# Homogenization of Leray’s flux problem for the steady-state Navier-Stokes
equations in a multiply-connected planar domain

Clara PATRIARCA - Gianmarco SPERONE111Research supported by MUR grant
Dipartimento di Eccellenza 2023-2027.

###### Abstract

The steady motion of a viscous incompressible fluid in a multiply-connected,
planar, bounded domain (perforated with a large number of small holes) is
modeled through the Navier-Stokes equations with non-homogeneous Dirichlet
boundary data satisfying the general outflow condition. Under either a
symmetry assumption on the data or under a smallness condition on each of the
boundary fluxes (therefore, no constraints on the magnitude of the boundary
velocity are imposed), we apply the classical energy method in homogenization
theory and study the asymptotic behavior of the solutions to this system as
the size of the perforations goes to zero: it is shown that the effective
equations remain unmodified in the limit. The main novelty of the present work
lies in the obtainment of the required uniform bounds, which are achieved by a
contradiction argument based on Bernoulli’s law for solutions of the
stationary Euler equations.

AMS Subject Classification: 76M50, 76D05, 35B27, 35G60, 35Q31.

Keywords: incompressible fluids, multiply-connected boundary, non-homogeneous
Dirichlet boundary conditions, homogenization, perforated domain.

##  1 Introduction and presentation of the problem

Let $M\geq 1$ be an integer and
$\Omega_{0},\Omega_{1},...,\Omega_{M}\subset\mathbb{R}^{2}$ be open, bounded
and simply connected domains having a $\mathcal{C}^{2}$-boundary, such that

$\bigcup_{i=1}^{M}\overline{\Omega_{i}}\subset\Omega_{0}\qquad\text{and}\qquad\Omega_{i}\cap\Omega_{j}=\emptyset\quad\forall
i,j\in\\{1,...,M\\}\,,\ i\neq j\,.$

Define the open set

$\Omega\doteq\Omega_{0}\setminus\bigcup_{i=1}^{M}\overline{\Omega_{i}}\,,$

whose boundary consists of $M+1$ connected components, and is given by

$\partial\Omega=\bigcup_{i=0}^{M}\partial\Omega_{i}\,.$

For any $\xi\in\mathbb{R}^{2}$ and $r>0$ we denote by
$D(\xi,r)\subset\mathbb{R}^{2}$ the open disk of radius $r$ with center at
$\xi$. Let $(K_{n})_{n\in\mathbb{N}}$ be a sequence of open, bounded and
simply connected sets with a $\mathcal{C}^{2}$-boundary such that $(0,0)\in
K_{n}$, for every $n\in\mathbb{N}$, and

$\sup_{n\in\mathbb{N}}|K_{n}|<\infty\,.$

Take $\varepsilon_{*}\in(0,1)$ such that $\varepsilon_{*}|K_{n}|<|\Omega|$,
$\forall n\in\mathbb{N}$. Given $\alpha>2$, we define the quantities

$a_{\varepsilon}\doteq
e^{-\frac{1}{\varepsilon^{\alpha}}}\qquad\text{and}\qquad\sigma_{\varepsilon}\doteq\varepsilon\sqrt{\left|\log\left(\dfrac{a_{\varepsilon}}{\varepsilon}\right)\right|}\qquad\forall\varepsilon\in(0,\varepsilon_{*}]\quad\Longrightarrow\quad\lim\limits_{\varepsilon\to
0^{+}}\sigma_{\varepsilon}=+\infty\,.$ (1.1)

Following [33, 34], given $\varepsilon\in(0,\varepsilon_{*}]$, suppose that
there exist an integer $N(\varepsilon)\geq 1$ and a collection of points
$\xi^{\varepsilon}_{1},...,\xi^{\varepsilon}_{N(\varepsilon)}\in\mathbb{R}^{2}$
such that

$\displaystyle\xi^{\varepsilon}_{n}+a_{\varepsilon}\overline{K_{n}}\subset
D(\xi^{\varepsilon}_{n},\delta_{0}a_{\varepsilon})\subset
D(\xi^{\varepsilon}_{n},\delta_{1}\varepsilon)\subset\Omega\qquad\forall
n\in\\{1,...,N(\varepsilon)\\}\,,$ (1.2) $\displaystyle\partial
D\left(\xi^{\varepsilon}_{n},\delta_{1}\varepsilon\right)\cap\partial
D\left(\xi^{\varepsilon}_{m},\delta_{1}\varepsilon\right)=\emptyset\qquad\forall
n,m\in\\{1,...,N(\varepsilon)\\}\,,\ n\neq m\,,$ $\displaystyle\partial
D\left(\xi^{\varepsilon}_{n},\delta_{1}\varepsilon\right)\cap\partial\Omega=\emptyset\qquad\forall
n\in\\{1,...,N(\varepsilon)\\}\,,$

for some constants $0<\delta_{0}<\delta_{1}$ that are independent of
$\varepsilon\in(0,\varepsilon_{*}]$. Setting
$K^{\varepsilon}_{n}\doteq\xi^{\varepsilon}_{n}+a_{\varepsilon}K_{n}$ for
every $n\in\\{1,...,N(\varepsilon)\\}$, we will refer to the family
$\\{K^{\varepsilon}_{n}\\}^{N(\varepsilon)}_{n=1}$ satisfying (1.2) as the
solid obstacles, while

$\Omega_{\varepsilon}\doteq\Omega\setminus\overline{K_{\varepsilon}}\doteq\Omega\setminus\bigcup^{N(\varepsilon)}_{n=1}\overline{K^{\varepsilon}_{n}}\,,$
(1.3)

represents the perforated fluid domain at the $\varepsilon$-level. We
emphasize that, given $\varepsilon\in(0,\varepsilon_{*}]$, the family of
obstacles $\\{K^{\varepsilon}_{n}\\}^{N(\varepsilon)}_{n=1}$ is built in such
a way that the size of each solid is proportional to $a_{\varepsilon}$, while
the mutual distance between any two consecutive holes is proportional to
$\varepsilon$. Moreover, since we only consider those obstacles that are
strictly contained in $\Omega$ (in the sense of (1.2)3), the following bound
on the number $N(\varepsilon)$ holds:

$N(\varepsilon)\leq\dfrac{|\Omega|}{\pi\delta_{1}^{2}\varepsilon^{2}}\,.$
(1.4)

Figure 1.1: Representation of the domain $\Omega$ (left) and the perforated
domain $\Omega_{\varepsilon}$ (right).

Notice, however, that the solids
$\\{K^{\varepsilon}_{n}\\}^{N(\varepsilon)}_{n=1}$ may have different shapes
and that they are not necessarily periodically distributed in $\Omega$
(compare this with Subsection 1.1). We decompose the boundary of
$\Omega_{\varepsilon}$ as

$\partial\Omega_{\varepsilon}=\partial\Omega\cup\partial
K_{\varepsilon}=\partial\Omega\cup\bigcup^{N(\varepsilon)}_{n=1}\partial
K^{\varepsilon}_{n}\,.$

The outward unit normal to $\partial\Omega_{\varepsilon}$ is denoted by $\nu$
(with some abuse of notation, as such vector also depends on $\varepsilon$).
Given $\varepsilon\in(0,\varepsilon_{*}]$, we analyze the steady motion of a
viscous incompressible fluid (having a constant kinematic viscosity $\eta>0$)
along $\Omega_{\varepsilon}$, which is characterized by its velocity vector
field $u_{\varepsilon}:\Omega_{\varepsilon}\longrightarrow\mathbb{R}^{3}$ and
its scalar pressure
$p_{\varepsilon}:\Omega_{\varepsilon}\longrightarrow\mathbb{R}$, under the
action of an external force $f:\Omega\longrightarrow\mathbb{R}^{3}$ and a
boundary velocity $v_{*}:\partial\Omega\longrightarrow\mathbb{R}^{3}$
satisfying the compatibility condition

$\int_{\partial\Omega}v_{*}\cdot\nu=0\,.$ (1.5)

Such stationary motion is modeled through the following boundary-value problem
(with non-homogeneous Dirichlet boundary conditions) associated to the steady-
state Navier-Stokes equations in $\Omega_{\varepsilon}$:

$\left\\{\begin{aligned} &-\eta\Delta
u_{\varepsilon}+(u_{\varepsilon}\cdot\nabla)u_{\varepsilon}+\nabla
p_{\varepsilon}=f\,,\quad\nabla\cdot u_{\varepsilon}=0\ \ \mbox{ in }\ \
\Omega_{\varepsilon}\,,\\\\[4.0pt] &u_{\varepsilon}=v_{*}\ \ \mbox{ on }\ \
\partial\Omega\,,\quad u_{\varepsilon}=0\ \ \mbox{ on }\ \ \partial
K_{\varepsilon}\,.\end{aligned}\right.$ (1.6)

###  1.1 Symmetric domain with periodic perforations

Special attention will be devoted to the case when each of the domains
$\Omega_{0},\Omega_{1},...,\Omega_{M}$ intersects the $x$-axis and is
symmetric with respect to the $x$-axis, meaning that

$(x,y)\in\Omega_{j}\quad\Longleftrightarrow\quad(x,-y)\in\Omega_{j}\qquad\forall
j\in\\{0,...,M\\}\,.$

Let $K\subset\mathbb{R}^{2}$ be an open, bounded and simply connected set with
a $\mathcal{C}^{2}$-boundary such that $(0,0)\in K$. Suppose there exists
$\lambda_{0}\in(0,1)$ for which

$\overline{K}\subset
D((0,0),\lambda_{0})\subset\overline{D((0,0),\lambda_{0})}\subset(0,1)\times(0,1)\,,$

and take $\varepsilon_{*}\in(0,1)$ verifying $\varepsilon_{*}|K|<|\Omega|$.
Given $\varepsilon\in(0,\varepsilon_{*}]$, for every $m\in\mathbb{Z}^{2}$ we
then have

$\overline{K^{\varepsilon}_{m}}\doteq a_{\varepsilon}(m+\overline{K})\subset
D(a_{\varepsilon}m,\lambda_{0}a_{\varepsilon})\subset\overline{D(a_{\varepsilon}m,\lambda_{0}a_{\varepsilon})}\subset
a_{\varepsilon}m+(0,\varepsilon)\times(0,\varepsilon)\,.$ (1.7)

Following [1, 9, 11], the periodically perforated fluid domain at the
$\varepsilon$-level is defined as

$\Omega_{\varepsilon}\doteq\Omega\setminus\overline{K_{\varepsilon}}\doteq\Omega\setminus\bigcup_{m\in\mathcal{M}_{\varepsilon}}\overline{K^{\varepsilon}_{m}}\qquad\text{where}\qquad\mathcal{M}_{\varepsilon}\doteq\\{m\in\mathbb{Z}^{2}\
|\ a_{\varepsilon}m+(0,\varepsilon)\times(0,\varepsilon)\subset\Omega\\}\,.$
(1.8)

Therefore, $\Omega_{\varepsilon}$ is obtained by removing from $\Omega$ all
translations (through vectors $a_{\varepsilon}m$, with $m\in\mathbb{Z}^{2}$)
of the representative hole $a_{\varepsilon}\overline{K}$ that are strictly
contained in $\Omega$ (see the last inclusion in (1.7)). Thus,

$|\mathcal{M}_{\varepsilon}|\leq\dfrac{|\Omega|}{\varepsilon^{2}}\,.$

Figure 1.2: The symmetric domain $\Omega$ (left) and the symmetric perforated
domain $\Omega_{\varepsilon}$ (right).

In particular, $\Omega_{\varepsilon}$ is also symmetric with respect to the
$x$-axis, for every $\varepsilon\in(0,\varepsilon_{*}]$.

In order to determine the existence of symmetric (with respect the $x$-axis)
generalized solutions to problem (1.6), the following definition is given (see
[4, Introduction] and [31, 37]):

######  Definition 1.1.

Let $Q\subset\mathbb{R}^{2}$ be a symmetric domain with respect to the
$x$-axis.

  * -

We say that a scalar function $g:Q\longrightarrow\mathbb{R}$ is symmetric
(with respect to the $x$-axis) if

$g(x,-y)=g(x,y)\qquad\forall(x,y)\in Q\,.$

  * -

We say that a vector field $G=(G_{1},G_{2}):Q\longrightarrow\mathbb{R}^{2}$ is
symmetric (with respect to the $x$-axis) if

$G_{1}(x,-y)=G_{1}(x,y)\qquad\text{and}\qquad
G_{2}(x,-y)=-G_{2}(x,y)\qquad\forall(x,y)\in Q\,.$

System (1.6) constitutes a particular instance of the celebrated Leray flux
problem [19, 21] in the context of homogenization theory. For a fixed
$\varepsilon\in(0,\varepsilon_{*}]$, the weak solvability of (1.6) under the
general outflow condition (1.5) (which, physically, allows for the presence of
interior sources and sinks in $\Omega_{\varepsilon}$) remained one of the most
outstanding open problems in Mathematical Fluid Mechanics between the years
1933 and 2015, listed among the Eleven Great Problems of Mathematical
Hydrodynamics by Yudovich [39]. In fact, assuming the stringent outflow
condition

$\int_{\partial\Omega_{j}}v_{*}\cdot\nu=0\qquad\forall j\in\\{0,...,M\\}\,,$

which rules out the presence of sources and sinks inside
$\Omega_{\varepsilon}$, Leray [23] showed in 1933 the existence of a
generalized solution to (1.6) employing a fixed-point technique, nowadays
known as the Leray-Schauder Principle [40, Chapter 6], combined with a
reductio ad absurdum argument. Independently from each other, Amick [4] and
Sazonov [37] recovered the result of Leray under the general outflow condition
(1.5), but imposing a symmetry assumption on both $\Omega_{\varepsilon}$ and
the data of the problem (external force and boundary velocity): while Amick
extended the original method of Leray, Sazonov built a symmetric solenoidal
extension of the boundary velocity satisfying the Leray-Hopf inequality [13].
Relaxing this symmetry restriction, Neustupa [35] reached the same outcome
under a smallness condition on each of the boundary fluxes. The fully two-
dimensional general result was proved in 2015 by Korobkov, Pileckas & Russo
[20], where the uniform bounds required by the Leray-Schauder Principle were
achieved by a contradiction argument based on Bernoulli’s law [17] for
solutions of the stationary Euler equations and a generalization of the Morse-
Sard Theorem for Sobolev functions given by Bourgain, Kristensen & Korobkov in
[6]. Nevertheless, as far as our knowledge goes, the corresponding
homogenization limit associated to Leray’s flux problem has not been tackled
before, and constitutes the core of the present article. Precisely, our goal
is to study the asymptotic behavior of the solutions of problem (1.6) as
$\varepsilon\to 0^{+}$ in two different settings:

  * -

in the general perforated domain (1.3) under a smallness condition on each of
the boundary fluxes;

  * -

in the symmetric perforated domain (1.8) under a symmetry condition on the
data of the problem.

