# Time-periodic weak solutions for an incompressible Newtonian fluid
interacting with an elastic plate.

Claudiu Mîndrilă  Department of Analysis, Faculty of Mathematics and Physics,
Charles University, Sokolovská 83, 18675, Prague, Czech Republic
<EMAIL_ADDRESS>and  Sebastian Schwarzacher  Department of
Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská
83, 18675, Prague, Czech Republic<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

Under the action of a time-periodic external forces we prove the existence of
at least one time-periodic weak solution for the interaction between a three-
dimensional incompressible fluid, governed by the Navier- Stokes equation and
a two dimensional elastic plate. The challenge is that the Eulerian domain for
the fluid changes in time and is a part of the solution. We introduce a two
fixed-point methodology: First we construct a time-periodic solutions for a
given variable time-periodic geometry. Then in a second step a (set-valued)
fixed point is performed w.r.t. the geometry of the domain. The existence
relies on newly developed a-priori estimates applicable for both coupled and
uncoupled variable geometries. Due to the expected weak regularity of the
solutions such Eulerian estimates are unavoidable. Note in particular, that
only the fluid is assumed to be dissipative. But the here produced a-priori
estimates show that its possible to exploit the dissipative effects of the
fluid also for the solid deformation. The existence of periodic solutions for
a given geometry is valid for arbitrary large data. The existence of periodic
coupled solutions to the fluid-structure interaction is valid for all data
that excludes a self-intersection a-priori.

Keywords: Navier-Stokes equations, Periodic solutions, Fluid-structure
interaction, Elastic plates

##  1\. Introduction

Periodic solution naturally appear in the context of fluid-structure
interactions. Applications are from various disciplines. Examples are from
engineering (turbines, steam engines, oscillating bridge-decks) or biology
(blood-flow). The latter one is the model example for this article: We
consider an elastic wall that interacts with an incompresible fluid modeled by
the unsteady Navier-Stokes equations in 3D. We seek for time-periodic weak
solutions. While many results for elastic plates or shells interacting with
fluids have been shown in the last decades, the relevant regime of time-
periodic solutions has rarely been touched. For an overview on applications
and on the state of the art for such fluid-structure interactions we refere to
the recent overview given in [6] and the references therein.

This article aims to fill this gap by providing a methodology that allows for
the construction of time-periodic weak solutions. Our main result (Theorem
1.22) is the existence of at least one time-periodic weak solution for given
periodic data. The constraints on the data are given only to exclude self-
intersections of the elastic structure a-priori. Observe that in 3 space
dimensions and for large data the frame of weak solutions is unavoidable (due
to the open problem of regularity for Navier-Stokes equations). However in
this setting merely energy estimates do not suffice to construct periodic
solutions. Hence the key step are new regularity estimates beyond the energy
estimates. In explicit, non-trivial estimates of the displacement variable of
the elastic wall, which is coupled to the fluid velocity via its time
derivative only.

###  1.1. State of the art

The study of the interaction between fluids and solids from a mathematical
point of view has received a lot of interest in the last years. Several
difficulties occur when studying these interactions. This includes among
others the changes of the Eulerian domain in time, the choice of appropriate
coupling (boundary) conditions or the respective concepts of weak solutions.
The interaction with elastic solids, in particular, turned out to be
particularly difficult due to the mixed parabolic-hyperbolic nature of the
system, see also [2, 29] and the references therein. In particular due to the
variable-in-time Eulerian geometry the matter of periodic solutions in that
frame-work seems rather untouched.

In contrast, the literature for cyindrical time-space domains the existence of
periodic solutions for incompressible Navier-Stokes systems is vast and we
refer to [16] where several results are surveyed. A first existence result was
shown in [28] . Here, the periodicity is obtained at the discrete (Galerkin)
level, by solving the corresponding ODE with a time-periodic condition; the
latter objective is achieved by noticing that such a solution appears as a
fixed point of the function that maps initial values to the endpoint values of
the corresponding solutions. This strategy has proved to be quite robust and
actually it was recently employed in [1] in order to prove weak time-periodic
solutions for a class of non-Newtonian fluids. The so-far available literature
on fluid-structure interactions seems to be restricted to rigid body motions
moving periodically inside a Navier-Stokes fluid. See the seminal works by
Galdi at all [17, 13, 14, 15] and the references therein. See also [3].

