where the positive constant $\rho_{*}>0$ has been defined in Proposition
3.1.3.

Next, we observe that

$\Big{[}\mathcal{E}\big{(}\rho(t,x),\widetilde{\varrho}_{\varepsilon}(x)\big{)}\Big{]}_{\rm{ess}}\,\sim\,\left[\rho-\widetilde{\varrho}_{\varepsilon}(x)\right]_{\rm{ess}}^{2}\qquad\quad\mbox{
and
}\qquad\quad\Big{[}\mathcal{E}\big{(}\rho(t,x),\widetilde{\varrho}_{\varepsilon}(x)\big{)}\Big{]}_{\rm{res}}\,\geq\,C\left(1\,+\,\big{[}\rho(t,x)\big{]}_{\rm{res}}^{\gamma}\right)\,,$

where $\widetilde{\varrho}_{\varepsilon}$ is the static density state
identified in Paragraph 3.1.1.1. Here above, the multiplicative constants are
all strictly positive and may depend on $\rho_{*}$, and we agree to write
$A\sim B$ whenever there exists a “universal” constant $c>0$ such that
$(1/c)B\leq A\leq c\,B$.

Thanks to the previous observations, we easily see that, under the assumptions
fixed in Section 3.1 on the initial data, the right-hand side of (3.1.18) is
_uniformly bounded_ for all $\varepsilon\in\,]0,1]$: specifically, we have

$\int_{\Omega}\frac{1}{2}\varrho_{0,\varepsilon}|\boldsymbol{u}_{0,\varepsilon}|^{2}\,\,{\rm
d}x+\frac{1}{\varepsilon^{2m}}\int_{\Omega}\mathcal{E}\left(\varrho_{0,\varepsilon},\,\widetilde{\varrho}_{\varepsilon}\right)\,\,{\rm
d}x\,\leq\,C\,.$

Owing to the previous inequalities and the finite energy condition (3.1.18) on
the family of weak solutions, it is quite standard to derive, for any time
$T>0$ fixed and any $\varepsilon\in\,]0,1]$, the following estimates, that we
recall here for the reader’s convenience:

$\displaystyle\sup_{t\in[0,T]}\|\sqrt{\varrho_{\varepsilon}}\boldsymbol{u}_{\varepsilon}\|_{L^{2}(\Omega;\,\mathbb{R}^{3})}\,$
$\displaystyle\leq\,c$ (3.2.1)
$\displaystyle\sup_{t\in[0,T]}\left\|\left[\dfrac{\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}}{\varepsilon^{m}}\right]_{\rm{ess}}(t)\right\|_{L^{2}(\Omega)}\,$
$\displaystyle\leq\,c$ (3.2.2)
$\displaystyle\sup_{t\in[0,T]}\int_{\Omega}\mathds{1}_{\mathcal{M}^{\varepsilon}_{\rm{res}}[t]}\,dx\,$
$\displaystyle\leq\,c\,\varepsilon^{2m}$ (3.2.3)
$\displaystyle\sup_{t\in[0,T]}\int_{\Omega}[\varrho_{\varepsilon}]^{\gamma}_{\rm{res}}(t)\,\,{\rm
d}x\,\,$ $\displaystyle\leq\,c\,\varepsilon^{2m}$ (3.2.4)
$\displaystyle\int_{0}^{T}\left\|\nabla_{x}\boldsymbol{u}_{\varepsilon}+\,^{t}\nabla_{x}\boldsymbol{u}_{\varepsilon}-\frac{2}{3}{\rm
div}\,\boldsymbol{u}_{\varepsilon}\,{\rm
Id}\,\right\|^{2}_{L^{2}(\Omega;\,\mathbb{R}^{3\times 3})}\,\,{\rm d}t\,$
$\displaystyle\leq\,c\,.$ (3.2.5)

We refer to [39] (see also [35], [34], [30]) for the details of the
computations.

Owing to (3.2.5) and a generalisation of the Korn-Poincaré inequality (see
Proposition B.8 in the Appendix), we gather that
$\big{(}\nabla\boldsymbol{u}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{2})$.
On the other hand, by arguing as in [35], we can use (3.2.1), (3.2.3) and
(3.2.4) to deduce that also
$\big{(}\boldsymbol{u}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{2})$.
Putting those bounds together, we finally infer that

$\int_{0}^{T}\left\|\boldsymbol{u}_{\varepsilon}\right\|^{2}_{H^{1}(\Omega;\,\mathbb{R}^{3})}\,\,{\rm
d}t\,\leq\,c\,.$ (3.2.6)

In particular, there exist $\boldsymbol{U}\,\in\,L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};H^{1}(\Omega;\mathbb{R}^{3})\big{)}$ such that, up
to a suitable extraction (not relabelled here), we have

$\boldsymbol{u}_{\varepsilon}\,\rightharpoonup\,\boldsymbol{U}\qquad\qquad\mbox{
in }\quad L^{2}_{\rm
loc}\big{(}\mathbb{R}_{+};H^{1}(\Omega;\mathbb{R}^{3})\big{)}\,.$ (3.2.7)

Let us move further and consider the density functions. The previous estimates
on the density tell us that we must find a finer decomposition for the
densities. As a matter of fact, for any time $T>0$ fixed, we have

$\|\varrho_{\varepsilon}-1\|_{L^{\infty}_{T}(L^{2}+L^{\gamma}+L^{\infty})}\,\leq\,c\,\varepsilon^{2(m-n)}\,.$
(3.2.8)

In order to see (3.2.8), to begin with, we write

$|\varrho_{\varepsilon}-1|\,\leq\,|\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}|+|\widetilde{\varrho}_{\varepsilon}-1|\,.$
(3.2.9)

From (3.2.2), we infer that
$\big{[}\varrho_{\varepsilon}\,-\,\widetilde{\varrho}_{\varepsilon}\big{]}_{\rm{ess}}$
is of order $O(\varepsilon^{m})$ in $L^{\infty}_{T}(L^{2})$. For the residual
part of the same term, we can use (3.2.4) to discover that it is of order
$O(\varepsilon^{2m/\gamma})$. Observe that, if $1<\gamma<2$, the higher order
is $O(\varepsilon^{m})$, whereas, in the case $\gamma\geq 2$, by use of
(3.2.4) and (3.2.3) again, it is easy to get

$\left\|\left[\varrho_{\varepsilon}\,-\,\widetilde{\varrho}_{\varepsilon}\right]_{\rm{res}}\right\|_{L^{\infty}_{T}(L^{2})}^{2}\,\leq\,C\,\varepsilon^{2m}\,.$
(3.2.10)

Finally, we apply Proposition 3.1.3 to control the last term in the right-hand
side of (3.2.9). In the end, estimate (3.2.8) is proved.

This having been established, and keeping in mind the notation introduced in
(3.1.10) and (3.1.11), we can introduce the density oscillation functions

$R_{\varepsilon}\,:=\,\frac{\varrho_{\varepsilon}-1}{\varepsilon^{2(m-n)}}\,=\,\widetilde{r}_{\varepsilon}\,+\,\varepsilon^{2n-m}\,\varrho_{\varepsilon}^{(1)}\,,$

where we have defined

$\varrho_{\varepsilon}^{(1)}(t,x)\,:=\,\frac{\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon}}{\varepsilon^{m}}\qquad\mbox{
and
}\qquad\widetilde{r}_{\varepsilon}(x)\,:=\,\frac{\widetilde{\varrho}_{\varepsilon}-1}{\varepsilon^{2(m-n)}}\,.$
(3.2.11)

Thanks again to (3.2.2), (3.2.4) and Proposition 3.1.3, we see that the
previous quantities verify the following uniform bounds, for any time $T>0$
fixed:

$\sup_{\varepsilon\in\,]0,1]}\left\|\varrho_{\varepsilon}^{(1)}\right\|_{L^{\infty}_{T}(L^{2}+L^{\gamma}({\Omega}))}\,\leq\,c\qquad\qquad\mbox{
and
}\qquad\qquad\sup_{\varepsilon\in\,]0,1]}\left\|\widetilde{r}_{\varepsilon}\right\|_{L^{\infty}(\Omega)}\,\leq\,c\,.$
(3.2.12)

In view of the previous properties, there exist $\varrho^{(1)}\in
L^{\infty}_{T}(L^{2}+L^{\gamma})$ and $\widetilde{r}\in L^{\infty}$ such that
(up to the extraction of a new suitable subsequence)

$\varrho_{\varepsilon}^{(1)}\,\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\,\varrho^{(1)}\qquad\qquad\mbox{
and
}\qquad\qquad\widetilde{r}_{\varepsilon}\,\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\,\widetilde{r}$
(3.2.13)

in the weak-$*$ topology of the respective spaces. In particular, we get

$R_{\varepsilon}\,\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\,\widetilde{r}\qquad\qquad\qquad\mbox{
weakly-$*$ in }\qquad L^{\infty}\bigl{(}[0,T];L^{\min\\{\gamma,2\\}}_{\rm
loc}(\Omega)\bigr{)}\,.$

###### Remark 3.2.1

Observe that, owing to (3.2.10), when $\gamma\geq 2$ we get

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

Therefore, in that case we actually have that
$\varrho^{(1)}\,\in\,L^{\infty}_{T}(L^{2})$ and
$\varrho_{\varepsilon}^{(1)}\,\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\,\varrho^{(1)}$
in that space.

Analogously, when $\gamma\geq 2$ we also get

$\|\varrho_{\varepsilon}-1\|_{L^{\infty}_{T}(L^{2}+L^{\infty})}\,\leq\,c\,\varepsilon^{2(m-n)}.$

#####  3.2.2 Constraints on the limit

In this subsection, we establish some properties that the limit points of the
family
$\bigl{(}\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon}\bigr{)}_{\varepsilon}$,
which have been identified here above, have to satisfy.

We first need a preliminary result about the decomposition of the pressure
function, which will be useful in the following computations.

###### Lemma 3.2.2

Let $(m,n)\in\mathbb{R}^{2}$ verify the condition $m+1\,\geq\,2n\,>\,m\geq 1$.
Let $p$ be the pressure term satisfying the structural hypotheses (3.1.5) and
(3.1.6). Then, for any $\varepsilon\in\,]0,1]$, one has

$\begin{split}\frac{1}{\varepsilon^{2m}}\,\nabla_{x}\Big{(}p(\varrho_{\varepsilon})\,-\,p(\widetilde{\varrho}_{\varepsilon})\Big{)}\,=\,\frac{1}{\varepsilon^{m}}\nabla_{x}\Big{(}p^{\prime}(1)\varrho_{\varepsilon}^{(1)}\Big{)}\,+\,\frac{1}{\varepsilon^{2n-m}}\,\nabla_{x}\Pi_{\varepsilon}\,,\end{split}$
(3.2.14)

where the functions $\varrho_{\varepsilon}^{(1)}$ have been introduced in
(3.2.11) and for all $T>0$ the family
$\big{(}\Pi_{\varepsilon}\big{)}_{\varepsilon}$ verifies the uniform bound

$\left\|\Pi_{\varepsilon}\right\|_{L^{\infty}_{T}(L^{1}+L^{2}+L^{\gamma})}\,\leq\,C\,.$
(3.2.15)

When $\gamma\geq 2$, one can dispense of the space $L^{\gamma}$ in the
previous control of $\big{(}\Pi_{\varepsilon}\big{)}_{\varepsilon}$.

Proof:  First of all, we write

$\begin{split}\frac{1}{\varepsilon^{2m}}\,\nabla_{x}\Big{(}p(\varrho_{\varepsilon})\,-\,p(\widetilde{\varrho}_{\varepsilon})\Big{)}\,&=\,\frac{1}{\varepsilon^{2m}}\,\nabla_{x}\Big{(}p(\varrho_{\varepsilon})\,-\,p(\widetilde{\varrho}_{\varepsilon})-p^{\prime}(\widetilde{\varrho}_{\varepsilon})(\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon})\Big{)}\\\
&\qquad\qquad+\,\frac{1}{\varepsilon^{m}}\,\nabla_{x}\Big{(}\big{(}p^{\prime}(\widetilde{\varrho}_{\varepsilon})\,-\,p^{\prime}(1)\big{)}\,\varrho_{\varepsilon}^{(1)}\Big{)}\,+\,\frac{1}{\varepsilon^{m}}\nabla_{x}\Big{(}p^{\prime}(1)\varrho_{\varepsilon}^{(1)}\Big{)}\,.\end{split}$
(3.2.16)

We start by analysing the first term on the right-hand side of (3.2.16). For
the essential part, we can employ a Taylor expansion to write

$\left[p(\varrho_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon})-p^{\prime}(\widetilde{\varrho}_{\varepsilon})(\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon})\right]_{\rm{ess}}=\left[p^{\prime\prime}(z_{\varepsilon})(\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon})^{2}\right]_{\rm{ess}}\,,$

where $z_{\varepsilon}$ is a suitable point between $\varrho_{\varepsilon}$
and $\widetilde{\varrho}_{\varepsilon}$. Thanks to the uniform bound (3.2.2),
we have that this term is of order $O(\varepsilon^{2m})$ in
$L^{\infty}_{T}(L^{1})$, for any $T>0$ fixed. For the residual part, we can
use (3.2.3) and (3.2.4), together with the boundedness of the profiles
$\widetilde{\varrho}_{\varepsilon}$ (keep in mind Proposition 3.1.3), to
deduce that

$\left\|\left[p(\varrho_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon})-p^{\prime}(\widetilde{\varrho}_{\varepsilon})(\varrho_{\varepsilon}-\widetilde{\varrho}_{\varepsilon})\right]_{\rm{res}}\right\|_{L^{\infty}_{T}(L^{1})}\,\leq
C\,\varepsilon^{2m}\,.$

We refer to e.g. Lemma 4.1 of [30] for details.

In a similar way, a Taylor expansion for the second term on the right-hand
side of (3.2.16) gives

$\big{(}p^{\prime}(\widetilde{\varrho}_{\varepsilon})-p^{\prime}(1)\big{)}\varrho^{(1)}_{\varepsilon}\,=\,p^{\prime\prime}(\eta_{\varepsilon})(\widetilde{\varrho}_{\varepsilon}-1)\varrho^{(1)}_{\varepsilon}\,,$

where $\eta_{\varepsilon}$ is a suitable point between
$\widetilde{\varrho}_{\varepsilon}$ and 1. Owing to Proposition 3.1.3 again
and to bound (3.2.12), we infer that this term is of order
$O(\varepsilon^{2(m-n)})$ in $L^{\infty}_{T}(L^{2}+L^{\gamma})$, for any time
$T>0$ fixed.

Then, defining

$\Pi_{\varepsilon}:=\frac{1}{\varepsilon^{2(m-n)}}\left[\frac{p(\varrho_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon})}{\varepsilon^{m}}-p^{\prime}(1)\varrho_{\varepsilon}^{(1)}\right]$
(3.2.17)

we have the control (3.2.15).

The final statement concerning the case $\gamma\geq 2$ easily follows from
Remark 3.2.1. This completes the proof of the lemma.

Notice that the last term appearing in (3.2.14) is singular in $\varepsilon$.
This is in stark contrast with the situation considered in previous works, see
e.g. [35], [34], [40] and [30]. However, its gradient structure will play a
fundamental role in the computations below.

This having been pointed out, we can now analyse the constraints on the weak-
limit points $\big{(}\varrho^{(1)},\boldsymbol{U}\big{)}$, identified in
relations (3.2.7) and (3.2.13) above.

######  3.2.2.1 The case of large values of the Mach number: $m>1$

We start by considering the case of anisotropic scaling, namely $m>1$ and
$m+1\geq 2n>m$. Notice that, in particular, one has $m>n$.

###### Proposition 3.2.3

Let $m>1$ and $m+1\geq 2n>m$ in
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$). Let
$\left(\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon}\right)_{\varepsilon}$
be a family of weak solutions, related to initial data
$\left(\varrho_{0,\varepsilon},\boldsymbol{u}_{0,\varepsilon}\right)_{\varepsilon}$
verifying the hypotheses of Paragraph 3.1.1.2. Let
$(\varrho^{(1)},\boldsymbol{U})$ be a limit point of the sequence
$\left(\varrho_{\varepsilon}^{(1)},\boldsymbol{u}_{\varepsilon}\right)_{\varepsilon}$,
as identified in Subsection 3.2.1. Then,

$\displaystyle\boldsymbol{U}\,=\,\,\Big{(}\boldsymbol{U}^{h}\,,\,0\Big{)}\,,\qquad\qquad\mbox{
with }\qquad\boldsymbol{U}^{h}\,=\,\boldsymbol{U}^{h}(t,x^{h})\quad\mbox{ and
}\quad{\rm div}\,_{\\!h}\,\boldsymbol{U}^{h}\,=\,0\,,$ (3.2.18)
$\displaystyle\nabla_{x}\varrho^{(1)}\,=\,0\qquad\qquad\mbox{ in
}\;\,\mathcal{D}^{\prime}(\mathbb{R}_{+}\times\Omega)\,.$ (3.2.19)

Proof:  First of all, let us consider the weak formulation of the mass
equation (NSC${}_{\varepsilon}^{1}$): for any test function $\varphi\in
C_{c}^{\infty}\bigl{(}\mathbb{R}_{+}\times\Omega\bigr{)}$, denoting
$[0,T]\times K\,:=\,{\rm Supp}\,\,\varphi$, with $\varphi(T,\cdot)\equiv 0$,
we have

$-\int^{T}_{0}\int_{K}\bigl{(}\varrho_{\varepsilon}-1\bigr{)}\,\partial_{t}\varphi\,{\rm
d}x{\rm
d}t\,-\,\int^{T}_{0}\int_{K}\varrho_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\,\cdot\,\nabla_{x}\varphi\,{\rm
d}x{\rm
d}t\,=\,\int_{K}\bigl{(}\varrho_{0,\varepsilon}-1\bigr{)}\,\varphi(0,\,\cdot\,)\,{\rm
d}x\,.$

We can easily pass to the limit in this equation, thanks to the strong
convergence $\varrho_{\varepsilon}\longrightarrow 1$, provided by (3.2.8), and
the weak convergence of $\boldsymbol{u}_{\varepsilon}$ in
$L_{T}^{2}\bigl{(}L^{6}_{\rm loc}\bigr{)}$, provided by (3.2.7) and Sobolev
embeddings. Notice that one always has $1/\gamma\,+\,1/6\,\leq\,1$. In this
way, we find

$-\,\int^{T}_{0}\int_{K}\boldsymbol{U}\,\cdot\,\nabla_{x}\varphi\,{\rm d}x{\rm
d}t\,=\,0\,,$

for any test function $\varphi\,\in
C_{c}^{\infty}\bigl{(}\mathbb{R}_{+}\times\Omega\bigr{)}$ taken as above. The
previous relation in particular implies

${\rm div}\,\boldsymbol{U}=0\qquad\qquad\mbox{ a.e. in
}\;\,\mathbb{R}_{+}\times\Omega\,.$ (3.2.20)

Next, we test the momentum equation (NSC${}_{\varepsilon}^{2}$) on
$\varepsilon^{m}\,\boldsymbol{\phi}$, for a smooth compactly supported
$\boldsymbol{\phi}$. Using the uniform bounds established in Subsection 3.2.1,
it is easy to see that the term presenting the derivative in time, the
viscosity term and the convective term converge to $0$, in the limit
$\varepsilon\rightarrow 0^{+}$. Since $m>1$, also the Coriolis term vanishes
when $\varepsilon\rightarrow 0^{+}$. It remains us to consider the pressure
and gravity terms in the weak formulation (3.1.17) of the momentum equation:
using relation (3.1.7), we see that we can couple them to write

$\displaystyle\frac{1}{\varepsilon^{2m}}\,\nabla_{x}p(\varrho_{\varepsilon})-\,\frac{1}{\varepsilon^{2n}}\,\varrho_{\varepsilon}\nabla_{x}G\,=\,\frac{1}{\varepsilon^{2m}}\nabla_{x}\Big{(}p(\varrho_{\varepsilon})\,-\,p(\widetilde{\varrho}_{\varepsilon})\Big{)}-\,\varepsilon^{m-2n}\varrho_{\varepsilon}^{(1)}\nabla_{x}G\,.$
(3.2.21)

By (3.2.12) and the fact that $m>n$, we readily see that the last term in the
right-hand side of (3.2.21) converges to $0$, when tested against any smooth
compactly supported $\varepsilon^{m}\,\boldsymbol{\phi}$. At this point, we
use Lemma 3.2.2 to treat the first term on the right-hand side of (3.2.21).
So, taking $\boldsymbol{\phi}\in C^{\infty}_{c}([0,T[\,\times\Omega)$ (for
some $T>0$), we test the momentum equation against
$\varepsilon^{m}\,\boldsymbol{\phi}$ and using (2.2.24), in the limit
$\varepsilon\rightarrow 0^{+}$ we find that

$\int_{0}^{T}\int_{\Omega}p^{\prime}(1)\varrho^{(1)}{\rm
div}\,\boldsymbol{\phi}\,{\rm d}x{\rm d}t=0\,.$

Recalling that $p^{\prime}(1)=1$, the previous relation implies (2.2.29) for
$\varrho^{(1)}$.