Notice, therefore, that neither of the above settings imposes a size
constraint on the boundary velocity, see Remark 2.1. To achieve our goal, in
Section 2 we derive uniform $\varepsilon$-independent bounds for the solutions
of (1.6). The proofs of Theorems 2.1-2.2, the most involved in this work,
adapt the contradiction argument of [20] previously described and,
additionally, employ several properties of the relative capacity of the
perforations inside $\Omega$ (see Lemma 2.1) and a uniform solenoidal
extension of the boundary velocity, see Lemma 2.2 and its symmetric
counterpart Corollary 2.1; in turn, Lemma 2.2 relies on the
$\varepsilon$-uniform boundedness of the Bogovskii operator in these
perforated domains, as shown by Nečasová et al. [33, 34] (see [10] for the
corresponding result in the three-dimensional case). Subsequently, applying
the classical energy method in homogenization theory [36, Appendix], in
Section 3 we show that, as $\varepsilon\to 0^{+}$, the effective or
homogenized equation remains unchanged in the limit: up to the extraction of a
subsequence, the sequence of solutions (indexed by the parameter
$\varepsilon$) of (1.6) converges strongly (in a sense made precise in
Theorems 3.1-3.2) to a weak solution of problem (1.6) in $\Omega$ as
$\varepsilon\to 0^{+}$. We point out that the statements of Theorems 3.1-3.2
reflect other existing results in the homogenization literature in the regime
of very tiny holes [2, 11, 24, 26, 27, 28]. Finally, Section 4 provides some
remarks concerning the possibility of recovering the results contained in this
article in the fully two-dimensional case, that is, dropping both the symmetry
condition and the smallness assumption on the boundary fluxes.

##  2 Boundary-value problem at the $\varepsilon$-level: uniform bounds

Let $\varepsilon\in I_{*}$ be a fixed parameter, with
$I_{*}\doteq(0,\varepsilon_{*}]$. Many of the results contained in the present
article exploit the concept of relative capacity of $K_{\varepsilon}$ with
respect to $\Omega$, defined as

$\mbox{Cap}_{\Omega}(K_{\varepsilon})\doteq\min_{v\in
H^{1}_{0}(\Omega)}\left\\{\int_{\Omega}|\nabla v|^{2}\ \Big{|}\ v=1\ \text{ in
}\ \overline{K_{\varepsilon}}\right\\}\,.$ (2.1)

The relative capacity potential of $K_{\varepsilon}$ with respect to $\Omega$,
that is, the scalar function $\phi_{\varepsilon}\in H_{0}^{1}(\Omega)$
achieving the minimum in (2.1), satisfies

$\Delta\phi_{\varepsilon}=0\ \text{ in }\
\Omega_{\varepsilon}\,,\qquad\phi_{\varepsilon}=0\ \text{ on }\
\partial\Omega\,,\qquad\phi_{\varepsilon}=1\ \text{ in }\
\overline{K_{\varepsilon}}\,,\qquad\mbox{Cap}_{\Omega}(K_{\varepsilon})=\|\nabla\phi_{\varepsilon}\|^{2}_{L^{2}(\Omega)}\,,$
(2.2)

see [30, Chapter 2] for more details. Further essential properties of the
relative capacity potential are collected in the following result, in the
spirit of [3, Proposition 4.3] and the examples of [8, Section 2]:

######  Lemma 2.1.

Let $\Omega_{\varepsilon}$ be as in (1.3) and $\phi_{\varepsilon}\in
H_{0}^{1}(\Omega)$ be the function satisfying (2.2). Then,
$\phi_{\varepsilon}\in H^{2}(\Omega_{\varepsilon})$ and the following
estimates hold

$\|1-\phi_{\varepsilon}\|_{L^{\infty}(\Omega_{\varepsilon})}\leq
C_{*}\,\qquad\text{and}\qquad\|\phi_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|\nabla\phi_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq\dfrac{C_{*}}{\sigma_{\varepsilon}}\,,$
(2.3)

for some constant $C_{*}>0$ that depends on $\Omega$ and
$\\{\delta_{0},\delta_{1}\\}$, but is independent of $\varepsilon\in I_{*}$.

Proof. In what follows, $C>0$ will always denote a generic constant that
depends on $\Omega$ and $\\{\delta_{0},\delta_{1}\\}$ (independently of
$\varepsilon\in I_{*}$), but that may change from line to line.
Since $\Omega_{\varepsilon}$ has a boundary of class $\mathcal{C}^{2}$,
standard elliptic regularity arguments show that $\phi_{\varepsilon}\in
H^{2}(\Omega_{\varepsilon})$. The first estimate in (2.3) follows directly
from the Maximum Principle. Concerning the second estimate in (2.3), given
$n\in\\{1,...,N(\varepsilon)\\}$, consider the function
$\varphi^{\varepsilon}_{n}:\Omega\longrightarrow\mathbb{R}$ defined by

$\varphi^{\varepsilon}_{n}(\xi)=\begin{cases}1&\quad\text{if}\ \
0\leq|\xi-\xi^{\varepsilon}_{n}|\leq\delta_{0}a_{\varepsilon}\,,\\\\[3.0pt]
\dfrac{\log(\delta_{1}\varepsilon)-\log(|\xi-\xi^{\varepsilon}_{n}|)}{\log(\delta_{1}\varepsilon)-\log(\delta_{0}a_{\varepsilon})}&\quad\text{if}\
\
\delta_{0}a_{\varepsilon}<|\xi-\xi^{\varepsilon}_{n}|\leq\delta_{1}\varepsilon\,,\\\\[3.0pt]
0&\quad\text{if}\ \
|\xi-\xi^{\varepsilon}_{n}|>\delta_{1}\varepsilon\,,\end{cases}$

so that, by the assumptions in (1.2)1, $\varphi^{\varepsilon}_{n}\in
H^{1}_{0}(\Omega)$ and $\varphi^{\varepsilon}_{n}=1$ in
$\overline{D(\xi^{\varepsilon}_{n},\delta_{0}a_{\varepsilon})}$. As the
relative capacity is an outer measure and is increasing with respect to domain
inclusion (see [30, Section 2.2]), we get

$\mbox{Cap}_{\Omega}(K_{\varepsilon})\leq\sum_{n=1}^{N(\varepsilon)}\mbox{Cap}_{\Omega}(K^{\varepsilon}_{n})\leq\sum_{n=1}^{N(\varepsilon)}\mbox{Cap}_{\Omega}(D(\xi^{\varepsilon}_{n},\delta_{0}a_{\varepsilon}))\leq\sum_{n=1}^{N(\varepsilon)}\int_{\Omega}|\nabla\varphi^{\varepsilon}_{n}|^{2}\leq\dfrac{C}{\sigma_{\varepsilon}^{2}}\,,$

so that

$\|\nabla\phi_{\varepsilon}\|_{L^{2}(\Omega)}\leq\dfrac{C}{\sigma_{\varepsilon}}\,.$

Since $\phi_{\varepsilon}\in H_{0}^{1}(\Omega)$, an application of Poincaré’s
inequality in $\Omega$ allows us to conclude the proof. $\Box$

Let $Q\subset\mathbb{R}^{2}$ be any bounded Lipschitz domain, and consider the
space of square-integrable functions in $Q$ having zero mean value:

$L^{2}_{0}(Q)\doteq\left\\{g\in L^{2}(Q)\ \Big{|}\ \int_{Q}g=0\right\\}\,.$

Another essential preliminary result concerns the construction of a uniform
solenoidal extension of the boundary velocity. To fix notation, given any
function (scalar or vector) $\psi\in L^{1}(\Omega_{\varepsilon})$, hereafter
we will denote by $\widetilde{\psi}\in L^{1}(\Omega)$ the function defined by

$\widetilde{\psi}\doteq\begin{cases}\psi&\quad\text{in}\ \
\Omega_{\varepsilon}\,,\\\\[3.0pt] 0&\quad\text{in}\ \
\overline{K_{\varepsilon}}\,.\end{cases}$

We then prove:

######  Lemma 2.2.

Let $\Omega_{\varepsilon}$ be as in (1.3) and $v_{*}\in
H^{3/2}(\partial\Omega)$ satisfying (1.5). There exists a vector field
$J_{\varepsilon}\in H^{1}(\Omega_{\varepsilon})$ such that

$\left\\{\begin{aligned} &\nabla\cdot J_{\varepsilon}=0\ \ \mbox{in}\ \
\Omega_{\varepsilon}\,;\qquad J_{\varepsilon}=v_{*}\ \ \mbox{on}\ \
\partial\Omega\,;\\\\[3.0pt] &J_{\varepsilon}=0\ \ \mbox{on}\ \ \partial
K_{\varepsilon}\,;\qquad\|\nabla
J_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C_{*}\|v_{*}\|_{H^{3/2}(\partial\Omega)}\,,\end{aligned}\right.$ (2.4)

for some constant $C_{*}>0$ that depends on $\Omega$ and
$\\{\delta_{0},\delta_{1}\\}$, but is independent of $\varepsilon\in I_{*}$.
In particular, there exists a vector field $\widehat{J}\in H^{1}(\Omega)$ such
that

$\nabla\cdot\widehat{J}=0\ \ \mbox{in}\ \
\Omega\qquad\text{and}\qquad\widehat{J}=v_{*}\ \ \mbox{on}\ \
\partial\Omega\,,$

and a (not relabeled) subsequence
$(\widetilde{J_{\varepsilon}})_{\varepsilon\in I_{*}}\subset H^{1}(\Omega)$
for which

$\widetilde{J_{\varepsilon}}\rightharpoonup\widehat{J}\ \ \ \text{weakly in}\
H^{1}(\Omega)\qquad\text{and}\qquad\widehat{J_{\varepsilon}}\to\widehat{J}\ \
\ \text{strongly in}\ L^{4}(\Omega)\ \ \ \text{as}\ \ \varepsilon\to 0^{+}\,.$
(2.5)

Proof. In what follows, $C>0$ will always denote a generic constant that
depends on $\Omega$ and $\\{\delta_{0},\delta_{1}\\}$ (independently of
$\varepsilon\in I_{*}$), but that may change from line to line.
Since $v_{*}\in H^{3/2}(\partial\Omega)$, there exists a vector field
$V_{*}\in H^{2}(\Omega)$ such that

$V_{*}=v_{*}\quad\text{on}\ \
\partial\Omega\qquad\text{and}\qquad\|V_{*}\|_{H^{2}(\Omega)}\leq
C\|v_{*}\|_{H^{3/2}(\partial\Omega)}\,,$ (2.6)

see [12, Teorema 1.I] or [14, Theorem II.4.3] as well. Let
$\phi_{\varepsilon}\in H^{2}(\Omega_{\varepsilon})\cap H_{0}^{1}(\Omega)$ be
the relative capacity potential of $K_{\varepsilon}$ with respect to $\Omega$,
see Lemma 2.1. Then, the vector field
$X_{\varepsilon}\doteq(1-\phi_{\varepsilon})V_{*}\in
H^{1}(\Omega_{\varepsilon})$ is such that $X_{\varepsilon}=0$ on $\partial
K_{\varepsilon}$ and $X_{\varepsilon}=v_{*}$ on $\partial\Omega$. Moreover

$\nabla X_{\varepsilon}=-\nabla\phi_{\varepsilon}\otimes
V_{*}+(1-\phi_{\varepsilon})\nabla V_{*}\quad\text{in}\ \
\Omega_{\varepsilon}\,,$

and therefore, in view of the Sobolev embedding
$H^{2}(\Omega)\subset\mathcal{C}(\overline{\Omega})$ and (2.3), we have

$\displaystyle\|\nabla
X_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq\|\nabla\phi_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\|V_{*}\|_{L^{\infty}(\Omega)}+\|1-\phi_{\varepsilon}\|_{L^{\infty}(\Omega_{\varepsilon})}\|\nabla
V_{*}\|_{L^{2}(\Omega)}\leq C\|V_{*}\|_{H^{2}(\Omega)}\,.$ (2.7)

Furthermore, from the Divergence Theorem and (1.5) we have

$\int_{\Omega_{\varepsilon}}\nabla\cdot
X_{\varepsilon}=\int_{\partial\Omega_{\varepsilon}}X_{\varepsilon}\cdot\nu=\int_{\partial\Omega}v_{*}\cdot\nu=0\,,$

that is, $\nabla\cdot X_{\varepsilon}\in L^{2}_{0}(\Omega_{\varepsilon})$.
Therefore, there exists a vector field $Y_{\varepsilon}\in
H^{1}_{0}(\Omega_{\varepsilon})$ such that

$\nabla\cdot Y_{\varepsilon}=-\nabla\cdot
X_{\varepsilon}\quad\text{in}\quad\Omega_{\varepsilon}\qquad\text{and}\qquad\|\nabla
Y_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C_{B}(\Omega_{\varepsilon})\|\nabla\cdot
X_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\,,$ (2.8)

see [5]. From [34, Proposition 1.7] we know that $C_{B}(\Omega_{\varepsilon})$
(the so-called Bogovskii constant of $\Omega_{\varepsilon}$) admits the
uniform bound

$C_{B}(\Omega_{\varepsilon})\leq C\,.$ (2.9)

We set $J_{\varepsilon}\doteq X_{\varepsilon}+Y_{\varepsilon}$ which, in view
of (2.6)-(2.7)-(2.8)-(2.9), is an element of $H^{1}(\Omega_{\varepsilon})$
satisfying (2.4). Notice that $\widetilde{J_{\varepsilon}}\in H^{1}(\Omega)$
is divergence-free separately in $\Omega_{\varepsilon}$ and $K_{\varepsilon}$.
Moreover,

$\|\nabla\widetilde{J_{\varepsilon}}\|_{L^{2}(\Omega)}=\|\nabla
J_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\,.$

The bound in (2.4)2 ensures the existence of $\widehat{J}\in H^{1}(\Omega)$
such that the convergences

$\widetilde{J_{\varepsilon}}\rightharpoonup\widehat{J}\ \ \ \text{weakly in}\
H^{1}(\Omega)\,;\quad\widetilde{J_{\varepsilon}}\to\widehat{J}\ \ \
\text{strongly in}\
L^{4}(\Omega)\,;\quad\widetilde{J_{\varepsilon}}\to\widehat{J}\ \ \
\text{strongly in}\ L^{2}(\partial\Omega)$ (2.10)

hold as $\varepsilon\to 0^{+}$, along a subsequence that is not being
relabeled, see also [32, Theorem 6.2]. Given any scalar function
$\phi\in\mathcal{C}_{0}^{\infty}(\Omega)$, an integration by parts and the
divergence-free condition in (2.4)1 imply