As indicated before we are concerned with _the interaction of a viscous and
incompressible fluid interacting with an elastic solid situated and being part
of the boundary of the fluid domain._ The first existence result for weak
solutions to the Cauchy problem of this subclass appeared in [9, 20]; there,
the existence was proved for the case of a 3D/2D coupling of a viscous and
incompressible fluid interacting with an _elastic plate_ and resp. with an
_elastic shell_. The motion of the fluid and shell is assumed to happen in the
normal direction and the initial geometry has a _flat_ moving part of the
boundary, where the shell is attached. The next steps towards non-flat moving
boundary and a linearized Koiter-type shell was considered in [23], while the
more recent work [27] deals with non-linear models of Koiter shells in the
spirit of [10]. While this paper follows the strategy of fixed point methods
there are also more constructibe approaches based on the A.L.E. method, see
[25]. Finally let us mention recent progress in compressible and/or heat
conducting fluids and shells [4, 30, 5].

There are also many results tackling the existence/uniqueness/regularity of
_strong solutions_. Among them let us refer to a recent work discussing the
existence of strong time-periodic solutions within a 2D/1D coupling with a
visco-elastic beam [7]. There an abstract semi-group theory is used that is
suitable for strong solutions that seems not to be suitable for the framework
of weak solutions. We also point out the recent work [26] where a maximal-in-
time well possedness result of strong solutions for an incompressible fluid
interacting with a damped shell situated on the boundary is proved. Let us
also mention the recent and excellent survey [6] where besides a vast
surveryed literature also several current open problems of _fluid-structure
interation_ were formulated.

###  1.2. Moving domains

Throughout this paper we consider a non-empty, open and bounded domain
$\Omega\subset\mathbb{R}^{3}$ of class $C^{4}$ whose boundary
$\partial\Omega=M\cup\Gamma$ consists of two parts: $\Gamma$ -the fixed part,
and $M$-the compact complement of $\Gamma$ which represents the moving part.
Since we consider the plate case here, we assume that $M$ is flat

$M=\omega\times\left\\{0\right\\}$

where $\omega\subset\mathbb{R}^{2}$ is an open and bounded Lipschitz domain.
Moreover we assume that the outer normal of $\Omega$ at $M$ is
$\mathbf{e}_{3}$. On the moving part we attach an elastic shell whose
displacement is denoted by $\eta:\omega\mapsto[a,b]$ and $\eta=0$ on
$\partial\omega$ where $a,b\in\mathbb{R}$. During this work we shall assume
that the elastic deformation is parametrised as a graph and that it will
change in $\mathbf{e}_{3}$ direction only. We define
$x^{\prime}=(x_{1},x_{2})$ and $z=x_{3}$.

$\phi_{\eta}(x^{\prime},0)=(x^{\prime},\eta(x^{\prime}))\text{ for
}(x^{\prime},0)\in M\text{ and }\phi_{\eta}(x^{\prime},z)=(x^{\prime},z)\text{
on }\Gamma,$

and

(1.1)
$\Omega_{\eta}:=(\Omega\cap\left\\{(x^{\prime},z):0>z>\eta\left(x^{\prime}\right)\right\\})\cup\left\\{(x^{\prime},z):0\leq
z<\eta\left(x^{\prime}\right)\right\\}.$

this implies by choosing $a,b$ appropriately that
$\phi_{\eta}:\partial\Omega\to\partial\Omega_{\eta}$ is a homeomorphism.

Assuming that $\eta\in C^{k}_{0}(\omega)$ it is possible to extend
$\phi_{\eta}:\Omega\to\Omega_{\eta}$, such that it is a $C^{k}$-diffeomorphism
for $k\in\\{0,1,2,3,4\\}$.

