepsilon}\|_{\mathcal{M}^{+}_{t,x}}$
and (2.2.3), we have that

$\left\|\frac{1}{\varepsilon^{2m}}\,\chi_{l}\,\Sigma_{\varepsilon}\right\|_{L^{\infty}_{T}(\mathcal{M}^{+})}\leq
c(l)\,,$

uniformly in $\varepsilon>0$. Next, we can write the following decomposition:

$\varrho_{\varepsilon}\,\chi_{l}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,=\,\frac{1}{\varepsilon^{m}}\,\chi_{l}\,\left(\varrho_{\varepsilon}\,s(\varrho_{\varepsilon},\vartheta_{\varepsilon})\,-\,\widetilde{\varrho}_{\varepsilon}\,s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})\right)\,-\,\chi_{l}\,\varrho_{\varepsilon}^{(1)}\,s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})\,,$

where the latter term in the right-hand side is bounded in
$L^{\infty}_{T}(L^{2}+L^{5/3})$ in view of (2.2.23) and Proposition 2.1.7.
Concerning the former term, we can write it as

$\frac{1}{\varepsilon^{m}}\chi_{l}\left(\varrho_{\varepsilon}s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-\varrho_{\varepsilon}s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})\right)=\frac{1}{\varepsilon^{m}}\chi_{l}\bigl{[}\varrho_{\varepsilon}s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-\varrho_{\varepsilon}s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})\bigr{]}_{{\rm{ess}}}+\frac{1}{\varepsilon^{m}}\chi_{l}\bigl{[}\varrho_{\varepsilon}s(\varrho_{\varepsilon},\vartheta_{\varepsilon})\bigr{]}_{{\rm{res}}}\,,$
(2.3.10)

since the support of
$\chi_{l}\varrho_{\varepsilon}s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})$
is contained in the essential set by Proposition 2.1.7, for small enough
$\varepsilon$ (depending on the fixed $l>0$). By (2.2.7), the last term on the
right-hand side of (2.3.10) is uniformly bounded in $L^{\infty}_{T}(L^{1})$;
as for the first term, a Taylor expansion at the first order, together with
inequality (2.2.4) and the structural restrictions on $s$, immediately yields
its uniform boundedness in $L^{\infty}_{T}(L^{2})$.

The lemma is hence completely proved.

In the next lemma, we establish bounds for the source terms in the system of
acoustic waves (2.3.9).

###### Lemma 2.3.2

For any $T>0$ fixed, let us define the following spaces:

  * •

$\mathcal{X}_{1}\,:=\,L^{2}\Bigl{(}[0,T];\big{(}L^{2}+L^{1}+L^{3/2}+L^{30/23}+L^{30/29}\big{)}(\Omega)\Bigr{)}$;

  * •

$\mathcal{X}_{2}\,:=\,L^{2}\Bigl{(}[0,T];\big{(}L^{2}+L^{1}+L^{4/3}\big{)}(\Omega)\Bigr{)}$;

  * •

$\mathcal{X}_{3}\,:=\,\mathcal{X}_{2}\,+\,L^{\infty}\Bigl{(}[0,T];\big{(}L^{2}+L^{5/3}+L^{1}\big{)}(\Omega)\Bigr{)}$;

  * •

$\mathcal{X}_{4}\,:=\,L^{\infty}\Bigl{(}[0,T];\big{(}L^{2}+L^{5/3}+L^{1}+\mathcal{M}^{+}\big{)}(\Omega)\Bigr{)}$.

Then, for any $l>0$ fixed, one has the following bounds, uniformly in
$\varepsilon\in\,]0,1]$:

$\left\|\boldsymbol{F}^{1}_{\varepsilon}\right\|_{\mathcal{X}_{1}}\,+\,\left\|F^{2}_{\varepsilon}\right\|_{\mathcal{X}_{1}}\,+\,\left\|\mathbb{G}^{1}_{\varepsilon}\right\|_{\mathcal{X}_{2}}\,+\,\left\|\boldsymbol{G}^{2}_{\varepsilon}\right\|_{\mathcal{X}_{3}}\,+\,\left\|G^{3}_{\varepsilon}\right\|_{\mathcal{X}_{4}}\,\leq\,C(l)\,.$

In particular, the sequences $\bigl{(}f_{\varepsilon}\bigr{)}_{\varepsilon}$
and $\bigl{(}\boldsymbol{G}_{\varepsilon}\bigr{)}_{\varepsilon}$, defined in
system (2.3.9), are uniformly bounded in the space
$L^{2}\big{(}[0,T];W^{-1,1}(\Omega)\big{)}$, thus in
$L^{2}\big{(}[0,T];H^{-s}(\Omega)\big{)}$, for all $s>5/2$.

Proof:  We start by dealing with $\boldsymbol{F}^{1}_{\varepsilon}$. By
relations (2.2.10) and (2.2.14), it is easy to see that

$\left\|\frac{1}{\varepsilon^{m}}\,\chi_{l}\,\frac{\kappa(\vartheta_{\varepsilon})}{\vartheta_{\varepsilon}}\,\nabla_{x}\vartheta_{\varepsilon}\right\|_{L^{2}_{T}(L^{2}+L^{1})}\,\leq\,c(l)\,.$

On the other hand, the analysis of the term

$\varrho_{\varepsilon}\,\chi_{l}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,\boldsymbol{u}_{\varepsilon}$

is based on an analogous decomposition as used in the proof of Lemma 2.3.1 and
on uniform bounds of Paragraph 2.2.1.2: these facts allow us to bound it in
$L^{2}_{T}(L^{3/2}+L^{30/23}+L^{30/29})$.

The bounds for $F^{2}_{\varepsilon}$ easily follow from the previous ones and
Lemma 2.3.1 (indeed, the analysis for $\boldsymbol{W}_{\varepsilon}$ applies
also to the terms of the form
$\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}$ which appear in the
definition of $F^{2}_{\varepsilon}$), provided we show that

$\frac{1}{\varepsilon^{m}}\,\left|\chi_{l}\,\nabla_{x}\widetilde{\varrho}_{\varepsilon}\right|\,\leq\,C(l)\,.$

The previous bound immediately follows from the equation

$\nabla_{x}\widetilde{\varrho}_{\varepsilon}\,=\,\frac{\widetilde{\varrho}_{\varepsilon}}{\partial_{\varrho}p(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}\,\left(\varepsilon^{2(m-1)}\,\nabla_{x}F\,+\,\varepsilon^{m}\,\nabla_{x}G\right)\,,$

which derives from (2.1.17). Hence, by Proposition 2.1.7 and the definitions
given in (2.1.7), we get

$\frac{1}{\varepsilon^{m}}\,\left|\chi_{l}\,\nabla_{x}\widetilde{\varrho}_{\varepsilon}\right|\,\leq\,C(l)\,\left(\varepsilon^{2(m-1)-m}+1\right)\,\leq\,C(l)\,.$

The bound on $\mathbb{G}^{1}_{\varepsilon}$ is an immediate consequence of
(2.2.9) and (2.2.2).

Let us focus now on the term $\boldsymbol{G}^{2}_{\varepsilon}$. The control
of the term $\,{}^{t}\mathbb{Y}^{1}_{\varepsilon}\cdot\nabla_{x}\chi_{l}$ is
the same as above. The control of $\chi_{l}\boldsymbol{Y}^{2}_{\varepsilon}$,
instead, gives rise to a bound in $L^{\infty}_{T}(L^{2}+L^{5/3})$: this is
easily seen once we write

$\chi_{l}\,\boldsymbol{Y}^{2}_{\varepsilon}\,=\,\chi_{l}\,\varrho_{\varepsilon}^{(1)}\,\nabla_{x}G\,+\,\varepsilon^{m-2}\,\chi_{l}\,\varrho_{\varepsilon}^{(1)}\,\nabla_{x}F$

and we use (2.2.23) and (2.1.7). Finally, we have the equality

$\begin{split}\nabla_{x}\chi_{l}\,\left(\frac{Z_{\varepsilon}}{\varepsilon^{m}}-Y^{3}_{\varepsilon}\right)&=\nabla_{x}\chi_{l}\,\left(\frac{p(\varrho_{\varepsilon},\vartheta_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\right)\\\
&=\nabla_{x}\chi_{l}\left[\frac{p(\varrho_{\varepsilon},\vartheta_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\right]_{\rm{ess}}+\nabla_{x}\chi_{l}\left[\frac{p(\varrho_{\varepsilon},\vartheta_{\varepsilon})}{\varepsilon^{m}}\right]_{\rm{res}}.\end{split}$

The second term in the last line is uniformly bounded in
$L^{\infty}_{T}(L^{1})$, in view of (2.2.8). For the first term, instead, we
can proceed as in (2.3.10).

At this point, we switch our attention to the term $G^{3}_{\varepsilon}$,
whose analysis is more involved. By definition, we have

$\begin{split}\chi_{l}\,Y^{3}_{\varepsilon}&:=\frac{1}{\varepsilon^{m}}\,\chi_{l}\,\left(\mathcal{A}\,\frac{\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}}{\varepsilon^{m}}\,+\mathcal{B}\,\varrho_{\varepsilon}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,-\,\mathcal{B}\,\frac{1}{\varepsilon^{m}}\Sigma_{\varepsilon}\,-\,\frac{p(\varrho_{\varepsilon},\vartheta_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\right)\\\
&=\frac{1}{\varepsilon^{m}}\,\chi_{l}\,\left(\mathcal{A}\,\frac{\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}}{\varepsilon^{m}}\,+\mathcal{B}\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,-\frac{p(\varrho_{\varepsilon},\vartheta_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\right)\\\
&\qquad-\,\mathcal{B}\,\frac{1}{\varepsilon^{2m}}\,\chi_{l}\,\Sigma_{\varepsilon}\,+\,\mathcal{B}\,\chi_{l}\,\left(\frac{\varrho_{\varepsilon}-1}{\varepsilon^{m}}\right)\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,,\end{split}$

with $\mathcal{A}$ and $\mathcal{B}$ (see definition (2.3.4) above) such that

$\mathcal{A}\,+\,\mathcal{B}\,\partial_{\varrho}s(1,\overline{\vartheta})\,=\,\partial_{\varrho}p(1,\overline{\vartheta})\qquad\mbox{
and
}\qquad\mathcal{B}\,\partial_{\vartheta}s(1,\overline{\vartheta})\,=\,\partial_{\vartheta}p(1,\overline{\vartheta})\,.$

Next, we use a Taylor expansion to write

$\begin{split}s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})&=s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(1,\overline{\vartheta})+s(1,\overline{\vartheta})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})\\\
&=\partial_{\varrho}\,s(1,\overline{\vartheta})\,(\varrho_{\varepsilon}-1)+\partial_{\vartheta}\,s(1,\overline{\vartheta})\,(\vartheta_{\varepsilon}-\overline{\vartheta})+\frac{1}{2}\,{\rm
Hess}(s)[\xi_{1},\eta_{1}]\begin{pmatrix}\varrho_{\varepsilon}-1\\\
\vartheta_{\varepsilon}-\overline{\vartheta}\end{pmatrix}\cdot\begin{pmatrix}\varrho_{\varepsilon}-1\\\
\vartheta_{\varepsilon}-\overline{\vartheta}\end{pmatrix}\\\
&+\partial_{\varrho}\,s(1,\overline{\vartheta})\,(1-\widetilde{\varrho}_{\varepsilon})+\frac{1}{2}\,\partial_{\varrho\varrho}^{2}\,s(\xi_{2},\overline{\vartheta})\,(\widetilde{\varrho}_{\varepsilon}-1)^{2}\\\
&=\partial_{\varrho}\,s(1,\overline{\vartheta})\,(\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon})+\partial_{\vartheta}\,s(1,\overline{\vartheta})\,(\vartheta_{\varepsilon}-\overline{\vartheta})\\\
&+\frac{1}{2}\left({\rm
Hess}(s)[\xi_{1},\eta_{1}]\begin{pmatrix}\varrho_{\varepsilon}-1\\\
\vartheta_{\varepsilon}-\overline{\vartheta}\end{pmatrix}\cdot\begin{pmatrix}\varrho_{\varepsilon}-1\\\
\vartheta_{\varepsilon}-\overline{\vartheta}\end{pmatrix}+\partial_{\varrho\varrho}^{2}\,s(\xi_{2},\overline{\vartheta})\,(\widetilde{\varrho}_{\varepsilon}-1)^{2}\right)\,,\end{split}$

where $\xi_{1},\xi_{2},\eta_{1}$ are suitable points between $1$ and
$\varrho_{\varepsilon}$, $1$ and $\widetilde{\varrho}_{\varepsilon}$,
$\overline{\vartheta}$ and $\vartheta_{\varepsilon}$ respectively, and we have
denoted by ${\rm Hess}(s)[\xi,\eta]$ the Hessian matrix of the function $s$
with respect to its variables $\big{(}\varrho,\vartheta\big{)}$, computed at
the point $(\xi,\eta)$. Analogously, for the pressure term we have

$\begin{split}p(\varrho_{\varepsilon},\vartheta_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})&=\partial_{\varrho}\,p(1,\overline{\vartheta})\,(\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon})+\partial_{\vartheta}\,p(1,\overline{\vartheta})\,(\vartheta_{\varepsilon}-\overline{\vartheta})\\\
&+\frac{1}{2}\left({\rm
Hess}(p)[\xi_{3},\eta_{2}]\begin{pmatrix}\varrho_{\varepsilon}-1\\\
\vartheta_{\varepsilon}-\overline{\vartheta}\end{pmatrix}\cdot\begin{pmatrix}\varrho_{\varepsilon}-1\\\
\vartheta_{\varepsilon}-\overline{\vartheta}\end{pmatrix}+\partial_{\varrho\varrho}^{2}\,p(\xi_{4},\overline{\vartheta})\,(\widetilde{\varrho}_{\varepsilon}-1)^{2}\right)\,,\end{split}$

where $\xi_{3},\xi_{4},\eta_{2}$ are still between $1$ and
$\varrho_{\varepsilon}$, $1$ and $\widetilde{\varrho}_{\varepsilon}$,
$\overline{\vartheta}$ and $\vartheta_{\varepsilon}$ respectively. Using now
(2.3.4), we find that the first order terms cancel out, and we are left with

$\begin{split}\chi_{l}\,Y^{3}_{\varepsilon}&=\frac{\mathcal{B}}{2\varepsilon^{2m}}\,\chi_{l}\,\left(\,{\rm
Hess}(s)[\xi_{1},\eta_{1}]\begin{pmatrix}\varrho_{\varepsilon}-1\\\
\vartheta_{\varepsilon}-\overline{\vartheta}\end{pmatrix}\cdot\begin{pmatrix}\varrho_{\varepsilon}-1\\\
\vartheta_{\varepsilon}-\overline{\vartheta}\end{pmatrix}+\partial_{\varrho\varrho}^{2}\,s(\xi_{2},\overline{\vartheta})\,(\widetilde{\varrho}_{\varepsilon}-1)^{2}\right)\\\
&-\,\frac{1}{2\varepsilon^{2m}}\,\chi_{l}\,\left({\rm
Hess}(p)[\xi_{3},\eta_{2}]\begin{pmatrix}\varrho_{\varepsilon}-1\\\
\vartheta_{\varepsilon}-\overline{\vartheta}\end{pmatrix}\cdot\begin{pmatrix}\varrho_{\varepsilon}-1\\\
\vartheta_{\varepsilon}-\overline{\vartheta}\end{pmatrix}+\partial_{\varrho\varrho}^{2}\,p(\xi_{4},\overline{\vartheta})\,(\widetilde{\varrho}_{\varepsilon}-1)^{2}\right)\\\
&-\,\frac{\mathcal{B}}{\varepsilon^{2m}}\,\chi_{l}\,\Sigma_{\varepsilon}\,+\,\mathcal{B}\,\chi_{l}\,\left(\frac{\varrho_{\varepsilon}-1}{\varepsilon^{m}}\right)\,\frac{s(\varrho_{\varepsilon},\vartheta_{\varepsilon})-s(\widetilde{\varrho}_{\varepsilon},\overline{\vartheta})}{\varepsilon^{m}}\,.\end{split}$

Thanks to the uniform bounds established in Paragraph 2.2.1.2 and the
decomposition into essential and residual parts, the claimed control in the
space $\mathcal{X}_{4}$ follows.