$\int_{\Omega}\widetilde{J_{\varepsilon}}\cdot\nabla\phi=\int_{\Omega_{\varepsilon}}J_{\varepsilon}\cdot\nabla\phi=\int_{\partial\Omega_{\varepsilon}}\phi(J_{\varepsilon}\cdot\nu)=0\qquad\forall\varepsilon\in
I_{*}\,,$

since $J_{\varepsilon}$ vanishes on $\partial K_{\varepsilon}$ and so does
$\phi$ on $\partial\Omega$. Then, the weak convergence in (2.10) yields

$\int_{\Omega}\widehat{J}\cdot\nabla\phi=-\int_{\Omega}\phi(\nabla\cdot\widehat{J})=0\qquad\forall\phi\in\mathcal{C}_{0}^{\infty}(\Omega;\mathbb{R})\,,$

thus proving that $\nabla\cdot\widehat{J}=0$ almost everywhere in $\Omega$.
Finally, the strong convergence in $L^{2}(\partial\Omega)$ in (2.10) also
guarantees that $\widehat{J}=v_{*}$ almost everywhere on $\partial\Omega$.
$\Box$

We introduce the functional spaces (of vector fields) that will be employed
hereafter:

$H_{\sigma}^{1}(\Omega_{\varepsilon})\doteq\left\\{v\in
H^{1}(\Omega_{\varepsilon})\ |\ \nabla\cdot v=0\ \ \mbox{in}\ \
\Omega_{\varepsilon}\right\\}\qquad\text{and}\qquad
H^{1}_{0,\sigma}(\Omega_{\varepsilon})\doteq\left\\{v\in
H_{0}^{1}(\Omega_{\varepsilon})\ |\ \nabla\cdot v=0\ \ \mbox{in}\ \
\Omega_{\varepsilon}\right\\}\,,$

which are Hilbert spaces if endowed with the scalar product of
$H^{1}(\Omega_{\varepsilon};\mathbb{R}^{2})$. We recall the standard
definition for the weak solutions of problem (1.6):

######  Definition 2.1.

A vector field $u\in H_{\sigma}^{1}(\Omega_{\varepsilon})$ is called a weak
solution of (1.6) if $u-J_{\varepsilon}\in
H_{0,\sigma}^{1}(\Omega_{\varepsilon})$ and

$\eta\int_{\Omega_{\varepsilon}}\nabla
u\cdot\nabla\varphi+\int_{\Omega_{\varepsilon}}(u\cdot\nabla)u\cdot\varphi=\int_{\Omega_{\varepsilon}}f\cdot\varphi\qquad\forall\varphi\in
H^{1}_{0,\sigma}(\Omega_{\varepsilon})\,.$

Given $v_{*}\in H^{3/2}(\partial\Omega)$ satisfying (1.5), we will denote by

$F_{j}\doteq\int_{\partial\Omega_{j}}v_{*}\cdot\nu\qquad\forall
j\in\\{0,...,M\\}$

the flux of $v_{*}$ across each of the connected components of
$\partial\Omega$. Given $j\in\\{0,...,M\\}$, let $\chi_{j}\in
H_{0}^{1}(\mathbb{R}^{2})$ be the harmonic capacity potential (relative to
$\mathbb{R}^{2}$) of $\Omega_{j}$, that is, the scalar function satisfying

$\Delta\chi_{j}=0\ \text{ in }\
\mathbb{R}^{2}\setminus\overline{\Omega_{j}}\,,\qquad\chi_{j}=1\ \text{ in }\
\overline{\Omega_{j}}\,,\qquad\lim_{|\xi|\to\infty}\chi_{j}(\xi)=0\,,$ (2.11)

and so we denote by

$\text{Cap}_{\mathbb{R}^{2}}(\Omega_{j})\doteq\int_{\partial\Omega_{j}}\dfrac{\partial\chi_{j}}{\partial\nu}=\int_{\mathbb{R}^{2}}|\nabla\chi_{j}|^{2}$
(2.12)

the harmonic capacity (relative to $\mathbb{R}^{2}$) of $\Omega_{j}$, see [30,
Chapter 2] for further details. The first main result of this section provides
uniform bounds (with respect to $\varepsilon\in I_{*}$) for the solutions of
problem (1.6).

######  Theorem 2.1.

Let $\Omega_{\varepsilon}$ be as in (1.3). For any $f\in H^{1}(\Omega)$ and
$v_{*}\in H^{3/2}(\partial\Omega)$ satisfying (1.5), there exists at least one
weak solution $u_{\varepsilon}\in H^{2}(\Omega_{\varepsilon})\cap
H^{1}_{\sigma}(\Omega_{\varepsilon})$ of problem (1.6), and an associated
pressure $p_{\varepsilon}\in H^{1}(\Omega_{\varepsilon})\cap
L^{2}_{0}(\Omega_{\varepsilon})$ such that the pair
$(u_{\varepsilon},p_{\varepsilon})$ solves (1.6) in strong form. Moreover, if

$\sum_{j=0}^{M}\dfrac{|F_{j}|}{\textup{Cap}_{\mathbb{R}^{2}}(\Omega_{j})}<\eta\,,$
(2.13)

then the uniform bound

$\sup_{\varepsilon\in I_{*}}\left(\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\right)\leq
C_{*}\,,$ (2.14)

holds for some constant $C_{*}>0$ that depends on $\Omega$, $\eta$, $f$,
$v_{*}$ and $\\{\delta_{0},\delta_{1}\\}$, but is independent of
$\varepsilon\in I_{*}$.

Proof. In what follows, $C>0$ will always denote a generic constant that
depends on $\Omega$, $\eta$ and $\\{\delta_{0},\delta_{1}\\}$ (independently
of $\varepsilon\in I_{*}$), but that may change from line to line.

Then, given any $f\in H^{1}(\Omega)$, $v_{*}\in H^{3/2}(\partial\Omega)$
satisfying (1.5) and $\varepsilon\in I_{*}$, [20, Theorem 1.1] (see also [20,
Remark 1.1]) ensures the existence of at least one weak solution
$u_{\varepsilon}\in H^{2}(\Omega_{\varepsilon})\cap
H^{1}_{\sigma}(\Omega_{\varepsilon})$ of problem (1.6) and an associated
pressure $p_{\varepsilon}\in H^{1}(\Omega_{\varepsilon})\cap
L^{2}_{0}(\Omega_{\varepsilon})$ such that the pair
$(u_{\varepsilon},p_{\varepsilon})$ satisfies (1.6) in strong form. Let
$J_{\varepsilon}\in H^{1}(\Omega_{\varepsilon})$ be the vector field arising
from Lemma (2.2), which satisfies (2.4). We set $w_{\varepsilon}\doteq
u_{\varepsilon}-J_{\varepsilon}\in H^{1}_{0,\sigma}(\Omega_{\varepsilon})$, so
that, in view of the bound in (2.4)2, it suffices to show that

$\sup_{\varepsilon\in I_{*}}\left(\|\nabla
w_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\right)\leq
C\,.$

The weak formulation in Definition (2.1) may be then re-written as

$\displaystyle\eta\int_{\Omega_{\varepsilon}}\nabla
w_{\varepsilon}\cdot\nabla\varphi+\int_{\Omega_{\varepsilon}}(w_{\varepsilon}\cdot\nabla)w_{\varepsilon}\cdot\varphi+\int_{\Omega_{\varepsilon}}(w_{\varepsilon}\cdot\nabla)J_{\varepsilon}\cdot\varphi+\int_{\Omega_{\varepsilon}}(J_{\varepsilon}\cdot\nabla)w_{\varepsilon}\cdot\varphi$
(2.15)
$\displaystyle=\int_{\Omega_{\varepsilon}}f\cdot\varphi-\eta\int_{\Omega_{\varepsilon}}\nabla
J_{\varepsilon}\cdot\nabla\varphi-\int_{\Omega_{\varepsilon}}(J_{\varepsilon}\cdot\nabla)J_{\varepsilon}\cdot\varphi\qquad\forall\varphi\in
H^{1}_{0,\sigma}(\Omega_{\varepsilon})\,.$

For later use, we also present the weak formulation of Definition (2.1)
incorporating the pressure term:

$\displaystyle\eta\int_{\Omega_{\varepsilon}}\nabla
w_{\varepsilon}\cdot\nabla\varphi+\int_{\Omega_{\varepsilon}}(w_{\varepsilon}\cdot\nabla)w_{\varepsilon}\cdot\varphi+\int_{\Omega_{\varepsilon}}(w_{\varepsilon}\cdot\nabla)J_{\varepsilon}\cdot\varphi+\int_{\Omega_{\varepsilon}}(J_{\varepsilon}\cdot\nabla)w_{\varepsilon}\cdot\varphi-\int_{\Omega_{\varepsilon}}p_{\varepsilon}(\nabla\cdot\varphi)$
(2.16)
$\displaystyle=\int_{\Omega_{\varepsilon}}f\cdot\varphi-\eta\int_{\Omega_{\varepsilon}}\nabla
J_{\varepsilon}\cdot\nabla\varphi-\int_{\Omega_{\varepsilon}}(J_{\varepsilon}\cdot\nabla)J_{\varepsilon}\cdot\varphi\qquad\forall\varphi\in
H^{1}_{0}(\Omega_{\varepsilon})\,.$

Notice that $\widetilde{p_{\varepsilon}}\in L_{0}^{2}(\Omega)$ and that
$\widetilde{w_{\varepsilon}}\in H_{0}^{1}(\Omega)$ is divergence-free
separately in $\Omega_{\varepsilon}$ and $K_{\varepsilon}$. Moreover,

$\|\nabla\widetilde{w_{\varepsilon}}\|_{L^{2}(\Omega)}=\|\nabla
w_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\qquad\text{and}\qquad\|\widetilde{p_{\varepsilon}}\|_{L^{2}(\Omega)}=\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\,.$
(2.17)

We take $\varphi=w_{\varepsilon}$ in (2.15) and integrate by parts, thereby
obtaining:

$\eta\|\nabla
w_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}=-\eta\int_{\Omega}\nabla\widetilde{w_{\varepsilon}}\cdot\nabla\widetilde{J_{\varepsilon}}+\int_{\Omega}(\widetilde{w_{\varepsilon}}\cdot\nabla)\widetilde{w_{\varepsilon}}\cdot\widetilde{J_{\varepsilon}}-\int_{\Omega}(\widetilde{J_{\varepsilon}}\cdot\nabla)\widetilde{J_{\varepsilon}}\cdot\widetilde{w_{\varepsilon}}+\int_{\Omega}f\cdot\widetilde{w_{\varepsilon}}\,.$
(2.18)

Now, since $p_{\varepsilon}\in L^{2}_{0}(\Omega_{\varepsilon})$, let
$P_{\varepsilon}\in H_{0}^{1}(\Omega_{\varepsilon})$ be a vector field such
that

$\nabla\cdot
P_{\varepsilon}=p_{\varepsilon}\quad\text{in}\quad\Omega_{\varepsilon}\qquad\text{and}\qquad\|\nabla
P_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\,,$ (2.19)

see (2.8)-(2.9). We multiply the first identity in (1.6)1 by $P_{\varepsilon}$
and integrate by parts in $\Omega_{\varepsilon}$ to obtain

$\eta\int_{\Omega_{\varepsilon}}\nabla u_{\varepsilon}\cdot\nabla
P_{\varepsilon}+\int_{\Omega_{\varepsilon}}(u_{\varepsilon}\cdot\nabla)u_{\varepsilon}\cdot
P_{\varepsilon}-\|p_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}=\int_{\Omega_{\varepsilon}}f\cdot
P_{\varepsilon}\,.$

Observing that $\widetilde{P_{\varepsilon}}\in H_{0}^{1}(\Omega)$ and that
$\|\nabla\widetilde{P_{\varepsilon}}\|_{L^{2}(\Omega)}=\|\nabla
P_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$, we apply Hölder’s
inequality, the Sobolev and Poincaré inequalities in $\Omega$, and ultimately
(2.17)-(2.19), in order to estimate

$\displaystyle\|p_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle=\eta\int_{\Omega_{\varepsilon}}\nabla
u_{\varepsilon}\cdot\nabla
P_{\varepsilon}+\int_{\Omega}(\widetilde{u_{\varepsilon}}\cdot\nabla)\widetilde{u_{\varepsilon}}\cdot\widetilde{P_{\varepsilon}}-\int_{\Omega}f\cdot\widetilde{P_{\varepsilon}}$
$\displaystyle\leq\eta\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\|\nabla
P_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|\nabla\widetilde{u_{\varepsilon}}\|_{L^{2}(\Omega)}\|\widetilde{u_{\varepsilon}}\|_{L^{4}(\Omega)}\|\widetilde{P_{\varepsilon}}\|_{L^{4}(\Omega)}+\|f\|_{L^{2}(\Omega)}\|\widetilde{P_{\varepsilon}}\|_{L^{2}(\Omega)}$
$\displaystyle\leq\eta\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\|\nabla
P_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+C\|\nabla\widetilde{u_{\varepsilon}}\|^{2}_{L^{2}(\Omega)}\|\nabla\widetilde{P_{\varepsilon}}\|_{L^{2}(\Omega)}+C\|f\|_{L^{2}(\Omega)}\|\nabla\widetilde{P_{\varepsilon}}\|_{L^{2}(\Omega)}$
$\displaystyle\leq C\left(\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|\nabla
u_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega)}\right)\|\nabla
P_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$ $\displaystyle\leq
C\left(1+\|\nabla
u_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega)}\right)\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\,,$

so that, after further applying (2.4)2, we get

$\displaystyle\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\leq C\left(1+\|\nabla
u_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega)}\right)$
(2.20) $\displaystyle\leq C\left(1+\|\nabla
w_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}+\|v_{*}\|^{2}_{H^{3/2}(\partial\Omega)}+\|f\|_{L^{2}(\Omega)}\right)\qquad\forall\varepsilon\in
I_{*}\,.$

By contradiction, suppose now that the norms $\|\nabla
w_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$ are not uniformly bounded
with respect to $\varepsilon\in I_{*}$. Then, there must exists a subsequence
(not being relabeled) such that

$\lim_{\varepsilon\to 0^{+}}\mathcal{Z}_{\varepsilon}=+\infty\quad\text{with}\
\ \mathcal{Z}_{\varepsilon}\doteq\|\nabla
w_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\quad\forall\varepsilon\in
I_{*}\,.$ (2.21)