In particular, that relation implies that $\varrho^{(1)}(t,x)\,=\,c(t)$ for
almost all $(t,x)\in\mathbb{R}_{+}\times\Omega$, for a suitable function
$c=c(t)$ depending only on time.

Now, in order to see effects due to the fast rotation in the limit, we need to
“filter out” the contribution coming from the low Mach number. To this end, we
test (NSC${}_{\varepsilon}^{2}$) on $\varepsilon\,\boldsymbol{\phi}$, where
this time we take $\boldsymbol{\phi}\,=\,{\rm curl}\,\boldsymbol{\psi}$, for
some smooth compactly supported $\boldsymbol{\psi}\,\in
C^{\infty}_{c}\bigl{(}[0,T[\,\times\Omega\bigr{)}$, with $T>0$. Once again, by
uniform bounds we infer that the $\partial_{t}$ term, the convective term and
the viscosity term all converge to $0$ when $\varepsilon\rightarrow 0^{+}$. As
for the pressure and the gravitational force, we argue as in (3.2.21): since
the structure of $\boldsymbol{\phi}$ kills any gradient term, we are left with
the convergence of the integral

$\int^{T}_{0}\int_{\Omega}\varepsilon^{m-2n+1}\varrho_{\varepsilon}^{(1)}\nabla_{x}G\cdot\boldsymbol{\phi}\,\,{\rm
d}x{\rm
d}t\,\longrightarrow\,\delta_{0}(m-2n+1)\int^{T}_{0}\int_{\Omega}\varrho^{(1)}\nabla_{x}G\cdot\boldsymbol{\phi}\,\,{\rm
d}x{\rm d}t\,,$

where $\delta_{0}(\zeta)\,=\,1$ if $\zeta=0$, $\delta_{0}(\zeta)\,=\,0$
otherwise. Finally, arguing as done for the mass equation, we see that the
Coriolis term converges to the integral
$\int^{T}_{0}\int_{\Omega}\boldsymbol{e}_{3}\times\boldsymbol{U}\cdot\boldsymbol{\phi}$.

Consider the case $m+1>2n$ for a while. Passing to the limit for
$\varepsilon\rightarrow 0^{+}$, we find that
$\mathbb{H}\left(\boldsymbol{e}_{3}\times\boldsymbol{U}\right)\,=\,0$, which
implies that $\boldsymbol{e}_{3}\times\boldsymbol{U}\,=\,\nabla_{x}\Phi$, for
some potential function $\Phi$. From this relation, one easily deduces 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. one has
$\boldsymbol{U}^{h}\,=\,\boldsymbol{U}^{h}(t,x^{h})$. Furthermore, since
$\boldsymbol{U}^{h}\,=\,-\,\nabla_{h}^{\perp}\Phi$, we get that ${\rm
div}\,_{\\!h}\,\boldsymbol{U}^{h}\,=\,0$. At this point, we combine this fact
with (3.2.20) to infer that $\partial_{3}U^{3}\,=\,0$; but, thanks to the
boundary condition (3.1.4), we must have
$\bigl{(}\boldsymbol{U}\cdot\boldsymbol{n}\bigr{)}_{|\partial\Omega}\,=\,0$,
which implies that $U^{3}$ has to vanish at the boundary of $\Omega$. Thus, we
finally deduce that $U^{3}\,\equiv\,0$, whence (3.2.18) follows (see also the
proof of Proposition 2.2.7 in this regard).

Now, let us focus on the case when $m+1=2n$. The previous computations show
that, when $\varepsilon\rightarrow 0^{+}$, we get

$\boldsymbol{e}_{3}\times\boldsymbol{U}+\varrho^{(1)}\nabla_{x}G\,=\,\nabla_{x}\Phi\qquad\qquad\mbox{
in }\;\mathcal{D}^{\prime}(\mathbb{R}_{+}\times\Omega)\,,$ (3.2.22)

for a new suitable function $\Phi$. However, owing to (3.2.19), we can say
that $\varrho^{(1)}\nabla_{x}G\,=\,\nabla_{x}\big{(}\varrho^{(1)}\,G\big{)}$;
hence, the previous relations can be recasted as
$\boldsymbol{e}_{3}\times\boldsymbol{U}\,=\,\nabla_{x}\widetilde{\Phi}$, for a
new scalar function $\widetilde{\Phi}$. Therefore, the same analysis as above
applies, allowing us to gather (3.2.18) also in the case $m+1=2n$.

######  3.2.2.2 The case $m=1$

Now we focus on the case $m=1$. In this case, the fast rotation and weak
compressibility effects are of the same order: this allows to reach the so-
called _quasi-geostrophic balance_ in the limit.

###### Proposition 3.2.4

Take $m=1$ and $1/2<n<1$ in system
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$). Let
$\left(\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon}\right)_{\varepsilon}$
be a family of weak solutions to
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$), associated with
initial data
$\left(\varrho_{0,\varepsilon},\boldsymbol{u}_{0,\varepsilon}\right)$
verifying the hypotheses fixed in Paragraph 3.1.1.2. Let
$(\varrho^{(1)},\boldsymbol{U})$ be a limit point of the sequence
$\left(\varrho^{(1)}_{\varepsilon},\boldsymbol{u}_{\varepsilon}\right)_{\varepsilon}$,
as identified in Subsection 3.2.1. Then,

$\displaystyle\varrho^{(1)}\,=\,\varrho^{(1)}(t,x^{h})\quad\mbox{ and
}\quad\boldsymbol{U}\,=\,\,\Big{(}\boldsymbol{U}^{h}\,,\,0\Big{)}\,,\quad\mbox{
with }\quad\boldsymbol{U}^{h}\,=\,\nabla^{\perp}_{h}\varrho^{(1)}\;\mbox{ a.e.
in }\;\mathbb{R}_{+}\times\mathbb{R}^{2}\,.$ (3.2.23)

In particular, one has $\boldsymbol{U}^{h}\,=\,\boldsymbol{U}^{h}(t,x^{h})$
and ${\rm div}\,_{\\!h}\boldsymbol{U}^{h}\,=\,0$.

Proof:  Arguing as in the proof of Proposition 3.2.3, it is easy to pass to
the limit in the continuity equation. In particular, we obtain again relation
(3.2.20) for $\boldsymbol{U}$.

Only the analysis of the momentum equation changes a bit with respect to the
previous case $m>1$. Now, since the most singular terms are of order
$\varepsilon^{-1}$ (keep in mind Lemma 3.2.2), we test the weak formulation
(3.1.17) of the momentum equation against $\varepsilon\,\boldsymbol{\phi}$,
where $\boldsymbol{\phi}$ is a smooth compactly supported function. Similarly
to what done above, the uniform bounds of Subsection 3.2.1 allow us to infer
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:
more precisely, using also Lemma 3.2.2, we have

$\boldsymbol{e}_{3}\times\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\,+\frac{\nabla_{x}\Big{(}p(\varrho_{\varepsilon})-p(\widetilde{\varrho}_{\varepsilon})\Big{)}}{\varepsilon}\,-\,\varepsilon^{2(1-n)}\varrho_{\varepsilon}^{(1)}\nabla_{x}G\,=\,\mathcal{O}(\varepsilon)\,,$

in the sense of $\mathcal{D}^{\prime}(\mathbb{R}_{+}\times\Omega)$. Following
the same computations performed in the proof of Proposition 3.2.3, in the
limit $\varepsilon\rightarrow 0^{+}$ it is easy to get that

$\boldsymbol{e}_{3}\times\boldsymbol{U}+\nabla_{x}\left(p^{\prime}(1)\varrho^{(1)}\right)\,=\,0\qquad\qquad\mbox{
in }\;\mathcal{D}^{\prime}\big{(}\mathbb{R}_{+}\times\Omega\big{)}\,.$

After recalling that $p^{\prime}(1)=1$, this equality can be equivalently
written as

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

Notice that $\boldsymbol{U}$ is in fact in $L^{2}_{\rm
loc}(\mathbb{R}_{+};L^{2})$, therefore so is $\nabla_{x}\varrho^{(1)}$; hence
the previous relation is in fact satisfied almost everywhere in
$\mathbb{R}_{+}\times\Omega$.

At this point, we can repeat the same argument used in the proof of
Proposition 3.2.3 to deduce (3.2.23). The proposition is thus proved.

####  3.3 Convergence in the case $m>1$

In this section, we complete the proof of Theorem 3.1.5. Namely, we show
convergence in the weak formulation of the primitive system, in the case when
$m>1$ and $m+1\geq 2n>m$.

In Proposition 3.2.3, we have already seen how passing to the limit in the
mass equation. However, problems arise when tackling the convergence in the
momentum equation. Indeed, the analysis carried out so far is not enough to
identify the weak limit of the convective term
$\varrho_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}$,
which is highly non-linear. For proving that this term converges to the
expected limit $\boldsymbol{U}\otimes\boldsymbol{U}$, the key point is to
control the fast oscillations in time of the solutions, generated by the
singular terms in the momentum equation. For this, we will use a compensated
compactness argument and we exploit the algebraic structure of the wave system
underlying the primitive equations
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$).

In Subsection 3.3.1, we start by giving a quite accurate description of those
fast oscillations. Then, using that description, we are able, in Subsection
3.3.2, to establish two fundamental properties: on the one hand, strong
convergence of a suitable quantity related to the velocity fields; on the
other hand, the other terms, which do not involve that quantity, tend to
vanish when $\varepsilon\rightarrow 0^{+}$. In turn, this allows us to
complete, in Subsection 3.3.3, the proof of the convergence.

#####  3.3.1 Analysis of the strong oscillations

The goal of the present subsection is to describe the fast oscillations in
time of the solutions. First of all, we recast our equations into a wave
system. Then, we establish uniform bounds for the quantities appearing in the
wave system. Finally, we apply a regularisation in space procedure for all the
quantities, which is preparatory in view of the computations of Subsection
3.3.2.

######  3.3.1.1 Formulation of the acoustic wave system

We introduce the quantity

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

Then, straightforward computations show that we can recast the continuity
equation in the form

$\varepsilon^{m}\,\partial_{t}\varrho^{(1)}_{\varepsilon}\,+\,{\rm
div}\,\boldsymbol{V}_{\varepsilon}\,=\,0\,,$ (3.3.1)

where $\varrho^{(1)}_{\varepsilon}$ is defined in (3.2.11). Next, thanks to
Lemma 3.2.2 and the static relation (3.1.7), we can derive the following form
of the momentum equation:

$\displaystyle\varepsilon^{m}\,\partial_{t}\boldsymbol{V}_{\varepsilon}\,+\,\varepsilon^{m-1}\,\boldsymbol{e}_{3}\times\boldsymbol{V}_{\varepsilon}\,+p^{\prime}(1)\,\nabla_{x}\varrho_{\varepsilon}^{(1)}\,$
$\displaystyle=\,\varepsilon^{2(m-n)}\left(\varrho_{\varepsilon}^{(1)}\nabla_{x}G\,-\,\nabla_{x}\Pi_{\varepsilon}\right)$
(3.3.2) $\displaystyle\qquad\qquad+\,\varepsilon^{m}\,\Big{(}{\rm
div}\,\mathbb{S}\\!\left(\nabla_{x}\boldsymbol{u}_{\varepsilon}\right)\,-\,{\rm
div}\,\\!\left(\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}\right)\Big{)}\,.$

Then, if we define

$\boldsymbol{f}_{\varepsilon}:={\rm
div}\,\big{(}\mathbb{S}\\!\left(\nabla_{x}\boldsymbol{u}_{\varepsilon}\right)\,-\,\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}\big{)}\qquad\mbox{
and
}\qquad\boldsymbol{g}_{\varepsilon}:=\varrho_{\varepsilon}^{(1)}\nabla_{x}G\,-\,\nabla_{x}\Pi_{\varepsilon}\,,$
(3.3.3)

recalling that we have normalised the pressure function so that
$p^{\prime}(1)=1$, we can rewrite the primitive system
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$) in the following form:

$\left\\{\begin{array}[]{l}\varepsilon^{m}\,\partial_{t}\varrho^{(1)}_{\varepsilon}\,+\,{\rm
div}\,\boldsymbol{V}_{\varepsilon}\,=\,0\\\\[4.30554pt]
\varepsilon^{m}\,\partial_{t}\boldsymbol{V}_{\varepsilon}\,+\,\nabla_{x}\varrho_{\varepsilon}^{(1)}\,+\,\varepsilon^{m-1}\,\boldsymbol{e}_{3}\times\boldsymbol{V}_{\varepsilon}\,=\,\varepsilon^{m}\,\boldsymbol{f}_{\varepsilon}+\varepsilon^{2(m-n)}\boldsymbol{g}_{\varepsilon}\,.\end{array}\right.$
(3.3.4)

We remark that system (3.3.4) has to be read in the weak sense: for any
$\varphi\in C_{c}^{\infty}\bigl{(}[0,T[\,\times\overline{\Omega}\bigr{)}$, one
has

$-\,\varepsilon^{m}\,\int^{T}_{0}\int_{\Omega}\varrho^{(1)}_{\varepsilon}\,\partial_{t}\varphi\,-\,\int^{T}_{0}\int_{\Omega}\boldsymbol{V}_{\varepsilon}\cdot\nabla_{x}\varphi\,=\,\varepsilon^{m}\int_{\Omega}\varrho^{(1)}_{0,\varepsilon}\,\varphi(0)\,\,,$

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

$\displaystyle-\,\varepsilon^{m}\,\int^{T}_{0}\int_{\Omega}\boldsymbol{V}_{\varepsilon}\cdot\partial_{t}\boldsymbol{\psi}\,-\,\int^{T}_{0}\int_{\Omega}\varrho^{(1)}_{\varepsilon}\,{\rm
div}\,\boldsymbol{\psi}\,+\,\varepsilon^{m-1}\int^{T}_{0}\int_{\Omega}\boldsymbol{e}_{3}\times\boldsymbol{V}_{\varepsilon}\cdot\boldsymbol{\psi}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad=\,\varepsilon^{m}\int_{\Omega}\varrho_{0,\varepsilon}\,\boldsymbol{u}_{0,\varepsilon}\cdot\boldsymbol{\psi}(0)\,+\,\varepsilon^{m}\,\int^{T}_{0}\int_{\Omega}\boldsymbol{f}_{\varepsilon}\cdot\boldsymbol{\psi}+\,\varepsilon^{2(m-n)}\,\int^{T}_{0}\int_{\Omega}\boldsymbol{g}_{\varepsilon}\cdot\boldsymbol{\psi}\,.$

Here we use estimates of Subsection 3.2.1 in order to show uniform bounds for
the solutions and the data in the wave equation (3.3.4). We start by dealing
with the “unknown” $\boldsymbol{V}_{\varepsilon}$. Splitting the term into
essential and residual parts, one can obtain for all $T>0$,

$\|\boldsymbol{V}_{\varepsilon}\|_{L^{\infty}_{T}(L^{2}+L^{2\gamma/(\gamma+1)})}\leq
c\,.$ (3.3.5)

In the next lemma, we establish bounds for the source terms in the system of
acoustic waves (3.3.4).

###### Lemma 3.3.1

Write $\boldsymbol{f}_{\varepsilon}\,=\,{\rm
div}\,\widetilde{\boldsymbol{f}}_{\varepsilon}$ and
$\boldsymbol{g}_{\varepsilon}\,=\,\boldsymbol{g}^{1}_{\varepsilon}\,-\,\nabla_{x}\Pi_{\varepsilon}$,
where we have defined the quantities
$\widetilde{\boldsymbol{f}}_{\varepsilon}:=\mathbb{S}\\!\left(\nabla_{x}\boldsymbol{u}_{\varepsilon}\right)-\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}$,
$\boldsymbol{g}^{1}_{\varepsilon}\,:=\,\varrho_{\varepsilon}^{(1)}\,\nabla_{x}G$
and the functions $\Pi_{\varepsilon}$ have been introduced in (3.2.17) of
Lemma 3.2.2.

For any $T>0$ fixed, one has the uniform embedding properties

$\big{(}\widetilde{\boldsymbol{f}}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{2}+L^{1})\qquad\mbox{
and
}\qquad\big{(}\boldsymbol{g}^{1}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(L^{2}+L^{\gamma})\,.$

In the case $\gamma\geq 2$, we may get rid of the space $L^{\gamma}$ in the
control of $\big{(}\boldsymbol{g}^{1}_{\varepsilon}\big{)}_{\varepsilon}$.