######  2.3.1.3 Regularization and description of the oscillations

Following [41] and [40] (see also [27]), it is convenient to reformulate our
problem (NSF${}_{\varepsilon}^{1}$) to (NSF${}_{\varepsilon}^{4}$),
supplemented with complete slip boundary conditions (2.1.8) and (2.1.9), in a
completely equivalent way, in the domain

$\widetilde{\Omega}_{\varepsilon}\,:=\,{B}_{L_{\varepsilon}}(0)\times\mathbb{T}^{1}\,,\qquad\qquad\mbox{
with }\qquad\mathbb{T}^{1}\,:=\,[-1,1]/\sim\,,$

where $\sim$ denotes the equivalence relation which identifies $-1$ and $1$.
For this, it is enough to extend $\varrho_{\varepsilon}$,
$\vartheta_{\varepsilon}$, and $\boldsymbol{u}_{\varepsilon}^{h}$ as even
functions with respect to $x^{3}$, $u_{\varepsilon}^{3}$ and $G$ as odd
functions.

Correspondingly, we consider also the wave system (2.3.9) to be satisfied in
the new domain $\widetilde{\Omega}_{\varepsilon}$. It goes without saying that
the uniform bounds established above hold true also when replacing $\Omega$
with $\widetilde{\Omega}$, where we have set

$\widetilde{\Omega}\,:=\,\mathbb{R}^{2}\times\mathbb{T}^{1}\,.$

Notice that the wave speed in (2.3.9) is proportional to $\varepsilon^{-m}$,
while, in view of assumption (2.1.2), the domains
$\widetilde{\Omega}_{\varepsilon}$ are expanding at speed proportional to
$\varepsilon^{-m-\delta}$, for some $\delta>0$. Therefore, no interactions of
the acoustic-Poincaré waves with the boundary of
$\widetilde{\Omega}_{\varepsilon}$ take place (see also Remark 2.1.3 in this
respect), for any finite time $T>0$ and sufficiently small $\varepsilon>0$.
Thanks to this fact and the spatial localisation given by the cut-off function
$\chi_{l}$, we can assume that (2.3.9) is satisfied (still in a weak sense) on
the whole $\widetilde{\Omega}$.

Now, for any $M\in\mathbb{N}$ let us consider the low-frequency cut-off
operator ${S}_{M}$ of a Littlewood-Paley decomposition, as introduced in
(A.1.1). We define

$\Lambda_{\varepsilon,M}={S}_{M}\Lambda_{\varepsilon}\qquad\qquad\text{ and
}\qquad\qquad\boldsymbol{W}_{\varepsilon,M}={S}_{M}\boldsymbol{W}_{\varepsilon}\,.$

The following result holds true. Recall that we are omitting from the notation
the dependence of all quantities on $l>0$, due to multiplication by the cut-
off function $\chi_{l}$ fixed above.

###### Proposition 2.3.3

For any $T>0$, we have the following convergence properties, in the limit
$M\rightarrow+\infty$:

$\begin{split}&\sup_{0<\varepsilon\leq
1}\,\left\|\Lambda_{\varepsilon}-\Lambda_{\varepsilon,M}\right\|_{L^{\infty}([0,T];H^{s})}\longrightarrow
0\quad\forall s<-3/2-\delta\\\ &\sup_{0<\varepsilon\leq
1}\,\left\|\boldsymbol{W}_{\varepsilon}-\boldsymbol{W}_{\varepsilon,M}\right\|_{L^{\infty}([0,T];H^{s})}\longrightarrow
0\quad\forall s<-4/5-\delta\,,\end{split}$ (2.3.11)

for any $\delta>0$. Moreover, for any $M>0$, the couple
$(\Lambda_{\varepsilon,M},\boldsymbol{W}_{\varepsilon,M})$ satisfies the
approximate wave equations

$\left\\{\begin{array}[]{l}\varepsilon^{m}\,\partial_{t}\Lambda_{\varepsilon,M}\,+\,\mathcal{A}\,{\rm
div}\,\boldsymbol{W}_{\varepsilon,M}\,=\,\varepsilon^{m}\,f_{\varepsilon,M}\\\\[4.30554pt]
\varepsilon^{m}\,\partial_{t}\boldsymbol{W}_{\varepsilon,M}\,+\varepsilon^{m-1}\,e_{3}\times\boldsymbol{W}_{\varepsilon,M}+\,\nabla_{x}\Lambda_{\varepsilon,M}\,=\,\varepsilon^{m}\,\boldsymbol{G}_{\varepsilon,M}\,,\end{array}\right.$
(2.3.12)

where $(f_{\varepsilon,M})_{\varepsilon}$ and
$(\boldsymbol{G}_{\varepsilon,M})_{\varepsilon}$ are families of smooth (in
the space variables) functions satisfying, for any $s\geq 0$, the uniform
bounds

$\sup_{0<\varepsilon\leq
1}\,\left\|f_{\varepsilon,M}\right\|_{L^{2}([0,T];H^{s})}\,+\,\sup_{0<\varepsilon\leq
1}\,\left\|\boldsymbol{G}_{\varepsilon,M}\right\|_{L^{2}([0,T];H^{s})}\,\leq\,C(l,s,M)\,,$
(2.3.13)

where the constant $C(l,s,M)$ depends on the fixed values of $l>0$, $s\geq 0$
and $M>0$, but not on $\varepsilon>0$.

Proof:  Thanks to Lemma A.5, properties (2.3.11) are straightforward
consequences of the uniform bounds establish in Subsection 2.3.1.2.

Next, applying the operator ${S}_{M}$ to (2.3.9) immediately gives us system
(2.3.12), where we have set

$f_{\varepsilon,M}:={S}_{M}\left({\rm
div}\,\boldsymbol{F}^{1}_{\varepsilon}\,+\,F^{2}_{\varepsilon}\right)\qquad\text{
and }\qquad\boldsymbol{G}_{\varepsilon,M}:={S}_{M}\left({\rm
div}\,\mathbb{G}^{1}_{\varepsilon}\,+\,\boldsymbol{G}^{2}_{\varepsilon}\,+\,\nabla_{x}G^{3}_{\varepsilon}\right)\,.$

Thanks to Lemma 2.3.2 and (A.1.2) (and also Lemma A.5), it is easy to verify
inequality (2.3.13).

We also have an important decomposition for the approximated velocity fields
and their ${\rm curl}\,$.

###### Proposition 2.3.4

For any $M>0$ and any $\varepsilon\in\,]0,1]$, the following decompositions
hold true:

$\boldsymbol{W}_{\varepsilon,M}\,=\,\varepsilon^{m}\boldsymbol{t}_{\varepsilon,M}^{1}+\boldsymbol{t}_{\varepsilon,M}^{2}\qquad\mbox{
and }\qquad{\rm
curl}\,\boldsymbol{W}_{\varepsilon,M}=\varepsilon^{m}\boldsymbol{T}_{\varepsilon,M}^{1}+\boldsymbol{T}_{\varepsilon,M}^{2}\,,$

where, for any $T>0$ and $s\geq 0$, one has

$\displaystyle\left\|\boldsymbol{t}_{\varepsilon,M}^{1}\right\|_{L^{2}([0,T];H^{s})}+\left\|\boldsymbol{T}_{\varepsilon,M}^{1}\right\|_{L^{2}([0,T];H^{s})}\leq
C(l,s,M)$
$\displaystyle\left\|\boldsymbol{t}_{\varepsilon,M}^{2}\right\|_{L^{2}([0,T];H^{1})}+\left\|\boldsymbol{T}_{\varepsilon,M}^{2}\right\|_{L^{2}\left([0,T];L^{2}\right)}\leq
C(l)\,,$

for suitable positive constants $C(l,s,M)$ and $C(l)$, which are uniform with
respect to $\varepsilon\in\,]0,1]$.

Proof:  We start by defining

$\boldsymbol{t}_{\varepsilon,M}^{1}\,:=\,{S}_{M}\left(\chi_{l}\left(\frac{\varrho_{\varepsilon}-1}{\varepsilon^{m}}\right)\boldsymbol{u}_{\varepsilon}\right)\qquad\mbox{
and
}\qquad\boldsymbol{t}_{\varepsilon,M}^{2}\,:=\,{S}_{M}\left(\chi_{l}\boldsymbol{u}_{\varepsilon}\right)\,.$
(2.3.14)

Then, it is apparent that
$\boldsymbol{W}_{\varepsilon,M}\,=\,\varepsilon^{m}\boldsymbol{t}_{\varepsilon,M}^{1}\,+\,\boldsymbol{t}_{\varepsilon,M}^{2}$.
The decomposition of ${\rm curl}\,\boldsymbol{W}_{\varepsilon,M}$ is also easy
to get, if we set $\boldsymbol{T}_{\varepsilon,M}^{j}\,:=\,{\rm
curl}\,\boldsymbol{t}_{\varepsilon,M}^{j}$, for $j=1,2$. We have to prove
uniform bounds for all those terms. But this is an easy verification, thanks
to the $L^{\infty}_{T}(L^{5/3}_{\rm loc})$ bound on $R_{\varepsilon}$ and the
$L^{2}_{T}(H^{1}_{\rm loc})$ bound on $\boldsymbol{u}_{\varepsilon}$, for any
fixed time $T>0$ (recall the estimates obtained in Subsection 2.2.1 above).

On the one hand, for the estimate

$\left\|\boldsymbol{t}_{\varepsilon,M}^{1}\right\|_{L_{T}^{2}(H^{s})}+\left\|\boldsymbol{T}_{\varepsilon,M}^{1}\right\|_{L_{T}^{2}(H^{s})}\leq
C(l,s,M)\,,$

it is sufficient to employ relation (A.1.2) and Lemma A.5.

On the other hand, we have

$\begin{split}\left\|S_{M}\left(\chi_{l}\boldsymbol{u}_{\varepsilon}\right)\right\|_{H^{1}}^{2}&\leq
C\sum_{j=-1}^{M-1}2^{2j}\left\|\Delta_{j}\left(\chi_{l}\boldsymbol{u}_{\varepsilon}\right)\right\|_{L^{2}}^{2}\\\
&\leq
C\|\chi_{l}\boldsymbol{u}_{\varepsilon}\|_{L^{2}}^{2}+C\sum_{j=0}^{M-1}\left\|\Delta_{j}\nabla_{x}\left(\chi_{l}\boldsymbol{u}_{\varepsilon}\right)\right\|_{L^{2}}^{2}\leq
C(l)\,,\end{split}$

and the estimate for $\boldsymbol{T}_{\varepsilon,M}^{2}$ follows from
analogous computations. This completes the proof.

#####  2.3.2 Convergence of the non-linear convective term

In this subsection we show convergence of the convective term, by using a
compensated compactness argument. Namely, we manipulate this term, by
performing algebraic computations on the wave system formulated above. As a
consequence, we derive two key pieces of information: on the one hand, we see
that some non-linear terms are small remainders (in the sense specified by
relations (2.3.15) and (2.3.17) below); on the other hand, we derive a
compactness property for a new quantity, called $\gamma_{\varepsilon,M}$.

The first step is to reduce the study to the case of smooth vector fields
$\boldsymbol{W}_{\varepsilon,M}$.

###### Lemma 2.3.5

Let $T>0$. For any $\boldsymbol{\psi}\in
C_{c}^{\infty}\bigl{(}[0,T[\,\times\widetilde{\Omega};\mathbb{R}^{3}\bigr{)}$,
we have

$\lim_{M\rightarrow+\infty}\limsup_{\varepsilon\rightarrow
0^{+}}\left|\int_{0}^{T}\int_{\widetilde{\Omega}}\varrho_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}\,\,{\rm
d}x{\rm
d}t-\int_{0}^{T}\int_{\widetilde{\Omega}}\boldsymbol{W}_{\varepsilon,M}\otimes\boldsymbol{W}_{\varepsilon,M}:\nabla_{x}\boldsymbol{\psi}\,\,{\rm
d}x{\rm d}t\right|=0\,.$

Proof:  Let $\boldsymbol{\psi}\in
C_{c}^{\infty}\bigl{(}\mathbb{R}_{+}\times\widetilde{\Omega};\mathbb{R}^{3}\bigr{)}$,
with ${\rm Supp}\,\boldsymbol{\psi}\subset[0,T]\times K$, for some compact set
$K\subset\widetilde{\Omega}$. Then, we take $l>0$ in (2.3.7) so large that
$K\subset\widetilde{\mathbb{B}}_{l}\,:=\,B_{l}(0)\times\mathbb{T}^{1}$.
Therefore, using (2.2.22), we get

$\int_{0}^{T}\int_{\widetilde{\Omega}}\varrho_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}\,=\,\int_{0}^{T}\int_{K}(\chi_{l}\,\boldsymbol{u}_{\varepsilon})\otimes\boldsymbol{u}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}\,+\varepsilon^{m}\int_{0}^{T}\int_{K}R_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}\,.$

As a consequence of the uniform bounds
$\big{(}\boldsymbol{u}_{\varepsilon}\big{)}_{\varepsilon}\subset
L^{2}_{T}(L^{6}_{\rm loc})$ and
$\big{(}R_{\varepsilon}\big{)}_{\varepsilon}\subset L^{\infty}_{T}(L_{\rm
loc}^{5/3})$ (recall (2.2.23) above), the second integral in the right-hand
side is of order $\varepsilon^{m}$. As for the first one, using (2.3.14), we
can write

$\int_{0}^{T}\int_{K}(\chi_{l}\,\boldsymbol{u}_{\varepsilon})\otimes\boldsymbol{u}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}\,=\,\int_{0}^{T}\int_{K}\boldsymbol{t}^{2}_{\varepsilon,M}\otimes\boldsymbol{u}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}+\int_{0}^{T}\int_{K}\,({\rm
Id}\,-{S}_{M})(\chi_{l}\,\boldsymbol{u}_{\varepsilon})\otimes\boldsymbol{u}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}\,.$

Observe that, in view of characterisation (A.1.2), one has the property (see
also Lemma A.5)

$\left\|({\rm
Id}\,-{S}_{M})(\chi_{l}\,\boldsymbol{u}_{\varepsilon})\right\|_{L_{T}^{2}(L^{2})}\,\leq\,C\,2^{-M}\,\left\|\nabla_{x}(\chi_{l}\,\boldsymbol{u}_{\varepsilon})\right\|_{L_{T}^{2}(L^{2})}\,\leq\,C(l)\,2^{-M}\,.$

Therefore, it is enough to consider the first term in the right-hand side of
the last relation: we have

$\int_{0}^{T}\int_{K}\boldsymbol{t}^{2}_{\varepsilon,M}\otimes\boldsymbol{u}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}\,=\,\int_{0}^{T}\int_{K}\boldsymbol{t}^{2}_{\varepsilon,M}\otimes\boldsymbol{t}^{2}_{\varepsilon,M}:\nabla_{x}\boldsymbol{\psi}\,+\,\int_{0}^{T}\int_{K}\,\boldsymbol{t}^{2}_{\varepsilon,M}\otimes({\rm
Id}\,-{S}_{M})(\chi_{l}\,\boldsymbol{u}_{\varepsilon}):\nabla_{x}\boldsymbol{\psi}\,,$

where, for the same reason as before, we gather that

$\lim_{M\rightarrow+\infty}\limsup_{\varepsilon\rightarrow
0^{+}}\left|\int_{0}^{T}\int_{K}\boldsymbol{t}^{2}_{\varepsilon,M}\otimes({\rm
Id}\,-{S}_{M})(\chi_{l}\,\boldsymbol{u}_{\varepsilon}):\nabla_{x}\boldsymbol{\psi}\right|=0\,.$

It remains us to consider the integral

$\int_{0}^{T}\int_{K}\boldsymbol{t}^{2}_{\varepsilon,M}\otimes\boldsymbol{t}^{2}_{\varepsilon,M}:\nabla_{x}\boldsymbol{\psi}\,=\,\int_{0}^{T}\int_{K}\boldsymbol{W}_{\varepsilon,M}\otimes\boldsymbol{t}^{2}_{\varepsilon,M}:\nabla_{x}\boldsymbol{\psi}-\varepsilon^{m}\int_{0}^{T}\int_{K}\boldsymbol{t}^{1}_{\varepsilon,M}\otimes\boldsymbol{t}^{2}_{\varepsilon,M}:\nabla_{x}\boldsymbol{\psi}\,,$

where we notice that, owing to Proposition 2.3.4, the latter term in the
right-hand side is of order $\varepsilon^{m}$, so it vanishes at the limit. As
a last step, we write

$\int_{0}^{T}\int_{K}\boldsymbol{W}_{\varepsilon,M}\otimes\boldsymbol{t}^{2}_{\varepsilon,M}:\nabla_{x}\boldsymbol{\psi}\,=\,\int_{0}^{T}\int_{K}\boldsymbol{W}_{\varepsilon,M}\otimes\boldsymbol{W}_{\varepsilon,M}:\nabla_{x}\boldsymbol{\psi}\,-\,\varepsilon^{m}\int_{0}^{T}\int_{K}\boldsymbol{W}_{\varepsilon,M}\otimes\boldsymbol{t}^{1}_{\varepsilon,M}:\nabla_{x}\boldsymbol{\psi}\,.$

Using Lemma 2.3.1 together with Bernstein’s inequalities of Lemma A.2, we see
that the latter integral in the right-hand side is of order $\varepsilon^{m}$.
This concludes the proof of the lemma.