The estimate (2.20) enables us to establish that, along this divergent
subsequence (2.21), the following sequences are uniformly bounded with respect
to $\varepsilon\in I_{*}$:

$(\widehat{w_{\varepsilon}})_{\varepsilon\in
I_{*}}\doteq\left(\dfrac{\widetilde{w_{\varepsilon}}}{\mathcal{Z}_{\varepsilon}}\right)_{\varepsilon\in
I_{*}}\subset
H_{0}^{1}(\Omega)\qquad\text{and}\qquad(\widehat{p_{\varepsilon}})_{\varepsilon\in
I_{*}}\doteq\left(\dfrac{\widetilde{p_{\varepsilon}}}{\mathcal{Z}^{2}_{\varepsilon}}\right)_{\varepsilon\in
I_{*}}\subset L^{2}(\Omega)\,.$

Then, there must exist $\widehat{w}\in H_{0}^{1}(\Omega)$ and $\widehat{p}\in
L^{2}(\Omega)$ such that the following convergences hold:

$\displaystyle\widehat{w_{\varepsilon}}\rightharpoonup\widehat{w}\ \ \
\text{weakly in}\ H^{1}(\Omega)\,;$
$\displaystyle\widehat{w_{\varepsilon}}\to\widehat{w}\ \ \ \text{strongly in}\
L^{4}(\Omega)\,;\qquad\widehat{p_{\varepsilon}}\rightharpoonup\widehat{p}\ \ \
\text{weakly in}\ L^{2}(\Omega)\,,$ (2.22)

as $\varepsilon\to 0^{+}$, along subsequences that are not being relabeled.
Clearly we have $\|\nabla\widehat{w}\|_{L^{2}(\Omega)}\leq 1$, and exactly as
in the final part of the proof of Lemma (2.2) we can easily deduce that
$\nabla\cdot\widehat{w}=0$ almost everywhere in $\Omega$, that is,
$\widehat{w}\in H_{0,\sigma}^{1}(\Omega)$. If we then divide identity (2.18)
by $\mathcal{Z}^{2}_{\varepsilon}$, we get

$\eta=-\dfrac{\eta}{\mathcal{Z}_{\varepsilon}}\int_{\Omega}\nabla\widehat{w_{\varepsilon}}\cdot\nabla\widetilde{J_{\varepsilon}}+\int_{\Omega}(\widehat{w_{\varepsilon}}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot\widetilde{J_{\varepsilon}}-\dfrac{1}{\mathcal{Z}_{\varepsilon}}\int_{\Omega}(\widetilde{J_{\varepsilon}}\cdot\nabla)\widetilde{J_{\varepsilon}}\cdot\widehat{w_{\varepsilon}}+\dfrac{1}{\mathcal{Z}_{\varepsilon}}\int_{\Omega}f\cdot\widehat{w_{\varepsilon}}\qquad\forall\varepsilon\in
I_{*}\,,$ (2.23)

along the subsequences (2.22). In order to handle the second term appearing at
the right-hand side of (2.23) we firstly notice that

$\int_{\Omega}(\widehat{w_{\varepsilon}}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot\widetilde{J_{\varepsilon}}=\int_{\Omega}(\widehat{w}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot\widehat{J}+\int_{\Omega}((\widehat{w_{\varepsilon}}-\widehat{w})\cdot\nabla)\widehat{w_{\varepsilon}}\cdot\widehat{J}+\int_{\Omega}(\widehat{w_{\varepsilon}}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot(\widetilde{J_{\varepsilon}}-\widehat{J})\qquad\forall\varepsilon\in
I_{*}\,.$ (2.24)

On one hand, the weak convergence in (2.22) implies

$\lim_{\varepsilon\to
0^{+}}\int_{\Omega}(\widehat{w}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot\widehat{J}=\int_{\Omega}(\widehat{w}\cdot\nabla)\widehat{w}\cdot\widehat{J}\,.$
(2.25)

On the other hand, the Hölder and Sobolev inequalities in $\Omega$, together
with the strong convergences in (2.5)-(2.22) entail that

$\displaystyle\left|\int_{\Omega}((\widehat{w_{\varepsilon}}-\widehat{w})\cdot\nabla)\widehat{w_{\varepsilon}}\cdot\widehat{J}+\int_{\Omega}(\widehat{w_{\varepsilon}}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot(\widetilde{J_{\varepsilon}}-\widehat{J})\right|$
(2.26)
$\displaystyle\leq\left|\int_{\Omega}((\widehat{w_{\varepsilon}}-\widehat{w})\cdot\nabla)\widehat{w_{\varepsilon}}\cdot\widehat{J}\right|+\left|\int_{\Omega}(\widehat{w_{\varepsilon}}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot(\widetilde{J_{\varepsilon}}-\widehat{J})\right|$
$\displaystyle\leq
C\left(\|\widehat{w_{\varepsilon}}-\widehat{w}\|_{L^{4}(\Omega)}+\|\widetilde{J_{\varepsilon}}-\widehat{J}\|_{L^{4}(\Omega)}\right)\to
0\quad\text{as}\ \ \varepsilon\to 0^{+}\,.$

Inserting (2.24)-(2.25)-(2.26) into (2.23) and letting $\varepsilon\to 0^{+}$,
again from (2.5)-(2.22) we deduce

$\eta=\int_{\Omega}(\widehat{w}\cdot\nabla)\widehat{w}\cdot\widehat{J}\,.$
(2.27)

A contradiction will be reached in (2.27) once we prove that

$\left|\int_{\Omega}(\widehat{w}\cdot\nabla)\widehat{w}\cdot\widehat{J}\right|<\eta\,.$

Given any vector field $\varphi\in\mathcal{C}^{\infty}_{0}(\Omega)$ (not
necessarily divergence-free) and the relative capacity potential
$\phi_{\varepsilon}\in H^{2}(\Omega_{\varepsilon})$ (see Lemma 2.1), we may
use $(1-\phi_{\varepsilon})\varphi\in H^{1}_{0}(\Omega_{\varepsilon})$ as a
test function in (2.16) and divide the resulting identity by
$\mathcal{Z}^{2}_{\varepsilon}$ to obtain

$\displaystyle\dfrac{\eta}{\mathcal{Z}_{\varepsilon}}\int_{\Omega}\nabla\widehat{w_{\varepsilon}}\cdot\nabla\varphi-\dfrac{\eta}{\mathcal{Z}_{\varepsilon}}\int_{\Omega}\nabla\widehat{w_{\varepsilon}}\cdot\nabla(\phi_{\varepsilon}\varphi)+\int_{\Omega}(\widehat{w_{\varepsilon}}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot\varphi-\int_{\Omega}(\widehat{w_{\varepsilon}}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot\phi_{\varepsilon}\varphi$
(2.28)
$\displaystyle+\dfrac{1}{\mathcal{Z}_{\varepsilon}}\int_{\Omega}(\widehat{w_{\varepsilon}}\cdot\nabla)\widetilde{J_{\varepsilon}}\cdot(1-\phi_{\varepsilon})\varphi+\dfrac{1}{\mathcal{Z}_{\varepsilon}}\int_{\Omega}(\widetilde{J_{\varepsilon}}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot(1-\phi_{\varepsilon})\varphi-\int_{\Omega}\widehat{p_{\varepsilon}}(\nabla\cdot\varphi)+\int_{\Omega}\widehat{p_{\varepsilon}}(\nabla\cdot\phi_{\varepsilon}\varphi)$
$\displaystyle=\dfrac{1}{\mathcal{Z}^{2}_{\varepsilon}}\left(\int_{\Omega}f\cdot(1-\phi_{\varepsilon})\varphi-\eta\int_{\Omega}\nabla\widetilde{J_{\varepsilon}}\cdot\nabla((1-\phi_{\varepsilon})\varphi)-\int_{\Omega}(\widetilde{J_{\varepsilon}}\cdot\nabla)\widetilde{J_{\varepsilon}}\cdot(1-\phi_{\varepsilon})\varphi\right)\,.$

for every $\varepsilon\in I_{*}$, along the subsequences (2.22). The
convergences in (2.22) guarantee that

$\lim_{\varepsilon\to
0^{+}}\int_{\Omega}(\widehat{w_{\varepsilon}}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot\varphi=\int_{\Omega}(\widehat{w}\cdot\nabla)\widehat{w}\cdot\varphi\qquad\text{and}\qquad\lim_{\varepsilon\to
0^{+}}\int_{\Omega}\widehat{p_{\varepsilon}}(\nabla\cdot\varphi)=\int_{\Omega}\widehat{p}(\nabla\cdot\varphi)\,.$
(2.29)

On the other hand, applying Hölder and Sobolev inequalities in $\Omega$, from
(2.3) we notice that

$\displaystyle\left|\int_{\Omega}\nabla\widehat{w_{\varepsilon}}\cdot(\nabla\phi_{\varepsilon}\otimes\varphi+\phi_{\varepsilon}\nabla\varphi)\right|\leq\dfrac{C}{\sigma_{\varepsilon}}\|\nabla\widehat{w_{\varepsilon}}\|_{L^{2}(\Omega)}\|\varphi\|_{W^{1,\infty}(\Omega)}\qquad\forall\varepsilon\in
I_{*}\,,$ (2.30)
$\displaystyle\left|\int_{\Omega}(\widehat{w_{\varepsilon}}\cdot\nabla)\widehat{w_{\varepsilon}}\cdot\phi_{\varepsilon}\varphi\right|\leq\dfrac{C}{\sigma_{\varepsilon}}\|\nabla\widehat{w_{\varepsilon}}\|_{L^{2}(\Omega)}\|\widehat{w_{\varepsilon}}\|_{L^{4}(\Omega)}\|\varphi\|_{L^{\infty}(\Omega)}\qquad\forall\varepsilon\in
I_{*}\,,$
$\displaystyle\left|\int_{\Omega}(\widehat{w_{\varepsilon}}\cdot\nabla)\widetilde{J_{\varepsilon}}\cdot(1-\phi_{\varepsilon})\varphi\right|\leq
C\|\nabla\widetilde{J_{\varepsilon}}\|_{L^{2}(\Omega)}\|\widehat{w_{\varepsilon}}\|_{L^{4}(\Omega)}\|\varphi\|_{L^{\infty}(\Omega)}\qquad\forall\varepsilon\in
I_{*}\,,$
$\displaystyle\left|\int_{\Omega}\widehat{p_{\varepsilon}}\left[\phi_{\varepsilon}(\nabla\cdot\varphi)+\nabla\phi_{\varepsilon}\cdot\varphi\right]\right|\leq\dfrac{C}{\sigma_{\varepsilon}}\|\widehat{p_{\varepsilon}}\|_{L^{2}(\Omega)}\|\varphi\|_{W^{1,\infty}(\Omega)}\qquad\forall\varepsilon\in
I_{*}\,,$
$\displaystyle\left|\int_{\Omega}f\cdot(1-\phi_{\varepsilon})\varphi\right|\leq
C\|f\|_{L^{2}(\Omega)}\|\varphi\|_{L^{\infty}(\Omega)}\qquad\forall\varepsilon\in
I_{*}\,,$

and the remaining terms appearing in (2.28) can be estimated in a similar
fashion. Observing (1.1)- (2.29)-(2.30), one can take the limit as
$\varepsilon\to 0^{+}$ in (2.28) to deduce that

$\int_{\Omega}(\widehat{w}\cdot\nabla)\widehat{w}\cdot\varphi-\int_{\Omega}\widehat{p}(\nabla\cdot\varphi)=0\qquad\forall\varphi\in\mathcal{C}^{\infty}_{0}(\Omega;\mathbb{R}^{2})\,,$

that is, the pair $(\widehat{w},\widehat{p})\in H^{1}_{0}(\Omega)\times
L^{2}(\Omega)$ satisfies in distributional form the steady-state Euler
equation

$(\widehat{w}\cdot\nabla)\widehat{w}+\nabla\widehat{p}=0\,,\quad\nabla\cdot\widehat{w}=0\
\ \mbox{ in }\ \ \Omega\,.$ (2.31)

Since $(\widehat{w}\cdot\nabla)\widehat{w}\in L^{3/2}(\Omega)$ (by Sobolev
embedding), (2.31) proves that actually $\widehat{p}\in W^{1,3/2}(\Omega)$, so
that the Euler equation (2.31) is satisfied in strong form. Since
$\widehat{w}=0$ on $\partial\Omega$, the Bernoulli law [17, Lemma 4] (see [4,
Theorem 2.2] and [18, Theorem 1] as well) states that $\widehat{p}$ must be
constant on each of the connected components of $\partial\Omega$. Thus, there
exist $\widehat{p}_{0},...,\widehat{p}_{M}\in\mathbb{R}$ such that

$\widehat{p}=\widehat{p}_{j}\quad\text{almost everywhere on}\ \
\partial\Omega_{j}\,,\ \forall j\in\\{0,...,M\\}\,.$ (2.32)

If we multiply the first equation in (2.31) by $\widehat{J}$, integrate by
parts in $\Omega$ and enforce (2.32), the following identity is obtained:

$\int_{\Omega}(\widehat{w}\cdot\nabla)\widehat{w}\cdot\widehat{J}=-\int_{\partial\Omega}\widehat{p}(\widehat{J}\cdot\nu)=-\sum_{j=0}^{M}\widehat{p}_{j}\,F_{j}\,.$
(2.33)

We now proceed as in the proof of [35, Lemma 4] (see also [19, Theorem 2.2]).
Define the functions $\widehat{W}\in H_{0}^{1}(\mathbb{R}^{2})$ and
$\widehat{P}\in W_{\text{loc}}^{1,3/2}(\mathbb{R}^{2})$ by

$\widehat{W}\doteq\begin{cases}\widehat{w}&\quad\text{in}\ \
\Omega\,,\\\\[4.0pt] 0&\quad\text{in}\ \
(\mathbb{R}^{2}\setminus\overline{\Omega_{0}})\cup\bigcup_{i=1}^{M}\Omega_{i}\,,\end{cases}\qquad\qquad\widehat{P}\doteq\begin{cases}\widehat{p}&\quad\text{in}\
\ \Omega\,,\\\\[4.0pt] \widehat{p}_{j}&\quad\text{in}\ \
\Omega_{j}\quad\forall j\in\\{1,...,M\\}\,,\\\\[4.0pt]
\widehat{p}_{0}&\quad\text{in}\ \
\mathbb{R}^{2}\setminus\overline{\Omega_{0}}\,,\end{cases}$

so that (2.31) implies that

$(\widehat{W}\cdot\nabla)\widehat{W}+\nabla\widehat{P}=0\,,\quad\nabla\cdot\widehat{W}=0\
\ \mbox{ in }\ \ \mathbb{R}^{2}\setminus\overline{\Omega_{j}}\,,\ \ \forall
j\in\\{0,...,M\\}\,.$ (2.34)