In particular, the sequences
$\bigl{(}\boldsymbol{f}_{\varepsilon}\bigr{)}_{\varepsilon}$ and
$\bigl{(}\boldsymbol{g}_{\varepsilon}\bigr{)}_{\varepsilon}$, defined in
system (3.3.4), are uniformly bounded in the space
$L^{2}\big{(}[0,T];H^{-s}(\Omega)\big{)}$, for all $s>5/2$.

Proof:  From (3.2.1), (3.2.5) and (3.2.6), we immediately infer the uniform
bound for the family
$\big{(}\widetilde{\boldsymbol{f}}_{\varepsilon}\big{)}_{\varepsilon}$ in
$L^{2}_{T}(L^{1}+L^{2})$, from which we deduce also the uniform boundedness of
$\big{(}\boldsymbol{f}_{\varepsilon}\big{)}_{\varepsilon}$ in
$L^{2}_{T}(H^{-s})$, for any $s>5/2$ (see Theorem B.2 in this respect).

Next, for bounding
$\big{(}\boldsymbol{g}^{1}_{\varepsilon}\big{)}_{\varepsilon}$ we simply use
(3.2.12), together with Remark 3.2.1 when $\gamma\geq 2$. Keeping in mind the
bounds established in Lemma 3.2.2, the uniform estimate for
$\big{(}\boldsymbol{g}_{\varepsilon}\big{)}_{\varepsilon}$ follows.

######  3.3.1.2 Regularization and description of the oscillations

As already mentioned in Remark 3.1.1, in order to apply the Littlewood-Paley
theory, it is convenient to reformulate problem
(NSC${}_{\varepsilon}^{1}$)-(NSC${}_{\varepsilon}^{2}$) in the new domain
(which we keep calling $\Omega$, with a little abuse of notation)

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

In addition, to avoid the appearing of the (irrelevant) multiplicative
constants on the computations, we suppose that the torus $\mathbb{T}^{1}$ has
been renormalised so that its Lebesgue measure is equal to 1.

Now, for any $M\in\mathbb{N}$ we consider the low-frequency cut-off operator
${S}_{M}$ of a Littlewood-Paley decomposition, as introduced in (A.1.1) of
Section A.1. Then, we define

$\varrho^{(1)}_{\varepsilon,M}={S}_{M}\varrho^{(1)}_{\varepsilon}\qquad\qquad\text{
and
}\qquad\qquad\boldsymbol{V}_{\varepsilon,M}={S}_{M}\boldsymbol{V}_{\varepsilon}\,.$
(3.3.6)

The previous regularised quantities satisfy the following properties.

###### Proposition 3.3.2

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

$\begin{split}&\sup_{0<\varepsilon\leq
1}\,\left\|\varrho^{(1)}_{\varepsilon}-\varrho^{(1)}_{\varepsilon,M}\right\|_{L^{\infty}([0,T];H^{-s})}\longrightarrow
0\qquad\forall\,s>\max\left\\{0,3\left(\frac{1}{\gamma}\,-\,\frac{1}{2}\right)\right\\}\\\
&\sup_{0<\varepsilon\leq
1}\,\left\|\boldsymbol{V}_{\varepsilon}-\boldsymbol{V}_{\varepsilon,M}\right\|_{L^{\infty}([0,T];H^{-s})}\longrightarrow
0\qquad\forall\,s>\frac{3}{2\,\gamma}\,.\end{split}$ (3.3.7)

Moreover, for any $M>0$, the couple
$(\varrho^{(1)}_{\varepsilon,M},\boldsymbol{V}_{\varepsilon,M})$ satisfies the
approximate wave equations

$\left\\{\begin{array}[]{l}\varepsilon^{m}\,\partial_{t}\varrho^{(1)}_{\varepsilon,M}\,+\,\,{\rm
div}\,\boldsymbol{V}_{\varepsilon,M}\,=\,0\\\\[4.30554pt]
\varepsilon^{m}\,\partial_{t}\boldsymbol{V}_{\varepsilon,M}\,+\varepsilon^{m-1}\,e_{3}\times\boldsymbol{V}_{\varepsilon,M}+\,\nabla_{x}\varrho^{(1)}_{\varepsilon,M}\,=\,\varepsilon^{m}\,\boldsymbol{f}_{\varepsilon,M}\,+\varepsilon^{2(m-n)}\boldsymbol{g}_{\varepsilon,M}\,,\end{array}\right.$
(3.3.8)

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

$\sup_{0<\varepsilon\leq
1}\,\left\|\boldsymbol{f}_{\varepsilon,M}\right\|_{L^{2}([0,T];H^{s})}\,+\,\sup_{0<\varepsilon\leq
1}\,\left\|\boldsymbol{g}_{\varepsilon,M}\right\|_{L^{\infty}([0,T];H^{s})}\,\leq\,C(s,M)\,,$
(3.3.9)

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

Proof:  Thanks to characterization (A.1.2) of $H^{s}$, properties (3.3.7) are
straightforward consequences of the uniform bounds establish in Subsection
3.2.1. For instance, let us consider the functions
$\varrho^{(1)}_{\varepsilon}$: when $\gamma\geq 2$, owing to Remark 3.2.1 one
has
$\big{(}\varrho^{(1)}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{\infty}_{T}(L^{2})$,
and then we use estimate (A.1.3) from Section A.1. When $1<\gamma<2$, instead,
we first apply the dual Sobolev embedding (see Theorem B.2) to infer that
$\big{(}\varrho^{(1)}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{\infty}_{T}(H^{-\sigma})$,
with $\sigma\,=\,\sigma(\gamma)\,=\,3\big{(}1/\gamma-1/2\big{)}$, and then we
use (A.1.3) again. The bounds for the momentum
$\big{(}\boldsymbol{V}_{\varepsilon}\big{)}_{\varepsilon}$ can be deduced by a
similar argument, after observing that $2\gamma/(\gamma+1)<2$ always.

Next, applying the operator ${S}_{M}$ to (3.3.4) immediately gives us system
(3.3.8), where we have set

$\boldsymbol{f}_{\varepsilon,M}:={S}_{M}\boldsymbol{f}_{\varepsilon}\qquad\text{
and
}\qquad\boldsymbol{g}_{\varepsilon,M}:={S}_{M}\boldsymbol{g}_{\varepsilon}\,.$

Thanks to Lemma 3.3.1 and (A.1.2), it is easy to verify inequality (3.3.9).

At this point, we will need also the following important decomposition for the
momentum vector fields $\boldsymbol{V}_{\varepsilon,M}$ and their ${\rm
curl}\,$.

###### Proposition 3.3.3

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

$\boldsymbol{V}_{\varepsilon,M}\,=\,\varepsilon^{2(m-n)}\boldsymbol{t}_{\varepsilon,M}^{1}+\boldsymbol{t}_{\varepsilon,M}^{2}\qquad\mbox{
and }\qquad{\rm
curl}\,\boldsymbol{V}_{\varepsilon,M}=\varepsilon^{2(m-n)}\boldsymbol{T}_{\varepsilon,M}^{1}+\boldsymbol{T}_{\varepsilon,M}^{2}\,,$

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

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

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

Proof:  We decompose
$\boldsymbol{V}_{\varepsilon,M}\,=\,\varepsilon^{2(m-n)}\boldsymbol{t}_{\varepsilon,M}^{1}\,+\,\boldsymbol{t}_{\varepsilon,M}^{2}$,
where we define

$\boldsymbol{t}_{\varepsilon,M}^{1}\,:=\,{S}_{M}\left(\frac{\varrho_{\varepsilon}-1}{\varepsilon^{2(m-n)}}\,\boldsymbol{u}_{\varepsilon}\right)\qquad\mbox{
and
}\qquad\boldsymbol{t}_{\varepsilon,M}^{2}\,:=\,{S}_{M}\boldsymbol{u}_{\varepsilon}\,.$
(3.3.10)

The decomposition of ${\rm curl}\,\boldsymbol{V}_{\varepsilon,M}$ follows
after setting $\boldsymbol{T}_{\varepsilon,M}^{j}\,:=\,{\rm
curl}\,\boldsymbol{t}_{\varepsilon,M}^{j}$, for $j=1,2$.

We have to prove uniform bounds for all those terms, by using the estimates
established in Subsection 3.2.1 above. First of all, we have that
$\big{(}\boldsymbol{u}_{\varepsilon}\big{)}_{\varepsilon}\,\subset\,L^{2}_{T}(H^{1})$,
for any $T>0$ fixed. Then, we immediately gather the sought bounds for the
vector fields $\boldsymbol{t}_{\varepsilon,M}^{2}$ and
$\boldsymbol{T}_{\varepsilon,M}^{2}$.

For the families of $\boldsymbol{t}_{\varepsilon,M}^{1}$ and
$\boldsymbol{T}_{\varepsilon,M}^{1}$, instead, we have to use the bounds
provided by (3.2.8) and (when $\gamma\geq 2$) Remark 3.2.1. In turn, we see
that for any $T>0$,

$\left(\frac{\varrho_{\varepsilon}-1}{\varepsilon^{2(m-n)}}\,\boldsymbol{u}_{\varepsilon}\right)\,\subset\,L^{2}_{T}(L^{1}+L^{2}+L^{6\gamma/(\gamma+6)})\,\hookrightarrow\,L^{2}_{T}(H^{-\sigma})\,,$

for some $\sigma>0$ large enough. Therefore, the claimed bounds follow thanks
to the regularising effect of the operators $S_{M}$. The proof of the
proposition is thus completed.

#####  3.3.2 Analysis of the convection

In this subsection we show the convergence of the convective term. The first
step is to reduce its analysis to the case of smooth vector fields
$\boldsymbol{V}_{\varepsilon,M}$.

###### Lemma 3.3.4

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

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

Proof:  The proof is very similar to the one of Lemma 2.3.5 from Chapter 2,
for this reason we just outline it.

One starts by using the decomposition
$\varrho_{\varepsilon}\,=\,1\,+\,\varepsilon^{2(m-n)}\,R_{\varepsilon}$ to
reduce (owing to the uniform bounds of Subsection 3.2.1) the convective term
to the “homogeneous counterpart”: for any test function $\boldsymbol{\psi}\in
C^{\infty}_{c}\big{(}\mathbb{R}_{+}\times\Omega;\mathbb{R}^{3}\big{)}$, one
has

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

Notice that, here, one has to use that $\gamma\geq 3/2$.

After that, we write
$\boldsymbol{u}_{\varepsilon}\,=\,S_{M}\boldsymbol{u}_{\varepsilon}\,+\,({\rm
Id}\,-S_{M})\boldsymbol{u}_{\varepsilon}\,=\,\boldsymbol{t}^{2}_{\varepsilon,M}\,+\,({\rm
Id}\,-S_{M})\boldsymbol{u}_{\varepsilon}$. Using Proposition 3.3.3 and the
fact that $\left\|({\rm
Id}\,-{S}_{M})\,\boldsymbol{u}_{\varepsilon}\right\|_{L_{T}^{2}(L^{2})}\leq
C\,2^{-M}\|\nabla_{x}\boldsymbol{u}_{\varepsilon}\|_{L^{2}_{T}(L^{2})}\,\leq
C\,2^{-M}$, which holds in view of estimate (A.1.3) from Section A.1 and the
uniform bound (3.2.6), one can conclude.

From now on, for notational convenience, we generically denote by
$\mathcal{R}_{\varepsilon,M}$ any remainder term, that again is any term
satisfying the property

$\lim_{M\rightarrow+\infty}\limsup_{\varepsilon\rightarrow
0^{+}}\left|\int_{0}^{T}\int_{\Omega}\mathcal{R}_{\varepsilon,M}\cdot\boldsymbol{\psi}\,\,{\rm
d}x{\rm d}t\right|=0\,,$ (3.3.11)

for all test functions $\boldsymbol{\psi}\in
C_{c}^{\infty}\bigl{(}[0,T[\,\times\Omega;\mathbb{R}^{3}\bigr{)}$ lying in the
kernel of the singular perturbation operator, namely (in view of Proposition
3.2.3) such that

$\boldsymbol{\psi}\in
C_{c}^{\infty}\big{(}[0,T[\,\times\Omega;\mathbb{R}^{3}\big{)}\qquad\qquad\mbox{
with }\qquad{\rm div}\,\boldsymbol{\psi}=0\quad\mbox{ and
}\quad\partial_{3}\boldsymbol{\psi}=0\,.$ (3.3.12)

Notice that, in order to pass to the limit in the weak formulation of the
momentum equation and derive the limit system, it is enough to use test
functions $\boldsymbol{\psi}$ as above.

Thus, for $\boldsymbol{\psi}$ as in (3.3.12), we have to pass to the limit in
the term

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

Notice that the integration by parts above is well-justified, since all the
quantities inside the integrals are now smooth with respect to the space
variable. Owing to the structure of the test function, and resorting to the
notation (OSC) setted in the introductory part, we remark that we can write

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

where we have defined the terms

$\mathcal{T}^{1}_{\varepsilon,M}\,:=\,{\rm
div}_{h}\left(\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\otimes\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\right)\qquad\mbox{
and }\qquad\mathcal{T}^{2}_{\varepsilon,M}\,:=\,{\rm
div}_{h}\left(\langle\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\otimes\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\rangle\right)\,.$
(3.3.13)

In the next two paragraphs, we will deal with each one of those terms
separately. We borrow most of the arguments from Chapter 2 (see also [34],
[30] for a similar approach). However, the special structure of the gravity
force will play a key role here, in order (loosely speaking) to compensate the
stronger singularity due to our scaling $2n>m$. Finally, we point out that, in
what follows, all the equalities (which will involve the derivative in time)
will hold in the sense of distributions.

######  3.3.2.1 Convergence of the vertical averages

We start by dealing with $\mathcal{T}^{1}_{\varepsilon,M}$. It is standard to
write

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

Notice that the second term is a perfect gradient, so it vanishes when tested
against divergence-free test functions. Hence, we can treat it as a remainder,
in the sense of (3.3.11).

For the first term in the second line of (3.3.14), instead, we take advantage
of system (3.3.8): averaging the first equation with respect to $x^{3}$ and
multiplying it by $\langle\boldsymbol{V}^{h}_{\varepsilon,M}\rangle$, we
arrive at

${\rm
div}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,=\,-\varepsilon^{m}\partial_{t}\langle\varrho^{(1)}_{\varepsilon,M}\rangle\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,=\,\varepsilon^{m}\langle\varrho^{(1)}_{\varepsilon,M}\rangle\partial_{t}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle+\mathcal{R}_{\varepsilon,M}\,.$

We remark that the term presenting the total derivative in time is in fact a
remainder, thanks to the factor $\varepsilon^{m}$ in front of it. Now, we use
the horizontal part of (3.3.8), where we first take the vertical average and
then multiply by $\langle\varrho^{(1)}_{\varepsilon,M}\rangle$: since $m>1$,
we gather

$\displaystyle\varepsilon^{m}\langle\varrho^{(1)}_{\varepsilon,M}\rangle\partial_{t}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle$
$\displaystyle\qquad\quad=-\langle\varrho^{(1)}_{\varepsilon,M}\rangle\nabla_{h}\langle\varrho^{(1)}_{\varepsilon,M}\rangle-\varepsilon^{m-1}\langle\varrho^{(1)}_{\varepsilon,M}\rangle\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle^{\perp}+\varepsilon^{m}\langle\varrho^{(1)}_{\varepsilon,M}\rangle\langle\boldsymbol{f}_{\varepsilon,M}^{h}\rangle+\varepsilon^{2(m-n)}\langle\varrho^{(1)}_{\varepsilon,M}\rangle\langle\boldsymbol{g}_{\varepsilon,M}^{h}\rangle$
$\displaystyle\qquad\quad=-\varepsilon^{m-1}\langle\varrho^{(1)}_{\varepsilon,M}\rangle\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle^{\perp}+\mathcal{R}_{\varepsilon,M}\,,$

where we have repeatedly exploited the properties proved in Proposition 3.3.2
and we have included in the remainder term also the perfect gradient.
Inserting this relation into (3.3.14) yields

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

Observe that the first addendum appearing in the right-hand side of the
previous relation is bilinear. Thus, in order to pass to the limit in it, one
needs some strong convergence properties. As a matter of fact, in the next
computations we will work on the regularised wave system (3.3.8) to show that
the quantity

$\Gamma_{\varepsilon,M}:={\rm
curl}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,-\,\varepsilon^{m-1}\langle\varrho^{(1)}_{\varepsilon,M}\rangle$

is _compact_ in some suitable space. In particular, as $m>1$, also ${\rm
curl}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle$ is compact.

In order to see this, we write the vertical average of the first equation in
(3.3.8) as

$\varepsilon^{2m-1}\,\partial_{t}\langle\varrho^{(1)}_{\varepsilon,M}\rangle\,+\,\varepsilon^{m-1}{\rm
div}\,_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,=0\,.$

Next, we take the vertical average of the horizontal components of (3.3.8) and
then we apply ${\rm curl}_{h}$: one obtains

$\varepsilon^{m}\,\partial_{t}{\rm
curl}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,+\varepsilon^{m-1}\,{\rm
div}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,=\,\varepsilon^{m}{\rm
curl}_{h}\langle\boldsymbol{f}_{\varepsilon,M}^{h}\rangle+\varepsilon^{2(m-n)}{\rm
curl}_{h}\langle\boldsymbol{g}_{\varepsilon,M}^{h}\rangle\,.$

At this point, we recall the definition (3.3.3) of
$\boldsymbol{g}_{\varepsilon}$, and we see that ${\rm
curl}_{h}\langle\boldsymbol{g}_{\varepsilon,M}^{h}\rangle\,\equiv\,0$. This
property is absolutely fundamental, since it allows to erase the last term in
the previous relation, which otherwise would have represented an obstacle to
get compactness for $\Gamma_{\varepsilon,M}$. Indeed, thanks to this
observation, we can sum up the last two equations to get

$\partial_{t}\Gamma_{\varepsilon,M}\,=\,{\rm
curl}_{h}\langle\boldsymbol{f}_{\varepsilon,M}^{h}\rangle\,.$ (3.3.15)

Using estimate (3.3.9) in Proposition 3.3.2, 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. the space $L_{T}^{2}(L^{2})$.
On the other hand, we have that, again for any $M>0$ fixed, the sequence
$(\Gamma_{\varepsilon,M})_{\varepsilon}$ is uniformly bounded (with respect to
$\varepsilon$) e.g. 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 implies that, for any $M>0$ fixed, the family
$(\Gamma_{\varepsilon,M})_{\varepsilon}$ is compact in $L_{T}^{2}(L_{\rm
loc}^{2})$. Then, up to extraction of a suitable subsequence (not relabelled
here), that family converges strongly to a tempered distribution $\Gamma_{M}$
in the same space.