From now on, in order to avoid the appearance of (irrelevant) multiplicative
constants everywhere, we suppose that the torus $\mathbb{T}^{1}$ has been
normalised so that its Lebesgue measure is equal to $1$.

In view of the previous lemma and of Proposition 2.2.7, for any test-function

$\boldsymbol{\psi}\in
C_{c}^{\infty}\big{(}[0,T[\,\times\widetilde{\Omega};\mathbb{R}^{3}\big{)}\qquad\qquad\mbox{
such that }\qquad{\rm div}\,\boldsymbol{\psi}=0\quad\mbox{ and
}\quad\partial_{3}\boldsymbol{\psi}=0\,,$ (2.3.15)

we have to pass to the limit in the term

$\displaystyle-\int_{0}^{T}\int_{\widetilde{\Omega}}\boldsymbol{W}_{\varepsilon,M}\otimes\boldsymbol{W}_{\varepsilon,M}:\nabla_{x}\boldsymbol{\psi}\,$
$\displaystyle=\,\int_{0}^{T}\int_{\widetilde{\Omega}}{\rm
div}\,\left(\boldsymbol{W}_{\varepsilon,M}\otimes\boldsymbol{W}_{\varepsilon,M}\right)\cdot\boldsymbol{\psi}\,.$

Notice that the integration by parts above is well-justified, since all the
quantities inside the integrals are smooth. At this point, we observe that,
resorting to the notation in (OSC) presented in the introductory part, we can
write

$\int_{0}^{T}\int_{\widetilde{\Omega}}{\rm
div}\,\left(\boldsymbol{W}_{\varepsilon,M}\otimes\boldsymbol{W}_{\varepsilon,M}\right)\cdot\boldsymbol{\psi}\,=\,\int_{0}^{T}\int_{\mathbb{R}^{2}}\left(\mathcal{T}_{\varepsilon,M}^{1}+\mathcal{T}_{\varepsilon,M}^{2}\right)\cdot\boldsymbol{\psi}^{h}\,,$

where we have defined the terms

$\mathcal{T}^{1}_{\varepsilon,M}\,:=\,{\rm
div}_{h}\left(\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\otimes\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\right)\qquad\mbox{
and }\qquad\mathcal{T}^{2}_{\varepsilon,M}\,:=\,{\rm
div}_{h}\left(\langle\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}\otimes\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}\rangle\right)\,.$
(2.3.16)

So, it is enough to focus on each of them separately. For notational
convenience, from now on we will generically denote by
$\mathcal{R}_{\varepsilon,M}$ any remainder term, that is any term satisfying
the property

$\lim_{M\rightarrow+\infty}\limsup_{\varepsilon\rightarrow
0^{+}}\left|\int_{0}^{T}\int_{\widetilde{\Omega}}\mathcal{R}_{\varepsilon,M}\cdot\boldsymbol{\psi}\,\,{\rm
d}x{\rm d}t\right|=0\,,$ (2.3.17)

for all test functions $\boldsymbol{\psi}\in
C_{c}^{\infty}\bigl{(}[0,T[\,\times\widetilde{\Omega};\mathbb{R}^{3}\bigr{)}$
as in (2.3.15).

######  2.3.2.1 The analysis of the $\mathcal{T}_{\varepsilon,M}^{1}$ term

We start by dealing with $\mathcal{T}^{1}_{\varepsilon,M}$. Standard
computations give

$\displaystyle\mathcal{T}_{\varepsilon,M}^{1}\,$ $\displaystyle=\,{\rm
div}_{h}\left(\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\otimes\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\right)={\rm
div}_{h}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\,\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle+\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\cdot\nabla_{h}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle$
(2.3.18) $\displaystyle={\rm
div}_{h}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\,\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle+\frac{1}{2}\,\nabla_{h}\left(\left|\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\right|^{2}\right)+{\rm
curl}_{h}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\,\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle^{\perp}\,.$

Notice that we can forget about the second term, because it is a perfect
gradient and we are testing against divergence-free test functions. For the
first term, we take advantage of system (2.3.12): averaging the first equation
with respect to $x^{3}$ and multiplying it by
$\langle\boldsymbol{W}^{h}_{\varepsilon,M}\rangle$, we arrive at

${\rm
div}_{h}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\,\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\,=\,-\frac{\varepsilon^{m}}{\mathcal{A}}\partial_{t}\langle\Lambda_{\varepsilon,M}\rangle\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle+\frac{\varepsilon^{m}}{\mathcal{A}}\langle
f_{\varepsilon,M}^{h}\rangle\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\,=\,\frac{\varepsilon^{m}}{\mathcal{A}}\langle\Lambda_{\varepsilon,M}\rangle\partial_{t}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle+\mathcal{R}_{\varepsilon,M}\,.$

We remark that the term presenting the total derivative in time is in fact a
remainder. We use now the horizontal part of (2.3.12), where we take the
vertical average and then multiply by $\langle\Lambda_{\varepsilon,M}\rangle$:
we gather

$\displaystyle\frac{\varepsilon^{m}}{\mathcal{A}}\langle\Lambda_{\varepsilon,M}\rangle\partial_{t}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle$
$\displaystyle=-\frac{1}{\mathcal{A}}\langle\Lambda_{\varepsilon,M}\rangle\nabla_{h}\langle\Lambda_{\varepsilon,M}\rangle+\frac{\varepsilon^{m}}{\mathcal{A}}\langle\Lambda_{\varepsilon,M}\rangle\langle\boldsymbol{G}_{\varepsilon,M}^{h}\rangle-\frac{\varepsilon^{m-1}}{\mathcal{A}}\langle\Lambda_{\varepsilon,M}\rangle\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle^{\perp}$
$\displaystyle=-\frac{\varepsilon^{m-1}}{\mathcal{A}}\langle\Lambda_{\varepsilon,M}\rangle\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle^{\perp}-\frac{1}{2\mathcal{A}}\nabla_{h}\left(\left|\langle\Lambda_{\varepsilon,M}\rangle\right|^{2}\right)+\mathcal{R}_{\varepsilon,M}$
$\displaystyle=-\frac{\varepsilon^{m-1}}{\mathcal{A}}\langle\Lambda_{\varepsilon,M}\rangle\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle^{\perp}+\mathcal{R}_{\varepsilon,M}\,,$

where we repeatedly exploited the properties proved in Proposition 2.3.3 and
we included in the remainder term also the perfect gradient. Inserting this
relation into (2.3.18), we find

$\mathcal{T}_{\varepsilon,M}^{1}=\gamma_{\varepsilon,M}\,\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle^{\perp}+\mathcal{R}_{\varepsilon,M}\,,\qquad\qquad\mbox{
with }\qquad\gamma_{\varepsilon,M}:={\rm
curl}_{h}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\,-\,\frac{\varepsilon^{m-1}}{\mathcal{A}}\langle\Lambda_{\varepsilon,M}\rangle\,.$

We observe that, for passing to the limit in
$\mathcal{T}_{\varepsilon,M}^{1}$, there is no other way than finding some
strong convergence property for $\boldsymbol{W}_{\varepsilon,M}$. Such a
property is in fact hidden in the structure of the wave system (2.3.12): in
order to exploit it, some work on the term $\gamma_{\varepsilon,M}$ is needed.
We start by rewriting the vertical average of the first equation in (2.3.12)
as

$\frac{\varepsilon^{2m-1}}{\mathcal{A}}\,\partial_{t}\langle\Lambda_{\varepsilon,M}\rangle\,+\,\varepsilon^{m-1}{\rm
div}\,_{h}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\,=\,\frac{\varepsilon^{2m-1}}{\mathcal{A}}\,\langle
f_{\varepsilon,M}^{h}\rangle\,.$

On the other hand, taking the vertical average of the horizontal components of
(2.3.12) and then applying ${\rm curl}_{h}$, we obtain the relation

$\varepsilon^{m}\,\partial_{t}{\rm
curl}_{h}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\,+\varepsilon^{m-1}\,{\rm
div}_{h}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\,=\,\varepsilon^{m}{\rm
curl}_{h}\langle\boldsymbol{G}_{\varepsilon,M}^{h}\rangle\,.$

Summing up the last two equations, we discover that

$\partial_{t}\gamma_{\varepsilon,M}\,=\,{\rm
curl}_{h}\langle\boldsymbol{G}_{\varepsilon,M}^{h}\rangle\,-\,\frac{\varepsilon^{m-1}}{\mathcal{A}}\,\langle
f_{\varepsilon,M}^{h}\rangle\,.$ (2.3.19)

Thanks to estimate (2.3.13) in Proposition 2.3.3, we discover that (for any
$M>0$ fixed) the family
$\left(\partial_{t}\,\gamma_{\varepsilon,M}\right)_{\varepsilon}$ is uniformly
bounded (with respect to $\varepsilon$) in e.g. $L_{T}^{2}(L^{2})$. On the
other hand, thanks to Lemma 2.3.1 and Sobolev embeddings, we have that (for
any $M>0$ fixed) the sequence $(\gamma_{\varepsilon,M})_{\varepsilon}$ is
uniformly bounded (with respect to $\varepsilon$) in the space
$L_{T}^{2}(H^{1})$. Since the embedding $H_{\rm loc}^{1}\hookrightarrow L_{\rm
loc}^{2}$ is compact, the Aubin-Lions Theorem (see again the Appendix B)
implies that, for any $M>0$ fixed, the family
$(\gamma_{\varepsilon,M})_{\varepsilon}$ is compact in $L_{T}^{2}(L_{\rm
loc}^{2})$. Then, it converges strongly (up to extracting a subsequence) to a
tempered distribution $\gamma_{M}$ in the same space. Of course, by definition
of $\gamma_{\varepsilon,M}$ (and whenever $m>1$), this tells us that also
$\big{(}{\rm
curl}_{h}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle\big{)}_{\varepsilon}$
is compact in $L^{2}_{T}(L^{2}_{\rm loc})$.

Now, we have that $\gamma_{\varepsilon,M}$ converges strongly to $\gamma_{M}$
in $L_{T}^{2}(L_{\rm loc}^{2})$ and
$\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle$ converges weakly to
$\langle\boldsymbol{W}_{M}^{h}\rangle$ in $L_{T}^{2}(L_{\rm loc}^{2})$ (owing
to Proposition 2.3.4, for instance). Then, we deduce that

$\gamma_{\varepsilon,M}\langle\boldsymbol{W}_{\varepsilon,M}^{h}\rangle^{\perp}\longrightarrow\gamma_{M}\langle\boldsymbol{W}_{M}^{h}\rangle^{\perp}\qquad\text{
in
}\qquad\mathcal{D}^{\prime}\big{(}\mathbb{R}_{+}\times\mathbb{R}^{2}\big{)}\,.$

Observe that, by definition of $\gamma_{\varepsilon,M}$, we must have
$\gamma_{M}={\rm curl}_{h}\langle\boldsymbol{W}_{M}^{h}\rangle$. On the other
hand, by Proposition 2.3.4 and (2.3.14), we know that
$\langle\boldsymbol{W}_{M}^{h}\rangle=\langle{S}_{M}(\chi_{l}\boldsymbol{U}^{h})\rangle$.

In the end, we have proved that, for any $T>0$ and any test-function
$\boldsymbol{\psi}$ as in (2.3.15), one has the convergence (at any
$M\in\mathbb{N}$ fixed, when $\varepsilon\rightarrow 0^{+}$)

$\int_{0}^{T}\int_{\mathbb{R}^{2}}\mathcal{T}_{\varepsilon,M}^{1}\cdot\boldsymbol{\psi}^{h}\,\,{\rm
d}x^{h}\,{\rm d}t\,\longrightarrow\,\int^{T}_{0}\int_{\mathbb{R}^{2}}{\rm
curl}_{h}\langle{S}_{M}(\chi_{l}\boldsymbol{U}^{h})\rangle\;\langle{S}_{M}\big{(}\chi_{l}(\boldsymbol{U}^{h})^{\perp}\big{)}\rangle\cdot\boldsymbol{\psi}^{h}\,\,{\rm
d}x^{h}\,{\rm d}t\,.$ (2.3.20)

######  2.3.2.2 Dealing with the term $\mathcal{T}_{\varepsilon,M}^{2}$

Let us now consider the term $\mathcal{T}_{\varepsilon,M}^{2}$, defined in
(2.3.16). By the same computation as above, we infer that

$\displaystyle\mathcal{T}_{\varepsilon,M}^{2}\,$ $\displaystyle=\,\langle{\rm
div}_{h}(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h})\;\;\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}\rangle+\frac{1}{2}\,\langle\nabla_{h}|\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}|^{2}\rangle+\langle{\rm
curl}_{h}\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}\,\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}\right)^{\perp}\rangle\,.$
(2.3.21)

Let us now introduce now the quantities

$\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\,:=\,(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h})^{\perp}-\partial_{3}^{-1}\nabla_{h}^{\perp}\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{3}\qquad\mbox{
and }\qquad\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\,:=\,{\rm
curl}_{h}\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}\,.$

Then we can write

$\left({\rm
curl}\,\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}\right)^{h}\,=\,\partial_{3}\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\qquad\text{
and }\qquad\left({\rm
curl}\,\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}\right)^{3}\,=\,\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\,.$

In addition, from the momentum equation in (2.3.12), where we take the mean-
free part and then the ${\rm curl}\,$, we deduce the equations

$\begin{cases}\varepsilon^{m}\partial_{t}\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}-\varepsilon^{m-1}\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}=\varepsilon^{m}\left(\partial_{3}^{-1}{\rm
curl}\,\accentset{\approx}{\boldsymbol{G}}_{\varepsilon,M}\right)^{h}\\\\[4.30554pt]
\varepsilon^{m}\partial_{t}\accentset{\approx}{\omega}_{\varepsilon,M}^{3}+\varepsilon^{m-1}{\rm
div}_{h}\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}=\varepsilon^{m}\,{\rm
curl}_{h}\accentset{\approx}{\boldsymbol{G}}_{\varepsilon,M}^{h}\,.\end{cases}$
(2.3.22)

Making use of the relations above and of Propositions 2.3.3 and 2.3.4, we get

$\begin{split}{\rm
curl}_{h}\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}\;\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}\right)^{\perp}&=\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}\right)^{\perp}\,=\,\varepsilon\partial_{t}\\!\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\accentset{\approx}{\omega}_{\varepsilon,M}^{3}-\varepsilon\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\left(\left(\partial_{3}^{-1}{\rm
curl}\,\accentset{\approx}{\boldsymbol{G}}_{\varepsilon,M}\right)^{h}\right)^{\perp}\\\
&=-\varepsilon\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\partial_{t}\accentset{\approx}{\omega}_{\varepsilon,M}^{3}+\mathcal{R}_{\varepsilon,M}=\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\,{\rm
div}_{h}\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}+\mathcal{R}_{\varepsilon,M}\,.\end{split}$