To simplify the notation, we assume in the following:

(1.2) $\displaystyle\text{If }{\left\|{\eta}\right\|}_{\infty}\leq\kappa\text{
and }\eta\in C^{k}_{0}(\omega)\text{, then }\phi_{\eta}\text{ is a $C^{k}$
diffeomorphism}.$

By denoting

(1.3) $I:=(0,T)\ \text{and}\ I\times\Omega_{\eta}:=\bigcup_{t\in
I}\left\\{t\right\\}\times\Omega_{\eta\left(t\right)}$

we can now introduce the Lebesgue and Sobolev spaces adapted to the moving
domains $\Omega_{\eta}$. Namely, for $1\leq p,\ q\leq\infty$ we have

$\displaystyle L^{p}\left(I;L^{q}\left(\Omega_{\eta}\right)\right):=$
$\displaystyle\left\\{\mathbf{v}\in
L^{1}\left(I\times\Omega_{\eta}\right):\mathbf{v}\left(t,\cdot\right)\in
L^{q}\left(\Omega_{\eta\left(t\right)}\right)\ \text{for a .e.}\ t\in I,\
\left\|\mathbf{v}\left(t,\cdot\right)\right\|_{L^{q}\left(\Omega_{\eta\left(t\right)}\right)}\in
L^{p}\left(I\right)\right\\}$ $\displaystyle
L^{p}\left(I;W^{1,q}\left(\Omega_{\eta}\right)\right):=$
$\displaystyle\left\\{\mathbf{v}\in
L^{1}\left(I\times\Omega_{\eta}\right):\mathbf{v}\left(t,\cdot\right)\in
W^{1,q}\left(\Omega_{\eta\left(t\right)}\right)\ \text{for a .e.}\ t\in I,\
\left\|\mathbf{v}\left(t,\cdot\right)\right\|_{W^{1,q}\left(\Omega_{\eta\left(t\right)}\right)}\in
L^{p}\left(I\right)\right\\}.$

This spaces admit the usual _trace_ and _extension_ operators as long as
$a<\eta<b$ and as long as $\eta\in C^{0,1}$; see [9, Lemma 1] for further
details.

###  1.3. The Koiter elastic energy

With the previous notations we can now introduce the _change of metric tensor_

(1.4)
$\mathbb{G}_{ij}\left(\eta\right):=\partial_{i}\eta\cdot\partial_{j}\eta,\quad
i,j=1,2$

and, after introducing the normal vector $\nu_{\eta}$ to the deformed surface
via

(1.5) $\nu_{\eta}:=\partial_{1}\phi_{\eta}\times\partial_{2}\phi_{\eta}$

the _change of curvature tensor_ as

(1.6) $\mathbb{R}_{ij}^{\sharp}:=\partial_{ij}\eta,\quad i,j=1,2.$

All-in-all, if $h$ denotes the thickness of the shell, then the _Koiter
energy_ of the plate reads as follows:

(1.7)
$K\left(\eta\right):=K\left(\eta,\eta\right):=\frac{h}{2}\int_{\omega}\mathbb{A}:\mathbb{G}\left(\eta\right)\otimes\mathbb{G}\left(\eta\right)dx^{\prime}+\frac{h^{3}}{6}\int_{\omega}\mathbb{A}:\mathbb{R}^{\sharp}\left(\eta\right)\otimes\mathbb{R}^{\sharp}\left(\eta\right)dx^{\prime}.$

Here

(1.8)
$\mathbb{A}^{ijkl}=\frac{4\lambda_{s}u}{\lambda_{s}+2\mu_{s}}\mathbb{A}^{ij}\mathbb{A}^{kl}+4\mu_{s}\left(\mathbb{A}^{ik}\mathbb{A}^{jl}+\mathbb{A}^{il}\mathbb{A}^{jk}\right)$

is a fourth-order tensor known as _the elasticity (or stiffness) tensor_ , and
$\lambda_{s},\mu_{s}$ are the Lamé coefficients; see [10, p. 162]. A crucial
feature of the Koiter energy $K$ is the $H^{2}$ coercivity. More precisely,
there exists a constant $c_{0}=c(\lambda_{s},\mu_{s},h)$ such that

(1.9) $K\left(\eta\right)\geq
c_{0}\left\|\nabla^{2}\eta\right\|_{L^{2}\left(\omega\right)}^{2}\quad\forall\eta\in
H_{0}^{2}(\omega)$