Now, we have that $\Gamma_{\varepsilon,M}$ converges strongly to $\Gamma_{M}$
in $L_{T}^{2}(L_{\rm loc}^{2})$ and
$\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle$ converges weakly to
$\langle\boldsymbol{V}_{M}^{h}\rangle$ in $L_{T}^{2}(L_{\rm loc}^{2})$ (owing
to Proposition 3.3.3, for instance). Then, we deduce

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

Observe that, by definition of $\Gamma_{\varepsilon,M}$, we must have
$\Gamma_{M}={\rm curl}_{h}\langle\boldsymbol{V}_{M}^{h}\rangle$. On the other
hand, owing to Proposition 3.3.3 and (3.3.10), we know that
$\langle\boldsymbol{V}_{M}^{h}\rangle=\langle{S}_{M}\boldsymbol{U}^{h}\rangle$.
Therefore, in the end we have proved that, for $m>1$ and $m+1\geq 2n>m$, one
has the convergence (at any $M\in\mathbb{N}$ fixed, when
$\varepsilon\rightarrow 0^{+}$)

$\int_{0}^{T}\int_{\mathbb{R}^{2}}\mathcal{T}_{\varepsilon,M}^{1}\cdot\boldsymbol{\psi}^{h}\,{\rm
d}x^{h}\,{\rm d}t\,\longrightarrow\,\int^{T}_{0}\int_{\mathbb{R}^{2}}{\rm
curl}_{h}\langle{S}_{M}\boldsymbol{U}^{h}\rangle\;\langle{S}_{M}(\boldsymbol{U}^{h})^{\perp}\rangle\cdot\boldsymbol{\psi}^{h}\,{\rm
d}x^{h}\,{\rm d}t\,,$ (3.3.16)

for any $T>0$ and for any test-function $\boldsymbol{\psi}$ as in (3.3.12).

######  3.3.2.2 Vanishing of the oscillations

We now focus on the term $\mathcal{T}_{\varepsilon,M}^{2}$, defined in
(3.3.13). Recall that $m>1$. In what follows, we consider separately the two
cases $m+1>2n$ and $m+1=2n$. As a matter of fact, in the case when $m+1=2n$, a
bilinear term involving $\boldsymbol{g}_{\varepsilon,M}$ has no power of
$\varepsilon$ in front of it, so it is not clear that it converges to $0$ and,
in fact, it might persist in the limit, giving rise to an additional term in
the target system. To overcome this issue and show that this actually does not
happen, we deeply exploit the structure of the wave system to recover a
quantitative smallness for that term (namely, in terms of positive powers of
$\varepsilon$).

###### The case $m+1>2n$

Starting from the definition of $\mathcal{T}_{\varepsilon,M}^{2}$, the same
computations as above yield

$\displaystyle\mathcal{T}_{\varepsilon,M}^{2}\,$ $\displaystyle=\,\langle{\rm
div}_{h}(\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h})\;\;\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\rangle+\frac{1}{2}\,\langle\nabla_{h}|\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}|^{2}\rangle+\langle{\rm
curl}_{h}\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\,\left(\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\right)^{\perp}\rangle\,.$
(3.3.17)

Let us now introduce the quantities

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

Then we can write

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

In addition, from the momentum equation in (3.3.8), where we take the mean-
free part and then the ${\rm curl}\,$, we deduce the equations

$\begin{cases}\varepsilon^{m}\partial_{t}\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}-\varepsilon^{m-1}\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}=\varepsilon^{m}\left(\partial_{3}^{-1}{\rm
curl}\,\accentset{\approx}{\boldsymbol{f}}_{\varepsilon,M}\right)^{h}+\varepsilon^{2(m-n)}\left(\partial_{3}^{-1}{\rm
curl}\,\accentset{\approx}{\boldsymbol{g}}_{\varepsilon,M}\right)^{h}\\\\[4.30554pt]
\varepsilon^{m}\partial_{t}\accentset{\approx}{\omega}_{\varepsilon,M}^{3}+\varepsilon^{m-1}{\rm
div}_{h}\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}=\varepsilon^{m}\,{\rm
curl}_{h}\accentset{\approx}{\boldsymbol{f}}_{\varepsilon,M}^{h}\,.\end{cases}$
(3.3.18)

Making use of the relations above, recalling the definitions in (3.3.3), and
thanks to Propositions 3.3.2 and 3.3.3, we can write

$\begin{split}{\rm
curl}_{h}\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\;\left(\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\right)^{\perp}&=\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\left(\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\right)^{\perp}\,\\\
&=\varepsilon\partial_{t}\\!\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\accentset{\approx}{\omega}_{\varepsilon,M}^{3}-\varepsilon\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\left(\left(\partial_{3}^{-1}{\rm
curl}\,\accentset{\approx}{\boldsymbol{f}}_{\varepsilon,M}\right)^{h}\right)^{\perp}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad-\varepsilon^{m+1-2n}\,\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\left(\left(\partial_{3}^{-1}{\rm
curl}\,\accentset{\approx}{\boldsymbol{g}}_{\varepsilon,M}\right)^{h}\right)^{\perp}\\\
&=-\varepsilon\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\partial_{t}\accentset{\approx}{\omega}_{\varepsilon,M}^{3}+\mathcal{R}_{\varepsilon,M}=\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\,{\rm
div}_{h}\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}+\mathcal{R}_{\varepsilon,M}\,.\end{split}$
(3.3.19)

We point out that, thanks to the scaling $m+1>2n$, we could include in the
remainder also the last term appearing in the second equality, which was of
order $O(\varepsilon^{m+1-2n})$.

Hence, putting the gradient term into $\mathcal{R}_{\varepsilon,M}$, from
(3.3.17) we arrive at

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

At this point, the computations mainly follow the same lines of [34] (see also
[30]). First of all, we notice that, in the last line, the second term on the
right-hand side is 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

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

Next, in order to deal with the first term, we use the first equation in
(3.3.8) to obtain

$\begin{split}{\rm
div}\,\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}\left(\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)&=-\varepsilon^{m}\partial_{t}\accentset{\approx}{\varrho}^{(1)}_{\varepsilon,M}\left(\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)+\mathcal{R}_{\varepsilon,M}\\\
&=\varepsilon^{m}\accentset{\approx}{\varrho}_{\varepsilon,M}^{(1)}\,\partial_{t}\left(\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}+\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\right)+\mathcal{R}_{\varepsilon,M}\,.\end{split}$

Now, equations (3.3.8) and (3.3.18) immediately yield that

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

This relation finally implies that
$\mathcal{T}_{\varepsilon,M}^{2}\,=\,\mathcal{R}_{\varepsilon,M}$ is a
remainder, in the sense of relation (3.3.11): for any $T>0$ and any test-
function $\boldsymbol{\psi}$ as in (3.3.12), one has the convergence (at any
$M\in\mathbb{N}$ fixed, when $\varepsilon\rightarrow 0^{+}$)

$\int_{0}^{T}\int_{\mathbb{R}^{2}}\mathcal{T}_{\varepsilon,M}^{2}\cdot\boldsymbol{\psi}^{h}\,{\rm
d}x^{h}\,{\rm d}t\,\longrightarrow\,0\,.$ (3.3.20)

###### The case $m+1=2n$

In the case $m+1=2n$, most of the previous computations may be reproduced
exactly in the same way. The only (fundamental) change concerns relation
(3.3.19): since now $m+1-2n=0$, that equation reads

$\begin{split}{\rm
curl}_{h}\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\;\left(\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\right)^{\perp}\,=\,\left(\accentset{\approx}{\Phi}_{\varepsilon,M}^{h}\right)^{\perp}\,{\rm
div}_{h}\accentset{\approx}{\boldsymbol{V}}_{\varepsilon,M}^{h}\,-\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\left(\left(\partial_{3}^{-1}{\rm
curl}\,\accentset{\approx}{\boldsymbol{g}}_{\varepsilon,M}\right)^{h}\right)^{\perp}+\mathcal{R}_{\varepsilon,M}\,,\end{split}$
(3.3.21)

and, repeating the same computations performed for
$\mathcal{T}^{2}_{\varepsilon,M}$ in the previous paragraph, we have

$\mathcal{T}^{2}_{\varepsilon,M}=\mathcal{R}_{\varepsilon,M}-\langle\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\left(\left(\partial_{3}^{-1}{\rm
curl}\,\accentset{\approx}{\boldsymbol{g}}_{\varepsilon,M}\right)^{h}\right)^{\perp}\rangle\,.$

Hence, the main difference with respect to the previous case is that we have
to take care of the term
$\widetilde{\omega}_{\varepsilon,M}^{3}\left(\left(\partial_{3}^{-1}{\rm
curl}\,\accentset{\approx}{\boldsymbol{g}}_{\varepsilon,M}\right)^{h}\right)^{\perp}$,
which is non-linear and of order $O(1)$, so it may potentially give rise to
oscillations which persist in the limit.

In order to show that this does not happen, we make use of definition (3.3.3)
of $\boldsymbol{g}_{\varepsilon}$ to compute

$\displaystyle\left({\rm
curl}\,\accentset{\approx}{\boldsymbol{g}}_{\varepsilon,M}\right)^{h,\perp}\,$
$\displaystyle=\,\left({\rm
curl}\,\left(\accentset{\approx}{\varrho}_{\varepsilon,M}^{(1)}\nabla_{x}G-\nabla_{x}\accentset{\approx}{\Pi}_{\varepsilon,M}\right)\right)^{h,\perp}$
$\displaystyle=\,\begin{pmatrix}-\partial_{2}\accentset{\approx}{\varrho}^{(1)}_{\varepsilon,M}\\\
\partial_{1}\accentset{\approx}{\varrho}^{(1)}_{\varepsilon,M}\\\
0\end{pmatrix}^{h,\perp}\,=-\,\nabla_{h}\accentset{\approx}{\varrho}_{\varepsilon,M}^{(1)}\,.$

From this relation, in turn we get

$\mathcal{T}^{2}_{\varepsilon,M}\,=\,\mathcal{R}_{\varepsilon,M}\,+\,\langle\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\,\partial_{3}^{-1}\nabla_{h}\accentset{\approx}{\varrho}_{\varepsilon,M}^{(1)}\rangle\,.$
(3.3.22)

Now, we have to employ the potential part of the momentum equation in (3.3.8),
which has not been used so far. Taking the oscillating component of the
solutions, we obtain

$\nabla_{h}\accentset{\approx}{\varrho}_{\varepsilon,M}^{(1)}\,=-\,\varepsilon^{m}\,\partial_{t}\accentset{\approx}{\boldsymbol{V}}^{h}_{\varepsilon,M}\,-\varepsilon^{m-1}(\accentset{\approx}{\boldsymbol{V}}^{h}_{\varepsilon,M})^{\perp}+\varepsilon^{m}\,\accentset{\approx}{\boldsymbol{f}}^{h}_{\varepsilon,M}\,+\varepsilon^{2(m-n)}\accentset{\approx}{\boldsymbol{g}}^{h}_{\varepsilon,M}=-\,\varepsilon^{m}\,\partial_{t}\accentset{\approx}{\boldsymbol{V}}^{h}_{\varepsilon,M}\,+\mathcal{R}_{\varepsilon,M}\,.$

Inserting this relation into (3.3.22) and using (3.3.18), we finally gather

$\mathcal{T}^{2}_{\varepsilon,M}=-\varepsilon^{m}\langle\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\,\partial_{t}\partial_{3}^{-1}\accentset{\approx}{\boldsymbol{V}}^{h}_{\varepsilon,M}\rangle+\mathcal{R}_{\varepsilon,M}=\varepsilon^{m}\langle\partial_{t}\accentset{\approx}{\omega}_{\varepsilon,M}^{3}\,\partial_{3}^{-1}\accentset{\approx}{\boldsymbol{V}}^{h}_{\varepsilon,M}\rangle+\mathcal{R}_{\varepsilon,M}=\mathcal{R}_{\varepsilon,M}\,,$

because we have taken $m>1$.

This relation finally implies that, also in the case when $m+1=2n$,
$\mathcal{T}_{\varepsilon,M}^{2}$ is a remainder: for any $T>0$ and any test-
function $\boldsymbol{\psi}$ as in (3.3.12), one has the convergence (3.3.20).

#####  3.3.3 The limit system

Thanks to the computations of the previous subsections, we can now pass to the
limit in equation (3.1.17). Recall that $m>1$ and $m+1\geq 2n>m$ here.

To begin with, we take a test-function $\boldsymbol{\psi}$ as in (3.3.12),
specifically

$\boldsymbol{\psi}=\big{(}\nabla_{h}^{\perp}\phi,0\big{)}\,,\qquad\qquad\mbox{
with }\qquad\phi\in
C_{c}^{\infty}\big{(}[0,T[\,\times\mathbb{R}^{2}\big{)}\,,\quad\phi=\phi(t,x^{h})\,.$
(3.3.23)

We point out that since all the integrals will be made on $\mathbb{R}^{2}$ (in
view of the choice of the test functions in (3.3.23) above), we can work on
the domain $\Omega=\mathbb{R}^{2}\times\,]0,1[\,$.

In addition, for such $\boldsymbol{\psi}$ as in (3.3.23), all the gradient
terms vanish identically, as well as all the contributions due to the vertical
component of the equation. In particular, we do not see any contribution of
the pressure and gravity terms: equation (3.1.17) becomes

$\displaystyle\int_{0}^{T}\\!\\!\\!\int_{\Omega}$
$\displaystyle\left(-\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{h}\cdot\partial_{t}\boldsymbol{\psi}^{h}-\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{h}\otimes\boldsymbol{u}_{\varepsilon}^{h}:\nabla_{h}\boldsymbol{\psi}^{h}+\frac{1}{\varepsilon}\varrho_{\varepsilon}\big{(}\boldsymbol{u}_{\varepsilon}^{h}\big{)}^{\perp}\cdot\boldsymbol{\psi}^{h}\right)\,\,{\rm
d}x{\rm d}t$ (3.3.24)
$\displaystyle\qquad\qquad\qquad\qquad=-\int_{0}^{T}\\!\\!\\!\int_{\Omega}\mathbb{S}(\nabla_{x}\boldsymbol{\boldsymbol{u}_{\varepsilon}}):\nabla_{x}\boldsymbol{\psi}\,{\rm
d}x{\rm
d}t+\int_{\Omega}\varrho_{0,\varepsilon}\boldsymbol{u}_{0,\varepsilon}\cdot\boldsymbol{\psi}(0,\cdot)\,{\rm
d}x\,.$

Making use of the uniform bounds of Subsection 3.2.1, we can pass to the limit
in the $\partial_{t}$ term and in the viscosity term. Moreover, our
assumptions imply that
$\varrho_{0,\varepsilon}\boldsymbol{u}_{0,\varepsilon}\rightharpoonup\boldsymbol{u}_{0}$
in e.g. $L_{\rm loc}^{2}$. Next, the Coriolis term can be arranged in a
standard way: using the structure of $\boldsymbol{\psi}$ and the mass equation
(3.1.16), we can write

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

which of course converges to $0$ when $\varepsilon\rightarrow 0^{+}$.

It remains us to tackle the convective term
$\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{h}\otimes\boldsymbol{u}_{\varepsilon}^{h}$.
For it, we take advantage of Lemma 3.3.4 and relations (3.3.16) and (3.3.20),
but we still have to take care of the convergence for $M\rightarrow+\infty$ in
(3.3.16). We start by performing equalities (3.3.14) backwards in the term on
the right-hand side of (3.3.16): thus, we have to pass to the limit for
$M\rightarrow+\infty$ in

$\int^{T}_{0}\int_{\mathbb{R}^{2}}\boldsymbol{U}_{M}^{h}\otimes\boldsymbol{U}_{M}^{h}:\nabla_{h}\boldsymbol{\psi}^{h}\,\,{\rm
d}x^{h}\,\,{\rm d}t\,.$

Now, we remark that, since $\boldsymbol{U}^{h}\in L^{2}_{T}(H^{1})$ by
(3.2.7), from (A.1.3) we gather the strong convergence
$S_{M}\boldsymbol{U}^{h}\longrightarrow\boldsymbol{U}^{h}$ in
$L_{T}^{2}(H^{s})$ for any $s<1$, in the limit for $M\rightarrow+\infty$.
Then, passing to the limit for $M\rightarrow+\infty$ in the previous relation
is an easy task: we finally get, 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, we have shown that, letting $\varepsilon\rightarrow 0^{+}$ in
(3.3.24), one obtains

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

for any test-function $\boldsymbol{\psi}$ as in (3.3.12). This implies
(3.1.19), concluding the proof of Theorem 3.1.5.

####  3.4 Proof of the convergence for $m=1$

In the present section, we complete the proof of the convergence in the case
$m=1$ and $1/2<n<1$. We will use again the compensated compactness argument
depicted in Subsection 3.3.2, and in fact most of the computations apply also
in this case.

#####  3.4.1 The acoustic-Poincaré waves system

When $m=1$, the wave system (3.3.4) takes the form

$\left\\{\begin{array}[]{l}\varepsilon\,\partial_{t}\varrho_{\varepsilon}^{(1)}\,+\,{\rm
div}\,\boldsymbol{V}_{\varepsilon}\,=\,0\\\\[4.30554pt]
\varepsilon\,\partial_{t}\boldsymbol{V}_{\varepsilon}\,+\,\nabla_{x}\varrho^{(1)}_{\varepsilon}\,+\,\,\boldsymbol{e}_{3}\times\boldsymbol{V}_{\varepsilon}\,=\,\varepsilon\,\boldsymbol{f}_{\varepsilon}+\varepsilon^{2(1-n)}\boldsymbol{g}_{\varepsilon}\,,\end{array}\right.$
(3.4.1)

where $\bigl{(}\varrho^{(1)}_{\varepsilon}\bigr{)}_{\varepsilon}$ and
$\bigl{(}\boldsymbol{V}_{\varepsilon}\bigr{)}_{\varepsilon}$ are defined as in
Paragraph 3.3.1.1. This system is supplemented with the initial datum
$\big{(}\varrho^{(1)}_{0,\varepsilon},\varrho_{0,\varepsilon}\boldsymbol{u}_{0,\varepsilon}\big{)}$.

Next, we regularise all the quantities, by applying the Littlewood-Paley cut-
off operator $S_{M}$ to (3.4.1): we deduce that
$\varrho^{(1)}_{\varepsilon,M}$ and $\boldsymbol{V}_{\varepsilon,M}$, defined
as in (3.3.6), satisfy the regularised wave system

$\left\\{\begin{array}[]{l}\varepsilon\,\partial_{t}\varrho_{\varepsilon,M}^{(1)}\,+\,{\rm
div}\,\boldsymbol{V}_{\varepsilon,M}\,=\,0\\\\[4.30554pt]
\varepsilon\,\partial_{t}\boldsymbol{V}_{\varepsilon,M}\,+\,\nabla_{x}\varrho^{(1)}_{\varepsilon,M}\,+\,\,\boldsymbol{e}_{3}\times\boldsymbol{V}_{\varepsilon,M}\,=\,\varepsilon\,\boldsymbol{f}_{\varepsilon,M}+\varepsilon^{2(1-n)}\boldsymbol{g}_{\varepsilon,M}\,,\end{array}\right.$
(3.4.2)

in the domain $\mathbb{R}_{+}\times\Omega$, where we recall that
$\boldsymbol{f}_{\varepsilon,M}:=S_{M}\boldsymbol{f}_{\varepsilon}$ and
$\boldsymbol{g}_{\varepsilon,M}:=S_{M}\boldsymbol{g}_{\varepsilon}$. It goes
without saying that a result similar to Proposition 3.3.2 holds true also in
this case.