Hence, including also the gradient term into the remainders, from (2.3.21) we
arrive at

$\displaystyle\mathcal{T}_{\varepsilon,M}^{2}\,$ $\displaystyle=\,\langle{\rm
div}_{h}\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}\,\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)\rangle+\mathcal{R}_{\varepsilon,M}$
$\displaystyle=\,\langle{\rm
div}\,\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)\rangle-\langle\partial_{3}\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{3}\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)\rangle+\mathcal{R}_{\varepsilon,M}\,.$

The second term on the right-hand side of the last line is actually another
remainder. Indeed, using the definition of the function
$\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}$ and the fact that the test
function $\boldsymbol{\psi}$ does not depend on $x^{3}$, one has

$\begin{split}\partial_{3}\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{3}\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)&=\partial_{3}\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{3}\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)\right)-\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{3}\,\partial_{3}\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)\\\
&=\mathcal{R}_{\varepsilon,M}-\frac{1}{2}\nabla_{h}\left|\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{3}\right|^{2}=\mathcal{R}_{\varepsilon,M}\,.\end{split}$

As for the first term, instead, we use the first equation in (2.3.12) to
obtain

$\begin{split}{\rm
div}\,\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)&=-\frac{\varepsilon^{m}}{\mathcal{A}}\partial_{t}\accentset{\approx}{\Lambda}_{\varepsilon,M}\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)+\mathcal{R}_{\varepsilon,M}\\\
&=\frac{\varepsilon^{m}}{\mathcal{A}}\accentset{\approx}{\Lambda}_{\varepsilon,M}\,\partial_{t}\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)+\mathcal{R}_{\varepsilon,M}\,.\end{split}$

Now, equations (2.3.12) and (2.3.22) immediately yield that

$\frac{\varepsilon^{m}}{\mathcal{A}}\accentset{\approx}{\Lambda}_{\varepsilon,M}\,\partial_{t}\left(\accentset{\approx}{\boldsymbol{W}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)=\mathcal{R}_{\varepsilon,M}-\frac{1}{\mathcal{A}}\accentset{\approx}{\Lambda}_{\varepsilon,M}\,\nabla_{h}\left(\accentset{\approx}{\Lambda}_{\varepsilon,M}\right)=\mathcal{R}_{\varepsilon,M}-\frac{1}{2\mathcal{A}}\nabla_{h}\left|\accentset{\approx}{\Lambda}_{\varepsilon,M}\right|^{2}=\mathcal{R}_{\varepsilon,M}\,.$

This relation finally implies that
$\mathcal{T}_{\varepsilon,M}^{2}\,=\,\mathcal{R}_{\varepsilon,M}$ is a
remainder, in the sense of relation (2.3.17): for any $T>0$ and any test-
function $\boldsymbol{\psi}$ as in (2.3.15), one has the convergence (at any
$M\in\mathbb{N}$ fixed, when $\varepsilon\rightarrow 0^{+}$)

$\int_{0}^{T}\int_{\mathbb{R}^{2}}\mathcal{T}_{\varepsilon,M}^{2}\cdot\boldsymbol{\psi}^{h}\,\,{\rm
d}x^{h}\,{\rm d}t\,\longrightarrow\,0\,.$ (2.3.23)

###### Remark 2.3.6

Due to the presence of the term $\boldsymbol{Y}^{2}_{\varepsilon}$ in (2.3.5),
the choice $m\geq 2$ is fundamental. However, as soon as $F=0$, our analysis
applies also in the case when $1<m<2$.

#####  2.3.3 The limit dynamics

With the convergences established in (2.2.25) to (2.2.27) and in Subsection
2.3.2, we can pass to the limit in equation (2.1.28). Since all the integrals
will be made on $\mathbb{R}^{2}$ (in view of the choice of the test functions
in (2.3.15) above), we can safely come back to the notation on $\Omega$
instead of $\widetilde{\Omega}$.

To begin with, we take a test-function $\boldsymbol{\psi}$ as in (2.3.15),
specifically

$\boldsymbol{\psi}=\big{(}\nabla_{h}^{\perp}\phi,0\big{)}\,,\qquad\qquad\mbox{
with }\qquad\phi\in
C_{c}^{\infty}\big{(}[0,T[\,\times\mathbb{R}^{2}\big{)}\,,\quad\phi=\phi(t,x^{h})\,.$

For such a $\boldsymbol{\psi}$, all the gradient terms vanish identically, as
well as all the contributions due to the vertical component of the equation.
Hence, after using also (2.1.17), equation (2.1.28) becomes

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{\Omega}$
$\displaystyle\left(-\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{h}\cdot\partial_{t}\boldsymbol{\psi}^{h}-\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{h}\otimes\boldsymbol{u}_{\varepsilon}^{h}:\nabla_{h}\boldsymbol{\psi}^{h}+\frac{1}{\varepsilon}\varrho_{\varepsilon}\big{(}\boldsymbol{u}_{\varepsilon}^{h}\big{)}^{\perp}\cdot\boldsymbol{\psi}^{h}\right)\,\,{\rm
d}x{\rm d}t$ (2.3.24)
$\displaystyle=\int_{0}^{T}\\!\\!\\!\int_{\Omega}\left(-\mathbb{S}(\vartheta_{\varepsilon},\nabla_{x}\boldsymbol{\boldsymbol{u}_{\varepsilon}}):\nabla_{x}\boldsymbol{\psi}+\frac{1}{\varepsilon^{2}}(\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon})\,\nabla_{x}F\cdot\boldsymbol{\psi}\right)\,\,{\rm
d}x{\rm
d}t+\int_{\Omega}\varrho_{0,\varepsilon}\boldsymbol{u}_{0,\varepsilon}\cdot\boldsymbol{\psi}(0,\cdot)\,{\rm
d}x\,.$

Making use of the uniform bounds of Subsection 2.2.1, we can pass to the limit
in the $\partial_{t}$ term, in the viscosity term and in the centrifugal
force. Moreover, our assumptions imply that
$\varrho_{0,\varepsilon}\boldsymbol{u}_{0,\varepsilon}\rightharpoonup\boldsymbol{u}_{0}$
in $L_{\rm loc}^{2}$.

Let us consider now the Coriolis term. We can write

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{\Omega}\frac{1}{\varepsilon}\varrho_{\varepsilon}\big{(}\boldsymbol{u}_{\varepsilon}^{h}\big{)}^{\perp}\cdot\boldsymbol{\psi}^{h}\,$
$\displaystyle=\,\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\frac{1}{\varepsilon}\langle\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{h}\rangle\cdot\nabla_{h}\phi\,=\,-\varepsilon^{m-1}\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\langle
R_{\varepsilon}\rangle\,\partial_{t}\phi\,-\,\varepsilon^{m-1}\int_{\mathbb{R}^{2}}\langle
R_{0,\varepsilon}\rangle\,\phi(0,\cdot)\,,$

which of course converges to $0$ when $\varepsilon\rightarrow 0^{+}$. Notice
that the second equality derives from the mass equation (2.1.27), tested
against $\phi$: namely,

$-\varepsilon^{m}\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\langle\frac{\varrho_{\varepsilon}-1}{\varepsilon^{m}}\rangle\,\partial_{t}\phi\,-\,\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\langle\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{h}\rangle\cdot\nabla_{h}\phi\,=\,\varepsilon^{m}\int_{\mathbb{R}^{2}}\langle\frac{\varrho_{0,\varepsilon}-1}{\varepsilon^{m}}\rangle\,\phi(0,\cdot)\,.$

It remains to deal with the convective term
$\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{h}\otimes\boldsymbol{u}_{\varepsilon}^{h}$.
For this, we take advantage of Lemma 2.3.5 and relations (2.3.20) and
(2.3.23). Next, we remark that, since $\boldsymbol{U}^{h}\in L^{2}_{T}(H_{\rm
loc}^{1})$ by (2.2.27), from (A.1.3) we gather the strong convergence
$S_{M}(\chi_{l}\boldsymbol{U}^{h})\longrightarrow\chi_{l}\boldsymbol{U}^{h}$
in $L_{T}^{2}(H^{s})$ for any $s<1$ and any $l>0$ fixed, in the limit for
$M\rightarrow+\infty$. Therefore, in the term on the right-hand side of
(2.3.20), we can perform equalities (2.3.18) backwards, and then pass to the
limit also for $M\rightarrow+\infty$. Using that $\chi_{l}\equiv 1$ on ${\rm
Supp}\,\boldsymbol{\psi}$ by construction, we finally get the convergence (for
$\varepsilon\rightarrow 0^{+}$)

$\int_{0}^{T}\int_{\Omega}\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{h}\otimes\boldsymbol{u}_{\varepsilon}^{h}:\nabla_{h}\boldsymbol{\psi}^{h}\,\longrightarrow\,\int_{0}^{T}\int_{\mathbb{R}^{2}}\boldsymbol{U}^{h}\otimes\boldsymbol{U}^{h}:\nabla_{h}\boldsymbol{\psi}^{h}\,.$

In the end, letting $\varepsilon\rightarrow 0^{+}$ in (2.3.24), we may infer
that

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\left(\boldsymbol{U}^{h}\cdot\partial_{t}\boldsymbol{\psi}^{h}+\boldsymbol{U}^{h}\otimes\boldsymbol{U}^{h}:\nabla_{h}\boldsymbol{\psi}^{h}\right)\,\,{\rm
d}x^{h}\,{\rm d}t$
$\displaystyle\qquad\qquad=\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\left(\mu(\overline{\vartheta})\nabla_{h}\boldsymbol{U}^{h}:\nabla_{h}\boldsymbol{\psi}^{h}-\delta_{2}(m)\langle\varrho^{(1)}\rangle\nabla_{h}F\cdot\boldsymbol{\psi}^{h}\right)\,\,{\rm
d}x^{h}\,{\rm
d}t-\int_{\mathbb{R}^{2}}\langle\boldsymbol{u}_{0}^{h}\rangle\cdot\boldsymbol{\psi}^{h}(0,\cdot)\,{\rm
d}x^{h}\,,$

where $\delta_{2}(m)=1$ if $m=2$, $\delta_{2}(m)=0$ otherwise. At this point,
Remark 2.2.8 applied to the case $m=2$ yields the equality
$\partial_{\varrho}p(1,\overline{\vartheta})\,\nabla_{h}\langle\widetilde{r}\rangle\,=\,\nabla_{h}F$.
Therefore, keeping in mind that $R=\varrho^{(1)}+\widetilde{r}$, we get

$\langle\varrho^{(1)}\rangle\nabla_{h}F\,=\,\langle
R\rangle\nabla_{h}F\,-\,\langle\widetilde{r}\rangle\nabla_{h}F\,=\,\langle
R\rangle\nabla_{h}F\,-\,\frac{\partial_{\varrho}p(1,\overline{\vartheta})}{2}\,\nabla_{h}\left|\langle\widetilde{r}\rangle\right|^{2}\,.$

Of course, the perfect gradient disappears from the weak formulation. Using
this observation in the target momentum equation written above, we finally
deduce (2.1.34). This completes the proof of Theorem 2.1.10, in the case when
$m\geq 2$ and $F\neq 0$.

When $m>1$ and $F=0$, most of the arguments above still apply. We refer to the
next section for more details.

####  2.4 Proof of the convergence in the case when $F=0$

In the present section we prove the convergence result in the case $F=0$. For
the sake of brevity, we focus on the case $m=1$, completing in this way the
proof to Theorem 2.1.12. The case $m>1$ follows by a similar analysis, using
at the end the compensated compactness argument depicted in Subsection 2.3.2
(recall also Remark 2.3.6 above).

#####  2.4.1 Analysis of the acoustic-Poincaré waves

We start by remarking that system (NSF${}_{\varepsilon}^{1}$) to
(NSF${}_{\varepsilon}^{4}$) can be recasted in the form (2.3.5), with $m=1$:
with the same notation introduced in Paragraph 2.3.1.1, and after setting
$X_{\varepsilon}\,:=\,{\rm
div}\,\boldsymbol{X}^{1}_{\varepsilon}\,+\,X^{2}_{\varepsilon}$ and
$\boldsymbol{Y}_{\varepsilon}\,:=\,{\rm
div}\,\mathbb{Y}^{1}_{\varepsilon}\,+\,\boldsymbol{Y}^{2}_{\varepsilon}\,+\,\nabla_{x}Y^{3}_{\varepsilon}$,
we have

$\left\\{\begin{array}[]{l}\varepsilon\,\partial_{t}Z_{\varepsilon}\,+\,\mathcal{A}\,{\rm
div}\,\boldsymbol{V}_{\varepsilon}\,=\,\varepsilon\,X_{\varepsilon}\\\\[4.30554pt]
\varepsilon\,\partial_{t}\boldsymbol{V}_{\varepsilon}\,+\,\nabla_{x}Z_{\varepsilon}\,+\,\,\boldsymbol{e}_{3}\times\boldsymbol{V}_{\varepsilon}\,=\,\varepsilon\,\boldsymbol{Y}_{\varepsilon}\,,\qquad\qquad\big{(}\boldsymbol{V}_{\varepsilon}\cdot\boldsymbol{n}\big{)}_{|\partial\Omega_{\varepsilon}}\,=\,0\,,\end{array}\right.$
(2.4.1)

where $\bigl{(}Z_{\varepsilon}\bigr{)}_{\varepsilon}$ and
$\bigl{(}\boldsymbol{V}_{\varepsilon}\bigr{)}_{\varepsilon}$ are defined as in
Paragraph 2.3.1.1. This system is supplemented with the initial datum
$\big{(}Z_{0,\varepsilon},\boldsymbol{V}_{0,\varepsilon}\big{)}$, where these
two functions are defined as in relation (2.3.6) above.

######  2.4.1.1 Uniform bounds and regularisation

In the next lemma, we establish uniform bounds for $Z_{\varepsilon}$ and
$\boldsymbol{V}_{\varepsilon}$. Its proof is an easy adaptation of the one
given in Lemma 2.3.1, hence omitted. One has to use the fact that, since
$F=0$, all the bounds obtained in the previous sections hold now on the whole
$\Omega_{\varepsilon}$, with constants which are uniform in
$\varepsilon\in\,]0,1]$; therefore, one can abstain from using the cut-off
functions $\chi_{l}$.

###### Lemma 2.4.1

For any $T>0$ and all $\varepsilon\in\,]0,1]$, we have

$\sup_{\varepsilon\in\,]0,1]}\|Z_{\varepsilon}\|_{L^{\infty}_{T}((L^{2}+L^{5/3}+L^{1}+\mathcal{M}^{+})(\Omega_{\varepsilon}))}\leq
c\,,\quad\quad\sup_{\varepsilon\in\,]0,1]}\|\boldsymbol{V}_{\varepsilon}\|_{L^{2}_{T}((L^{2}+L^{30/23})(\Omega_{\varepsilon}))}\leq
c\,.$

Now, we state the analogous of Lemma 2.3.2 for $m=1$ and $F=0$.

###### Lemma 2.4.2

For $\varepsilon\in\,]0,1]$, let us introduce the following spaces:

  1. (i)

$\mathcal{X}^{\varepsilon}_{1}\,:=\,L^{2}_{\rm
loc}\Bigl{(}\mathbb{R}_{+};\big{(}L^{2}+L^{1}+L^{3/2}+L^{30/23}+L^{30/29}\big{)}(\Omega_{\varepsilon})\Bigr{)}$;

  2. (ii)

$\mathcal{X}^{\varepsilon}_{2}\,:=\,L^{2}_{\rm
loc}\Bigl{(}\mathbb{R}_{+};\big{(}L^{2}+L^{1}+L^{4/3}\big{)}(\Omega_{\varepsilon})\Bigr{)}$;

  3. (iii)

$\mathcal{X}^{\varepsilon}_{3}\,:=\,L^{\infty}_{\rm
loc}\Bigl{(}\mathbb{R}_{+};\big{(}L^{2}+L^{5/3}\big{)}(\Omega_{\varepsilon})\Bigr{)}$;

  4. (iv)

$\mathcal{X}^{\varepsilon}_{4}\,:=\,L^{\infty}_{\rm
loc}\Bigl{(}\mathbb{R}_{+};\big{(}L^{2}+L^{5/3}+L^{1}+\mathcal{M}^{+}\big{)}(\Omega_{\varepsilon})\Bigr{)}$.