###  1.4. The coupled system

We assume that the domain $\Omega_{\eta(t)}$ is filled at any time $t\in(0,T)$
an _incompressible_ fluid of viscosity $\mu$ and density $\rho_{f}$. Its
velocity flow denoted by

$\mathbf{u}:I\times\Omega_{\eta}\mapsto\mathbb{R}^{3}$

solves the system

(1.10)
$\begin{cases}\rho_{f}(\partial_{t}\mathbf{u}+\left(\mathbf{u}\cdot\nabla\right)\mathbf{u})=\text{div}\sigma\left(\mathbf{u},p\right)+f&\text{in}\
I\times\Omega_{\eta(t)}\\\ \text{div}\left(\mathbf{u}\right)=0&\text{in}\
I\times\Omega_{\eta(t)}\end{cases}$

where $\sigma(\mathbf{u},p)$ is the _Cauchy stress tensor_

(1.11) $\sigma(\mathbf{u},p)=-pI_{3}+2\mu D(\mathbf{u})$

where $D(\mathbf{u})$ denotes the symmetric gradient

(1.12) $D(\mathbf{u}):=\frac{\nabla\mathbf{u}+\nabla\mathbf{u}^{T}}{2}.$

The density of the external forces is given by

(1.13) $\mathbf{f}:\mathbb{R}\times\mathbb{R}^{3}\mapsto\mathbb{R}^{3},\
\mathbf{f}\left(t,\cdot\right)=\mathbf{f}\left(t+T,\cdot\right)\ \text{for
a.e.}\ t\in I$

Now, concerning the elastic shell, we denote its density by $\rho_{s}$, its
thickness by $h$ and its displacement w.r.t. the reference configuration by

(1.14) $\eta:I\times\omega\mapsto\mathbb{R}.$

Then $\eta$ solves the following hyperbolic system

(1.15)
$\begin{cases}h\rho_{s}\partial_{tt}\eta+K^{\prime}\left(\eta\right)=g+\mathbf{F}\cdot\mathbf{e}_{3}&\text{in}\
I\times\omega\\\ \eta=\nabla\eta=0&\text{on}\
I\times\partial\omega\end{cases}$

where $K^{\prime}(\eta)$ is the $L^{2}$-gradient of the Koiter energy $K$.
Here $\mathbf{F}$ denotes the force exerted by the fluid on the shell, that is

(1.16)
$\mathbf{F}:=-\sigma\left(\mathbf{u},p\right)\left(t,\phi_{\eta}\left(t,x^{\prime}\right)\right)\nu_{\eta}\cdot\mathbf{e}_{3},\
\left(x^{\prime}\right)\in\omega$

which is the stress tensor evaluated in normal direction. The distribution of
external forces acting on the shell is denoted by

(1.17) $g:\mathbb{R}\times\omega\mapsto\mathbb{R},\
g\left(t,\cdot\right)=g\left(t+T,\cdot\right)\ \text{for a.e.}\ t\in I$

Regarding the boundary conditions, we assume that the fluid and the shell move
with the same speed which means we assume the _no-slip boundary condition_
$\text{tr}_{\eta}\mathbf{u}=\partial_{t}\eta\mathbf{e}_{3}\ $, or, more
explicitly

(1.18)
$\mathbf{u}\left(t,x^{\prime},\eta\left(t,x^{\prime}\right)\right)=\left(0,0,\partial_{t}\eta\left(t,x^{\prime}\right)\right),\quad
x^{\prime}\in\omega\text{ and }\mathbf{u}=0\text{ in }\Gamma.$

the full system that will be studied reads as follows:

(1.19)
$\begin{cases}\rho_{f}(\partial_{t}\mathbf{u}+\left(\mathbf{u}\cdot\nabla\right)\mathbf{u})=\text{div}\sigma\left(\mathbf{u},p\right)+\mathbf{f}&\text{\
in}\ I\times\Omega_{\eta}\\\ \text{div}\mathbf{u}=0&\text{\ in}\
I\times\Omega_{\eta}\\\ \mathbf{u}=0&\ \text{on\
}I\times\left(\partial\Omega\setminus\omega\right)\\\
\mathbf{u}\circ\phi_{\eta}=\partial_{t}\eta\mathbf{e}_{3}&\ \text{in}\
I\times\omega\\\
\rho_{s}h\partial_{tt}\eta+K^{\prime}\left(\eta\right)=-\sigma\left(\mathbf{u},p\right)\left(t,\phi_{\eta\left(t\right)}\left(t,y\right)\right)\nu_{\eta}\cdot\mathbf{e}_{3}+g&\
\text{in}\ I\times\omega\\\ \eta=\nabla\eta=0&\text{\ on\
}I\times\partial\omega\\\ \eta(0,\cdot)=\eta(T,\cdot),\
\partial_{t}\eta(0,\cdot)=\partial_{t}\eta(T,\cdot)&\text{\ in}\ \omega\\\
\mathbf{u}(0,\cdot)=\mathbf{u}(T,\cdot)&\ \text{in}\
\Omega_{\eta\left(0\right)}.\end{cases}$

In order to reduce the notation we assume (without loss of generality) that
$\rho_{f}=\rho_{s}h=\mu=1$ from now on .Throughout this work, we shall denote
the surface measure of $\omega$ by $dx^{\prime}$.

######  Remark 1.20 (On the restrictions of the reference geometry).

Note that since $\mathbf{e}_{3}=(0,0,1)$ , from (1.18) and from the
incompresiblity condition $\text{div}\mathbf{u}=0$ we find that the volume of
the whole cavity $\int_{\omega}\eta(t,x)dx^{\prime}$ has to remain constant.
That is

(1.21) $\int_{\omega}\eta\left(t,\cdot\right)dx^{\prime}=:m\quad\forall t\in
I.$

In the time-periodic setting this mean value $m$ is an independent constraint
that can be given a-priori. In fact, the constant mean-value is a notorious
issue in fluid-structure interaction: there is a coupling of a deforming and
potentially stretching solid with an incompressible fluid that conserves
volume. If the prescribed direction of deformation is non orthogonal (or more
explicit if the reference geometry has a non-zero Gauss curvature) equation
(1.21) becomes non-linear w.r.t. the deformation (along the reference
coordinate direction). That is the reason why in the theory for the existence
of incompressible fluids the non-orthogonal deformations where an open problem
for some time. Indeed, the existence for the Cauchy problem was onbly shown
recently in [23] by introducing respective non-linear correctors. We wish to
emphasize that it is this key difficulty that reduces the the set-up of this
paper to flat geometries and excludes non-flat Koiter shells.

###  1.5. The main result

The main theorem of this paper is the following111For the precise defintion of
the function spaces please see Subsection 2.3

######  Theorem 1.22.

There exists a constant $C_{0}=C_{0}(c_{0},\Omega,\alpha,M,\kappa,T)$ such
that for any $m\in\mathbb{R}$ and any given forces

$\left(\mathbf{f},g\right)\in\left(L_{\text{per}}^{2}\left(0,T\right);L^{2}\left(\mathbb{R}^{3}\right)\right)\times\left(L_{\text{per}}^{2}\left(0,T\right);L^{2}\left(\omega\right)\right)$

such that

$m^{2}+\int_{0}^{T}\int_{\mathbb{R}^{3}}\left|\mathbf{f}\right|^{2}dxdt+\int_{0}^{T}\int_{\omega}\left|g\right|^{2}dx^{\prime}dt\leq
C_{0}$

the problem (1.19) admits at least one weak time-periodic solution
$(\mathbf{u},\eta)$ in the sense of Definition 2.12, with

(1.23) $\int_{\omega}\eta(t,\cdot)dx^{\prime}=m\quad\forall t\in I.$

Furthermore it holds that

(1.24)
$\left\|\mathbf{u}\right\|_{L^{\infty}\left(I;L^{2}\left(\Omega_{\eta}\right)\right)\cap
L^{2}\left(I;W_{0,\text{div}}^{1,2}\left(\Omega_{\eta}\right)\right)}+\left\|\eta\right\|_{L^{\infty}\left(I;H_{0}^{2}\left(\omega\right)\right)\cap
W^{1,\infty}\left(I;L^{2}\left(\omega\right)\right)}\leq
c\left(C_{0}^{2}+C_{0}\right)$

with $c$ depending on $\Omega$, $\kappa$, $c_{0}$, $T$ only. Additionally, the
following energy inequality holds