As it is apparent from the wave system (3.4.1) and its regularised version,
when $m=1$ the pressure term and the Coriolis term are in balance, since they
are of the same order. This represents the main change with respect to the
case $m>1$, and it comes into play in the compensated compactness argument.
Therefore, despite most of the computations may be repeated identical as in
the previous section, let us present the main points of the argument.

#####  3.4.2 Handling the convective term when $m=1$

Let us take care of the convergence of the convective term when $m=1$.

First of all, it is easy to see that Lemma 3.3.4 still holds true. Therefore,
given a test-function $\boldsymbol{\psi}\in
C_{c}^{\infty}\big{(}[0,T[\,\times\Omega;\mathbb{R}^{3}\big{)}$ such that
${\rm div}\,\boldsymbol{\psi}=0$ and $\partial_{3}\boldsymbol{\psi}=0$, we
have to pass to the limit in the term

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

where we agree again that the torus $\mathbb{T}^{1}$ has been normalised so
that its Lebesgue measure is equal to $1$ and we have adopted the same
notation as in (3.3.13).

At this point, we notice that the analysis of
$\mathcal{T}_{\varepsilon,M}^{2}$ can be performed as in Paragraph 3.3.2.2,
because we have $m+1>2n$, i.e. $n<1$. _Mutatis mutandis_ , we find relation
(3.3.20) also in the case $m=1$.

Let us now deal with the term $\mathcal{T}_{\varepsilon,M}^{1}$. Arguing as in
Paragraph 3.3.2.1, we may write it as

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

Now we use the horizontal part of (3.4.2): averaging it with respect to the
vertical variable and applying the operator ${\rm curl}_{h}$, we find

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

Taking the difference of this equation with the first one in (3.4.2), we
discover that

$\partial_{t}\widetilde{\Gamma}_{\varepsilon,M}\,=\,{\rm
curl}_{h}\langle\boldsymbol{f}_{\varepsilon,M}^{h}\rangle\,,\qquad\qquad\mbox{
where }\qquad\widetilde{\Gamma}_{\varepsilon,M}:={\rm
curl}_{h}\langle\boldsymbol{V}_{\varepsilon,M}^{h}\rangle\,-\,\langle\varrho^{(1)}_{\varepsilon,M}\rangle\,.$

An argument analogous to the one used after (3.3.15) above, based on Aubin-
Lions Theorem, shows that
$\big{(}\widetilde{\Gamma}_{\varepsilon,M}\big{)}_{\varepsilon}$ is compact in
e.g. $L_{T}^{2}(L_{\rm loc}^{2})$. Then, this sequence converges strongly (up
to extraction of a suitable subsequence, not relabelled here) to a tempered
distribution $\widetilde{\Gamma}_{M}$ in the same space.

Using the previous property, we may deduce that

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

where we have $\langle\boldsymbol{V}_{M}^{h}\rangle=\langle
S_{M}\boldsymbol{U}^{h}\rangle$ and $\widetilde{\Gamma}_{M}={\rm
curl}_{h}\langle S_{M}\boldsymbol{U}^{h}\rangle-\langle
S_{M}\varrho^{(1)}\rangle$.

Owing to the regularity of the target velocity $\boldsymbol{U}^{h}$, we can
pass to the limit also for $M\rightarrow+\infty$, as detailed in Subsection
3.3.3 above. Thus, we find

$\int^{T}_{0}\\!\\!\\!\int_{\Omega}\varrho_{\varepsilon}\,\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon}:\nabla_{x}\boldsymbol{\psi}\,\,{\rm
d}x{\rm
d}t\,\longrightarrow\,\int^{T}_{0}\\!\\!\\!\int_{\mathbb{R}^{2}}\big{(}\boldsymbol{U}^{h}\otimes\boldsymbol{U}^{h}:\nabla_{h}\boldsymbol{\psi}^{h}\,-\,\varrho^{(1)}\,(\boldsymbol{U}^{h})^{\perp}\cdot\boldsymbol{\psi}^{h}\big{)}\,{\rm
d}x^{h}\,{\rm d}t,$ (3.4.3)

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
$\boldsymbol{U}^{h}=\nabla_{h}^{\perp}\varrho^{(1)}$ when $m=1$ (keep in mind
Proposition 3.2.4), the last term in the integral on the right-hand side is
actually zero.

#####  3.4.3 End of the study

Thanks to the previous analysis, we are now ready to pass to the limit in
equation (3.1.17) also in the case when $m=1$. For this, we take a test-
function $\boldsymbol{\psi}$ as in (3.3.23); notice in particular that ${\rm
div}\,\boldsymbol{\psi}=0$ and $\partial_{3}\boldsymbol{\psi}=0$. Then, once
again all the gradient terms and all the contributions coming from the
vertical component of the momentum equation vanish identically, when tested
against such a $\boldsymbol{\psi}$. Recall that all the integrals will be
performed on $\mathbb{R}^{2}$. So, equation (3.1.17) reduces to

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

For the rotation term, we can test the first equation in (3.4.1) against
$\phi$ to get

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

whence we deduce that

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

In addition, the convergence of the convective term has been performed in
(3.4.3), and for the other terms, we can argue as in Subsection 3.3.3. Hence,
letting $\varepsilon\rightarrow 0^{+}$ in the equation above, we get

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

which is the weak formulation of equation (3.1.20). In the end, also Theorem
3.1.6 is proved.

##  Part II The Euler equations

###  Chapter 4 The fast rotation limit for Euler

> We conclude this thesis describing the behaviour of a fluid which, in
> contrast with Chapters 2 and 3, evolves far from the physical boundaries. An
> example of such fluid flows is represented by currents in the middle of the
> oceans, i.e. far enough from the surface and the bottom. In this context, we
> can assume the following physical approximations:
>
>   * •
>
> the fluid is incompressible;
>
>   * •
>
> the density is a small perturbation of a constant profile (the Boussinesq
> approximation);
>
>   * •
>
> the domain is the $\mathbb{R}^{2}$ plane: the motion of a $3$-D highly
> rotating fluid is, in a first approximation, planar (the Taylor-Proudman
> theorem).
>
>

>
> Then, the system reads:
>
> $\begin{cases}\partial_{t}\varrho+{\rm div}\,(\varrho\boldsymbol{u})=0\\\
> \partial_{t}(\varrho\boldsymbol{u})+{\rm
> div}\,(\varrho\boldsymbol{u}\otimes\boldsymbol{u})+\dfrac{1}{Ro}\varrho\boldsymbol{u}^{\perp}+\nabla
> p=0\\\ {\rm div}\,\boldsymbol{u}=0\end{cases}$ (4.0.1)
>
> in the domain $\Omega=\mathbb{R}^{2}$. Since in this chapter there will be
> no more competition between the horizontal and vertical scales, for
> notational convenience, we will drop everywhere (in the differential
> operators) the subscript $x$.
>
> With respect to the previous systems (see (2.0.1) and (3.0.1) in this
> respect), the Euler equations (4.0.1) is an incompressible system without
> the viscosity effects and with a hyperbolic structure. For that reason, we
> will need different analysis techniques (see Appendix A) and in addition the
> functional framework will change (now we will be in regular spaces) in order
> to preserve the initial regularity.
>
> The topics presented here are part of the pre-print [65].
>
> Let us now give a summary of the chapter.
>
> In Section 4.1, we collect our assumptions and we state our main results. In
> Section 4.2, we investigate the well-posedness issues in the Sobolev spaces
> $H^{s}$ for any $s>2$. In Section 4.3, we study the singular perturbation
> problem, establishing constraints that the limit points have to satisfy and
> proving the convergence to the quasi-homogeneous Euler system thanks to a
> compensated compactness technique. In Section 4.4 we review, for the quasi-
> homogeneous limiting system, the results presented in [18] and [16], and we
> explicitly derive the lifespan of solutions to the limit equations (see
> relation (4.4.34) in this regard).
>
> In the last section, we deal with the lifespan analysis for system (4.0.1)
> and we point out some consequences of the continuation criterion we have
> established (see in particular Subsection 4.5.1).

####  4.1 The density-dependent Euler problem

In this section, we formulate our working setting (Subsection 4.1.1) and we
state the main results (Subsection 4.1.2).

#####  4.1.1 Formulation

In this subsection, we present the rescaled density-dependent Euler equations
with the Coriolis force, which we are going to consider in our study, and we
formulate the main working hypotheses.

To begin with, let us introduce the “primitive system”, that is the rescaled
incompressible Euler system (4.0.1), supplemented with the scaling
$Ro=\varepsilon$, where $\varepsilon\in\,]0,1]$ is a small parameter. Thus,
the system consists of continuity equation (conservation of mass), the
momentum equation and the divergence-free condition: respectively

$\begin{cases}\partial_{t}\varrho_{\varepsilon}+{\rm
div}\,(\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon})=0\\\
\partial_{t}(\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon})+{\rm
div}\,(\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}\otimes\boldsymbol{u}_{\varepsilon})+\dfrac{1}{\varepsilon}\varrho_{\varepsilon}\boldsymbol{u}_{\varepsilon}^{\perp}+\dfrac{1}{\varepsilon}\nabla\Pi_{\varepsilon}=0\\\
{\rm div}\,\boldsymbol{u}_{\varepsilon}=0\,.\end{cases}$ (4.1.1)

We point out that, here above in (4.1.1), the domain $\Omega$ is the plane
$\mathbb{R}^{2}$ and the unknowns are $\varrho_{\varepsilon}\in\mathbb{R}_{+}$
and $\boldsymbol{u}_{\varepsilon}\in\mathbb{R}^{2}$.

In (4.1.1), the pressure term has to scale like $1/\varepsilon$, since it is
the only force that allows to compensate the effect of fast rotation, at the
geophysical scale.

From now on, in order to make the Lipschitz condition (A.3.1) holds, we fix

$s>2\,.$

We assume that the initial density is a small perturbation of a constant
profile. Namely, we consider initial densities of the following form:

$\varrho_{0,\varepsilon}=1+\varepsilon\,R_{0,\varepsilon}\,,$ (4.1.2)

where we suppose $R_{0,\varepsilon}$ to be a bounded measurable function
satisfying the controls

$\displaystyle\sup_{\varepsilon\in\,]0,1]}\left\|R_{0,\varepsilon}\right\|_{L^{\infty}(\mathbb{R}^{2})}\,\leq\,C$
(4.1.3) $\displaystyle\sup_{\varepsilon\in\,]0,1]}\left\|\nabla
R_{0,\varepsilon}\right\|_{H^{s-1}(\mathbb{R}^{2})}\,\leq\,C$ (4.1.4)

and the initial mass density is bounded and bounded away from zero, i.e. for
all $\varepsilon\in\;]0,1]$:

$0<\underline{\varrho}\leq\varrho_{0,\varepsilon}(x)\,\leq\,\overline{\varrho}\,,\qquad
x\in\mathbb{R}^{2}$ (4.1.5)

where $\underline{\varrho},\overline{\varrho}>0$ are positive constants.

As for the initial velocity fields, due to framework needed for the well-
posedness issues, we require the following uniform bound

$\sup_{\varepsilon\in\,]0,1]}\left\|\boldsymbol{u}_{0,\varepsilon}\right\|_{H^{s}(\mathbb{R}^{2})}\,\leq\,C\,.$
(4.1.6)

Thanks to the previous uniform estimates, we can assume (up to passing to
subsequences) that there exist $R_{0}\in W^{1,\infty}(\mathbb{R}^{2})$, with
$\nabla R_{0}\in H^{s-1}(\mathbb{R}^{2})$, and $\boldsymbol{u}_{0}\in
H^{s}(\mathbb{R}^{2})$ such that

$\begin{split}R_{0}:=\lim_{\varepsilon\rightarrow
0}R_{0,\varepsilon}\quad&\text{in}\quad L^{\infty}(\mathbb{R}^{2})\\\ \nabla
R_{0}:=\lim_{\varepsilon\rightarrow 0}\nabla
R_{0,\varepsilon}\quad&\text{in}\quad H^{s-1}(\mathbb{R}^{2})\\\
\boldsymbol{u}_{0}:=\lim_{\varepsilon\rightarrow
0}\boldsymbol{u}_{0,\varepsilon}\quad&\text{in}\quad
H^{s}(\mathbb{R}^{2})\,,\end{split}$ (4.1.7)

where we agree that the previous limits are taken in the corresponding
weak-$\ast$ topology.

#####  4.1.2 Main statements

We can now state our main results. We recall the notation
$\big{(}f_{\varepsilon}\big{)}_{\varepsilon}\subset X$ to denote that the
family $\big{(}f_{\varepsilon}\big{)}_{\varepsilon}$ is uniformly (in
$\varepsilon$) bounded in $X$.

The following theorem establishes the local well-posedness of system (4.1.1)
in the Sobolev spaces $B^{s}_{2,2}\equiv H^{s}$ (see Section 4.2) and gives a
lower bound for the lifespan of solutions (see Section 4.5).

###### Theorem 4.1.1

For any $\varepsilon\in\,]0,1]$, let initial densities
$\varrho_{0,\varepsilon}$ be as in (4.1.2) and satisfy the controls (4.1.3) to
(4.1.5). Let $\boldsymbol{u}_{0,\varepsilon}$ be divergence-free vector fields
such that $\boldsymbol{u}_{0,\varepsilon}\in H^{s}(\mathbb{R}^{2})$ for $s>2$.
Then, for any $\varepsilon>0$, there exists a time $T_{\varepsilon}^{\ast}>0$
such that the system (4.1.1) has a unique solution
$(\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon},\nabla\Pi_{\varepsilon})$
where

  * •

$\varrho_{\varepsilon}$ belongs to the space
$C^{0}([0,T_{\varepsilon}^{\ast}]\times\mathbb{R}^{2})$ with
$\nabla\varrho_{\varepsilon}\in
C^{0}([0,T_{\varepsilon}^{\ast}];H^{s-1}(\mathbb{R}^{2}))$;

  * •

$\boldsymbol{u}_{\varepsilon}$ belongs to the space
$C^{0}([0,T_{\varepsilon}^{\ast}];H^{s}(\mathbb{R}^{2}))$;

  * •

$\nabla\Pi_{\varepsilon}$ belongs to the space
$C^{0}([0,T_{\varepsilon}^{\ast}];H^{s}(\mathbb{R}^{2}))$.

Moreover, the lifespan $T_{\varepsilon}^{\ast}$ of the solution to the two-
dimensional density-dependent incompressible Euler equations with the Coriolis
force is bounded from below by

$\frac{C}{\|\boldsymbol{u}_{0,\varepsilon}\|_{H^{s}}}\log\left(\log\left(\frac{C\,\|\boldsymbol{u}_{0,\varepsilon}\|_{H^{s}}}{\max\\{\mathcal{A}_{\varepsilon}(0),\,\varepsilon\,\mathcal{A}_{\varepsilon}(0)\,\|\boldsymbol{u}_{0,\varepsilon}\|_{H^{s}}\\}}+1\right)+1\right)\,,$
(4.1.8)

where $\mathcal{A}_{\varepsilon}(0):=\|\nabla
R_{0,\varepsilon}\|_{H^{s-1}}+\varepsilon\,\|\nabla
R_{0,\varepsilon}\|_{H^{s-1}}^{\lambda+1}$, for some suitable $\lambda\geq 1$.

In particular, one has

$\inf_{\varepsilon>0}T_{\varepsilon}^{\ast}>0\,.$

Looking at (4.1.8), we stress the fact that the fast rotational effects are
not enough to state a global well-posedness result for system (4.1.1), in the
sense that $T^{\ast}_{\varepsilon}$ does not tend to $+\infty$ when
$\varepsilon\rightarrow 0^{+}$.

Now, once we have stated the local in time well-posedness for system (4.1.1)
in the Sobolev spaces $H^{s}$, in Section 4.3 we address the singular
perturbation problem describing, in a rigorous way, the limit dynamics
depicted by the quasi-homogeneous incompressible Euler system (4.1.9) below.

###### Theorem 4.1.2

Let $s>2$. 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 4.1.1, and let
$\left(\varrho_{\varepsilon},\boldsymbol{u}_{\varepsilon}\right)$ be a
corresponding solution to system (4.1.1). Let
$\left(R_{0},\boldsymbol{u}_{0}\right)$ be defined as in (4.1.7).

Then, one has the following convergence properties:

$\displaystyle\varrho_{\varepsilon}\rightarrow 1\qquad\qquad\mbox{ in }\qquad$
$\displaystyle
L^{\infty}\big{(}[0,T^{\ast}];L^{\infty}(\mathbb{R}^{2})\big{)}\,,$
$\displaystyle
R_{\varepsilon}:=\frac{\varrho_{\varepsilon}-1}{\varepsilon}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}R\qquad\qquad\mbox{
in }\qquad$ $\displaystyle
L^{\infty}\bigl{(}[0,T^{\ast}];L^{\infty}(\mathbb{R}^{2})\bigr{)}\,,$
$\displaystyle\nabla
R_{\varepsilon}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\nabla
R\qquad\qquad\mbox{ in }\qquad$ $\displaystyle
L^{\infty}\bigl{(}[0,T^{\ast}];H^{s-1}(\mathbb{R}^{2})\bigr{)}\,,$
$\displaystyle\boldsymbol{u}_{\varepsilon}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\boldsymbol{u}\qquad\qquad\mbox{
in }\qquad$ $\displaystyle
L^{\infty}\big{(}[0,T^{\ast}];H^{s}(\mathbb{R}^{2})\big{)}\,.$

In addition, $\Big{(}R\,,\,\boldsymbol{u}\Big{)}$ is a solution to the quasi-
homogeneous incompressible Euler system in $[0,T^{\ast}]\times\mathbb{R}^{2}$:

$\begin{cases}\partial_{t}R+{\rm div}\,(R\boldsymbol{u})=0\\\
\partial_{t}\boldsymbol{u}+{\rm
div}\,\left(\boldsymbol{u}\otimes\boldsymbol{u}\right)+R\boldsymbol{u}^{\perp}+\nabla\Pi=0\\\
{\rm div}\,\boldsymbol{u}\,=\,0\,,\end{cases}$ (4.1.9)

where $\nabla\Pi$ is a suitable pressure term belonging to
$L^{\infty}\big{(}[0,T^{\ast}];H^{s}(\mathbb{R}^{2})\big{)}$.

###### Remark 4.1.3

Due to the fact that the system (4.1.9) is well-posed in the previous
functional setting (see Theorem 4.1.4 below), we get the convergence of the
whole sequence of weak solutions to the solutions of the target equations on
the large time interval where the weak solutions to the primitive equations
exist.