Then, one has the following uniform bound, for a constant $C>0$ independent of
$\varepsilon\in\,]0,1]$:

$\left\|\boldsymbol{X}^{1}_{\varepsilon}\right\|_{\mathcal{X}_{1}^{\varepsilon}}\,+\,\left\|X^{2}_{\varepsilon}\right\|_{\mathcal{X}_{1}^{\varepsilon}}\,+\,\left\|\mathbb{Y}^{1}_{\varepsilon}\right\|_{\mathcal{X}_{2}^{\varepsilon}}\,+\,\left\|\boldsymbol{Y}^{2}_{\varepsilon}\right\|_{\mathcal{X}_{3}^{\varepsilon}}\,+\,\left\|Y^{3}_{\varepsilon}\right\|_{\mathcal{X}_{4}^{\varepsilon}}\,\leq\,C\,.$

In particular, one has that the sequences $(X_{\varepsilon})_{\varepsilon}$
and $(\boldsymbol{Y}_{\varepsilon})_{\varepsilon}$, defined in system (2.4.1),
verify111For any $s\in\mathbb{R}$, we denote by $\lfloor s\rfloor$ the entire
part of $s$, i.e. the greatest integer smaller than or equal to $s$.

$\left\|X_{\varepsilon}\right\|_{L^{2}_{T}(H^{-\lfloor
s\rfloor-1}(\Omega_{\varepsilon}))}\,+\,\left\|\boldsymbol{Y}_{\varepsilon}\right\|_{L^{2}_{T}(H^{-\lfloor
s\rfloor-1}(\Omega_{\varepsilon}))}\,\leq\,C,$

for all $s>5/2$ and for a constant $C>0$ independent of
$\varepsilon\in\,]0,1]$.

Proof:  The proof follows the main lines of the proof of Lemma 2.3.2. Here, we
limit ourselves to point out that we have a slightly better control on
$\boldsymbol{Y}^{2}_{\varepsilon}\,=\,\varrho_{\varepsilon}^{(1)}\,\nabla_{x}G$,
whose boundedness in $\mathcal{X}^{\varepsilon}_{3}$ follows from (2.1.7) and
the estimate analogous to (2.2.23) for the case $F=0$.

The next step consists in regularising all the terms appearing in (2.4.1).
Here we have to pay attention: since the domains $\Omega_{\varepsilon}$ are
bounded, we cannot use the Littlewood-Paley operators $S_{M}$ directly. Rather
than multiplying by a cut-off function $\chi_{l}$ as done in the previous
section (a procedure which would create more complicated forcing terms in the
wave system), we use here the arguments of Chapter 8 of [39] (see also [44],
[71]), based on finite propagation speed properties for (2.4.1).

First of all, similarly to Paragraph 2.3.1.3 above, we extend our domains
$\Omega_{\varepsilon}$ and $\Omega$ by periodicity in the third variable and
denote

$\widetilde{\Omega}_{\varepsilon}\,:=\,{B}_{L_{\varepsilon}}(0)\times\mathbb{T}^{1}\qquad\qquad\mbox{
and
}\qquad\qquad\widetilde{\Omega}\,:=\,\mathbb{R}^{2}\times\mathbb{T}^{1}\,.$

Thanks to the complete slip boundary conditions (2.1.8) and (2.1.9), system
(NSF${}_{\varepsilon}^{1}$) to (NSF${}_{\varepsilon}^{4}$) can be equivalently
reformulated in $\widetilde{\Omega}_{\varepsilon}$. Analogously, the wave
system (2.4.1) can be recasted in $\widetilde{\Omega}_{\varepsilon}$ in a
completely equivalent way. From now on, we will focus on the equations
satisfied on the domain $\widetilde{\Omega}_{\varepsilon}$.

Next, we fix a smooth radially decreasing function
$\omega\in{C}^{\infty}_{c}(\mathbb{R}^{3})$, such that $0\leq\omega\leq 1$,
$\omega(x)=0$ for $|x|\geq 1$ and $\int_{\mathbb{R}^{3}}\omega(x)\,{\rm
d}x=1$. Next, we define the mollifying kernel
$\big{(}\omega_{M}\big{)}_{M\in\mathbb{N}}$ by the formula

$\omega_{M}(x)\,:=\,2^{3M}\,\,\omega\\!\left(2^{M}x\right)\qquad\qquad\text{for
any}\;\,M\in\mathbb{N}\,\;\text{and any}\;\,x\in\mathbb{R}^{3}\,.$

Then, for any tempered distribution $\mathfrak{S}=\mathfrak{S}(t,x)$ on
$\mathbb{R}_{+}\times\widetilde{\Omega}$ and any $M\in\mathbb{N}$, we define

$\mathfrak{S}_{M}\,:=\,\omega_{M}\,*\,\mathfrak{S}\,,$

where the convolution is taken only with respect to the space variables.
Applying the mollifier $\omega_{M}$ to (2.4.1), we deduce that
$Z_{\varepsilon,M}\,:=\,\omega_{M}*Z_{\varepsilon}$ and
$\boldsymbol{V}_{\varepsilon,M}\,:=\,\omega_{M}*\boldsymbol{V}_{\varepsilon}$
satisfy the regularised wave system

$\left\\{\begin{array}[]{l}\varepsilon\,\partial_{t}Z_{\varepsilon,M}\,+\,\mathcal{A}\,{\rm
div}\,\boldsymbol{V}_{\varepsilon,M}\,=\,\varepsilon\,X_{\varepsilon,M}\\\\[4.30554pt]
\varepsilon\,\partial_{t}\boldsymbol{V}_{\varepsilon,M}\,+\,\nabla_{x}Z_{\varepsilon,M}\,+\,\,\boldsymbol{e}_{3}\times\boldsymbol{V}_{\varepsilon,M}\,=\,\varepsilon\,\boldsymbol{Y}_{\varepsilon,M}\end{array}\right.$
(2.4.2)

in the domain $\mathbb{R}_{+}\times\widetilde{\Omega}_{\varepsilon,M}$, where
we have defined

$\widetilde{\Omega}_{\varepsilon,M}\,:=\,\left\\{x\in\widetilde{\Omega}_{\varepsilon}\;:\quad{\rm
dist}(x,\partial\widetilde{\Omega}_{\varepsilon})\geq 2^{-M}\right\\}\,.$
(2.4.3)

Since the mollification commutes with standard derivatives, we notice that
$X_{\varepsilon,M}\,=\,{\rm
div}\,\boldsymbol{X}^{1}_{\varepsilon,M}\,+\,X^{2}_{\varepsilon,M}$ and
$\boldsymbol{Y}_{\varepsilon,M}\,=\,{\rm
div}\,\mathbb{Y}^{1}_{\varepsilon,M}\,+\,\boldsymbol{Y}^{2}_{\varepsilon,M}\,+\,\nabla_{x}Y^{3}_{\varepsilon,M}$.
Moreover, system (2.4.2) is supplemented with the initial data

$Z_{0,\varepsilon,M}\,:=\,\omega_{M}*Z_{0,\varepsilon}\qquad\qquad\mbox{ and
}\qquad\qquad\boldsymbol{V}_{0,\varepsilon,M}\,:=\,\omega_{M}*\boldsymbol{V}_{0,\varepsilon}\,.$

In accordance with Lemmas 2.4.1 and 2.4.2, by standard properties of
mollifying kernels (see Theorem B.3), we get the following properties: for all
$k\in\mathbb{N}$, one has

$\displaystyle\left\|Z_{\varepsilon,M}\right\|_{L^{\infty}_{T}(H^{k}(\widetilde{\Omega}_{\varepsilon,M}))}\,+\,\left\|\boldsymbol{V}_{\varepsilon,M}\right\|_{L^{2}_{T}(H^{k}(\widetilde{\Omega}_{\varepsilon,M}))}\,\leq\,C(k,M)$
$\displaystyle\left\|X_{\varepsilon,M}\right\|_{L^{2}_{T}(H^{k}(\widetilde{\Omega}_{\varepsilon,M}))}\,+\,\left\|\boldsymbol{Y}_{\varepsilon,M}\right\|_{L^{2}_{T}(H^{k}(\widetilde{\Omega}_{\varepsilon,M}))}\,\leq\,C(k,M)\,,$

for some positive constants $C(k,M)$, only depending on the fixed $k$ and $M$.
Of course, the constants blow up when $M\rightarrow+\infty$, but they are
uniform for $\varepsilon\in\,]0,1]$.

We have the following statement, analogous to Proposition 2.3.4 above. Its
proof is also similar, hence omitted. In addition, we notice that the strong
convergence follows from standard properties of the mollifying kernel.

###### Proposition 2.4.3

For any $M>0$ and any $\varepsilon\in\,]0,1]$, we have

$\boldsymbol{V}_{\varepsilon,M}\,=\,\varepsilon\,\boldsymbol{v}_{\varepsilon,M}^{(1)}\,+\,\boldsymbol{v}_{\varepsilon,M}^{(2)}\,,$

together with the following bounds: for any $T>0$, any compact set
$K\subset\widetilde{\Omega}$ and any $s\in\mathbb{N}$, one has (for
$\varepsilon>0$ small enough, depending only on $K$)

$\displaystyle\left\|\boldsymbol{v}_{\varepsilon,M}^{(1)}\right\|_{L^{2}([0,T];H^{s}(K))}\,\leq\,C(K,s,M)\qquad\qquad\mbox{
and
}\qquad\qquad\left\|\boldsymbol{v}_{\varepsilon,M}^{(2)}\right\|_{L^{2}([0,T];H^{1}(K))}\,\leq\,C(K)\,,$

for suitable positive constants $C(K,s,M)$ and $C(K)$ depending only on the
quantities in the brackets, but uniform with respect to
$\varepsilon\in\,]0,1]$.

In particular, we deduce the following fact: for any $T>0$ and any compact
$K\subset\widetilde{\Omega}$, there exist $\varepsilon_{K}>0$ and
$M_{K}\in\mathbb{N}$ such that, for all $\varepsilon\in\,]0,\varepsilon_{K}]$
and all $M\geq M_{K}$, there are positive constants $C(K)$ and $C(K,M)$ for
which

$\left\|\boldsymbol{V}_{\varepsilon}\,-\,\boldsymbol{V}_{\varepsilon,M}\right\|_{L^{2}_{T}(L^{2}(K))}\,\leq\,C(K,M)\,\varepsilon\,+\,C(K)\,2^{-M}\,.$
(2.4.4)

######  2.4.1.2 Finite propagation speed and consequences

In this paragraph we show that, for the scopes of our study, we can safely
assume that system (2.4.2) is set in the whole $\widetilde{\Omega}$ and it is
supplemented with compactly supported initial data and external forces.

Take smooth initial data $\mathcal{Z}_{0}$ and $\boldsymbol{\mathcal{V}_{0}}$
and forces $\mathfrak{X}$ and $\boldsymbol{\mathcal{Y}}$. Consider, in
$\mathbb{R}_{+}\times\widetilde{\Omega}$, the wave system

$\left\\{\begin{array}[]{l}\varepsilon\,\partial_{t}\mathcal{Z}\,+\,\mathcal{A}\,{\rm
div}\,\boldsymbol{\mathcal{V}}\,=\,\varepsilon\,\mathfrak{X}\\\\[4.30554pt]
\varepsilon\,\partial_{t}\boldsymbol{\mathcal{V}}\,+\,\nabla_{x}\mathcal{Z}\,+\,\,\boldsymbol{e}_{3}\times\boldsymbol{\mathcal{V}}\,=\,\varepsilon\,\boldsymbol{\mathcal{Y}}\,,\end{array}\right.$
(2.4.5)

supplemented with initial data $\mathcal{Z}_{|t=0}\,=\,\mathcal{Z}_{0}$ and
$\boldsymbol{\mathcal{V}}_{|t=0}\,=\,\boldsymbol{\mathcal{V}_{0}}$.

System (2.4.5) is a symmetrizable (in the sense of Friedrichs) first-order
hyperbolic system with a skew-symmetric $0$-th order term. Therefore,
classical arguments based on energy methods (see e.g. Chapter 3 of [60] and
Chapter 7 of [1]) allow to establish the properties of finite propagation
speed and domain of dependence for solutions to (2.4.5).

Namely, set $\lambda\,:=\,\sqrt{\mathcal{A}}/\varepsilon$ to be the
propagation speed of acoustic-Poincaré waves. Let $\mathfrak{B}$ be a cylinder
included in $\widetilde{\Omega}$. Then one has the following two properties.

  1. (i)

_Domain of dependence_ : assume that

${\rm Supp}\,\mathcal{Z}_{0}\,,\;{\rm
Supp}\,\boldsymbol{\mathcal{V}_{0}}\,\subset\,\mathfrak{B}\,,\qquad\qquad\qquad{\rm
Supp}\,\mathfrak{X}(t)\,,\;{\rm
Supp}\,\boldsymbol{\mathcal{Y}}(t)\,\subset\,\mathfrak{B}\quad\mbox{ for a.a.
}t\in[0,T]\,;$

then the corresponding solution
$\big{(}\mathcal{Z},\boldsymbol{\mathcal{V}}\big{)}$ to (2.4.5) is
_identically zero_ outside the cone

$\Big{\\{}(t,x)\in\,]0,T[\,\times\,\widetilde{\Omega}\;:\quad{\rm
dist}\big{(}x,\mathfrak{B}\big{)}\,<\,\lambda\,t\Big{\\}}\,.$

  2. (ii)

_Finite propagation speed_ : define the set

$\mathfrak{B}_{\lambda T}\,:=\,\Big{\\{}x\in\widetilde{\Omega}\;:\quad{\rm
dist}\big{(}x,\mathfrak{B}\big{)}\,<\,\lambda\,T\Big{\\}}$

and assume that

${\rm Supp}\,\mathcal{Z}_{0}\,,\;{\rm
Supp}\,\boldsymbol{\mathcal{V}_{0}}\,\subset\,\mathfrak{B}_{\lambda
T}\,,\qquad\qquad\qquad{\rm Supp}\,\mathfrak{X}(t)\,,\;{\rm
Supp}\,\boldsymbol{\mathcal{Y}}(t)\,\subset\,\mathfrak{B}_{\lambda
T}\quad\mbox{ for a.a. }t\in[0,T]\,;$

then the solution $\big{(}\mathcal{Z},\boldsymbol{\mathcal{V}}\big{)}$ is
_uniquely determined_ by the data inside the cone

$\mathcal{C}_{\lambda
T}\,:=\,\Big{\\{}(t,x)\in\,]0,T[\,\times\mathfrak{B}_{\lambda T}\;:\quad{\rm
dist}\big{(}x,\partial\mathfrak{B}_{\lambda
T}\big{)}\,>\,\lambda\,t\Big{\\}}\,,$

and in particular in the space-time cylinder $\,]0,T[\,\times\,\mathfrak{B}$.