Given $j\in\\{0,...,M\\}$, a standard integration by parts and the properties
of $\chi_{j}$ in (2.11)-(2.12) give us

$\int_{\mathbb{R}^{2}\setminus\overline{\Omega_{j}}}\nabla\widehat{P}\cdot\nabla\chi_{j}=\int_{\partial\Omega_{j}}\widehat{P}\,\dfrac{\partial\chi_{j}}{\partial\nu}=\widehat{p}_{j}\,\text{Cap}_{\mathbb{R}^{2}}(\Omega_{j})\,,$
(2.35)

and also

$\int_{\mathbb{R}^{2}\setminus\overline{\Omega_{j}}}(\widehat{W}\cdot\nabla)\widehat{W}\cdot\nabla\chi_{j}=-\int_{\mathbb{R}^{2}\setminus\overline{\Omega_{j}}}\chi_{j}\left(\nabla\widehat{W}\cdot(\nabla\widehat{W})^{\top}\right)=-\int_{\Omega}\chi_{j}\left(\nabla\widehat{w}\cdot(\nabla\widehat{w})^{\top}\right)\,.$
(2.36)

From (2.34)-(2.35)-(2.36) and the Maximum Principle we immediately deduce that

$\left|\widehat{p}_{j}\,\text{Cap}_{\mathbb{R}^{2}}(\Omega_{j})\right|=\left|\int_{\Omega}\chi_{j}\left(\nabla\widehat{w}\cdot(\nabla\widehat{w})^{\top}\right)\right|\leq\|\nabla\widehat{w}\|^{2}_{L^{2}(\Omega)}\leq
1\qquad\forall j\in\\{0,...,M\\}\,.$

Once inserted into (2.33), this yields

$\left|\int_{\Omega}(\widehat{w}\cdot\nabla)\widehat{w}\cdot\widehat{J}\right|\leq\sum_{j=0}^{M}\left|\widehat{p}_{j}\,F_{j}\right|\leq\sum_{j=0}^{M}\dfrac{|F_{j}|}{\textup{Cap}_{\mathbb{R}^{2}}(\Omega_{j})}<\eta$

as a consequence of (2.13), in contradiction with (2.27). This concludes the
proof. $\Box$

Theorem 2.1 deserves some remarks and comments.

######  Remark 2.1.

Inequality (2.13) imposes a smallness condition on the flux of $v_{*}\in
H^{3/2}(\partial\Omega)$ across each of the connected components of
$\partial\Omega$, but it does not necessarily enforce a bound on the size of
$v_{*}$. As an example we can consider the two-dimensional Taylor-Couette
problem for the steady motion of a viscous incompressible fluid in the region
between two concentric disks (the inner one at rest and the outer one rotating
with arbritarily large and constant angular speed $\omega>0$, see [22, Chapter
II]).

######  Remark 2.2.

The hypothesis $f\in H^{1}(\Omega)$ of Theorem 2.1 is needed in [20, Theorem
1.1] to ensure the unrestricted weak solvabilty of problem (1.6) under the
general outflow condition (1.5). Notice that the inequality (2.13) is imposed
here to achieve the $\varepsilon$-uniform bound (2.14), whereas it can also be
assumed in order to find a generalized solution to (1.6) relaxing the
regularity constraints on the data of the problem (see [35, Theorem 1] or [19,
Theorem 2.2]).

######  Remark 2.3.

The uniform bound (2.14) of Theorem 2.1 can be easily achieved under a
smallness assumption on the data (boundary velocity and external force). More
precisely, if $\Omega_{\varepsilon}$ is the perforated domain (1.3), it
follows from [14, Theorem IX.4.1] the existence of a constant $\delta_{*}>0$
(depending only on $\Omega$, $\eta$, and $\\{\delta_{0},\delta_{1}\\}$,
independent of $\varepsilon\in I_{*}$) such that, whenever

$\|v_{*}\|_{H^{3/2}(\Omega)}+\|f\|_{H^{1}(\Omega)}<\delta_{*}\,,$

then problem (1.6) has a unique weak solution $u_{\varepsilon}\in
H^{1}_{\sigma}(\Omega_{\varepsilon})$ which, moreover, admits the bound

$\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq C_{*}\,,$

for some constant $C_{*}>0$ that depends on $\Omega$, $\eta$, $v_{*}$, $f$ and
$\\{\delta_{0},\delta_{1}\\}$ (independent of $\varepsilon\in I_{*}$).

###  2.1 $\varepsilon$-Uniform bounds in the symmetric case

Let $\varepsilon\in I_{*}$ be a fixed parameter. In this subsection we suppose
that each of the domains $\Omega_{0},\Omega_{1},...,\Omega_{M}$ intersects the
$x$-axis and is symmetric with respect to the $x$-axis. The symmetric version
of Lemma 2.2 can be easily derived as a corollary, and it reads:

######  Corollary 2.1.

Let $\Omega_{\varepsilon}$ be the symmetric perforated domain (1.8) and
$v_{*}\in H^{3/2}(\partial\Omega)$ a symmetric vector field satisfying (1.5).
There exists a symmetric vector field $J_{\varepsilon}\in
H^{1}(\Omega_{\varepsilon})$ such that

$\left\\{\begin{aligned} &\nabla\cdot J_{\varepsilon}=0\ \ \mbox{in}\ \
\Omega_{\varepsilon}\,;\qquad J_{\varepsilon}=v_{*}\ \ \mbox{on}\ \
\partial\Omega\,;\\\\[3.0pt] &J_{\varepsilon}=0\ \ \mbox{on}\ \ \partial
K_{\varepsilon}\,;\qquad\|\nabla
J_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C_{*}\|v_{*}\|_{H^{3/2}(\partial\Omega)}\,,\end{aligned}\right.$ (2.37)

for some constant $C_{*}>0$ that depends on $\Omega$ and $\lambda_{0}$, but is
independent of $\varepsilon\in I_{*}$. In particular, there exists a symmetric
vector field $\widehat{J}\in H^{1}(\Omega)$ such that

$\nabla\cdot\widehat{J}=0\ \ \mbox{in}\ \
\Omega\qquad\text{and}\qquad\widehat{J}=v_{*}\ \ \mbox{on}\ \
\partial\Omega\,,$

and a (not relabeled) subsequence
$(\widetilde{J_{\varepsilon}})_{\varepsilon\in I_{*}}\subset H^{1}(\Omega)$
for which

$\widetilde{J_{\varepsilon}}\rightharpoonup\widehat{J}\ \ \ \text{weakly in}\
H^{1}(\Omega)\qquad\text{and}\qquad\widehat{J_{\varepsilon}}\to\widehat{J}\ \
\ \text{strongly in}\ L^{4}(\Omega)\ \ \ \text{as}\ \ \varepsilon\to 0^{+}\,.$
(2.38)

Proof. Let
$\Psi_{\varepsilon}=(\Psi_{1}^{\varepsilon},\Psi_{2}^{\varepsilon})\in
H^{1}(\Omega_{\varepsilon})$ be the vector field arising from Lemma (2.2),
which satisfies (2.4) (but is not necessarily symmetric). We then define the
vector field $J_{\varepsilon}\in H^{1}(\Omega_{\varepsilon})$ by

$J_{\varepsilon}(x,y)\doteq\dfrac{1}{2}(\Psi_{1}^{\varepsilon}(x,y)+\Psi_{1}^{\varepsilon}(x,-y),\Psi_{2}^{\varepsilon}(x,y)-\Psi_{2}^{\varepsilon}(x,-y))\qquad\text{for
a.e.}\ (x,y)\in\Omega_{\varepsilon}\,,$

which is symmetric and satisfies all the properties listed in (2.37). To see
this, a direct computation (involving Young’s inequality) shows that $\|\nabla
J_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq\|\nabla\Psi_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$.
The rest of the proof follows exactly the lines of the proof of Lemma (2.2),
where, in addition, we point out that the limit vector field $\widehat{J}\in
H^{1}(\Omega)$ is symmetric as a consequence of the strong convergence in
(2.38). $\Box$

The second main result of this section provides uniform bounds (with respect
to $\varepsilon\in I_{*}$) for the symmetric solutions of problem (1.6).

######  Theorem 2.2.

Let $\Omega_{\varepsilon}$ be the symmetric perforated domain (1.8). For any
symmetric vector fields $f\in L^{2}(\Omega)$ and $v_{*}\in
H^{3/2}(\partial\Omega)$ such that (1.5) holds, there exists at least one
symmetric weak solution $u_{\varepsilon}\in H^{2}(\Omega_{\varepsilon})\cap
H^{1}_{\sigma}(\Omega_{\varepsilon})$ of problem (1.6), and an associated
symmetric pressure $p_{\varepsilon}\in H^{1}(\Omega_{\varepsilon})\cap
L^{2}_{0}(\Omega_{\varepsilon})$ such that the pair
$(u_{\varepsilon},p_{\varepsilon})$ satisfies (1.6) in strong form. Moreover,
the uniform bound

$\sup_{\varepsilon\in I_{*}}\left(\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\right)\leq
C_{*}\,,$ (2.39)

holds for some constant $C_{*}>0$ that depends on $\Omega$, $\eta$, $f$,
$v_{*}$ and $\lambda_{0}$, but is independent of $\varepsilon\in I_{*}$.

Proof. In what follows, $C>0$ will always denote a generic constant that
depends on $\Omega$, $\eta$ and $\lambda_{0}$ (independently of
$\varepsilon\in I_{*}$), but that may change from line to line.

Then, given any symmetric vector fields $f\in L^{2}(\Omega)$, $v_{*}\in
H^{3/2}(\partial\Omega)$ satisfying (1.5) and $\varepsilon\in I_{*}$, a direct
extension of [4, Theorem 1.1] ensures the existence of at least one symmetric
weak solution $u_{\varepsilon}\in H^{2}(\Omega_{\varepsilon})\cap
H^{1}_{\sigma}(\Omega_{\varepsilon})$ of problem (1.6) and an associated
symmetric pressure $p_{\varepsilon}\in H^{1}(\Omega_{\varepsilon})\cap
L^{2}_{0}(\Omega_{\varepsilon})$ such that the pair
$(u_{\varepsilon},p_{\varepsilon})$ solves (1.6) in strong form. This follows
simply from the fact that the sought solution is required to have zero flux
across each of the holes $K^{\varepsilon}_{m}$, for
$m\in\mathcal{M}_{\varepsilon}$. Let $J_{\varepsilon}\in
H^{1}(\Omega_{\varepsilon})$ be the symmetric vector field arising from
Corollary (2.1), which satisfies (2.37). Set $w_{\varepsilon}\doteq
u_{\varepsilon}-J_{\varepsilon}\in H^{1}_{0,\sigma}(\Omega_{\varepsilon})$
(which is a symmetric vector field), so that, in view of the bound in (2.37)2,
it suffices to show that

$\sup_{\varepsilon\in I_{*}}\left(\|\nabla
w_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\right)\leq
C\,.$

Exactly as in the proof of Theorem 2.1 we derive identity (2.18) and the
estimate (2.20). By contradiction, suppose now that the norms $\|\nabla
w_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$ are not uniformly bounded
with respect to $\varepsilon\in I_{*}$. Then, there must exists a subsequence
(not being relabeled) such that

$\lim_{\varepsilon\to 0^{+}}\mathcal{Z}_{\varepsilon}=+\infty\quad\text{with}\
\ \mathcal{Z}_{\varepsilon}\doteq\|\nabla
w_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\quad\forall\varepsilon\in
I_{*}\,.$ (2.40)

The estimate (2.20) enables us to establish that, along this divergent
subsequence (2.40), the following sequences are uniformly bounded with respect
to $\varepsilon\in I_{*}$:

$(\widehat{w_{\varepsilon}})_{\varepsilon\in
I_{*}}\doteq\left(\dfrac{\widetilde{w_{\varepsilon}}}{\mathcal{Z}_{\varepsilon}}\right)_{\varepsilon\in
I_{*}}\subset
H_{0}^{1}(\Omega)\qquad\text{and}\qquad(\widehat{p_{\varepsilon}})_{\varepsilon\in
I_{*}}\doteq\left(\dfrac{\widetilde{p_{\varepsilon}}}{\mathcal{Z}^{2}_{\varepsilon}}\right)_{\varepsilon\in
I_{*}}\subset L^{2}(\Omega)\,.$

Then, there must exist $\widehat{w}\in H_{0}^{1}(\Omega)$ and $\widehat{p}\in
L^{2}(\Omega)$ such that the following convergences hold:

$\displaystyle\widehat{w_{\varepsilon}}\rightharpoonup\widehat{w}\ \ \
\text{weakly in}\ H^{1}(\Omega)\,;$
$\displaystyle\widehat{w_{\varepsilon}}\to\widehat{w}\ \ \ \text{strongly in}\
L^{4}(\Omega)\,;\qquad\widehat{p_{\varepsilon}}\rightharpoonup\widehat{p}\ \ \
\text{weakly in}\ L^{2}(\Omega)\,,$ (2.41)

as $\varepsilon\to 0^{+}$, along subsequences that are not being relabeled.
Clearly we have $\|\nabla\widehat{w}\|_{L^{2}(\Omega)}\leq 1$, and as in the
final part of the proof of Lemma (2.2) we deduce that
$\nabla\cdot\widehat{w}=0$ almost everywhere in $\Omega$, that is,
$\widehat{w}\in H_{0,\sigma}^{1}(\Omega)$. Moreover, the strong convergence in
(2.41) ensures that $\widehat{w}$ is a symmetric vector field in $\Omega$.
Imitating the steps of the proof of Theorem 2.1 we also recover identity
(2.27) and infer that the pair $(\widehat{w},\widehat{p})\in
H^{1}_{0}(\Omega)\times W^{1,3/2}(\Omega)$ satisfies in strong form the
stationary Euler equation (2.31) in $\Omega$. Again, the Bernoulli law
enforces that $\widehat{p}$ must be constant on each of the connected
components of $\partial\Omega$: there exist
$\widehat{p}_{0},...,\widehat{p}_{M}\in\mathbb{R}$ such that

$\widehat{p}=\widehat{p}_{j}\quad\text{almost everywhere on}\ \
\partial\Omega_{j}\,,\ \forall j\in\\{0,...,M\\}\,.$ (2.42)