Then, at the limit, we have found that the dynamics is prescribed by the
quasi-homogeneous incompressible Euler system (4.1.9), for which we state the
local well-posedness in $H^{s}$ (see Section 4.4). It is worth to remark that
the global well-posedness issue for this system is still an open problem.

###### Theorem 4.1.4

Take $s>2$. Let $\big{(}R_{0},u_{0}\big{)}$ be initial data such that
$R_{0}\in L^{\infty}(\mathbb{R}^{2})$ and $\boldsymbol{u}_{0}\,\in
H^{s}(\mathbb{R}^{2})$, with $\nabla R_{0}\in H^{s-1}(\mathbb{R}^{2})$ and
${\rm div}\,\boldsymbol{u}_{0}\,=\,0$.

Then, there exists a time $T^{\ast}>0$ such that, on
$[0,T^{\ast}]\times\mathbb{R}^{2}$, problem (4.1.9) has a unique solution
$(R,\boldsymbol{u},\nabla\Pi)$ with the following properties:

  * •

$R\in C^{0}\big{(}[0,T^{\ast}]\times\mathbb{R}^{2}\big{)}$ and $\nabla R\in
C^{0}\big{(}[0,T^{\ast}];H^{s-1}(\mathbb{R}^{2})\big{)}$;

  * •

$\boldsymbol{u}$ belongs to
$C^{0}\big{(}[0,T^{\ast}];H^{s}(\mathbb{R}^{2})\big{)}$;

  * •

the pressure term $\nabla\Pi$ is in
$C^{0}\big{(}[0,T^{\ast}];H^{s}(\mathbb{R}^{2})\big{)}$.

In addition, the lifespan $T^{\ast}>0$ of the solution
$(R,\boldsymbol{u},\nabla\Pi)$ to the $2$-D quasi-homogeneous Euler system
(4.1.9) enjoys the following lower bound:

$T^{\ast}\geq\frac{C}{\|\boldsymbol{u}_{0}\|_{H^{s}}}\log\left(\log\left(C\frac{\|\boldsymbol{u}_{0}\|_{H^{s}}}{\|R_{0}\|_{L^{\infty}}+\|\nabla
R_{0}\|_{H^{s-1}}}+1\right)+1\right)\,,$ (4.1.10)

where $C>0$ is a “universal” constant, independent of the initial datum.

The proof of the previous “asymptotically global” well-posedness result is
presented in Subsection 4.4.3.

####  4.2 Well-posedness for the original problem

This section is devoted to the well-posedness issue in the $H^{s}$ spaces
stated in Theorem 4.1.1. We recall that, due to the Littlewood-Paley theory,
we have the equivalence between $H^{s}$ and $B^{s}_{2,2}$ spaces.

We also underline that in this section we keep $\varepsilon\in\;]0,1]$ fixed.
However, we will take care of it, explicitly pointing out the dependence to
the Rossby number in all the computations in order to get controls that are
uniform with respect to the $\varepsilon$-parameter. The choice in keeping
explicit the dependence on the rotational parameter is motivated by the fact
that we will perform the fast rotation limit (see Section 4.3 below).

First of all, since $\varrho_{\varepsilon}$ is a small perturbation of a
constant profile, we set

$\alpha_{\varepsilon}:=\frac{1}{\varrho_{\varepsilon}}-1=\varepsilon
a_{\varepsilon}\quad\text{with}\quad
a_{\varepsilon}:=-R_{\varepsilon}/\varrho_{\varepsilon}\,.$ (4.2.1)

The choice of looking at $\alpha_{\varepsilon}$ is dictated by the fact that
we will extensively exploit the elliptic equation

$-{\rm
div}\,\left(\alpha_{\varepsilon}\nabla\Pi_{\varepsilon}\right)=\,\varepsilon\,{\rm
div}\,\left(\boldsymbol{u}_{\varepsilon}\cdot\nabla\boldsymbol{u}_{\varepsilon}+\frac{1}{\varepsilon}\boldsymbol{u}_{\varepsilon}^{\perp}\right)\,.$
(4.2.2)

Now, using the divergence-free condition, we can rewrite the system (4.1.1) in
the following way (see also Lemma 3 in [22]):

$\begin{cases}\partial_{t}a_{\varepsilon}+\boldsymbol{u}_{\varepsilon}\cdot\nabla
a_{\varepsilon}=0\\\
\partial_{t}\boldsymbol{u}_{\varepsilon}+\boldsymbol{u}_{\varepsilon}\cdot\nabla\boldsymbol{u}_{\varepsilon}+\dfrac{1}{\varepsilon}\boldsymbol{u}_{\varepsilon}^{\perp}+(1+\varepsilon
a_{\varepsilon})\dfrac{1}{\varepsilon}\nabla\Pi_{\varepsilon}=0\\\ {\rm
div}\,\boldsymbol{u}_{\varepsilon}=0\,,\end{cases}$ (4.2.3)

with the initial condition
$(a_{\varepsilon},\boldsymbol{u}_{\varepsilon})_{|t=0}=(a_{0,\varepsilon},\boldsymbol{u}_{0,\varepsilon})$.

We start by presenting the proof of existence of solutions at the claimed
regularity. For that scope, we follow a standard procedure: first, we
construct a sequence of smooth approximate solutions. Next, we deduce uniform
bounds (with respect to the approximation parameter and also to $\varepsilon$)
for those regular solutions. Finally, by use of those uniform bounds and an
energy method, together with an interpolation argument, we are able to take
the limit in the approximation parameter and gather the existence of a
solution to the original problem.

We end this Section 4.2, proving uniqueness of solutions in the claimed
functional setting, by using a relative entropy method.

#####  4.2.1 Construction of smooth approximate solutions

For any $n\in\mathbb{N}$, let us define

$(a_{0,\varepsilon}^{n},\boldsymbol{u}_{0,\varepsilon}^{n}):=(S_{n}a_{0,\varepsilon},S_{n}\boldsymbol{u}_{0,\varepsilon})\,,$

where $S_{n}$ is the low frequency cut-off operator introduced in (A.1.1) in
Section A.1. We stress also the fact that $a_{0,\varepsilon}\in C^{0}_{\rm
loc}$, since $a_{0,\varepsilon}$ and $\nabla a_{0,\varepsilon}$ are in
$L^{\infty}$.

Then, for any $n\in\mathbb{N}$, we have the density functions
$a_{0,\varepsilon}^{n}\in L^{\infty}$. Moreover, one has that $\nabla
a_{0,\varepsilon}^{n}$ and $\boldsymbol{u}_{0,\varepsilon}^{n}$ belong to
$H^{\infty}:=\bigcap_{\sigma\in\mathbb{R}}H^{\sigma}$ which is embedded (for a
suitable topology on $H^{\infty}$) in the space $C_{b}^{\infty}$ of
$C^{\infty}$ functions which are globally bounded together with all their
derivatives.

In addition, by standard properties of mollifiers (see Section B.2 of the
Appendix B), one has the following strong convergences

$\begin{split}a^{n}_{0,\varepsilon}\rightarrow
a_{0,\varepsilon}\quad&\text{in}\quad C_{\rm loc}^{0}\\\ \nabla
a^{n}_{0,\varepsilon}\rightarrow\nabla a_{0,\varepsilon}\quad&\text{in}\quad
H^{s-1}\\\
\boldsymbol{u}_{0,\varepsilon}^{n}\rightarrow\boldsymbol{u}_{0,\varepsilon}\quad&\text{in}\quad
H^{s}\,.\end{split}$ (4.2.4)

This having been established, we are going to define a sequence of approximate
solutions to system (4.2.3) by induction. First of all, we set
$(a_{\varepsilon}^{0},\boldsymbol{u}_{\varepsilon}^{0},\nabla\Pi_{\varepsilon}^{0})=(a^{0}_{0,\varepsilon},\boldsymbol{u}^{0}_{0,\varepsilon},0)$.
Then, for all $\sigma\in\mathbb{R}$, we have that $\nabla
a_{\varepsilon}^{0},\boldsymbol{u}_{\varepsilon}^{0}\in H^{\sigma}$ and
$a^{0}_{\varepsilon}\in L^{\infty}$ with ${\rm
div}\,\boldsymbol{u}_{\varepsilon}^{0}=0$. Next, assume that the couple
$(a_{\varepsilon}^{n},\boldsymbol{u}_{\varepsilon}^{n})$ is given such that,
for all $\sigma\in\mathbb{R}$,

$a^{n}_{\varepsilon}\in C^{0}(\mathbb{R}_{+};L^{\infty})\quad\nabla
a_{\varepsilon}^{n},\boldsymbol{u}_{\varepsilon}^{n}\in
C^{0}(\mathbb{R}_{+};H^{\sigma})\quad\quad\text{and}\quad{\rm
div}\,\boldsymbol{u}_{\varepsilon}^{n}=0\,.$

First of all, we define $a_{\varepsilon}^{n+1}$ as the unique solution to the
linear transport equation

$\partial_{t}a_{\varepsilon}^{n+1}+\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla
a_{\varepsilon}^{n+1}=0\quad\text{with}\quad{(a_{\varepsilon}^{n+1})}_{|t=0}=a_{0,\varepsilon}^{n+1}\,.$
(4.2.5)

Since, by inductive hypothesis and embeddings,
$\boldsymbol{u}_{\varepsilon}^{n}$ is divergence-free, smooth and uniformly
bounded with all its derivatives, we can deduce that $a_{\varepsilon}^{n+1}\in
L^{\infty}(\mathbb{R}_{+};L^{\infty})$. Moreover, from

$\partial_{t}\,\partial_{i}a_{\varepsilon}^{n+1}+\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\,\partial_{i}a_{\varepsilon}^{n+1}=-\partial_{i}\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla
a_{\varepsilon}^{n+1}\quad\text{with}\quad{(\partial_{i}a_{\varepsilon}^{n+1})}_{|t=0}=\partial_{i}a_{0,\varepsilon}^{n+1}\quad\text{for}\;i=1,2$

and thanks to the Theorem A.15, we can propagate all the $H^{\sigma}$ norms of
the initial datum. We deduce that $a_{\varepsilon}^{n+1}\in
C^{0}(\mathbb{R}_{+};L^{\infty})$ and $\nabla a_{\varepsilon}^{n+1}\in
C^{0}(\mathbb{R}_{+};H^{\sigma})$ for any $\sigma\in\mathbb{R}$. Next, we
consider the approximate linear iteration

$\begin{cases}\partial_{t}\boldsymbol{u}_{\varepsilon}^{n+1}+\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n+1}+\dfrac{1}{\varepsilon}\boldsymbol{u}_{\varepsilon}^{\perp,n+1}+(1+\varepsilon
a_{\varepsilon}^{n+1})\dfrac{1}{\varepsilon}\nabla\Pi_{\varepsilon}^{n+1}=0\\\
{\rm div}\,\boldsymbol{u}_{\varepsilon}^{n+1}=0\\\
(\boldsymbol{u}_{\varepsilon}^{n+1})_{|t=0}=\boldsymbol{u}_{0,\varepsilon}^{n+1}\,.\end{cases}$
(4.2.6)

At this point, one can solve the previous linear problem finding a unique
solution $\boldsymbol{u}_{\varepsilon}^{n+1}\in
C^{0}(\mathbb{R}_{+};H^{\sigma})$ for any $\sigma\in\mathbb{R}$ and the
pressure term $\nabla\Pi_{\varepsilon}^{n+1}$ can be uniquely determined (we
refer to [20] for details in this respect).

#####  4.2.2 Uniform estimates for the approximate solutions

We now have to show (by induction) uniform bounds for the sequence
$(a_{\varepsilon}^{n},\boldsymbol{u}_{\varepsilon}^{n},\nabla\Pi_{\varepsilon}^{n})_{n\in\mathbb{N}}$
we have constructed above.

We start by finding uniform estimates for $a_{\varepsilon}^{n+1}$. Thanks to
equation (4.2.5) and the divergence-free condition on
$\boldsymbol{u}_{\varepsilon}^{n}$, we can propagate the $L^{\infty}$ norm for
any $t\geq 0$:

$\|a_{\varepsilon}^{n+1}(t)\|_{L^{\infty}}\leq\|a_{0,\varepsilon}^{n+1}\|_{L^{\infty}}\leq
C\|a_{0,\varepsilon}\|_{L^{\infty}}\,.$ (4.2.7)

At this point we want to estimate $\nabla a_{\varepsilon}^{n+1}$ in $H^{s-1}$.
We have for $i=1,2$:

$\partial_{t}\,\partial_{i}a_{\varepsilon}^{n+1}+\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\,\partial_{i}a_{\varepsilon}^{n+1}=-\partial_{i}\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla
a_{\varepsilon}^{n+1}\,.$

Taking the non-homogeneous dyadic blocks $\Delta_{j}$, we obtain

$\partial_{t}\Delta_{j}\,\partial_{i}a_{\varepsilon}^{n+1}+\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\Delta_{j}\,\partial_{i}a_{\varepsilon}^{n+1}=[\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla,\Delta_{j}]\,\partial_{i}a_{\varepsilon}^{n+1}-\Delta_{j}\left(\partial_{i}\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla
a_{\varepsilon}^{n+1}\right)\,.$

Multiplying the previous relation by
$\Delta_{j}\,\partial_{i}a_{\varepsilon}^{n+1}$, we have

$\|\Delta_{j}\,\partial_{i}a_{\varepsilon}^{n+1}(t)\|_{L^{2}}\leq\|\Delta_{j}\,\partial_{i}a_{0,\varepsilon}^{n+1}\|_{L^{2}}+C\int_{0}^{t}\left(\left\|[\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla,\Delta_{j}]\,\partial_{i}a_{\varepsilon}^{n+1}\right\|_{L^{2}}+\|\Delta_{j}\left(\partial_{i}\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla
a_{\varepsilon}^{n+1}\right)\|_{L^{2}}\right)\,\,{\rm d}\tau\,.$

We apply now the second commutator estimate stated in Lemma A.13 to the former
term in the integral on the right-hand side, getting

$2^{j(s-1)}\left\|[\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla,\Delta_{j}]\,\partial_{i}a_{\varepsilon}^{n+1}\right\|_{L^{2}}\leq
C\,c_{j}(t)\left(\|\nabla\boldsymbol{u}_{\varepsilon}^{n}\|_{L^{\infty}}\|\partial_{i}a_{\varepsilon}^{n+1}\|_{H^{s-1}}+\|\nabla\boldsymbol{u}_{\varepsilon}^{n}\|_{H^{s-1}}\|\partial_{i}a_{\varepsilon}^{n+1}\|_{L^{\infty}}\right)\,,$

where $(c_{j}(t))_{j\geq-1}$ is a sequence in the unit ball of $\ell^{2}$.

Instead, the latter term can be bounded in the following way:

$2^{j(s-1)}\|\Delta_{j}\left(\partial_{i}\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla
a_{\varepsilon}^{n+1}\right)\|_{L^{2}}\leq
C\,c_{j}(t)\,\|\nabla\boldsymbol{u}_{\varepsilon}^{n}\|_{H^{s-1}}\|\nabla
a_{\varepsilon}^{n+1}\|_{H^{s-1}}\,.$

Then, due to the embedding $H^{\sigma}(\mathbb{R}^{2})\hookrightarrow
L^{\infty}(\mathbb{R}^{2})$ for $\sigma>1$, one has

$2^{j(s-1)}\|\Delta_{j}\,\nabla a_{\varepsilon}^{n+1}(t)\|_{L^{2}}\leq
2^{j(s-1)}\|\Delta_{j}\nabla
a_{0,\varepsilon}^{n+1}\|_{L^{2}}+\int_{0}^{t}C\,c_{j}(\tau)\left(\|\boldsymbol{u}_{\varepsilon}^{n}\|_{H^{s}}\|\nabla
a_{\varepsilon}^{n+1}\|_{H^{s-1}}\right)\,\,{\rm d}\tau\,.$

At this point, after summing on indices $j\geq-1$, thanks to the Minkowski
inequality (for which we refer to Proposition B.6) combined with a Grönwall
type argument (see Section B.6), we finally obtain

$\sup_{0\leq t\leq T}\|\nabla a_{\varepsilon}^{n+1}(t)\|_{H^{s-1}}\leq\|\nabla
a_{0,\varepsilon}^{n+1}\|_{H^{s-1}}\,\exp\left(\int_{0}^{T}C\,\|\boldsymbol{u}_{\varepsilon}^{n}(t)\|_{H^{s}}\,\,{\rm
d}t\right)\,.$ (4.2.8)

Now, we have to estimate the velocity field
$\boldsymbol{u}_{\varepsilon}^{n+1}$ and for that purpose we start with the
$L^{2}$ estimate. We take the momentum equation in the original form:

$\varrho_{\varepsilon}^{n+1}\left(\partial_{t}\boldsymbol{u}_{\varepsilon}^{n+1}+\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n+1}\right)+\frac{1}{\varepsilon}\varrho_{\varepsilon}^{n+1}\boldsymbol{u}_{\varepsilon}^{\perp,n+1}+\frac{1}{\varepsilon}\nabla\Pi_{\varepsilon}^{n+1}=0\,,$
(4.2.9)

where we construct $\varrho_{\varepsilon}^{n+1}:=1/(1+\varepsilon
a_{\varepsilon}^{n+1})$ starting from $a_{\varepsilon}^{n+1}$. Notice that
$\varrho_{\varepsilon}^{n+1}$ satisfies the transport equation

$\partial_{t}\varrho_{\varepsilon}^{n+1}+\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\varrho_{\varepsilon}^{n+1}=0\,.$

At this point, we test equation (4.2.9) against
$\boldsymbol{u}_{\varepsilon}^{n+1}$. We integrate by parts on
$\mathbb{R}^{2}$, deriving the following estimate:

$\int_{\mathbb{R}^{2}}\varrho_{\varepsilon}^{n+1}\partial_{t}|\boldsymbol{u}_{\varepsilon}^{n+1}|^{2}+\int_{\mathbb{R}^{2}}\varrho_{\varepsilon}^{n+1}\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla|\boldsymbol{u}_{\varepsilon}^{n+1}|^{2}=0\,,$

which implies (making use of the transport equation for
$\varrho_{\varepsilon}^{n+1}$)

$\left\|\sqrt{\varrho_{\varepsilon}^{n+1}(t)}\,\boldsymbol{u}_{\varepsilon}^{n+1}(t)\right\|_{L^{2}}\leq\left\|\sqrt{\varrho_{0,\varepsilon}^{n+1}}\,\boldsymbol{u}_{0,\varepsilon}^{n+1}\right\|_{L^{2}}\,.$

From the previous bound, due to the assumption (4.1.5), we can deduce the
preservation of the $L^{2}$ norm for the velocity field
$\boldsymbol{u}_{\varepsilon}^{n+1}$:

$\left\|\boldsymbol{u}_{\varepsilon}^{n+1}(t)\right\|_{L^{2}}\leq
C\left\|\boldsymbol{u}_{0,\varepsilon}^{n+1}\right\|_{L^{2}}\leq
C\left\|\boldsymbol{u}_{0,\varepsilon}\right\|_{L^{2}}\,.$

Taking now the operator $\Delta_{j}$ in the momentum equation in (4.2.6), we
obtain

$\partial_{t}\Delta_{j}\boldsymbol{u}_{\varepsilon}^{n+1}+\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\Delta_{j}\boldsymbol{u}_{\varepsilon}^{n+1}=[\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla,\Delta_{j}]\boldsymbol{u}_{\varepsilon}^{n+1}-\frac{1}{\varepsilon}\Delta_{j}\boldsymbol{u}_{\varepsilon}^{\perp,n+1}-\Delta_{j}\left[\left(1+\varepsilon
a_{\varepsilon}^{n+1}\right)\frac{1}{\varepsilon}\nabla\Pi_{\varepsilon}^{n+1}\right]$

and multiplying again by $\Delta_{j}\boldsymbol{u}_{\varepsilon}^{n+1}$, we
have cancellations so that

$\left\|\Delta_{j}\boldsymbol{u}_{\varepsilon}^{n+1}(t)\right\|_{L^{2}}\leq\left\|\Delta_{j}\boldsymbol{u}_{0,\varepsilon}^{n+1}\right\|_{L^{2}}+C\int_{0}^{t}\left(\left\|[\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla,\Delta_{j}]\boldsymbol{u}_{\varepsilon}^{n+1}\right\|_{L^{2}}+\left\|\Delta_{j}\left(a_{\varepsilon}^{n+1}\nabla\Pi_{\varepsilon}^{n+1}\right)\right\|_{L^{2}}\right)\,\,{\rm
d}\tau\,.$ (4.2.10)

As done before, we employ here the commutator estimates of Lemma A.13 in order
to have

$\begin{split}2^{js}\left\|[\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla,\Delta_{j}]\boldsymbol{u}_{\varepsilon}^{n+1}\right\|_{L^{2}}&\leq
C\,c_{j}\,\left(\|\nabla\boldsymbol{u}_{\varepsilon}^{n}\|_{L^{\infty}}\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{H^{s}}+\|\nabla\boldsymbol{u}_{\varepsilon}^{n+1}\|_{L^{\infty}}\|\boldsymbol{u}_{\varepsilon}^{n}\|_{H^{s}}\right)\\\
&\leq
C\,c_{j}\,\left(\|\boldsymbol{u}_{\varepsilon}^{n}\|_{H^{s}}\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{H^{s}}\right)\,.\end{split}$

For the latter term on the right-hand side of (4.2.10), we take advantage of
the Bony decomposition (see Paragraph A.2) and we apply Proposition A.10. We
may infer that

$\begin{split}\left\|a_{\varepsilon}^{n+1}\nabla\Pi_{\varepsilon}^{n+1}\right\|_{H^{s}}\leq
C\left(\|a_{\varepsilon}^{n+1}\|_{L^{\infty}}+\|\nabla
a_{\varepsilon}^{n+1}\|_{H^{s-1}}\right)\|\nabla\Pi_{\varepsilon}^{n+1}\|_{H^{s}}\,.\end{split}$

To finish with, we have to find a uniform bound for the pressure term. For
that, we apply the ${\rm div}\,$ operator in (4.2.6). Thus, we aim at solving
the elliptic problem

$-{\rm div}\,\left(\left(1+\varepsilon
a_{\varepsilon}^{n+1}\right)\nabla\Pi_{\varepsilon}^{n+1}\right)=\,\varepsilon\,{\rm
div}\,(\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n+1})-{\rm
curl}\,\boldsymbol{u}_{\varepsilon}^{n+1}\,.$ (4.2.11)

Thanks to the assumption (4.1.5) and Lemma B.10, we can obtain

$\begin{split}\|\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}&\leq
C\left(\varepsilon\,\|\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n+1}\|_{L^{2}}+\|\boldsymbol{u}_{\varepsilon}^{\perp,n+1}\|_{L^{2}}\right)\\\
&\leq
C\left(\varepsilon\,\|\boldsymbol{u}_{\varepsilon}^{n}\|_{L^{2}}\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{H^{s}}+\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{L^{2}}\right)\,.\end{split}$
(4.2.12)

Now, we apply the spectral cut-off operator $\Delta_{j}$ to (4.2.11). We get

$-\,{\rm
div}\,\left(A_{\varepsilon}\Delta_{j}\nabla\Pi_{\varepsilon}^{n+1}\right)={\rm
div}\,\left(\left[\Delta_{j},A_{\varepsilon}\right]\nabla\Pi_{\varepsilon}^{n+1}\right)+\,{\rm
div}\,\Delta_{j}F_{\varepsilon}\,,$

for all $j\geq 0$ and where we have defined
$A_{\varepsilon}:=\left(1+\varepsilon a_{\varepsilon}^{n+1}\right)$ and
$F_{\varepsilon}:=\varepsilon\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n+1}+\boldsymbol{u}_{\varepsilon}^{\perp,n+1}$.
Hence multiplying both sides by $\Delta_{j}\Pi_{\varepsilon}^{n+1}$ and
integrating over $\mathbb{R}^{2}$, we have

$-\int_{\mathbb{R}^{2}}\Delta_{j}\Pi_{\varepsilon}^{n+1}{\rm
div}\,\left(A_{\varepsilon}\Delta_{j}\nabla\Pi_{\varepsilon}^{n+1}\right)\,\,{\rm
d}x=\int_{\mathbb{R}^{2}}\Delta_{j}\Pi_{\varepsilon}^{n+1}{\rm
div}\,\left(\left[\Delta_{j},A_{\varepsilon}\right]\nabla\Pi_{\varepsilon}^{n+1}\right)\,\,{\rm
d}x+\int_{\mathbb{R}^{2}}\Delta_{j}\Pi_{\varepsilon}^{n+1}{\rm
div}\,\Delta_{j}F_{\varepsilon}\,\,{\rm d}x.$

Since for $j\geq 0$ we have
$\|\Delta_{j}\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}\sim
2^{j}\|\Delta_{j}\Pi_{\varepsilon}^{n+1}\|_{L^{2}}$ (according to Lemma A.2)
and using Hölder’s inequality (see Proposition B.5) for the right-hand side,
we obtain for all $j\geq 0$:

$2^{j}\|\Delta_{j}\nabla\Pi_{\varepsilon}^{n+1}\|^{2}_{L^{2}}\leq
C\|\Delta_{j}\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}\left(\|{\rm
div}\,\left[\Delta_{j},A_{\varepsilon}\right]\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}+\|{\rm
div}\,\Delta_{j}F_{\varepsilon}\|_{L^{2}}\right)\,.$

To deal with the former term on the right-hand side, we take advantage of the
following commutator estimate (see Lemma A.12):

$\|{\rm
div}\,\left[\Delta_{j},A_{\varepsilon}\right]\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}\leq
C\,c_{j}\,2^{-j(s-1)}\|\nabla
A_{\varepsilon}\|_{H^{s-1}}\|\nabla\Pi_{\varepsilon}^{n+1}\|_{H^{s-1}}\,,$

for a suitable sequence $(c_{j})_{j}$ belonging to the unit sphere of
$\ell^{2}$.

After multiplying by $2^{j(s-1)}$, we get

$2^{js}\|\Delta_{j}\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}\leq
C\left(c_{j}\,\|\nabla
A_{\varepsilon}\|_{H^{s-1}}\|\nabla\Pi_{\varepsilon}^{n+1}\|_{H^{s-1}}+2^{j(s-1)}\|{\rm
div}\,\Delta_{j}F_{\varepsilon}\|_{L^{2}}\right)\,.$

Taking the $\ell^{2}$ norm of both sides and summing up the low frequency
blocks related to $\Delta_{-1}\nabla\Pi_{\varepsilon}^{n+1}$, we may have

$\|\nabla\Pi_{\varepsilon}^{n+1}\|_{H^{s}}\leq C\left(\|\nabla
A_{\varepsilon}\|_{H^{s-1}}\|\nabla\Pi_{\varepsilon}^{n+1}\|_{H^{s-1}}+\|{\rm
div}\,F_{\varepsilon}\|_{H^{s-1}}+\|\Delta_{-1}\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}\right)\,.$

We observe that $\|\Delta_{-1}\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}\leq
C\|\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}$ and

$\|\nabla\Pi_{\varepsilon}^{n+1}\|_{H^{s-1}}\leq
C\|\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}^{1/s}\|\nabla\Pi_{\varepsilon}^{n+1}\|_{H^{s}}^{1-1/s}\,.$

Therefore,

$\|\nabla\Pi_{\varepsilon}^{n+1}\|_{H^{s}}\leq C\left(\|\nabla
A_{\varepsilon}\|_{H^{s-1}}\|\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}^{1/s}\|\nabla\Pi_{\varepsilon}^{n+1}\|_{H^{s}}^{1-1/s}+\|{\rm
div}\,F_{\varepsilon}\|_{H^{s-1}}+\|\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}\right)\,.$

Then applying Young’s inequality (see Proposition B.4) we finally infer that

$\|\nabla\Pi_{\varepsilon}^{n+1}\|_{H^{s}}\leq C\left(\left(1+\|\nabla
A_{\varepsilon}\|_{H^{s-1}}\right)^{s}\|\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{2}}+\|{\rm
div}\,F_{\varepsilon}\|_{H^{s-1}}\right)\,.$ (4.2.13)

It remains to analyse the term ${\rm div}\,F_{\varepsilon}$ where
$F_{\varepsilon}:=\varepsilon\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n+1}+\boldsymbol{u}_{\varepsilon}^{\perp,n+1}$.
Due to the divergence-free conditions, we can write

${\rm
div}\,(\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n+1})=\nabla\boldsymbol{u}_{\varepsilon}^{n}:\nabla\boldsymbol{u}_{\varepsilon}^{n+1}$

and as $H^{s-1}$ is an algebra, the term ${\rm
div}\,(\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n+1})$
is in $H^{s-1}$, with

$\|{\rm
div}\,(\boldsymbol{u}_{\varepsilon}^{n}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n+1})\|_{H^{s-1}}\leq
C\|\boldsymbol{u}_{\varepsilon}^{n}\|_{H^{s}}\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{H^{s}}\,.$
(4.2.14)

Putting (4.2.12) and (4.2.14) into (4.2.13), we find that

$\begin{split}\|\nabla\Pi_{\varepsilon}^{n+1}\|_{H^{s}}&\leq
C\left(1+\varepsilon\|\nabla
a_{\varepsilon}^{n+1}\|_{H^{s-1}}\right)^{s}\left(\varepsilon\|\boldsymbol{u}_{\varepsilon}^{n}\|_{L^{2}}\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{H^{s}}+\|\boldsymbol{u}_{\varepsilon}^{\perp,n+1}\|_{L^{2}}\right)\\\
&+C\,\left(\varepsilon\|\boldsymbol{u}_{\varepsilon}^{n}\|_{H^{s}}\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{H^{s}}+\|\boldsymbol{u}_{\varepsilon}^{\perp,n+1}\|_{H^{s}}\right)\\\
&\leq C\left(1+\varepsilon\|\nabla
a_{\varepsilon}^{n+1}\|_{H^{s-1}}\right)^{s}\left(\varepsilon\|\boldsymbol{u}_{\varepsilon}^{n}\|_{H^{s}}+1\right)\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{H^{s}}\,,\end{split}$
(4.2.15)

which implies the $L^{\infty}_{T}(H^{s})$ estimate for the pressure term:

$\|\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{\infty}_{T}H^{s}}\leq
C\left(1+\varepsilon\|\nabla
a_{\varepsilon}^{n+1}\|_{L^{\infty}_{T}H^{s-1}}\right)^{s}\left(\varepsilon\|\boldsymbol{u}_{\varepsilon}^{n}\|_{L^{\infty}_{T}H^{s}}+1\right)\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{L^{\infty}_{T}H^{s}}\,.$
(4.2.16)

Combining all the previous estimates together with a Grönwall type inequality
(again we refer to Section B.6), we finally obtain an estimate for the
velocity field:

$\sup_{0\leq t\leq
T}\|\boldsymbol{u}_{\varepsilon}^{n+1}(t)\|_{H^{s}}\leq\|\boldsymbol{u}_{0,\varepsilon}^{n+1}\|_{H^{s}}\exp\left(\int_{0}^{T}A_{n}(t)\,\,{\rm
d}t\right)\,,$ (4.2.17)

where

$\begin{split}A_{n}(t)=C\left(\|a_{\varepsilon}^{n+1}(t)\|_{L^{\infty}}+\|\nabla
a_{\varepsilon}^{n+1}(t)\|_{H^{s-1}}\right)\left(1+\varepsilon\|\nabla
a_{\varepsilon}^{n+1}(t)\|_{H^{s-1}}\right)^{s}\left(\varepsilon\|\boldsymbol{u}_{\varepsilon}^{n}(t)\|_{H^{s}}+1\right)+C\|\boldsymbol{u}_{\varepsilon}^{n}(t)\|_{H^{s}}\,.\end{split}$

We point out that the above constants $C$ do not depend on $n$ nor on
$\varepsilon$.

The scope in what follows is to obtain uniform estimates by induction. Thanks
to the assumptions stated in Paragraph 4.1.1, we can suppose that the initial
data satisfy

$\|a_{0,\varepsilon}\|_{L^{\infty}}\leq\frac{C_{0}}{2}\,,\quad\quad\|\nabla
a_{0,\varepsilon}\|_{H^{s-1}}\leq\frac{C_{1}}{2}\quad\quad\text{and}\quad\quad\|\boldsymbol{u}_{0,\varepsilon}\|_{H^{s}}\leq\frac{C_{2}}{2}\,,$

for some $C_{0},C_{1},C_{2}>0$. Due to the relation (4.2.7) we immediately
infer that, for all $n\geq 0$,

$\|a_{\varepsilon}^{n+1}\|_{L^{\infty}_{t}L^{\infty}}\leq
C\|a_{0,\varepsilon}\|_{L^{\infty}}\leq C\,C_{0}\quad\text{for all
}t\in\mathbb{R}_{+}.$

At this point, the aim is to show (by induction) that the following uniform
bounds hold for all $n\geq 0$:

$\begin{split}&\|\nabla
a_{\varepsilon}^{n+1}\|_{L^{\infty}_{T^{\ast}}H^{s-1}}\leq C_{1}\\\
&\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{L^{\infty}_{T^{\ast}}H^{s}}\leq
C_{2}\\\ &\|\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{\infty}_{T^{\ast}}H^{s}}\leq
C_{3}\,,\end{split}$ (4.2.18)

provided that $T^{\ast}$ is sufficiently small.

The previous estimates in (4.2.18) obviously hold for $n=0$. At this point, we
will prove them for $n+1$, supposing that the controls in (4.2.18) are true
for $n$. From (4.2.8), (4.2.17) and (4.2.16) we obtain

$\displaystyle\|\nabla
a_{\varepsilon}^{n+1}\|_{L^{\infty}_{T}H^{s-1}}\leq\frac{C_{1}}{2}\exp\Big{(}CTC_{2}\Big{)}$
$\displaystyle\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{L^{\infty}_{T}H^{s}}\leq\frac{C_{2}}{2}\exp\Big{(}CT(C_{0}+C_{1})\left(1+\varepsilon
C_{1}\right)^{s}(\varepsilon C_{2}+1)C_{2}\Big{)}$
$\displaystyle\|\nabla\Pi_{\varepsilon}^{n+1}\|_{L^{\infty}_{T}H^{s}}\leq
C(\varepsilon C_{2}+1)\left(1+\varepsilon
C_{1}\right)^{s}\|\boldsymbol{u}_{\varepsilon}^{n+1}\|_{L^{\infty}_{T}H^{s}}\,.$

So we can choose $T^{\ast}$ such that
$\exp\Big{(}\max\\{C_{0}+C_{1},\,1\\}\,CT\left(1+C_{1}\right)^{s}(1+C_{2})\,C_{2}\Big{)}\leq
2$. Notice that $T^{\ast}$ does not depend on $\varepsilon$.

Thus, by induction, (4.2.18) holds for the step $n+1$, and therefore it is
true for any $n\in\mathbb{N}$.

#####  4.2.3 Convergence

To prove the convergence, we will estimate the difference between two
iterations at different steps. First of all, let us define

$\widetilde{a}_{\varepsilon}^{n}:=a_{\varepsilon}^{n}-a^{n}_{0,\varepsilon}\,,$

that satisfies the transport equation

$\begin{cases}\partial_{t}\widetilde{a}^{n}_{\varepsilon}+\boldsymbol{u}_{\varepsilon}^{n-1}\cdot\nabla\widetilde{a}_{\varepsilon}^{n}=-\boldsymbol{u}_{\varepsilon}^{n-1}\cdot\nabla
a_{0,\varepsilon}^{n}\\\
\widetilde{a}{{}_{\varepsilon}^{n}}_{|t=0}=0\,.\end{cases}$

Hence, since the right-hand side is definitely uniformly bounded (with respect
to $n$) in $L^{1}_{\rm loc}(\mathbb{R}_{+};L^{2})$, from classical results for
transport equations we get that
$(\widetilde{a}^{n}_{\varepsilon})_{n\in\mathbb{N}}$ is uniformly bounded in
$C^{0}([0,T];L^{2})$. Now, we want to prove that the sequence
$(\widetilde{a}^{n}_{\varepsilon},\boldsymbol{u}_{\varepsilon}^{n},\nabla\Pi^{n}_{\varepsilon})_{n\in\mathbb{N}}$
is a Cauchy sequence in $C^{0}([0,T];L^{2})$. So, let us define, for
$(n,l)\in\mathbb{N}^{2}$, the following quantities,

$\displaystyle\delta{a}_{\varepsilon}^{n,l}:=a_{\varepsilon}^{n+l}-a_{\varepsilon}^{n}$
$\displaystyle\delta\widetilde{a}_{\varepsilon}^{n,l}:=\widetilde{a}_{\varepsilon}^{n+l}-\widetilde{a}_{\varepsilon}^{n}=\delta
a_{\varepsilon}^{n,l}-\delta a_{0,\varepsilon}^{n,l}\,,\quad\text{ where
}\quad\delta{a}_{0,\varepsilon}^{n,l}:=a_{0,\varepsilon}^{n+l}-a_{0,\varepsilon}^{n}$
$\displaystyle\delta\boldsymbol{u}_{\varepsilon}^{n,l}:=\boldsymbol{u}_{\varepsilon}^{n+l}-\boldsymbol{u}_{\varepsilon}^{n}$
(4.2.19)
$\displaystyle\delta\Pi_{\varepsilon}^{n,l}:=\Pi_{\varepsilon}^{n+l}-\Pi_{\varepsilon}^{n}\,.$

Of course, we have that ${\rm
div}\,\delta\boldsymbol{u}_{\varepsilon}^{n,l}=0$ for any
$(n,l)\in\mathbb{N}^{2}$.