Next, fix any test-function $\boldsymbol{\psi}\in
C^{\infty}_{c}\big{(}\mathbb{R}_{+}\times\widetilde{\Omega};\mathbb{R}^{3}\big{)}$,
and let $T>0$ and the compact set $K\subset\widetilde{\Omega}$ be such that
${\rm Supp}\,\boldsymbol{\psi}\subset[0,T[\,\times K$. Take a cylindrical
neighborhood $\mathfrak{B}$ of $K$ in $\widetilde{\Omega}$. It goes without
saying that there exist an
$\varepsilon_{0}=\varepsilon_{0}(\mathfrak{B})\in\,]0,1]$ and a
$M_{0}=M_{0}(\mathfrak{B})\in\mathbb{N}$ such that

$\overline{\mathfrak{B}}\,\subset\subset\,\widetilde{\Omega}_{\varepsilon,M}\qquad\qquad\mbox{
for all }\qquad 0<\varepsilon\leq\varepsilon_{0}\quad\mbox{ and }\quad M\geq
M_{0}\,,$ (2.4.6)

where the set $\widetilde{\Omega}_{\varepsilon,M}$ has been defined in (2.4.3)
above. Take now a cut-off function $\mathfrak{h}\in
C^{\infty}_{c}(\widetilde{\Omega})$ such that $\mathfrak{h}\equiv 1$ on
$\mathfrak{B}$ (and hence on $K$), and solve problem (2.4.5) with compactly
supported data

$\mathcal{Z}_{0}\,=\,\mathfrak{h}\,Z_{0,\varepsilon,M}\,,\qquad\boldsymbol{\mathcal{V}_{0}}\,=\,\mathfrak{h}\,\boldsymbol{V}_{0,\varepsilon,M}\,,\qquad\mathfrak{X}\,=\,\mathfrak{h}\,X_{\varepsilon,M}\,,\qquad\boldsymbol{\mathcal{Y}}\,=\,\mathfrak{h}\,\boldsymbol{Y}_{\varepsilon,M}\,.$

We point out that all the data are now localised around the compact set $K$.
Owing to assumption (2.1.2), the domains $\widetilde{\Omega}_{\varepsilon,M}$
are expanding at speed proportional to $\varepsilon^{-(1+\delta)}$, whereas,
in view of finite propagation speed, the support of the solution is expanding
at speed proportional to $\varepsilon^{-1}$ (keep in mind also Remark 2.1.3).
Thus, thanks to the inclusion (2.4.6), the previous discussion implies that,
up to take a smaller $\varepsilon_{0}$, for any
$\varepsilon\leq\varepsilon_{0}$ the corresponding solution
$\big{(}\mathcal{Z},\boldsymbol{\mathcal{V}}\big{)}$ of (2.4.5) has support
inside a cylinder
$\widetilde{\mathbb{B}}_{L}\,:=\,B_{L}(0)\times\mathbb{T}^{1}\subset\widetilde{\Omega}_{\varepsilon}$,
for some $L=L(T,K,\lambda)>0$, and it must coincide with the solution
$\big{(}Z_{\varepsilon,M},\boldsymbol{V}_{\varepsilon,M}\big{)}$ of (2.4.2) on
the set $\,]0,T[\,\times\,\mathfrak{B}$, for all
$0<\varepsilon\leq\varepsilon_{0}$ and all $M\geq M_{0}$. In particular, for
all $0<\varepsilon\leq\varepsilon_{0}$ and all $M\geq M_{0}$ we have

$\mathcal{Z}\,\equiv\,Z_{\varepsilon,M}\quad\mbox{ and
}\quad\boldsymbol{\mathcal{V}}\,\equiv\,\boldsymbol{V}_{\varepsilon,M}\qquad\qquad\mbox{
on }\qquad{\rm Supp}\,\boldsymbol{\psi}\,.$

The previous argument shows that, without loss of generality, we can assume
that the regularised wave system (2.4.2) is verified on the whole
$\widetilde{\Omega}$, with compactly supported initial data and forces, and
with solutions supported on some cylinder $\widetilde{\mathbb{B}}_{L}$. In
particular, we can safely work with system (2.4.2) and its smooth solutions
$\big{(}Z_{\varepsilon,M},\boldsymbol{V}_{\varepsilon,M}\big{)}$ in the
computations below.

#####  2.4.2 Convergence of the convective term

Here we tackle the convergence of the convective term, employing again a
compensated compactness argument. The first step is to reduce the study to the
case of smooth vector fields $\boldsymbol{V}_{\varepsilon,M}$. Arguing as in
Lemma 2.3.5, and using Proposition 2.4.3 and property (2.4.4), one can easily
prove the following approximation result. Again, the proof is omitted.

###### Lemma 2.4.4

Let $T>0$. For any $\boldsymbol{\psi}\in
C_{c}^{\infty}\bigl{(}[0,T[\,\times\widetilde{\Omega};\mathbb{R}^{3}\bigr{)}$,
we have

$\lim_{M\rightarrow+\infty}\limsup_{\varepsilon\rightarrow
0^{+}}\left|\int_{0}^{T}\int_{\widetilde{\Omega}}\varrho_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}\,\,{\rm
d}x{\rm
d}t-\int_{0}^{T}\int_{\widetilde{\Omega}}\boldsymbol{V}_{\varepsilon,M}\otimes\boldsymbol{V}_{\varepsilon,M}:\nabla_{x}\boldsymbol{\psi}\,\
dxdt\right|=0\,.$

Assume now $\boldsymbol{\psi}\in
C_{c}^{\infty}\big{(}[0,T[\,\times\widetilde{\Omega};\mathbb{R}^{3}\big{)}$
such that ${\rm div}\,\boldsymbol{\psi}=0$ and
$\partial_{3}\boldsymbol{\psi}=0$. Thanks to the previous lemma, it is enough
to pass to the limit in the smooth term

$\displaystyle-\int_{0}^{T}\int_{\widetilde{\Omega}}\boldsymbol{V}_{\varepsilon,M}\otimes\boldsymbol{V}_{\varepsilon,M}:\nabla_{x}\boldsymbol{\psi}\,$
$\displaystyle=\,\int_{0}^{T}\int_{\widetilde{\Omega}}{\rm
div}\,\left(\boldsymbol{V}_{\varepsilon,M}\otimes\boldsymbol{V}_{\varepsilon,M}\right)\cdot\boldsymbol{\psi}\,=\,\int_{0}^{T}\int_{\mathbb{R}^{2}}\left(\mathcal{T}_{\varepsilon,M}^{1}+\mathcal{T}_{\varepsilon,M}^{2}\right)\cdot\boldsymbol{\psi}^{h}\,,$

where, for simplicity, we agree that the torus $\mathbb{T}^{1}$ has been
normalised so that its Lebesgue measure is equal to $1$ and, analogously to
(2.3.16), we have introduced the quantities

$\mathcal{T}^{1}_{\varepsilon,M}\,:=\,{\rm
div}_{h}\left(\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\otimes\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\right)\qquad\mbox{
and }\qquad\mathcal{T}^{2}_{\varepsilon,M}\,:=\,{\rm
div}_{h}\left(\langle\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\otimes\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\rangle\right)\,.$

We notice that the analysis of $\mathcal{T}_{\varepsilon,M}^{2}$ is similar to
the one performed in Paragraph 2.3.2.2, up to taking $m=1$ and replacing
$\boldsymbol{W}_{\varepsilon,M}$ and $\Lambda_{\varepsilon,M}$ by
$\boldsymbol{V}_{\varepsilon,M}$ and $Z_{\varepsilon,M}$ respectively. Indeed,
it deeply relies on system (2.3.22), which remains unchanged when $m=1$. Also
in this case, we find (2.3.23).

Therefore, we can focus on the term $\mathcal{T}_{\varepsilon,M}^{1}$ only.
Its study presents some differences with respect to Paragraph 2.3.2.1, so let
us give the full details. To begin with, like in (2.3.18), we have

$\mathcal{T}_{\varepsilon,M}^{1}\,=\,{\rm
div}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\;\;\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle+\frac{1}{2}\,\nabla_{h}\left(\left|\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\right|^{2}\right)+{\rm
curl}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\;\;\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle^{\perp}\,.$

Of course, we can forget about the second term, because it is a perfect
gradient. For the first term, we use system (2.4.2): averaging the first
equation with respect to $x^{3}$ and multiplying it by
$\langle\boldsymbol{V}^{h}_{\varepsilon,M}\rangle$, we get

${\rm
div}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\;\;\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,=\,-\frac{\varepsilon}{\mathcal{A}}\partial_{t}\langle
Z_{\varepsilon,M}\rangle\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle+\frac{\varepsilon}{\mathcal{A}}\langle
X_{\varepsilon,M}\rangle\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,=\,\frac{\varepsilon}{\mathcal{A}}\langle
Z_{\varepsilon,M}\rangle\partial_{t}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle+\mathcal{R}_{\varepsilon,M}\,.$

We now use the horizontal part of (2.4.2), multiplied by $\langle
Z_{\varepsilon,M}\rangle$, and we gather

$\begin{split}\frac{\varepsilon}{\mathcal{A}}\langle
Z_{\varepsilon,M}\rangle\partial_{t}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle&=-\frac{1}{\mathcal{A}}\langle
Z_{\varepsilon,M}\rangle\nabla_{h}\langle
Z_{\varepsilon,M}\rangle-\frac{1}{\mathcal{A}}\langle
Z_{\varepsilon,M}\rangle\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle^{\perp}+\frac{\varepsilon}{\mathcal{A}}\langle
Z_{\varepsilon,M}\rangle\langle\boldsymbol{Y}_{\varepsilon,M}^{h}\rangle\\\
&=-\frac{1}{\mathcal{A}}\langle
Z_{\varepsilon,M}\rangle\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle^{\perp}+\mathcal{R}_{\varepsilon,M}\,.\end{split}$

This latter relation yields that

$\mathcal{T}_{\varepsilon,M}^{1}\,=\,\left({\rm
curl}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle-\frac{1}{\mathcal{A}}\langle
Z_{\varepsilon,M}\rangle\right)\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle^{\perp}+\mathcal{R}_{\varepsilon,M}.$

Now we use the horizontal part of (2.4.2): averaging it with respect to the
vertical variable and applying the operator ${\rm curl}_{h}$, we find

$\varepsilon\,\partial_{t}{\rm
curl}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,+\,{\rm
div}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,=\,\varepsilon\,{\rm
curl}_{h}\langle\boldsymbol{Y}_{\varepsilon,M}^{h}\rangle\,.$

Taking the difference of this equation with the first one in (2.4.2), we
discover that

$\partial_{t}\gamma_{\varepsilon,M}\,=\,{\rm
curl}_{h}\langle\boldsymbol{Y}_{\varepsilon,M}^{h}\rangle\,-\,\frac{1}{\mathcal{A}}\,\langle
X_{\varepsilon,M}\rangle\,,\qquad\qquad\mbox{ with
}\qquad\gamma_{\varepsilon,M}:={\rm
curl}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,-\,\frac{1}{\mathcal{A}}\langle
Z_{\varepsilon,M}\rangle\,.$

An argument analogous to the one used after (2.3.19) above, based on Aubin-
Lions Theorem, shows also in this case that
$(\gamma_{\varepsilon,M})_{\varepsilon}$ is compact in $L_{T}^{2}(L_{\rm
loc}^{2})$. Then, this sequence converges strongly (up to extraction of a
suitable subsequence) to a tempered distribution $\gamma_{M}$ in the same
space.

Since $\gamma_{\varepsilon,M}\longrightarrow\gamma_{M}$ strongly in
$L_{T}^{2}(L_{\rm loc}^{2})$ and
$\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\rightharpoonup\langle\boldsymbol{V}_{M}^{h}\rangle$
weakly in $L_{T}^{2}(L_{\rm loc}^{2})$, we deduce that

$\gamma_{\varepsilon,M}\,\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle^{\perp}\,\longrightarrow\,\gamma_{M}\,\langle\boldsymbol{V}_{M}^{h}\rangle^{\perp}\qquad\text{
in
}\qquad\mathcal{D}^{\prime}\big{(}\mathbb{R}_{+}\times\mathbb{R}^{2}\big{)},$

where
$\langle\boldsymbol{V}_{M}^{h}\rangle=\langle{\omega}_{M}*\boldsymbol{U}^{h}\rangle$
and $\gamma_{M}={\rm
curl}_{h}\langle{\omega}_{M}*\boldsymbol{U}^{h}\rangle-(1/\mathcal{A})\langle
Z_{M}\rangle$. Notice that, in view of (2.2.24), (2.2.26), (2.2.3),
Proposition 2.2.5 and the definitions given in (2.3.4), we have

$Z_{M}\,=\,\partial_{\varrho}p(1,\overline{\vartheta})\,\omega_{M}*\varrho^{(1)}\,+\,\partial_{\vartheta}p(1,\overline{\vartheta})\,\omega_{M}*\Theta\,=\,\omega_{M}*q\,,$

where $q$ is the quantity defined in (2.2.39). Owing to the regularity of the
target velocity $\boldsymbol{U}^{h}$, we can pass to the limit also for
$M\rightarrow+\infty$, thus finding that

$\int^{T}_{0}\\!\\!\\!\int_{\widetilde{\Omega}}\varrho_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}\,\,{\rm
d}x{\rm
d}t\,\longrightarrow\,\int^{T}_{0}\\!\\!\\!\int_{\mathbb{R}^{2}}\big{(}\boldsymbol{U}^{h}\otimes\boldsymbol{U}^{h}:\nabla_{h}\boldsymbol{\psi}^{h}\,+\,\frac{1}{\mathcal{A}}\,q\,(\boldsymbol{U}^{h})^{\perp}\cdot\boldsymbol{\psi}^{h}\big{)}\,{\rm
d}x^{h}\,{\rm d}t\,,$ (2.4.7)

for all test functions $\boldsymbol{\psi}$ such that ${\rm
div}\,\boldsymbol{\psi}=0$ and $\partial_{3}\boldsymbol{\psi}=0$. Recall the
convention $|\mathbb{T}^{1}|=1$. Notice that, since
$\boldsymbol{U}^{h}=\nabla_{h}^{\perp}q$, the last term in the integral on the
right-hand side is actually zero.

#####  2.4.3 End of the proof

Thanks to the previous analysis, we are now ready to pass to the limit in
equation (2.1.28). As done above, we take a test-function $\boldsymbol{\psi}$
such that

$\boldsymbol{\psi}=\big{(}\nabla_{h}^{\perp}\phi,0\big{)}\,,\qquad\qquad\mbox{
with }\qquad\phi\in
C_{c}^{\infty}\big{(}[0,T[\,\times\mathbb{R}^{2}\big{)}\,,\quad\phi=\phi(t,x^{h})\,.$

Notice that ${\rm div}\,\boldsymbol{\psi}=0$ and
$\partial_{3}\boldsymbol{\psi}=0$. Then, all the gradient terms and all the
contributions coming from the vertical component of the momentum equation
vanish identically, when tested against such a $\boldsymbol{\psi}$. In
particular, we have

$\int_{0}^{T}\int_{\Omega}\frac{1}{\varepsilon}\varrho_{\varepsilon}\nabla_{x}G\cdot\boldsymbol{\psi}\,{\rm
d}x{\rm d}t\equiv 0\,.$

So, equation (2.1.28) reduces to222Remark that, in view of our choice of the
test-functions, we can safely come back to the notation on $\Omega$ instead of
$\widetilde{\Omega}$.

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{\Omega}\left(-\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\cdot\partial_{t}\boldsymbol{\psi}-\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}:\nabla\boldsymbol{\psi}+\frac{1}{\varepsilon}\varrho_{\varepsilon}\big{(}\boldsymbol{u}_{\varepsilon}^{h}\big{)}^{\perp}\cdot\boldsymbol{\psi}^{h}+\mathbb{S}(\vartheta_{\varepsilon},\nabla_{x}\boldsymbol{\boldsymbol{u}_{\varepsilon}}):\nabla_{x}\boldsymbol{\psi}\right)=\int_{\Omega}\varrho_{0,\varepsilon}\boldsymbol{u}_{0,\varepsilon}\cdot\boldsymbol{\psi}(0,\cdot)\,.$

As done in Subsection 2.3.3, we can limit ourselves to consider the rotation
and convective terms only. As for the former term, we start by using the mass
equation in (NSF${}_{\varepsilon}^{1}$) tested against $\phi$: we get
(recalling also (2.2.22))

$\begin{split}-\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\left(\langle
R_{\varepsilon}\rangle\,\partial_{t}\phi+\frac{1}{\varepsilon}\,\langle\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{h}\rangle\cdot\nabla_{h}\phi\right)=\int_{\mathbb{R}^{2}}\langle
R_{0,\varepsilon}\rangle\,\phi(0,\cdot)\,,\end{split}$

whence we deduce that

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{\Omega}\frac{1}{\varepsilon}\varrho_{\varepsilon}\big{(}\boldsymbol{u}_{\varepsilon}^{h}\big{)}^{\perp}\cdot\boldsymbol{\psi}^{h}\,$
$\displaystyle=\,\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\frac{1}{\varepsilon}\langle\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{h}\rangle\cdot\nabla_{h}\phi\,$
$\displaystyle=\,-\,\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\langle
R_{\varepsilon}\rangle\,\partial_{t}\phi\,-\,\int_{\mathbb{R}^{2}}\langle
R_{0,\varepsilon}\rangle\,\phi(0,\cdot)\,.$

Letting now $\varepsilon\rightarrow 0^{+}$, thanks to the previous relation
and (2.4.7), we finally gather

$\displaystyle-\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\left(\boldsymbol{U}^{h}\cdot\partial_{t}\nabla_{h}^{\perp}\phi+\boldsymbol{U}^{h}\otimes\boldsymbol{U}^{h}:\nabla_{h}(\nabla_{h}^{\perp}\phi)+\langle
R\rangle\,\partial_{t}\phi\right)\,\,{\rm d}x^{h}\,{\rm d}t$
$\displaystyle=-\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\mu(\overline{\vartheta})\nabla_{h}\boldsymbol{U}^{h}:\nabla_{h}(\nabla_{h}^{\perp}\phi)\,\,{\rm
d}x^{h}\,\,{\rm
d}t+\int_{\mathbb{R}^{2}}\left(\langle\boldsymbol{u}_{0}^{h}\rangle\cdot\nabla_{h}^{\perp}\phi(0,\cdot)+\langle
R_{0}\rangle\,\phi(0,\cdot)\right)\,\,{\rm d}x^{h}\,.$

Now, keeping in mind the relation for $R$ in Remark 2.1.13, we have

$\begin{split}R\,&=\,-\,\frac{1}{\beta}\,\big{(}\partial_{\vartheta}p(1,\overline{\vartheta})\,\Upsilon\,-\,\partial_{\vartheta}s(1,\overline{\vartheta})\,q\,-\,\partial_{\vartheta}s(1,\overline{\vartheta})\,G\big{)}\\\
&=-\frac{1}{\mathcal{A}}\left(\mathcal{B}\Upsilon-q-G\right)\,,\end{split}$

where we have also employed the definitions of $\mathcal{A}$ and $\mathcal{B}$
in (2.3.4).