(1.25)
$\int_{0}^{T}\int_{\Omega_{\eta}\left(t\right)}\left|\nabla\mathbf{u}\right|^{2}dxdt\leq\int_{0}^{T}\int_{\Omega_{\eta}\left(t\right)}\mathbf{f}\cdot\mathbf{u}dxdt+\int_{\omega}g\cdot\partial_{t}\eta
dx^{\prime}dt.$

######  Remark 1.26 (The 2D/1D case).

Note that our proof can be adapted to the 2D coupling without of the fluid
with a 1D coupling of the shell without any change.

######  Remark 1.27 (On the constraints in Theorem 1.22).

One critical and non-trivial fact about fluid-structure interactions (in
general) is that the solution prescribes the geometry of the fluid domain.
Generally, weak solutions can be constructed up to the point of self-
intersection of the solid. The key obstacle to produce periodic solutions
(where the initial condition is the unknown!) is to exclude self-intersection
over the full time-interval a-priori. It is only this necessity that
determines the size of the constraint $C_{0}$. Certainly the theorem hence
provides the existence of periodic solutions under the assumption that the
applied forces are small. However, since it is the deformation that restricts
the geometric flexibility and no other part of the solution construction it is
also possible to assume that the elastic resistance of the shell is large
enough to withstand the given forces. Actually, as can be checked, the
assumption on the constant $C_{0}$ are strongly connected to the
$H^{2}$-coercivity of the shell $c_{0}$. This coercivity again is connected to
the thickness of the elastic plate and respectively its resistance against
deformations. Hence one consequence of the theorem above shows that periodic
weak-solutions for arbitrary large periodic forces (and arbitrary large time-
intervals) can be constructed as long as the structure is ”thick” enough. For
some applications these assumptions seem to be appropriate. Indeed, the
stronger the periodic forces (stemming from a pump for instance) are acting on
the structure, the more resistant the structure (the pipe) should be.

######  Remark 1.28 (On the assumed boundary values).

With regard of the importance of applications like pipelines or blood-vessels
inflow/outflow boundary conditions are very natural to be considered. In that
regard observe that the here provided existence theory can be addapted to
periodic-in-space boundary conditions on some part of the domain without
substantial effort (as was considered by many authors [9, 6]). More general
inflow and outflow conditions will be the subject to a follow-up-work of this
paper.

######  Remark 1.29 (Key novelty).

Please observe that the a-priori estimate provided in (1.24) is not a trivial
consequence of the energy estimate (1.25). Actually, we consider the estimate
on the deformation to be the main technical contribution of the present paper.
Formally this non-trivial estimate is explained in Subsection 2.1.

###  1.6. Outline of the paper

Our paper is structured as follows: in Section 2 we introduce the periodic
spaces and the concept of weak solutions for which we will show the existence.
Before we motivate both by justifying formally how to obtain the encessary
a-priori estimates which are _independent_ of the initial data.In Section 3 we
prove our main result.We perform two fixed-points: One at the discrete
(Galerkin) level, employing Schäffer’s fixed point Theorem- which ensures the
periodicity of $\mathbf{u}_{n}$, $\eta_{n}$,$\partial_{t}\eta_{n}$ for a given
geometry. At this level also the critical a-priori estimates will be derived
that eventually will withstand all further analytical manipulations. The
second fixed-point is performed at the continuous level for a regularized set-
up w.r.t. the geometry. Here we employ the set-valued fixed point Theorem 4.10
due to Kakutani-Glicksberg-Fan (since the uniqueness of solutions can not be
guaranteed). Finally, the last step consists in letting $\varepsilon\to 0$
which provides our main result- Theorem 1.22. The paper ends with an Appenix-
Section 4 -where we collect some known results, essentially needed in the
proof.