The previous quantities defined in (4.2.19) solve the system

$\begin{cases}\partial_{t}\delta\widetilde{a}_{\varepsilon}^{n,l}+\boldsymbol{u}_{\varepsilon}^{n+l-1}\cdot\nabla\delta\widetilde{a}_{\varepsilon}^{n,l}=-\delta\boldsymbol{u}_{\varepsilon}^{n-1,l}\cdot\nabla
a_{\varepsilon}^{n}-\boldsymbol{u}_{\varepsilon}^{n+l-1}\cdot\nabla\delta
a^{n,l}_{0,\varepsilon}\\\\[4.30554pt]
\partial_{t}\delta\boldsymbol{u}_{\varepsilon}^{n,l}+\boldsymbol{u}_{\varepsilon}^{n+l-1}\cdot\nabla\delta\boldsymbol{u}_{\varepsilon}^{n,l}=-\delta\boldsymbol{u}_{\varepsilon}^{n-1,l}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n}\\\\[2.15277pt]
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\dfrac{1}{\varepsilon}(\delta\boldsymbol{u}_{\varepsilon}^{n,l})^{\perp}-(1+\varepsilon
a_{\varepsilon}^{n+l})\dfrac{1}{\varepsilon}\nabla\delta\Pi_{\varepsilon}^{n,l}-\delta
a_{\varepsilon}^{n,l}\nabla\Pi_{\varepsilon}^{n}\\\\[8.61108pt] {\rm
div}\,\delta\boldsymbol{u}_{\varepsilon}^{n,l}=0\\\\[4.30554pt]
(\delta\widetilde{a}_{\varepsilon}^{n,l},\delta\boldsymbol{u}_{\varepsilon}^{n,l})_{|t=0}=(0,\delta\boldsymbol{u}_{0,\varepsilon}^{n,l})\,.\end{cases}$
(4.2.20)

We perform an energy estimate for the first equation in (4.2.20), getting

$\|\delta\widetilde{a}_{\varepsilon}^{n,l}(t)\|_{L^{2}}\leq
C\int_{0}^{t}\left(\|\nabla
a_{\varepsilon}^{n}\|_{L^{\infty}}\|\delta\boldsymbol{u}_{\varepsilon}^{n-1,l}\|_{L^{2}}+\|\boldsymbol{u}_{\varepsilon}^{n+l-1}\|_{L^{2}}\|\nabla\delta
a_{0,\varepsilon}^{n,l}\|_{L^{\infty}}\right)\,\,{\rm d}\tau\,.$

Moreover, from the momentum equation multiplied by
$\delta\boldsymbol{u}_{\varepsilon}^{n,l}$, integrating by parts over
$\mathbb{R}^{2}$, we obtain

$\begin{split}\int_{\mathbb{R}^{2}}\frac{1}{2}\partial_{t}\,|\delta\boldsymbol{u}_{\varepsilon}^{n,l}|^{2}=-\int_{\mathbb{R}^{2}}(\delta\boldsymbol{u}_{\varepsilon}^{n-1,l}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n})\cdot\delta\boldsymbol{u}_{\varepsilon}^{n,l}+\int_{\mathbb{R}^{2}}(a_{\varepsilon}^{n+l}\,\nabla\delta\Pi_{\varepsilon}^{n,l})\cdot\delta\boldsymbol{u}_{\varepsilon}^{n}+\int_{\mathbb{R}^{2}}(\delta
a_{\varepsilon}^{n,l}\,\nabla\Pi_{\varepsilon}^{n})\cdot\delta\boldsymbol{u}_{\varepsilon}^{n,l}\,,\end{split}$

which implies

$\begin{split}\|\delta\boldsymbol{u}_{\varepsilon}^{n,l}(t)\|_{L^{2}}&\leq
C\|\delta\boldsymbol{u}_{0,\varepsilon}^{n,l}\|_{L^{2}}+C\int_{0}^{t}\|\nabla\boldsymbol{u}_{\varepsilon}^{n}(\tau)\|_{L^{\infty}}\|\delta\boldsymbol{u}_{\varepsilon}^{n-1,l}(\tau)\|_{L^{2}}+\|a_{\varepsilon}^{n+l}(\tau)\|_{L^{\infty}}\|\nabla\delta\Pi_{\varepsilon}^{n,l}(\tau)\|_{L^{2}}\,\,{\rm
d}\tau\\\
&+C\int_{0}^{t}\left(\|\delta\widetilde{a}_{\varepsilon}^{n,l}(\tau)\|_{L^{2}}+\|\delta
a_{0,\varepsilon}^{n,l}\|_{L^{\infty}}\right)\|\nabla\Pi_{\varepsilon}^{n}(\tau)\|_{L^{2}\cap
L^{\infty}}\,\,{\rm d}\tau\,,\end{split}$

where we have also employed the fact that $\delta
a_{\varepsilon}^{n,l}=\delta\widetilde{a}_{\varepsilon}^{n,l}+\delta
a_{0,\varepsilon}^{n,l}$.

Finally, for the pressure term we take the ${\rm div}\,$ operator in the
momentum equation of system (4.2.20), obtaining

$-{\rm div}\,\left((1+\varepsilon
a_{\varepsilon}^{n+l})\frac{1}{\varepsilon}\nabla\delta\Pi_{\varepsilon}^{n,l}\right)={\rm
div}\,\left(-\delta\boldsymbol{u}_{\varepsilon}^{n,l}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n+l-1}+\delta\boldsymbol{u}_{\varepsilon}^{n-1,l}\cdot\nabla\boldsymbol{u}_{\varepsilon}^{n}+\frac{1}{\varepsilon}(\delta\boldsymbol{u}_{\varepsilon}^{n,l})^{\perp}+\delta
a_{\varepsilon}^{n,l}\nabla\Pi_{\varepsilon}^{n}\right),$

so that we have

$\begin{split}\|\nabla\delta\Pi_{\varepsilon}^{n,l}\|_{L^{2}}&\leq
C\varepsilon\left(\|\delta\boldsymbol{u}_{\varepsilon}^{n-1,l}\|_{L^{2}}\|\nabla\boldsymbol{u}_{\varepsilon}^{n}\|_{L^{\infty}}+\|\delta
a_{\varepsilon}^{n,l}\,\nabla\Pi_{\varepsilon}^{n}\|_{L^{2}}\right)\\\
&+C\varepsilon\|\delta\boldsymbol{u}_{\varepsilon}^{n,l}\|_{L^{2}}\|\nabla\boldsymbol{u}_{\varepsilon}^{n+l-1}\|_{L^{\infty}}+C\|(\delta\boldsymbol{u}_{\varepsilon}^{n,l})^{\perp}\|_{L^{2}}\\\
&\leq
C\varepsilon\left(\|\delta\boldsymbol{u}_{\varepsilon}^{n-1,l}\|_{L^{2}}\|\nabla\boldsymbol{u}_{\varepsilon}^{n}\|_{L^{\infty}}+\|\delta\widetilde{a}_{\varepsilon}^{n,l}\|_{L^{2}}\|\nabla\Pi_{\varepsilon}^{n}\|_{L^{\infty}}+\|\delta
a_{0,\varepsilon}^{n,l}\|_{L^{\infty}}\|\nabla\Pi_{\varepsilon}^{n}\|_{L^{2}}\right)\\\
&+C\varepsilon\|\delta\boldsymbol{u}_{\varepsilon}^{n,l}\|_{L^{2}}\|\nabla\boldsymbol{u}_{\varepsilon}^{n+l-1}\|_{L^{\infty}}+C\|\delta\boldsymbol{u}_{\varepsilon}^{n,l}\|_{L^{2}}\,.\end{split}$
(4.2.21)

At this point, applying Grönwall lemma (see Lemma B.15) and using the bounds
established in Paragraph 4.2.2, we thus argue that for $t\in[0,T^{\ast}]$:

$\begin{split}\|\delta\widetilde{a}_{\varepsilon}^{n,l}(t)\|_{L^{2}}+\|\delta\boldsymbol{u}_{\varepsilon}^{n,l}(t)\|_{L^{2}}&\leq
C_{T^{\ast}}\left(\|\nabla\delta
a_{0,\varepsilon}^{n,l}\|_{L^{\infty}}+\|\delta
a_{0,\varepsilon}^{n,l}\|_{L^{\infty}}+\|\delta\boldsymbol{u}_{0,\varepsilon}^{n,l}\|_{L^{2}}\right)\\\
&+C_{T^{\ast}}\int_{0}^{t}\left(\|\delta\widetilde{a}_{\varepsilon}^{n-1,l}(\tau)\|_{L^{2}}+\|\delta\boldsymbol{u}_{\varepsilon}^{n-1,l}(\tau)\|_{L^{2}}\right)\,{\rm
d}\tau\,,\end{split}$

where the constant $C_{T^{\ast}}$ depends on $T^{\ast}$ and on the initial
data.

After setting

$F_{0}^{n}:=\sup_{l\geq 0}\left(\|\nabla\delta
a_{0,\varepsilon}^{n,l}\|_{L^{\infty}}+\|\delta
a_{0,\varepsilon}^{n,l}\|_{L^{\infty}}+\|\delta\boldsymbol{u}_{0,\varepsilon}^{n,l}\|_{L^{2}}\right)$

and

$G^{n}(t):=\sup_{l\geq
0}\sup_{[0,t]}\left(\|\delta\widetilde{a}_{\varepsilon}^{n,l}(\tau)\|_{L^{2}}+\|\delta\boldsymbol{u}_{\varepsilon}^{n,l}(\tau)\|_{L^{2}}\right),$

by induction we may conclude that, for all $t\in[0,T^{\ast}]$,

$\begin{split}G^{n}(t)\leq
C_{T^{\ast}}\sum_{k=0}^{n-1}\frac{(C_{T^{\ast}}T^{\ast})^{k}}{k!}F_{0}^{n-k}+\frac{\left(C_{T^{\ast}}T^{\ast}\right)^{n}}{n!}G^{0}(t)\,.\end{split}$

Now, bearing (4.2.4) in mind, we have

$\lim_{n\rightarrow+\infty}F_{0}^{n}=0\,.$

Hence, we may infer that

$\lim_{n\rightarrow+\infty}\sup_{l\geq
0}\sup_{t\in[0,T^{\ast}]}\left(\|\delta\widetilde{a}_{\varepsilon}^{n,l}(t)\|_{L^{2}}+\|\delta\boldsymbol{u}_{\varepsilon}^{n,l}(t)\|_{L^{2}}\right)=0\,.$
(4.2.22)

Property (4.2.22) implies that both
$(\widetilde{a}_{\varepsilon}^{n})_{n\in\mathbb{N}}$ and
$(\boldsymbol{u}_{\varepsilon}^{n})_{n\in\mathbb{N}}$ are Cauchy sequences in
$C^{0}([0,T^{\ast}];L^{2})$: therefore, such sequences converge to some
functions $\widetilde{a}_{\varepsilon}$ and $\boldsymbol{u}_{\varepsilon}$ in
the same space. Taking advantage of previous computations in (4.2.21), we have
also that $(\nabla\Pi_{\varepsilon}^{n})_{n\in\mathbb{N}}$ converge to a
function $\nabla\Pi_{\varepsilon}$ in $C^{0}([0,T^{\ast}];L^{2})$.

Now, we define
$a_{\varepsilon}:=a_{0,\varepsilon}+\widetilde{a}_{\varepsilon}$. Hence,
$a_{\varepsilon}-a_{0,\varepsilon}$ is in $C^{0}([0,T^{\ast}];L^{2})$.
Moreover, as $(\nabla a_{\varepsilon}^{n})_{n\in\mathbb{N}}$ is uniformly
bounded in $L^{\infty}([0,T^{\ast}];H^{s-1})$ and Sobolev spaces have the
Fatou property (we refer to Proposition A.4 in this respect), we deduce that
$\nabla a_{\varepsilon}$ belongs to the same space. Moreover, since
$(a^{n}_{\varepsilon})_{n\in\mathbb{N}}$ is uniformly bounded in
$L^{\infty}([0,T^{\ast}]\times\mathbb{R}^{2})$, we also have that
$a_{\varepsilon}\in L^{\infty}([0,T^{\ast}]\times\mathbb{R}^{2})$.
Analogously, as $(\boldsymbol{u}_{\varepsilon}^{n})_{n\in\mathbb{N}}$ and
$(\nabla\Pi_{\varepsilon}^{n})_{n\in\mathbb{N}}$ are uniformly bounded in
$L^{\infty}([0,T^{\ast}];H^{s})$, we deduce that
$\boldsymbol{u}_{\varepsilon}$ and $\nabla\Pi_{\varepsilon}$ belong to
$L^{\infty}([0,T^{\ast}];H^{s})$.

Due to an interpolation argument, we see that the above sequences converge
strongly in every intermediate $C^{0}([0,T^{\ast}];H^{\sigma})$ for all
$\sigma<s$. This is enough to pass to the limit in the equations satisfied by
$(a_{\varepsilon}^{n},\boldsymbol{u}_{\varepsilon}^{n},\nabla\Pi_{\varepsilon}^{n})_{n\in\mathbb{N}}$.
Hence,
$(a_{\varepsilon},\boldsymbol{u}_{\varepsilon},\nabla\Pi_{\varepsilon})$
satisfies the original problem (4.2.3).

This having been established, we look at the time continuity of
$a_{\varepsilon}$. We exploit the transport equation:

$\partial_{t}a_{\varepsilon}=-\boldsymbol{u}_{\varepsilon}\cdot\nabla
a_{\varepsilon}\,,$

noticing that the term on the right-hand side belongs to
$L^{\infty}_{T^{\ast}}(L^{\infty})$. Thus, we can deduce that
$\partial_{t}a_{\varepsilon}\in L^{\infty}_{T^{\ast}}(L^{\infty})$. Moreover,
by embeddings, we already know that $\nabla a_{\varepsilon}\in
L^{\infty}_{T^{\ast}}(L^{\infty})$. The previous two relations imply that
$a_{\varepsilon}\in W^{1,\infty}_{T^{\ast}}(L^{\infty})\cap
L^{\infty}_{T^{\ast}}(W^{1,\infty})$. That give us the desired regularity
property $a_{\varepsilon}\in C^{0}([0,T^{\ast}]\times\mathbb{R}^{2})$. In
addition, looking at the momentum equation in (4.2.3) and employing Theorem
A.15, one obtains the claimed time regularity property for
$\boldsymbol{u}_{\varepsilon}$. At this point, the time regularity for the
pressure term $\nabla\Pi_{\varepsilon}$ is recovered from the elliptic problem
(4.2.2).

#####  4.2.4 Uniqueness

We conclude this section showing the uniqueness of solutions in our framework.

We start by stating a uniqueness result, that is a consequence of a standard
stability result based on energy methods. Since the proof is similar to the
convergence argument of the previous paragraph, we will omit it (see e.g. [21]
for details). We recall that, in what follows, the parameter $\varepsilon>0$
is fixed.

###### Theorem 4.2.1

Let
$\left(\varrho_{\varepsilon}^{(1)},\boldsymbol{u}_{\varepsilon}^{(1)},\nabla\Pi_{\varepsilon}^{(1)}\right)$
and
$\left(\varrho_{\varepsilon}^{(2)},\boldsymbol{u}_{\varepsilon}^{(2)},\nabla\Pi_{\varepsilon}^{(2)}\right)$
be two solutions to the Euler system (4.1.1) associated with the initial data
$\left(\varrho_{0,\varepsilon}^{(1)},\boldsymbol{u}_{0,\varepsilon}^{(1)}\right)$
and
$\left(\varrho_{0,\varepsilon}^{(2)},\boldsymbol{u}_{0,\varepsilon}^{(2)}\right)$.
Assume that, for some $T>0$, one has the following properties:

  1. (i)

the densities $\varrho_{\varepsilon}^{(1)}$ and $\varrho_{\varepsilon}^{(2)}$
are bounded and bounded away from zero;

  2. (ii)

the quantities
$\delta\varrho_{\varepsilon}:=\varrho_{\varepsilon}^{(2)}-\varrho_{\varepsilon}^{(1)}$
and
$\delta\boldsymbol{u}_{\varepsilon}:=\boldsymbol{u}_{\varepsilon}^{(2)}-\boldsymbol{u}_{\varepsilon}^{(1)}$
belong to the space $C^{1}\big{(}[0,T];L^{2}(\mathbb{R}^{2})\big{)}$;

  3. (iii)

$\nabla\boldsymbol{u}_{\varepsilon}^{(1)}$,
$\nabla\varrho_{\varepsilon}^{(1)}$ and $\nabla\Pi_{\varepsilon}^{(1)}$ belong
to $L^{1}\big{(}[0,T];L^{\infty}(\mathbb{R}^{2})\big{)}$.

Then, for all $t\in[0,T]$, we have the stability inequality:

$\|\delta\varrho_{\varepsilon}(t)\|_{L^{2}}+\left\|\sqrt{\varrho_{\varepsilon}^{(2)}(t)}\,\delta\boldsymbol{u}_{\varepsilon}(t)\right\|_{L^{2}}\leq\left(\|\delta\varrho_{0,\varepsilon}\|_{L^{2}}+\left\|\sqrt{\varrho_{0,\varepsilon}^{(2)}}\,\delta\boldsymbol{u}_{0,\varepsilon}\right\|_{L^{2}}\right)\,e^{CA(t)}\,,$
(4.2.23)

for a universal constant $C>0$, where we have defined

$A(t):=\int_{0}^{t}\left(\left\|\frac{\nabla\varrho_{\varepsilon}^{(1)}}{\sqrt{\varrho_{\varepsilon}^{(2)}}}\right\|_{L^{\infty}}+\left\|\frac{\nabla\Pi_{\varepsilon}^{(1)}}{\varrho_{\varepsilon}^{(1)}\sqrt{\varrho_{\varepsilon}^{(2)}}}\right\|_{L^{\infty}}+\|\nabla\boldsymbol{u}_{\varepsilon}^{(1)}\|_{L^{\infty}}\right)\,\,{\rm
d}\tau\,.$

It is worth to notice that, adapting the relative entropy arguments presented
in Subsection 4.3 of [18], we can replace (in the statement above) the
$C^{1}_{T}(L^{2})$ requirement for $\delta\varrho_{\varepsilon}$ and
$\delta\boldsymbol{u}_{\varepsilon}$ with the $C^{0}_{T}(L^{2})$ regularity.
However, one needs to pay an additional $L^{2}$ assumption on the densities.
In this way, we will have a weak-strong uniqueness type result and we will
prove it in the next theorem.

Concerning weak-strong results for density-dependent fluids, we refer to [48],
where Germain exhibited a weak-strong uniqueness property within a class of
(weak) solutions to the compressible Navier-Stokes system satisfying a
relative entropy inequality with respect to a (hypothetical) strong solution