Thus, we can write

$\begin{split}-\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\langle
R\rangle\,\partial_{t}\phi\,\,{\rm d}x^{h}\,{\rm
d}t&=\frac{1}{\mathcal{A}}\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\left(\mathcal{B}\,\langle\Upsilon\rangle-q+\frac{1}{2}\right)\,\partial_{t}\phi\,\,{\rm
d}x^{h}\,{\rm d}t\\\
&=\frac{1}{\mathcal{A}}\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\Big{(}\mathcal{B}\,\langle\Upsilon\rangle-q\Big{)}\,\partial_{t}\phi\,\,{\rm
d}x^{h}\,{\rm
d}t-\int_{\mathbb{R}^{2}}\frac{1}{2\mathcal{A}}\,\phi(0,\cdot)\,{\rm
d}x^{h}\,.\end{split}$

At the end, noticing that $\Upsilon$ solves (in the sense of distributions)
the transport-diffusion equation (2.1.40), we get

$\displaystyle-\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\left(\boldsymbol{U}^{h}\cdot\partial_{t}\nabla_{h}^{\perp}\phi+\boldsymbol{U}^{h}\otimes\boldsymbol{U}^{h}:\nabla_{h}(\nabla_{h}^{\perp}\phi)+\frac{1}{\mathcal{A}}\,q\,\partial_{t}\phi\right)\,\,{\rm
d}x^{h}\,{\rm d}t$
$\displaystyle=-\int_{0}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\mu(\overline{\vartheta})\nabla_{h}\boldsymbol{U}^{h}:\nabla_{h}(\nabla_{h}^{\perp}\phi)\,\,{\rm
d}x^{h}\,\,{\rm
d}t+\frac{1}{\mathcal{A}}\int_{0}^{T}\int_{\mathbb{R}^{2}}\langle
X\rangle\,\phi\,{\rm d}x^{h}\,{\rm d}t$
$\displaystyle+\int_{\mathbb{R}^{2}}\left(\langle\boldsymbol{u}_{0}^{h}\rangle\cdot\nabla_{h}^{\perp}\phi(0,\cdot)+\left(\langle
R_{0}\rangle+\frac{1}{2\mathcal{A}}\right)\phi(0,\cdot)\right)\,\,{\rm
d}x^{h}\,,$

where we have defined $X$ as in (2.1.42).

Theorem 2.1.12 is finally proved.

###  Chapter 3 On the influence of gravity

> In this chapter, we continue the investigation we began in Chapter 2,
> regarding the multi-scale analysis of mathematical models for geophysical
> flows. Our focus here is on the effect of gravity in regimes of _low
> stratification_ , but which go beyond the choice of the scaling that, in
> light of previous results, we call “critical”. For clarity of exposition, we
> consider the barotropic Navier-Stokes system with the Coriolis force, i.e.
>
> $\begin{cases}\partial_{t}\varrho+{\rm div}\,(\varrho\boldsymbol{u})=0\
> \\\\[8.61108pt] \partial_{t}(\varrho\boldsymbol{u})+{\rm
> div}\,(\varrho\boldsymbol{u}\otimes\boldsymbol{u})+\dfrac{\boldsymbol{e}_{3}\times\varrho\boldsymbol{u}}{Ro}\,+\dfrac{1}{Ma^{2}}\nabla_{x}p(\varrho)={\rm
> div}\,\mathbb{S}(\nabla_{x}\boldsymbol{u})+\dfrac{\varrho}{Fr^{2}}\nabla_{x}G\,,\end{cases}$
> (3.0.1)
>
> where we will take $Ro=\varepsilon$, $Ma=\varepsilon^{m}$ and
> $Fr=\varepsilon^{n}$ with $m$ and $n$ in suitable ranges (see (3.1.1) below
> in this respect).
>
> The results presented in this chapter are contained in [25].
>
> Before moving on, let us give a recapitulation of the contents of chapter.
>
> In Section 3.1 we collect our assumptions and we state the main theorems. In
> Section 3.2 we show the main consequences of the finite energy condition on
> the family of weak solutions we are going to consider. Namely, we derive
> uniform bounds in suitable norms, which allow us to extract weak-limit
> points, and we explore the constraints those limit points have to satisfy.
> In Sections 3.3 and 3.4, we complete the proof of our main results, showing
> convergence in the weak formulation of the equations in the cases $m>1$ and
> $m=1$, respectively.

####  3.1 Setting of the problem and main results

In this section, we introduce the primitive system and formulate our working
framework (see Subsection 3.1.1), then we state our main results (in
Subsection 3.1.2).

#####  3.1.1 The barotropic Navier-Stokes-Coriolis system

As already said in the introductory part, in this chapter we assumed that the
motion of the fluid is described by system (2.0.1) with constant density and
without the centrifugal force.

Thus, given a small parameter $\varepsilon\in\,]0,1]$, the barotropic Navier-
Stokes system with Coriolis and gravitational forces (see system (3.0.1) in
this respect) reads as follows:

$\displaystyle\partial_{t}\varrho_{\varepsilon}+{\rm
div}\,(\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon})=0$
(NSC${}_{\varepsilon}^{1}$)
$\displaystyle\partial_{t}(\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon})+{\rm
div}\,(\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon})+\frac{1}{\varepsilon}\,\boldsymbol{e}_{3}\times\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}+\frac{1}{\varepsilon^{2m}}\nabla_{x}p(\varrho_{\varepsilon})={\rm
div}\,\mathbb{S}(\nabla_{x}\boldsymbol{u}_{\varepsilon})+\frac{\varrho_{\varepsilon}}{\varepsilon^{2n}}\nabla_{x}G\,,$
(NSC${}_{\varepsilon}^{2}$)

where we recall that $m$ and $n$ are taken

$\mbox{ either }\qquad m\,>\,1\quad\mbox{ and }\quad
m\,<\,2\,n\,\leq\,m+1\,,\qquad\qquad\mbox{ or }\qquad m\,=\,1\quad\mbox{ and
}\quad\frac{1}{2}\,<\,n\,<\,1\,.$ (3.1.1)

As in the previous Chapter 2, here the unknowns in equations
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$) are the density
$\varrho_{\varepsilon}=\varrho_{\varepsilon}(t,x)\geq 0$ of the fluid and its
velocity field
$\boldsymbol{u}_{\varepsilon}=\boldsymbol{u}_{\varepsilon}(t,x)\in\mathbb{R}^{3}$,
where $t\in\mathbb{R}_{+}$ but now
$x\in\Omega:=\mathbb{R}^{2}\times\;]0,1[\,$. The viscous stress tensor in
(NSC${}_{\varepsilon}^{2}$) is given by Newton’s rheological law, that we
recall here,

$\mathbb{S}(\nabla_{x}\boldsymbol{u}_{\varepsilon})=\mu\left(\nabla_{x}\boldsymbol{u}_{\varepsilon}+\,^{t}\nabla_{x}\boldsymbol{u}_{\varepsilon}-\frac{2}{3}{\rm
div}\,\boldsymbol{u}_{\varepsilon}\,{\rm Id}\,\right)+\eta\,{\rm
div}\,\boldsymbol{u}_{\varepsilon}\,{\rm Id}\,\,,$ (3.1.2)

where $\mu>0$ and $\eta\geq 0$ now don’t depend on the temperature variations.
As for the gravitational force, we recall its formulation (see (2.1.7) in this
regard):

$G(x)=-x^{3}\,.$ (3.1.3)

The precise expression of $G$ will be useful in some computations below,
although some generalisations are certainly possible.

As done in the previous Chapter 2, the system is supplemented with _complete
slip boundary conditions_ , namely

$\displaystyle\big{(}\boldsymbol{u}_{\varepsilon}\cdot\boldsymbol{n}\big{)}_{|\partial\Omega}=0\quad$
$\displaystyle\mbox{ and
}\quad\bigl{(}[\mathbb{S}(\nabla_{x}\boldsymbol{u}_{\varepsilon})\boldsymbol{n}]\times\boldsymbol{n}\bigr{)}_{|\partial\Omega}=0\,,$
(3.1.4)

where $\boldsymbol{n}$ denotes the outer normal to the boundary
$\partial\Omega\,=\,\\{x_{3}=0\\}\cup\\{x_{3}=1\\}$. Notice that this is a
true simplification, because it avoids complications due to the presence of
Ekman boundary layers, when passing to the limit $\varepsilon\rightarrow
0^{+}$.

###### Remark 3.1.1

As it is well-known (see e.g. Subsection 2.3.1.3 and [27]), the equations
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$), supplemented by the
complete slip boundary boundary conditions (3.1.4), can be recasted as a
periodic problem with respect to the vertical variable, in the new domain

$\Omega\,=\,\mathbb{R}^{2}\,\times\,\mathbb{T}^{1}\,,\qquad\qquad\mbox{ with
}\qquad\mathbb{T}^{1}\,:=\,[-1,1]/\sim\,,$

where $\sim$ denotes the equivalence relation which identifies $-1$ and $1$.
Indeed, the equations are invariant if we extend $\varrho$ and
$\boldsymbol{u}^{h}$ as even functions with respect to $x^{3}$, and $u^{3}$ as
an odd function.

In what follows, we will always assume that such modifications have been
performed on the initial data, and that the respective solutions keep the same
symmetry properties.

Now we need to impose structural restrictions on the pressure function $p$. We
assume that

$p\in C^{1}[0,\infty)\cap C^{2}(0,\infty),\qquad p(0)=0,\qquad
p^{\prime}(\varrho)>0\quad\mbox{ for all }\,\varrho\geq 0\,.$ (3.1.5)

Additionally to (3.1.5), we require that (remember also Remark 2.1.4)

$\text{there exists}\quad\;\gamma\,>\,\frac{3}{2}\quad\mbox{ such that
}\qquad\lim\limits_{\varrho\to+\infty}\frac{p^{\prime}(\varrho)}{\varrho^{\gamma-1}}=p_{\infty}>0\,.$
(3.1.6)

Without loss of generality, we can suppose that $p$ has been renormalised so
that $p^{\prime}(1)=1$.

###### Remark 3.1.2

For a more detailed discussion about the choice
$\gamma>\overline{\gamma}:=3/2$, which is fundamental for the existence
theory, we address the reader to [42] by Feireisl, Novotný and Petzeltová, and
references therein. In particular, we remark that in two space dimensions
$\overline{\gamma}$ can be improved up to $1$.

######  3.1.1.1 Equilibrium states

Next, we focus our attention on the so-called _equilibrium states_. For each
value of $\varepsilon\in\,]0,1]$ fixed, the equilibria of system
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$) consist of static
densities $\widetilde{\varrho}_{\varepsilon}$ satisfying

$\nabla_{x}p(\widetilde{\varrho}_{\varepsilon})=\varepsilon^{2(m-n)}\widetilde{\varrho}_{\varepsilon}\nabla_{x}G\qquad\mbox{
in }\;\Omega\,.$ (3.1.7)

Equation (3.1.7) identifies $\widetilde{\varrho}_{\varepsilon}$ up to an
additive constant: taking the target density to be $1$, we get

$H^{\prime}(\widetilde{\varrho}_{\varepsilon})=\,\varepsilon^{2(m-n)}G+H^{\prime}(1)\,,\qquad\qquad\mbox{
where }\qquad H(\varrho)=\varrho\int_{1}^{\varrho}\frac{p(z)}{z^{2}}{\rm
d}z\,.$ (3.1.8)

Notice that relation (3.1.8) implies that

$H^{\prime\prime}(\varrho)=\frac{p^{\prime}(\varrho)}{\varrho}\quad\text{ and
}\quad H^{\prime\prime}(1)=1\,.$

Therefore, we infer that, whenever $m\geq 1$ and $m>n$ as in the present
chapter, for any $x\in\Omega$ one has
$\widetilde{\varrho}_{\varepsilon}(x)\longrightarrow 1$ in the limit
$\varepsilon\rightarrow 0^{+}$. More precisely, the next statement collects
all the necessary properties of the static states. It corresponds to Lemma
2.1.5 and Proposition 2.1.7 of Chapter 2.

###### Proposition 3.1.3

Let the gravitational force $G$ be given by (3.1.3). Let
$\bigl{(}\widetilde{\varrho}_{\varepsilon}\bigr{)}_{0<\varepsilon\leq 1}$ be a
family of static solutions to equation (3.1.7) in $\Omega$.

Then, there exist an $\varepsilon_{0}>0$ and a $0<\rho_{*}<1$ such that
$\widetilde{\varrho}_{\varepsilon}\geq\rho_{*}$ for all
$\varepsilon\in\,]0,\varepsilon_{0}]$ and all $x\in\Omega$. In addition, for
any $\varepsilon\in\,]0,\varepsilon_{0}]$, one has:

$\left|\widetilde{\varrho}_{\varepsilon}(x)\,-\,1\right|\,\leq\,C\,\varepsilon^{2(m-n)}\,,$

for a constant $C>0$ which is uniform in $x\in\Omega$ and in
$\varepsilon\in\,]0,1]$.

Without loss of any generality, we can assume that $\varepsilon_{0}=1$ in
Proposition 3.1.3.

In light of this analysis, it is natural to try to solve system
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$) in $\Omega$,
supplemented with the _far field conditions_

$\varrho_{\varepsilon}\longrightarrow\widetilde{\varrho}_{\varepsilon}\qquad\mbox{
and }\qquad\boldsymbol{u}_{\varepsilon}\longrightarrow 0\qquad\qquad\text{ as
}\quad|x|\rightarrow+\infty\,.$ (3.1.9)

######  3.1.1.2 Initial data and finite energy weak solutions

In view of the boundary conditions (3.1.9) “at infinity”, we assume that the
initial data are close (in a suitable sense) to the equilibrium states
$\widetilde{\varrho}_{\varepsilon}$ that we have just identified. Namely, we
consider initial densities of the following form:

$\varrho_{0,\varepsilon}=\widetilde{\varrho}_{\varepsilon}+\varepsilon^{m}\varrho_{0,\varepsilon}^{(1)}.$
(3.1.10)

For later use, let us introduce also the following decomposition of the
initial densities:

$\varrho_{0,\varepsilon}\,=\,1\,+\,\varepsilon^{2(m-n)}\,R_{0,\varepsilon}\qquad\qquad\mbox{
with }\qquad
R_{0,\varepsilon}\,=\,\widetilde{r}_{\varepsilon}\,+\,\varepsilon^{2n-m}\,\varrho_{0,\varepsilon}^{(1)}\,,\qquad\widetilde{r}_{\varepsilon}\,:=\,\frac{\widetilde{\varrho}_{\varepsilon}-1}{\varepsilon^{2(m-n)}}\,,$
(3.1.11)

where again $\widetilde{r}_{\varepsilon}$ is a datum of the system.