If we multiply the first equation in (2.31) by $\widehat{J}$, integrate by
parts in $\Omega$ and enforce (2.42), the following identity is obtained:

$\int_{\Omega}(\widehat{w}\cdot\nabla)\widehat{w}\cdot\widehat{J}=-\int_{\partial\Omega}\widehat{p}(\widehat{J}\cdot\nu)=-\sum_{j=0}^{M}\widehat{p}_{j}\,F_{j}\,.$
(2.43)

A contradiction will be reached after proving that
$\widehat{p}_{0}=\widehat{p}_{1}=...=\widehat{p}_{M}$, as once inserted into
(2.43) and combined with (1.5), this would imply

$\int_{\Omega}(\widehat{w}\cdot\nabla)\widehat{w}\cdot\widehat{J}=0\,,$

thereby violating (2.27). In order to do so, we follow closely the proof of
[4, Theorem 2.3] (see also [19, Theorem 2.1]). Take
$\\{\alpha_{j}\\}^{M}_{j=0},\\{\beta_{j}\\}^{M}_{j=0}\subset\mathbb{R}$ such
that

$\alpha_{j}<\beta_{j}\qquad\text{and}\qquad\partial\Omega_{j}\cap\\{(x,y)\in\mathbb{R}^{2}\
|\ y=0\\}=\\{(\alpha_{j},0)\,,(\beta_{j},0)\\}\qquad\forall
j\in\\{0,...,M\\}\,.$

Without loss of generality, we may label the sets $\Omega_{0},...,\Omega_{M}$
in such a way that

$\alpha_{0}<\alpha_{1}<\beta_{1}<...<\alpha_{M}<\beta_{M}<\beta_{0}\,.$

It suffices to show that $\widehat{p}_{0}=\widehat{p}_{1}$, as the remaining
equalities can be proved in analogous way. Since $\Omega_{0}$ and $\Omega_{1}$
intersect the $x$-axis and they are symmetric with respect to the $x$-axis,
there exist $\lambda_{*}>0$ and two functions
$g_{0},g_{1}\in\mathcal{C}^{2}((-\lambda_{*},\lambda_{*});\mathbb{R})$ such
that, for every $\lambda\in(0,\lambda_{*})$, the following representations
hold:

$\displaystyle\partial\Omega_{0}=\\{(x,y)\in\mathbb{R}^{2}\ |\
y\in(-\lambda,\lambda)\,,\ \ x=g_{0}(y)\\}\quad\text{near the point}\
(\alpha_{0},0)\,,$ $\displaystyle\partial\Omega_{1}=\\{(x,y)\in\mathbb{R}^{2}\
|\ y\in(-\lambda,\lambda)\,,\ \ x=g_{1}(y)\\}\quad\text{near the point}\
(\alpha_{1},0)\,,$

and such that the following inclusion is observed:

$\mathcal{A}_{\lambda}\doteq\\{(x,y)\in\mathbb{R}^{2}\ |\ 0<y<\lambda\,,\ \
g_{0}(y)<x<g_{1}(y)\\}\subset\Omega\,.$

Write $\widehat{w}=(\widehat{w}_{1},\widehat{w}_{2})$ in $\Omega$ and define
the total-head pressure as

$\widehat{\Phi}\doteq\widehat{p}+\dfrac{1}{2}|\widehat{w}|^{2}\quad\text{in}\
\ \Omega\,,$

so that $\widehat{\Phi}\in W^{1,3/2}(\Omega)$ and it clearly verifies the
identity

$\dfrac{\partial\widehat{\Phi}}{\partial
x}=\widehat{w}_{2}\left(\dfrac{\partial\widehat{w}_{2}}{\partial
x}-\dfrac{\partial\widehat{w}_{1}}{\partial y}\right)\quad\text{in}\ \
\Omega\,.$

Integrating this last identity over $\mathcal{A}_{\lambda}$ and enforcing
(2.42), we obtain

$\lambda(\widehat{p}_{1}-\widehat{p}_{0})=\int_{\mathcal{A}_{\lambda}}\widehat{w}_{2}\left(\dfrac{\partial\widehat{w}_{2}}{\partial
x}-\dfrac{\partial\widehat{w}_{1}}{\partial y}\right)\,,$

so that

$|\widehat{p}_{1}-\widehat{p}_{0}|=\dfrac{1}{\lambda}\left|\int_{\mathcal{A}_{\lambda}}\widehat{w}_{2}\left(\dfrac{\partial\widehat{w}_{2}}{\partial
x}-\dfrac{\partial\widehat{w}_{1}}{\partial
y}\right)\right|\qquad\forall\lambda\in(0,\lambda_{*})\,.$ (2.44)

If we extend $\widehat{w}_{2}$ by zero to the whole set
$\mathbb{R}\times(0,\lambda)$, by symmetry we have $\widehat{w}_{2}(x,0)=0$
(in the sense of traces) for every $x\in\mathbb{R}$, so that Hardy’s
inequality can be applied to yield

$\int_{\mathcal{A}_{\lambda}}\dfrac{|\widehat{w}_{2}(x,y)|^{2}}{y^{2}}\,dx\,dy=\int\limits^{+\infty}_{-\infty}\int\limits^{\lambda}_{0}\dfrac{|\widehat{w}_{2}(x,y)|^{2}}{y^{2}}\,dy\,dx\leq
4\int\limits^{+\infty}_{-\infty}\int\limits^{\lambda}_{0}\left|\dfrac{\partial\widehat{w}_{2}}{\partial
y}(x,y)\right|^{2}dy\,dx\leq
4\int_{\mathcal{A}_{\lambda}}|\nabla\widehat{w}|^{2}\,.$ (2.45)

Combining (2.44)-(2.45) (together with Hölder’s inequality) leads to the
estimate

$\displaystyle|\widehat{p}_{1}-\widehat{p}_{0}|^{2}\leq\dfrac{1}{\lambda^{2}}\left(\int_{\mathcal{A}_{\lambda}}y^{2}\dfrac{|\widehat{w}_{2}(x,y)|^{2}}{y^{2}}\,dx\,dy\right)\left(\int_{\mathcal{A}_{\lambda}}\left|\dfrac{\partial\widehat{w}_{2}}{\partial
x}-\dfrac{\partial\widehat{w}_{1}}{\partial y}\right|^{2}\right)\leq
16\int_{\mathcal{A}_{\lambda}}|\nabla\widehat{w}|^{2}\to 0\quad\text{as}\ \
\lambda\to 0^{+}\,,$

thereby proving that $\widehat{p}_{1}=\widehat{p}_{0}$. This concludes the
proof. $\Box$

######  Remark 2.4.

The uniform bound (2.39) of Theorem 2.2 can be easily achieved under a
smallness assumption on the data (boundary velocity and external force). More
precisely, if $\Omega_{\varepsilon}$ is the symmetric perforated domain (1.8),
it follows from [14, Theorem IX.4.1] the existence of a constant
$\delta_{*}>0$ (depending only on $\Omega$, $\eta$, and $\lambda_{0}$,
independent of $\varepsilon\in I_{*}$) such that, whenever

$\|v_{*}\|_{H^{3/2}(\Omega)}+\|f\|_{L^{2}(\Omega)}<\delta_{*}\,,$

then (1.6) has a unique (symmetric) weak solution $u_{\varepsilon}\in
H^{1}_{\sigma}(\Omega_{\varepsilon})$ which, moreover, admits the bound

$\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq C_{*}\,,$

for some constant $C_{*}>0$ that depends on $\Omega$, $\eta$, $v_{*}$, $f$ and
$\lambda_{0}$ (independent of $\varepsilon\in I_{*}$).

##  3 Asymptotic behavior as $\varepsilon\to 0^{+}$: homogenized equations

By employing the renowned energy method of Tartar [36, Appendix] (see also
[38, Chapter 15]), in this section we derive the effective (or homogenized)
equations satisfied by the solutions of (1.6) as $\varepsilon\to 0^{+}$.

######  Theorem 3.1.

Let $(\Omega_{\varepsilon})_{\varepsilon\in I_{*}}$ be the family of
perforated domains (1.3). Given any $f\in H^{1}(\Omega)$ and $v_{*}\in
H^{3/2}(\partial\Omega)$ satisfying (1.5), let
$(u_{\varepsilon},p_{\varepsilon})\in
H_{\sigma}^{1}(\Omega_{\varepsilon})\times L_{0}^{2}(\Omega_{\varepsilon})$ be
a weak solution of (1.6). Then, assuming condition (2.13), up to the
extraction of a subsequence, the sequence of extended functions
$\\{(\widetilde{u_{\varepsilon}},\widetilde{p_{\varepsilon}})\\}_{\varepsilon\in
I_{*}}\subset H^{1}(\Omega)\times L_{0}^{2}(\Omega)$ converges strongly to a
weak solution $(u,p)\in H^{1}(\Omega)\times L_{0}^{2}(\Omega)$ of problem
(1.6) in $\Omega$ as $\varepsilon\to 0^{+}$. Furthermore, $(u,p)\in
H^{2}(\Omega)\times H^{1}(\Omega)$ and it satisfies in strong form the system

$\left\\{\begin{aligned} &-\eta\Delta u+(u\cdot\nabla)u+\nabla
p=f\,,\quad\nabla\cdot u=0\ \ \mbox{ in }\ \ \Omega\,,\\\\[4.0pt] &u=v_{*}\ \
\mbox{ on }\ \ \partial\Omega\,.\end{aligned}\right.$ (3.1)

Proof. In what follows, $C>0$ will always denote a generic constant that
depends on $\Omega$, $\eta$ and $\\{\delta_{0},\delta_{1}\\}$ (independently
of $\varepsilon\in I_{*}$), but that may change from line to line.

Given any $f\in H^{1}(\Omega)$, $v_{*}\in H^{3/2}(\partial\Omega)$ satisfying
(1.5) and $\varepsilon\in I_{*}$, let $(u_{\varepsilon},p_{\varepsilon})\in
H_{\sigma}^{1}(\Omega_{\varepsilon})\times L_{0}^{2}(\Omega_{\varepsilon})$ be
a weak solution of (1.6). From Theorem 2.1 we know that
$(u_{\varepsilon},p_{\varepsilon})\in H^{2}(\Omega_{\varepsilon})\times
H^{1}(\Omega_{\varepsilon})$ solves (1.6) in strong form and that
$\\{(\widetilde{u_{\varepsilon}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{p_{\varepsilon}}})\\}_{\varepsilon\in
I_{*}}\subset H^{1}(\Omega)\times L_{0}^{2}(\Omega)$. In particular, the weak
formulation of Definition (2.1) incorporating the pressure term reads:

$\eta\int_{\Omega_{\varepsilon}}\nabla
u_{\varepsilon}\cdot\nabla\varphi+\int_{\Omega_{\varepsilon}}(u_{\varepsilon}\cdot\nabla)u_{\varepsilon}\cdot\varphi-\int_{\Omega_{\varepsilon}}p_{\varepsilon}(\nabla\cdot\varphi)=\int_{\Omega_{\varepsilon}}f\cdot\varphi\qquad\forall\varphi\in
H^{1}_{0}(\Omega_{\varepsilon})\,.$ (3.2)

Now, given any scalar function $\phi\in\mathcal{C}_{0}^{\infty}(\Omega)$, an
integration by parts and the divergence-free condition in (1.6)1 imply that

$-\int_{\Omega}\phi(\nabla\cdot\widetilde{u_{\varepsilon}})=\int_{\Omega}\widetilde{u_{\varepsilon}}\cdot\nabla\phi=\int_{\Omega_{\varepsilon}}u_{\varepsilon}\cdot\nabla\phi=\int_{\partial\Omega_{\varepsilon}}\phi(u_{\varepsilon}\cdot\nu)=0\qquad\forall\varepsilon\in
I_{*}\,,$

since $u_{\varepsilon}$ vanishes on $\partial K_{\varepsilon}$ and so does
$\phi$ on $\partial\Omega$. This proves
$\nabla\cdot\widetilde{u_{\varepsilon}}=0$ almost everywhere in $\Omega$, that
is, $\widetilde{u_{\varepsilon}}\in H_{\sigma}^{1}(\Omega)$ for every
$\varepsilon\in I_{*}$. Moreover, (2.14)-(2.17) ensure that the sequences
$(\widetilde{u_{\varepsilon}})_{\varepsilon\in I_{*}}\subset
H_{\sigma}^{1}(\Omega)$ and $(\widetilde{p_{\varepsilon}})_{\varepsilon\in
I_{*}}\subset L_{0}^{2}(\Omega)$ are uniformly bounded, so there exist $u\in
H_{\sigma}^{1}(\Omega)$ and $p\in L_{0}^{2}(\Omega)$ such that the following
convergences hold as $\varepsilon\to 0^{+}$:

$\displaystyle\widetilde{u_{\varepsilon}}\rightharpoonup u\ \ \ \text{weakly
in}\ H^{1}(\Omega)\,;\qquad\widetilde{u_{\varepsilon}}\to u\ \ \
\text{strongly in}\ L^{4}(\Omega)\,;$ (3.3)
$\displaystyle\widetilde{u_{\varepsilon}}\to u\ \ \ \text{strongly in}\
L^{2}(\partial\Omega)\,;\qquad\widetilde{p_{\varepsilon}}\rightharpoonup p\ \
\ \text{weakly in}\ L^{2}(\Omega)\,,$

along subsequences that are not being relabeled. The strong convergence in
(3.3)2 ensures that $u=v_{*}$ almost everywhere on $\partial\Omega$. Given any
vector field $\varphi\in\mathcal{C}_{0}^{\infty}(\Omega)$ (not necessarily
divergence-free) and the relative capacity potential $\phi_{\varepsilon}\in
H^{2}(\Omega_{\varepsilon})$ (see Lemma 2.1), we may use
$(1-\phi_{\varepsilon})\varphi\in H^{1}_{0}(\Omega_{\varepsilon})$ as a test
function in (3.2) to obtain

$\displaystyle\eta\int_{\Omega}\nabla\widetilde{u_{\varepsilon}}\cdot\nabla\varphi-\eta\int_{\Omega}\nabla\widetilde{u_{\varepsilon}}\cdot\nabla(\phi_{\varepsilon}\varphi)+\int_{\Omega}(\widetilde{u_{\varepsilon}}\cdot\nabla)\widetilde{u_{\varepsilon}}\cdot\varphi-\int_{\Omega}(\widetilde{u_{\varepsilon}}\cdot\nabla)\widetilde{u_{\varepsilon}}\cdot\phi_{\varepsilon}\varphi$
(3.4)
$\displaystyle-\int_{\Omega}\widetilde{p_{\varepsilon}}(\nabla\cdot\varphi)+\int_{\Omega}\widetilde{p_{\varepsilon}}(\nabla\cdot\phi_{\varepsilon}\varphi)=\int_{\Omega}f\cdot(1-\phi_{\varepsilon})\varphi\,,$

for every $\varepsilon\in I_{*}$, along the subsequences (3.3). Observing
(2.3), notice that

$\left|\int_{\Omega}f\cdot(1-\phi_{\varepsilon})\varphi-\int_{\Omega}f\cdot\varphi\right|\leq\left|\int_{K_{\varepsilon}}f\cdot\varphi\right|+\dfrac{C}{\sigma_{\varepsilon}}\|f\|_{L^{2}(\Omega)}\|\varphi\|_{L^{\infty}(\Omega)}\to
0\quad\text{as}\ \ \varepsilon\to 0^{+}\,,$ (3.5)

because $f\cdot\varphi\in L^{1}(\Omega)$ and $|K_{\varepsilon}|\to 0$ as
$\varepsilon\to 0^{+}$. Also, with the help of the convergences in (3.3) (see
again (2.29)-(2.30)) we can easily prove that

$\displaystyle\lim_{\varepsilon\to
0^{+}}\int_{\Omega}\nabla\widetilde{u_{\varepsilon}}\cdot\nabla\varphi=\int_{\Omega}\nabla
u\cdot\nabla\varphi\,,\qquad\lim_{\varepsilon\to
0^{+}}\int_{\Omega}(\widetilde{u_{\varepsilon}}\cdot\nabla)\widetilde{u_{\varepsilon}}\cdot\varphi=\int_{\Omega}(u\cdot\nabla)u\cdot\varphi\,,$
(3.6) $\displaystyle\lim_{\varepsilon\to
0^{+}}\int_{\Omega}\nabla\widetilde{u_{\varepsilon}}\cdot(\nabla\phi_{\varepsilon}\otimes\varphi+\phi_{\varepsilon}\nabla\varphi)=\lim_{\varepsilon\to
0^{+}}\int_{\Omega}(\widetilde{u_{\varepsilon}}\cdot\nabla)\widetilde{u_{\varepsilon}}\cdot\phi_{\varepsilon}\varphi=0\,,$
$\displaystyle\lim_{\varepsilon\to
0^{+}}\int_{\Omega}\widetilde{p_{\varepsilon}}(\nabla\cdot\varphi)=\int_{\Omega}p(\nabla\cdot\varphi)\,,\qquad\lim_{\varepsilon\to
0^{+}}\int_{\Omega}\widetilde{p_{\varepsilon}}\left[\phi_{\varepsilon}(\nabla\cdot\varphi)+\nabla\phi_{\varepsilon}\cdot\varphi\right]=0\,.$

In view of (3.5)-(3.6), we then take the limit as $\varepsilon\to 0^{+}$ in
(3.4) to deduce

$\eta\int_{\Omega}\nabla
u\cdot\nabla\varphi+\int_{\Omega}(u\cdot\nabla)u\cdot\varphi-\int_{\Omega}p(\nabla\cdot\varphi)=\int_{\Omega}f\cdot\varphi\qquad\forall\varphi\in\mathcal{C}_{0}^{\infty}(\Omega;\mathbb{R}^{2})\,,$

so that, by density, $(u,p)\in H_{\sigma}^{1}(\Omega)\times L_{0}^{2}(\Omega)$
is a weak solution of (1.6) in $\Omega$ according to Definition 2.1. Since
$\Omega$ has a boundary of class $\mathcal{C}^{2}$, $f\in L^{2}(\Omega)$ and
$v_{*}\in H^{3/2}(\partial\Omega)$, the usual regularity results for the
steady-state Navier-Stokes equations with non-homogeneous Dirichlet boundary
conditions (see, for example, [14, Theorem IX.5.2]) then ensure that $(u,p)\in
H^{2}(\Omega)\times H^{1}(\Omega)$ satisfies in strong form the system (3.1).

In order to show the strong convergence in $H^{1}(\Omega)\times
L^{2}(\Omega)$, in view of the weak convergences in (3.3), it clearly suffices
to prove that

$\lim_{\varepsilon\to
0^{+}}\|\nabla\widetilde{u_{\varepsilon}}\|_{L^{2}(\Omega)}=\|\nabla
u\|_{L^{2}(\Omega)}\qquad\text{and}\qquad\lim_{\varepsilon\to
0^{+}}\|\widetilde{p_{\varepsilon}}\|_{L^{2}(\Omega)}=\|p\|_{L^{2}(\Omega)}\,.$
(3.7)

Let $\widehat{J}\in H^{1}(\Omega)$ be the vector field arising from Lemma 2.2.
We take $u_{\varepsilon}-(1-\phi_{\varepsilon})\widehat{J}\in
H^{1}_{0}(\Omega_{\varepsilon})$ as a test function in (3.2) to get

$\displaystyle\eta\|\nabla\widetilde{u_{\varepsilon}}\|^{2}_{L^{2}(\Omega)}-\eta\int_{\Omega}\nabla\widetilde{u_{\varepsilon}}\cdot\nabla\widehat{J}+\eta\int_{\Omega}\nabla\widetilde{u_{\varepsilon}}\cdot\nabla(\phi_{\varepsilon}\widehat{J})+\int_{\Omega_{\varepsilon}}(u_{\varepsilon}\cdot\nabla)u_{\varepsilon}\cdot
u_{\varepsilon}-\int_{\Omega}(\widetilde{u_{\varepsilon}}\cdot\nabla)\widetilde{u_{\varepsilon}}\cdot\widehat{J}$
(3.8)
$\displaystyle+\int_{\Omega}(\widetilde{u_{\varepsilon}}\cdot\nabla)\widetilde{u_{\varepsilon}}\cdot\phi_{\varepsilon}\widehat{J}-\int_{\Omega}\widetilde{p_{\varepsilon}}(\nabla\cdot\phi_{\varepsilon}\widehat{J})=\int_{\Omega}f\cdot\widetilde{u_{\varepsilon}}-\int_{\Omega}f\cdot(1-\phi_{\varepsilon})\widehat{J}\,,$

for every $\varepsilon\in I_{*}$, along the subsequences (3.3). A standard
integration by parts yields

$\int_{\Omega_{\varepsilon}}(u_{\varepsilon}\cdot\nabla)u_{\varepsilon}\cdot
u_{\varepsilon}=\dfrac{1}{2}\int_{\Omega}|v_{*}|^{2}(v_{*}\cdot\nu)\qquad\forall\varepsilon\in
I_{*}\,.$ (3.9)

Upon substitution of (3.9) into (3.8), and arguing as in (3.6)-(3.5), we can
take the limit as $\varepsilon\to 0^{+}$ in (3.8) to infer the identity

$\eta\lim_{\varepsilon\to
0^{+}}\|\nabla\widetilde{u_{\varepsilon}}\|^{2}_{L^{2}(\Omega)}=\eta\int_{\Omega}\nabla
u\cdot\nabla\widehat{J}-\dfrac{1}{2}\int_{\Omega}|v_{*}|^{2}(v_{*}\cdot\nu)+\int_{\Omega}(u\cdot\nabla)u\cdot\widehat{J}+\int_{\Omega}f\cdot(u-\widehat{J})\,.$
(3.10)

On the other hand, multiplying the first identity in (3.1)1 by
$u-\widehat{J}\in H^{1}_{0}(\Omega)$ and integrating by parts in $\Omega$
(arguing again as in (3.9)), entails

$\eta\|\nabla u\|^{2}_{L^{2}(\Omega)}=\eta\int_{\Omega}\nabla
u\cdot\nabla\widehat{J}-\dfrac{1}{2}\int_{\Omega}|v_{*}|^{2}(v_{*}\cdot\nu)+\int_{\Omega}(u\cdot\nabla)u\cdot\widehat{J}+\int_{\Omega}f\cdot(u-\widehat{J})\,,$
(3.11)

so that (3.10)-(3.11) secure the first equality in (3.7), that is,
$\widetilde{u_{\varepsilon}}\to u$ strongly in $H^{1}(\Omega)$ as
$\varepsilon\to 0^{+}$. Now, since $p_{\varepsilon}\in
L^{2}_{0}(\Omega_{\varepsilon})$, let $P_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon})$ be a vector field verifying (2.19). Observing
that $\widetilde{P_{\varepsilon}}\in H_{0}^{1}(\Omega)$ and that
$\|\nabla\widetilde{P_{\varepsilon}}\|_{L^{2}(\Omega)}=\|\nabla
P_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$, (2.14) then ensures that the
sequence $(\widetilde{P_{\varepsilon}})_{\varepsilon\in I_{*}}\subset
H_{0}^{1}(\Omega)$ is uniformly bounded, and so we deduce the existence of
$P\in H_{0}^{1}(\Omega)$ such that

$\widetilde{P_{\varepsilon}}\rightharpoonup P\ \ \ \text{weakly in}\
H^{1}(\Omega)\qquad\text{and}\qquad\widetilde{P_{\varepsilon}}\to P\ \ \
\text{strongly in}\ L^{4}(\Omega)\quad\text{as}\ \ \varepsilon\to 0^{+}\,,$
(3.12)

along a (not relabeled) subsequence. Given any scalar function
$\phi\in\mathcal{C}_{0}^{\infty}(\Omega)$, since $\nabla\cdot
P_{\varepsilon}=p_{\varepsilon}$ in $\Omega_{\varepsilon}$ and
$P_{\varepsilon}$ vanishes on $\partial\Omega_{\varepsilon}$ for any
$\varepsilon\in I_{*}$, an integration by parts gives us

$\int_{\Omega}\widetilde{P_{\varepsilon}}\cdot\nabla\phi=\int_{\Omega_{\varepsilon}}P_{\varepsilon}\cdot\nabla\phi=-\int_{\Omega_{\varepsilon}}\phi\,p_{\varepsilon}=-\int_{\Omega}\phi\,\widetilde{p_{\varepsilon}}\qquad\forall\varepsilon\in
I_{*}\,,$

along the subsequences (3.3)-(3.12). Taking the limit in this last identity as
$\varepsilon\to 0^{+}$ entails

$\int_{\Omega}P\cdot\nabla\phi=-\int_{\Omega}\phi\,p\qquad\forall\phi\in\mathcal{C}_{0}^{\infty}(\Omega;\mathbb{R})\,,$

that is, $\nabla\cdot P=p$ in $\Omega$. As in the proof of Theorem 2.1 we
derive the identity

$\|p_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}=\eta\int_{\Omega}\nabla\widetilde{u_{\varepsilon}}\cdot\nabla\widetilde{P_{\varepsilon}}+\int_{\Omega}(\widetilde{u_{\varepsilon}}\cdot\nabla)\widetilde{u_{\varepsilon}}\cdot\widetilde{P_{\varepsilon}}-\int_{\Omega}f\cdot\widetilde{P_{\varepsilon}}\qquad\forall\varepsilon\in
I_{*}\,,$ (3.13)

along the subsequences (3.3)-(3.12). Similarly, multiply the first identity in
(3.1)1 by $P$ and integrate by parts in $\Omega$ to provide

$\|p\|^{2}_{L^{2}(\Omega)}=\eta\int_{\Omega}\nabla u\cdot\nabla
P+\int_{\Omega}(u\cdot\nabla)u\cdot P-\int_{\Omega}f\cdot P\,.$

Recalling that $\widetilde{u_{\varepsilon}}\to u$ strongly in $H^{1}(\Omega)$
as $\varepsilon\to 0^{+}$, we can take the limit in (3.13) as $\varepsilon\to
0^{+}$ to reach the second equality in (3.7). This concludes the proof. $\Box$

Concerning the symmetric case, we have the following result, analogous to
Theorem 3.1, whose proof is omitted (for the sake of brevity, since it is very
similar to the proof of Theorem 3.1 with obvious minor modifications):

######  Theorem 3.2.

Let $(\Omega_{\varepsilon})_{\varepsilon\in I_{*}}$ be the family of symmetric
perforated domains (1.8). For any given symmetric vector fields $f\in
H^{1}(\Omega)$ and $v_{*}\in H^{3/2}(\partial\Omega)$ satisfying (1.5), let
$(u_{\varepsilon},p_{\varepsilon})\in
H_{\sigma}^{1}(\Omega_{\varepsilon})\times L_{0}^{2}(\Omega_{\varepsilon})$ be
a symmetric weak solution of (1.6). Then, up to the extraction of a
subsequence, the sequence of extended functions
$\\{(\widetilde{u_{\varepsilon}},\widetilde{p_{\varepsilon}})\\}_{\varepsilon\in
I_{*}}\subset H^{1}(\Omega)\times L_{0}^{2}(\Omega)$ converges strongly to a
symmetric weak solution $(u,p)\in H^{1}(\Omega)\times L_{0}^{2}(\Omega)$ of
problem (1.6) in $\Omega$ as $\varepsilon\to 0^{+}$. Furthermore, $(u,p)\in
H^{2}(\Omega)\times H^{1}(\Omega)$ and it solves in strong form the system

$\left\\{\begin{aligned} &-\eta\Delta u+(u\cdot\nabla)u+\nabla
p=f\,,\quad\nabla\cdot u=0\ \ \mbox{ in }\ \ \Omega\,,\\\\[4.0pt] &u=v_{*}\ \
\mbox{ on }\ \ \partial\Omega\,.\end{aligned}\right.$