We suppose the density perturbations $\varrho_{0,\varepsilon}^{(1)}$ to be
measurable functions and satisfy the control

$\displaystyle\sup_{\varepsilon\in\,]0,1]}\left\|\varrho_{0,\varepsilon}^{(1)}\right\|_{(L^{2}\cap
L^{\infty})(\Omega)}\,\leq\,c\,,$ (3.1.12)

together with the “mean-free condition”

$\int_{\Omega}\varrho_{0,\varepsilon}^{(1)}\,{\rm d}x=0\,.$

As for the initial velocity fields, we assume the following uniform bound

$\sup_{\varepsilon\in\,]0,1]}\left\|\sqrt{\widetilde{\varrho}_{\varepsilon}}\boldsymbol{u}_{0,\varepsilon}\right\|_{L^{2}(\Omega)}\,\leq\,c\,\quad\text{which
also
implies}\quad\sup_{\varepsilon\in\,]0,1]}\,\left\|\boldsymbol{u}_{0,\varepsilon}\right\|_{L^{2}(\Omega)}\,\leq\,c\,.$
(3.1.13)

Thanks to the previous uniform estimates, up to extraction, we can identify
the limit points

$\displaystyle\varrho^{(1)}_{0}\,:=\,\lim_{\varepsilon\rightarrow
0}\varrho^{(1)}_{0,\varepsilon}\qquad\text{ weakly-$\ast$ in }$
$\displaystyle\qquad L^{\infty}\cap L^{2}$ (3.1.14)
$\displaystyle\boldsymbol{u}_{0}\,:=\,\lim_{\varepsilon\rightarrow
0}\boldsymbol{u}_{0,\varepsilon}\qquad\text{ weakly in }$ $\displaystyle\qquad
L^{2}\,.$ (3.1.15)

At this point, let us specify better what we mean by _finite energy weak
solution_ (see [39] for details).

###### Definition 3.1.4

Let $\Omega=\mathbb{R}^{2}\times\,]0,1[$. Fix $T>0$ and $\varepsilon>0$. Let
$(\varrho_{0,\varepsilon},\boldsymbol{u}_{0,\varepsilon})$ be an initial datum
satisfying (3.1.10) to (3.1.13). We say that the couple
$(\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon})$ is a _finite energy
weak solution_ of the system
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$) in
$\,]0,T[\,\times\Omega$, supplemented with the boundary conditions (3.1.4) and
far field conditions (3.1.9), related to the initial datum
$(\varrho_{0,\varepsilon},\boldsymbol{u}_{0,\varepsilon})$, if the following
conditions hold:

  1. (i)

the functions $\varrho_{\varepsilon}$ and $\boldsymbol{u}_{\varepsilon}$
belong to the class

$\varrho_{\varepsilon}\geq
0\,,\;\;\;\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}\,\in
L^{\infty}\big{(}\,]0,T[\,;L^{2}+L^{\gamma}(\Omega)\big{)}\,,\;\;\;\boldsymbol{u}_{\varepsilon}\in
L^{2}\big{(}\,]0,T[\,;H^{1}(\Omega)\big{)},\;\;\;\big{(}\boldsymbol{u}_{\varepsilon}\cdot\boldsymbol{n}\big{)}_{|\partial\Omega}=0;$

  2. (ii)

the equations have to be satisfied in a distributional sense:

$-\int_{0}^{T}\int_{\Omega}\left(\varrho_{\varepsilon}\partial_{t}\varphi+\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\cdot\nabla_{x}\varphi\right)\,{\rm
d}x{\rm d}t=\int_{\Omega}\varrho_{0,\varepsilon}\varphi(0,\cdot)\,{\rm d}x$
(3.1.16)

for any $\varphi\in C^{\infty}_{c}([0,T[\,\times\overline{\Omega})$ and

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{\Omega}\left(-\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\cdot\partial_{t}\boldsymbol{\psi}-\varrho_{\varepsilon}[\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}]:\nabla_{x}\boldsymbol{\psi}+\frac{1}{\varepsilon}\,\boldsymbol{e}_{3}\times(\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon})\cdot\boldsymbol{\psi}-\frac{1}{\varepsilon^{2m}}p(\varrho_{\varepsilon}){\rm
div}\,\boldsymbol{\psi}\right)\,{\rm d}x{\rm d}t$ (3.1.17)
$\displaystyle=\int_{0}^{T}\\!\\!\\!\int_{\Omega}\left(-\mathbb{S}(\nabla_{x}\boldsymbol{u}_{\varepsilon}):\nabla_{x}\boldsymbol{\psi}+\frac{1}{\varepsilon^{2n}}\varrho_{\varepsilon}\nabla_{x}G\cdot\boldsymbol{\psi}\right)\,{\rm
d}x{\rm
d}t+\int_{\Omega}\varrho_{0,\varepsilon}\boldsymbol{u}_{0,\varepsilon}\cdot\boldsymbol{\psi}(0,\cdot)\,{\rm
d}x$

for any test function $\boldsymbol{\psi}\in
C^{\infty}_{c}([0,T[\,\times\overline{\Omega};\mathbb{R}^{3})$ such that
$\big{(}\boldsymbol{\psi}\cdot\boldsymbol{n}\big{)}_{|\partial{\Omega}}=0$;

  3. (iii)

the energy inequality holds for almost every $t\in(0,T)$:

$\displaystyle\int_{\Omega}\frac{1}{2}\varrho_{\varepsilon}|\boldsymbol{u}_{\varepsilon}|^{2}(t)\,{\rm
d}x\,+\,\frac{1}{\varepsilon^{2m}}\int_{\Omega}\mathcal{E}\left(\varrho_{\varepsilon},\widetilde{\varrho}_{\varepsilon}\right)(t)\,{\rm
d}x+\int_{0}^{t}\int_{\Omega}\mathbb{S}(\nabla_{x}\boldsymbol{u}_{\varepsilon}):\nabla_{x}\boldsymbol{u}_{\varepsilon}\,\,{\rm
d}x{\rm d}\tau$ (3.1.18)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,\leq\,\int_{\Omega}\frac{1}{2}\varrho_{0,\varepsilon}|\boldsymbol{u}_{0,\varepsilon}|^{2}\,{\rm
d}x\,+\,\frac{1}{\varepsilon^{2m}}\int_{\Omega}\mathcal{E}\left(\varrho_{0,\varepsilon},\widetilde{\varrho}_{\varepsilon}\right)\,{\rm
d}x\,,$

where
$\mathcal{E}\left(\rho,\widetilde{\varrho}_{\varepsilon}\right)\,:=\,H(\rho)-(\rho-\widetilde{\varrho}_{\varepsilon})\,H^{\prime}(\widetilde{\varrho}_{\varepsilon})-H(\widetilde{\varrho}_{\varepsilon})$
is the _relative internal energy_ of the fluid.

The solution is _global_ if the previous conditions hold for all $T>0$.

Under the assumptions fixed above, for any _fixed_ value of the parameter
$\varepsilon\in\,]0,1]$, the existence of a global in time finite energy weak
solution $(\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon})$ to system
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$), related to the
initial datum $(\varrho_{0,\varepsilon},\boldsymbol{u}_{0,\varepsilon})$, in
the sense of the previous definition, can be proved as in the classical case,
see e.g. [56], [33]. Notice that the mapping
$t\mapsto(\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon})(t,\cdot)$ is
weakly continuous, and one has
$(\varrho_{\varepsilon})_{|t=0}=\varrho_{0,\varepsilon}$ together with
$(\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon})_{|t=0}=\varrho_{0,\varepsilon}\boldsymbol{u}_{0,\varepsilon}$.

We remark also that, in view of (NSC${}_{\varepsilon}^{1}$), the total mass is
conserved in time: for almost every $t\in[0,+\infty[\,$, one has

$\int_{\Omega}\bigl{(}\varrho_{\varepsilon}(t)\,-\,\widetilde{\varrho}_{\varepsilon}\bigr{)}\,\,{\rm
d}x\,=\,0\,.$

To conclude, as already highlighted in Chapter 2, in our framework of finite
energy weak solutions, inequality (3.1.18) will be the only tool to derive
uniform estimates for the family of weak solutions we are going to consider.

#####  3.1.2 Main theorems

We can now state our main results. We point out that, due to the scaling
(3.1.1), the relation $m>n$ is always true, so we will always be in a low
stratification regime.

The first statement concerns the case when the effects linked to the pressure
term are predominant in the dynamics (with respect to the fast rotation), i.e.
$m>1$.

###### Theorem 3.1.5

Let $\Omega=\mathbb{R}^{2}\times\,]0,1[\,$ and $G\in W^{1,\infty}(\Omega)$ be
as in (3.1.3). Take $m>1$ and $m+1\geq 2n>m$. For any fixed value of
$\varepsilon\in\;]0,1]$, let initial data
$\left(\varrho_{0,\varepsilon},\boldsymbol{u}_{0,\varepsilon}\right)$ verify
the hypotheses fixed in Paragraph 3.1.1.2, and let
$\left(\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon}\right)$ be a
corresponding weak solution to system
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$), supplemented with the
structural hypotheses (3.1.2) on
$\mathbb{S}(\nabla_{x}\boldsymbol{u}_{\varepsilon})$ and with boundary
conditions (3.1.4) and far field conditions (3.1.9). Let $\boldsymbol{u}_{0}$
be defined as in (3.1.15).

Then, for any $T>0$, one has that

$\displaystyle\varrho_{\varepsilon}\rightarrow 1\qquad\qquad$
$\displaystyle\mbox{ strongly in }\qquad L^{\infty}\big{(}[0,T];L_{\rm
loc}^{\min\\{2,\gamma\\}}(\Omega)\big{)}$
$\displaystyle\boldsymbol{u}_{\varepsilon}\rightharpoonup\boldsymbol{U}\qquad\qquad$
$\displaystyle\mbox{ weakly in }\qquad
L^{2}\big{(}[0,T];H^{1}(\Omega)\big{)}\,,$

where $\boldsymbol{U}=(\boldsymbol{U}^{h},0)$, with
$\boldsymbol{U}^{h}=\boldsymbol{U}^{h}(t,x^{h})$ such that ${\rm
div}_{h}\boldsymbol{U}^{h}=0$. In addition, the vector field
$\boldsymbol{U}^{h}$ is a weak solution to the following homogeneous
incompressible Navier-Stokes system in $\mathbb{R}_{+}\times\mathbb{R}^{2}$,

$\displaystyle\partial_{t}\boldsymbol{U}^{h}+{\rm
div}_{h}\left(\boldsymbol{U}^{h}\otimes\boldsymbol{U}^{h}\right)+\nabla_{h}\Gamma-\mu\Delta_{h}\boldsymbol{U}^{h}=0\,,$
(3.1.19)

for a suitable pressure function
$\Gamma\in\mathcal{D}^{\prime}(\mathbb{R}_{+}\times\mathbb{R}^{2})$ and
related to the initial condition

$\boldsymbol{U}_{|t=0}=\mathbb{H}_{h}\left(\langle\boldsymbol{u}^{h}_{0}\rangle\right)\,.$

When $m=1$, the Mach and Rossby numbers have the same order of magnitude, and
they keep in the so-called _quasi-geostrophic balance_ at the limit. Namely,
the next statement is devoted to this isotropic case.

###### Theorem 3.1.6

Let $\Omega=\mathbb{R}^{2}\times\,]0,1[\,$ and $G\in W^{1,\infty}(\Omega)$ be
as in (3.1.3). Take $m=1$ and $1/2<n<1$. For any fixed value of
$\varepsilon\in\;]0,1]$, let initial data
$\left(\varrho_{0,\varepsilon},\boldsymbol{u}_{0,\varepsilon}\right)$ verify
the hypotheses fixed in Paragraph 3.1.1.2, and let
$\left(\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon}\right)$ be a
corresponding weak solution to system
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$), supplemented with the
structural hypotheses (3.1.2) on
$\mathbb{S}(\nabla_{x}\boldsymbol{u}_{\varepsilon})$ and with boundary
conditions (3.1.4) and far field conditions (3.1.9). Let
$\left(\varrho^{(1)}_{0},\boldsymbol{u}_{0}\right)$ be defined as in (3.1.14)
and (3.1.15).

Then, for any $T>0$, one has the following convergence properties:

$\displaystyle\varrho_{\varepsilon}\rightarrow 1\qquad\qquad$
$\displaystyle\mbox{ strongly in }\qquad L^{\infty}\big{(}[0,T];L_{\rm
loc}^{\min\\{2,\gamma\\}}(\Omega)\big{)}$
$\displaystyle\varrho^{(1)}_{\varepsilon}:=\frac{\varrho_{\varepsilon}-\widetilde{\varrho_{\varepsilon}}}{\varepsilon}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\varrho^{(1)}\qquad\qquad$
$\displaystyle\mbox{ weakly-$*$ in }\qquad
L^{\infty}\big{(}[0,T];L^{2}+L^{\gamma}(\Omega)\big{)}$
$\displaystyle\boldsymbol{u}_{\varepsilon}\rightharpoonup\boldsymbol{U}\qquad\qquad$
$\displaystyle\mbox{ weakly in }\qquad
L^{2}\big{(}[0,T];H^{1}(\Omega)\big{)}\,,$

where, as above, $\boldsymbol{U}=(\boldsymbol{U}^{h},0)$, with
$\boldsymbol{U}^{h}=\boldsymbol{U}^{h}(t,x^{h})$ such that ${\rm
div}_{h}\boldsymbol{U}^{h}=0$. Moreover, one has the balance
$\boldsymbol{U}^{h}=\nabla_{h}^{\perp}\varrho^{(1)}$, and $\varrho^{(1)}$
satisfies (in the weak sense) the quasi-geostrophic equation

$\displaystyle\partial_{t}\left(\varrho^{(1)}-\Delta_{h}\varrho^{(1)}\right)-\nabla_{h}^{\perp}\varrho^{(1)}\cdot\nabla_{h}\left(\Delta_{h}\varrho^{(1)}\right)+\mu\Delta_{h}^{2}\varrho^{(1)}\,=\,0\,,$
(3.1.20)

supplemented with the initial condition

$\left(\varrho^{(1)}-\Delta_{h}\varrho^{(1)}\right)_{|t=0}=\langle\varrho_{0}^{(1)}\rangle-{\rm
curl}_{h}\langle\boldsymbol{u}^{h}_{0}\rangle\,.$

####  3.2 Consequences of the energy inequality

In Definition 3.1.4, we have postulated that the family of weak solutions
$\big{(}\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon}\big{)}_{\varepsilon}$
considered in Theorems 3.1.5 and 3.1.6 satisfies the energy inequality
(3.1.18).

In this section, we take advantage of the energy inequality to infer uniform
bounds for
$\big{(}\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon}\big{)}_{\varepsilon}$:
this will be done in Subsection 3.2.1. Thanks to those bounds, we can extract
(in Subsection 3.2.2) weak-limit points of the sequence of solutions and
deduce some properties these limit points have to satisfy.

#####  3.2.1 Uniform bounds and weak limits

This subsection is devoted to establish uniform bounds on the sequence
$\bigl{(}\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon}\bigr{)}_{\varepsilon}$.
This can be done as in the classical case (see e.g. [39] for details), since
again the Coriolis term does not contribute to the total energy balance of the
system. However, for the reader’s convenience, let us present some details.

To begin with, let us recall the partition of the space domain $\Omega$ into
the so-called “essential” and “residual” sets. For this, for $t>0$ and for all
$\varepsilon\in\,]0,1]$, we define the sets

$\Omega_{\rm{ess}}^{\varepsilon}(t)\,:=\,\left\\{x\in\Omega\;\big{|}\quad\varrho_{\varepsilon}(t,x)\in\left[1/2\,\rho_{*}\,,\,2\right]\right\\}\,,\qquad\Omega^{\varepsilon}_{\rm{res}}(t)\,:=\,\Omega\setminus\Omega^{\varepsilon}_{\rm{ess}}(t)\,,$