##  4 Remarks on the fully general two-dimensional case

Let $\varepsilon\in I_{*}$ and $\Omega_{\varepsilon}$ be as in (1.3), where we
additionally suppose that the sequence of sets $(K_{n})_{n\in\mathbb{N}}$ is
uniformly Lipschitz ($\bigstar$). Given any $f\in H^{1}(\Omega)$ and $v_{*}\in
H^{3/2}(\partial\Omega)$ satisfying (1.5), [20, Theorem 1.1] (see also [20,
Remark 1.1]) ensures the existence of at least one weak solution
$u_{\varepsilon}\in H^{2}(\Omega_{\varepsilon})\cap
H^{1}_{\sigma}(\Omega_{\varepsilon})$ of problem (1.6) and an associated
pressure $p_{\varepsilon}\in H^{1}(\Omega_{\varepsilon})\cap
L^{2}_{0}(\Omega_{\varepsilon})$ such that the pair
$(u_{\varepsilon},p_{\varepsilon})$ solves (1.6) in strong form. In order to
prove this result (which does not impose any additional assumption on the data
of the problem or on the size of the boundary fluxes), the Authors of [20] use
the Leray-Schauder Principle, where the required uniform bounds (with respect
to some parameter $\lambda\in[0,1]$) are reached through a contradiction
argument that employs Bernoulli’s law for weak solutions of the stationary
Euler equations and a generalization of the Morse-Sard Theorem for Sobolev
functions. An essential ingredient in the proof of [20, Theorem 1.1] is the
uniform boundedness (with respect to $\lambda\in[0,1]$) of the
$W^{1,3/2}(\Omega_{\varepsilon})$-norm of the scalar pressure, see [20, Lemma
3.1]. This would indicate that, in order to obtain $\varepsilon$-uniform
bounds for the solutions of (1.6) following the method given in [20, Theorem
1.1], one would need to build a $W^{1,3/2}(\Omega)$-uniform extension for the
pressure $p_{\varepsilon}\in H^{1}(\Omega_{\varepsilon})\cap
L^{2}_{0}(\Omega_{\varepsilon})$ inside the holes $K_{\varepsilon}$. Inspired
by [16, Lemma 3.1] and [29, Lemma 1.7], the purpose of this section is
precisely to illustrate the fact that such a uniform extension cannot be
achieved in a simple way. A first explanation derives from the observation
that, even though $p_{\varepsilon}\in H^{1}(\Omega_{\varepsilon})$, the system
(1.6) does not provide any information concerning the behavior of
$p_{\varepsilon}$ on the boundary of the holes $\partial K_{\varepsilon}$. A
second, more convincing, explanation involves a microscopic analysis of the
boundary-value problem (1.6) near each single hole $K^{\varepsilon}_{n}$, with
$n\in\\{1,...,N(\varepsilon)\\}$. We will show that

$\|\nabla
p_{\varepsilon}\|_{L^{3/2}(\Omega_{\varepsilon})}\leq\dfrac{C_{*}}{a_{\varepsilon}}\left(1+\|f\|^{2}_{L^{2}(\Omega)}+\|\nabla
u_{\varepsilon}\|^{4}_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}+\|v_{*}\|^{4}_{H^{3/2}(\partial\Omega)}\right)\qquad\forall\varepsilon\in
I_{*}\,,$ (4.1)

for some constant $C_{*}>0$ independent of $\varepsilon\in I_{*}$. To do so,
after translation and rescaling, (1.2)1 implies

$\overline{K_{n}}\subset D((0,0),\delta_{0})\subset D((0,0),\delta_{1})\subset
D\left((0,0),\dfrac{\delta_{1}\varepsilon}{a_{\varepsilon}}\right)\qquad\forall
n\in\\{1,...,N(\varepsilon)\\}\,,$

since $a_{\varepsilon}\ll\varepsilon$ for every $\varepsilon\in I_{*}$. Notice
then that

$z\in
D\left(\xi^{\varepsilon}_{n},\delta_{0}a_{\varepsilon}\right)\setminus\overline{K_{n}^{\varepsilon}}\quad\Longleftrightarrow\quad\dfrac{z-\xi^{\varepsilon}_{n}}{a_{\varepsilon}}\in
D((0,0),\delta_{0})\setminus\overline{K_{n}}\qquad\forall
n\in\\{1,...,N(\varepsilon)\\}\,.$ (4.2)

Fix any $n\in\\{1,...,N(\varepsilon)\\}$. In what follows, $C>0$ will always
denote a generic constant that is independent of $\varepsilon\in I_{*}$ and
$n\in\\{1,...,N(\varepsilon)\\}$, but that may change from line to line. Set
$D^{n}_{j}\doteq D((0,0),\delta_{j})\setminus\overline{K_{n}}$, for
$j\in\\{0,1\\}$, and define $(U_{\varepsilon},P_{\varepsilon},F)\in
H^{2}(D^{n}_{1})\times H^{1}(D^{n}_{1})\times H^{1}(D^{n}_{1})$ by

$U_{\varepsilon}(z)\doteq
u_{\varepsilon}(a_{\varepsilon}z+\xi^{\varepsilon}_{n})\,,\quad
P_{\varepsilon}(z)\doteq
p_{\varepsilon}(a_{\varepsilon}z+\xi^{\varepsilon}_{n})\,,\quad F(z)\doteq
f(a_{\varepsilon}z+\xi^{\varepsilon}_{n})\qquad\forall z\in D^{n}_{1}\,,$

so that these functions satisfy the following Stokes-type system in
$D^{n}_{1}$:

$\left\\{\begin{aligned} &-\eta\Delta U_{\varepsilon}+a_{\varepsilon}\nabla
P_{\varepsilon}=a^{2}_{\varepsilon}F-a_{\varepsilon}(U_{\varepsilon}\cdot\nabla)U_{\varepsilon}\,,\quad\nabla\cdot
U_{\varepsilon}=0\ \ \mbox{ in }\ \ D^{n}_{1}\,,\\\\[4.0pt]
&U_{\varepsilon}=0\ \ \mbox{ on }\ \ \partial K_{n}\,.\end{aligned}\right.$

The usual local regularity estimates for the Stokes equations (see [7,
Teorema, page 311] or [14, Theorem IV.4.1]) entail

$\|\nabla P_{\varepsilon}\|^{3/2}_{L^{3/2}(D^{n}_{0})}\leq
C\left(\|a^{2}_{\varepsilon}F-a_{\varepsilon}(U_{\varepsilon}\cdot\nabla)U_{\varepsilon}\|^{3/2}_{L^{3/2}(D^{n}_{1})}+\|\nabla
U_{\varepsilon}\|^{3/2}_{L^{3/2}(D^{n}_{1})}+\|P_{\varepsilon}\|^{3/2}_{L^{3/2}(D^{n}_{1})}\right)\,,$
(4.3)

where we emphasize that, as a consequence of property ($\bigstar$), the
constant $C>0$ entering (4.3) can be bounded independently of
$n\in\\{1,...,N(\varepsilon)\\}$. In view of the same property, the Young,
Hölder and Sobolev inequalities in $D^{n}_{1}$ provide

$\displaystyle\|a^{2}_{\varepsilon}F-a_{\varepsilon}(U_{\varepsilon}\cdot\nabla)U_{\varepsilon}\|^{3/2}_{L^{3/2}(D^{n}_{1})}$
$\displaystyle\leq
C\left(a^{3}_{\varepsilon}\|F\|^{3/2}_{L^{3/2}(D^{n}_{1})}+a^{3/2}_{\varepsilon}\|(U_{\varepsilon}\cdot\nabla)U_{\varepsilon}\|^{3/2}_{L^{3/2}(D^{n}_{1})}\right)$
(4.4) $\displaystyle\leq
C\left(a^{3}_{\varepsilon}\|F\|^{3/2}_{L^{3/2}(D^{n}_{1})}+a^{3/2}_{\varepsilon}\|\nabla
U_{\varepsilon}\|^{3/2}_{L^{2}(D^{n}_{1})}\|U_{\varepsilon}\|^{3/2}_{L^{6}(D^{n}_{1})}\right)$
$\displaystyle\leq
C\left(a^{3}_{\varepsilon}\|F\|^{3/2}_{L^{3/2}(D^{n}_{1})}+a^{3/2}_{\varepsilon}\|\nabla
U_{\varepsilon}\|^{3}_{L^{2}(D^{n}_{1})}\right)\,.$

Since $a_{\varepsilon}\ll\varepsilon$ for every $\varepsilon\in I_{*}$, we
clearly have

$D^{n}_{1}\subset\widetilde{D^{n}_{1}}\doteq
D\left((0,0),\dfrac{\delta_{1}\varepsilon}{a_{\varepsilon}}\right)\setminus\overline{K_{n}}\,,$

so that successive applications of the change of variables (4.2) ensure that

$\displaystyle\|F\|^{3/2}_{L^{3/2}(D^{n}_{1})}\leq\dfrac{1}{a^{2}_{\varepsilon}}\|f\|^{3/2}_{L^{3/2}(D\left(\xi^{\varepsilon}_{n},\delta_{1}\varepsilon\right)\setminus\overline{K_{n}^{\varepsilon}})}\,,\qquad\|\nabla
U_{\varepsilon}\|_{L^{2}(D^{n}_{1})}\leq\|\nabla
u_{\varepsilon}\|_{L^{2}(D\left(\xi^{\varepsilon}_{n},\delta_{1}\varepsilon\right)\setminus\overline{K_{n}^{\varepsilon}})}\,,$
(4.5) $\displaystyle\|\nabla
U_{\varepsilon}\|^{3/2}_{L^{3/2}(D^{n}_{1})}\leq\dfrac{1}{\sqrt{a_{\varepsilon}}}\|\nabla
u_{\varepsilon}\|^{3/2}_{L^{3/2}(D\left(\xi^{\varepsilon}_{n},\delta_{1}\varepsilon\right)\setminus\overline{K_{n}^{\varepsilon}})}\,,\qquad\|P_{\varepsilon}\|^{3/2}_{L^{3/2}(D^{n}_{1})}\leq\dfrac{1}{a^{2}_{\varepsilon}}\|p_{\varepsilon}\|^{3/2}_{L^{3/2}(D\left(\xi^{\varepsilon}_{n},\delta_{1}\varepsilon\right)\setminus\overline{K_{n}^{\varepsilon}})}\,.$

Inserting (4.4)-(4.5) into (4.3) gives us

$\displaystyle\|\nabla
p_{\varepsilon}\|^{3/2}_{L^{3/2}(D\left(\xi^{\varepsilon}_{n},\delta_{0}a_{\varepsilon}\right)\setminus\overline{K_{n}^{\varepsilon}})}$
$\displaystyle\leq
C\Bigg{(}a^{3/2}_{\varepsilon}\|f\|^{3/2}_{L^{3/2}(D\left(\xi^{\varepsilon}_{n},\delta_{1}\varepsilon\right)\setminus\overline{K_{n}^{\varepsilon}})}+a^{2}_{\varepsilon}\|\nabla
u_{\varepsilon}\|^{3}_{L^{2}(D\left(\xi^{\varepsilon}_{n},\delta_{1}\varepsilon\right)\setminus\overline{K_{n}^{\varepsilon}})}$
(4.6) $\displaystyle\hskip 34.1433pt+\|\nabla
u_{\varepsilon}\|^{3/2}_{L^{3/2}(D\left(\xi^{\varepsilon}_{n},\delta_{1}\varepsilon\right)\setminus\overline{K_{n}^{\varepsilon}})}+\dfrac{1}{a^{3/2}_{\varepsilon}}\|p_{\varepsilon}\|^{3/2}_{L^{3/2}(D\left(\xi^{\varepsilon}_{n},\delta_{1}\varepsilon\right)\setminus\overline{K_{n}^{\varepsilon}})}\Bigg{)}\,.$

Following the proof of [25, Theorem 2.1], we decompose the perforated domain
as

$\Omega_{\varepsilon}=\left[\Omega_{\varepsilon}\setminus\left(\bigcup_{n=1}^{N(\varepsilon)}D\left(\xi^{\varepsilon}_{n},\delta_{0}a_{\varepsilon}\right)\right)\right]\cup\bigcup_{n=1}^{N(\varepsilon)}\left(D\left(\xi^{\varepsilon}_{n},\delta_{0}a_{\varepsilon}\right)\setminus\overline{K_{n}^{\varepsilon}}\right)\doteq\widetilde{\Omega_{\varepsilon}}\cup\bigcup_{n=1}^{N(\varepsilon)}\left(D\left(\xi^{\varepsilon}_{n},\delta_{0}a_{\varepsilon}\right)\setminus\overline{K_{n}^{\varepsilon}}\right)\,.$

On one hand, since the holes are mutually disjoint (see (1.2)3), from
(1.4)-(4.6) we directly obtain

$\displaystyle\|\nabla
p_{\varepsilon}\|^{3/2}_{L^{3/2}(\bigcup_{n=1}^{N(\varepsilon)}\left(D\left(\xi^{\varepsilon}_{n},\delta_{0}a_{\varepsilon}\right)\setminus\overline{K_{n}^{\varepsilon}}\right))}=\sum_{n=1}^{N(\varepsilon)}\|\nabla
p_{\varepsilon}\|^{3/2}_{L^{3/2}(D\left(\xi^{\varepsilon}_{n},\delta_{0}a_{\varepsilon}\right)\setminus\overline{K_{n}^{\varepsilon}})}$
(4.7) $\displaystyle\leq
C\Bigg{(}a^{3/2}_{\varepsilon}\|f\|^{3/2}_{L^{3/2}(\Omega)}+\dfrac{a^{2}_{\varepsilon}}{\varepsilon^{2}}\|\nabla
u_{\varepsilon}\|^{3}_{L^{2}(\Omega_{\varepsilon})}+\|\nabla
u_{\varepsilon}\|^{3/2}_{L^{3/2}(\Omega_{\varepsilon})}+\dfrac{1}{a^{3/2}_{\varepsilon}}\|p_{\varepsilon}\|^{3/2}_{L^{3/2}(\Omega_{\varepsilon})}\Bigg{)}$
$\displaystyle\leq\dfrac{C}{a^{3/2}_{\varepsilon}}\left(1+\|f\|^{2}_{L^{2}(\Omega)}+\|\nabla
u_{\varepsilon}\|^{4}_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}\right)^{3/2}\qquad\forall\varepsilon\in
I_{*}\,.$

On the other hand, since the points of the region
$\widetilde{\Omega_{\varepsilon}}$ are sufficiently far away from the holes
$K_{\varepsilon}$, we can argue as in [15, Theorem 3.3] to obtain the bound

$\|\nabla
p_{\varepsilon}\|_{L^{3/2}(\widetilde{\Omega_{\varepsilon}})}\leq\dfrac{C}{a_{\varepsilon}}\left(1+\|f\|^{2}_{L^{2}(\Omega)}+\|v_{*}\|^{4}_{H^{3/2}(\partial\Omega)}\right)\qquad\forall\varepsilon\in
I_{*}\,.$ (4.8)

Adding (4.7)-(4.8) gives us (4.1). It is therefore left open the possibility
of recovering the results of Theorems 2.1-3.1 without the smallness assumption
(2.13).

Acknowledgements. The Authors declare that there is no conflict of interest.
Data sharing not applicable to this article as no datasets were generated or
analyzed during the current study.

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Clara Patriarca

Département de Mathématique

Université Libre de Bruxelles

Boulevard du Triomphe 155

1050 Brussels - Belgium

E-mail<EMAIL_ADDRESS>
Gianmarco Sperone

Dipartimento di Matematica

Politecnico di Milano

Piazza Leonardo da Vinci 32

20133 Milan - Italy

E-mail<EMAIL_ADDRESS>