##  2\. Preliminary

###  2.1. Formal a-priori estimates

Assume during this subsection that all the involved functions are smooth. We
will obtain suitable a-priori estimates needed to prove our main result-
Theorem 1.22. Let us first use the test function $(\mathbf{u},\eta)$. That is
we take the scalar product with $\mathbf{u}$ and $\partial_{t}\eta$ in (1.10)
and in (1.15) respectively. We add the resulting equations and after recalling
the coupling $\mathbf{u}=\partial_{t}\eta\mathbf{e}_{3}$ on
$\partial\Omega_{\eta}$ and the definition of $\mathbf{u}$ we get

(2.1)
$\frac{d}{dt}E\left(t\right)+\int_{\Omega_{\eta\left(t\right)}}\left|\nabla\mathbf{u}\right|^{2}dx=\int_{\Omega_{\eta\left(t\right)}}\mathbf{f}\cdot\mathbf{u}dx+\int_{\omega}g\partial_{t}\eta
dx^{\prime}\quad\forall t\in\left[0,T\right]$

where we have denoted

$E\left(t\right):=\frac{1}{2}\int_{\Omega_{\eta\left(t\right)}}\left|\mathbf{u}\left(t,\cdot\right)\right|^{2}dxds+\frac{1}{2}\int_{\omega}\left|\partial_{t}\eta\left(t,\cdot\right)\right|^{2}dx^{\prime}+K\left(\eta\left(t\cdot\right)\right),\
t\in\left[0,T\right].$

Now since $\mathbf{u}$ vanishes on $\Gamma\subset\partial\Omega_{\eta(t)}$ and
$\mathbf{u}=\partial_{t}\eta\cdot\mathbf{e}_{3}$ on
$\partial\Omega_{\eta(t)}$, by using Poincare’s inequality (see Theorem 4.2)
and recalling the properties of the trace operator from Lemma 4.6 we obtain

(2.2)
$\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\left|\mathbf{u}\right|^{2}dxdt+\int_{\omega}\left|\partial_{t}\eta\right|^{2}dx^{\prime}dt\leq
c(\Omega,\kappa)\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\left|\nabla\mathbf{u}\right|^{2}dxdt.$

On the other hand, using the periodicity of $\mathbf{u},\eta,\partial_{t}\eta$
we also have $E(0)=E(T)$ and thus

(2.3)
$\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\left|\nabla\mathbf{u}\right|^{2}dxdt=\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\mathbf{f}\cdot\mathbf{u}dx+\int_{\omega}g\partial_{t}\eta
dx^{\prime}dt.$

From (2.2) and (2.3) we obtain

(2.4)
$\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\left|\mathbf{u}\right|^{2}+\left|\nabla\mathbf{u}\right|^{2}dxdt+\int_{\omega}\left|\partial_{t}\eta\right|^{2}dx^{\prime}dt\leq
cC\left(\mathbf{f},g\right).$

where $C(\mathbf{f},g)$ will always denote

(2.5)
$C\left(\mathbf{f},g\right):=\int_{I}\int_{\mathbb{R}^{3}}\left|\mathbf{f}\right|^{2}dxdt+\int_{I}\int_{\omega}\left|g\right|^{2}dx^{\prime}dt=C_{0}-m^{2}.$

It is important to remark that we will obtain the periodic solutions as a
fixed point of the mapping

$\left(\mathbf{u}_{0},\eta_{0},\eta_{1}\right)\mapsto\left(\mathbf{u}\left(T,\cdot\right),\eta\left(T,\cdot\right),\partial_{t}\eta\left(T,\cdot\right)\right)$

where $\left(u_{0},\eta_{0},\eta_{1}\right)$ is a set of initial
values.222This approach is by now standard–see for example [28, 16, 1] and the
references therein. This is why we need estimates of $E(T)$ which are
_independent_ on the initial value $E(0)$. From the mean value theorem there
is $t_{0}\in[0,T]$ for which

$E\left(t_{0}\right)=\frac{1}{T}\int_{0}^{T}E\left(t\right)dt=:\fint_{0}^{T}E\left(t\right)dt$

and so by integrating the equation (2.1) on $[t_{0},t]$ and taking the
supremum over $t$ we get that

(2.6) $\sup_{0\leq t\leq
T}E\left(t\right)\leq\fint_{0}^{T}E\left(t\right)dt+cC\left(\mathbf{f},g\right)\leq
cC\left(\mathbf{f},g\right)+\fint_{0}^{T}K\left(\eta\right)dt$

In order to estimate the last term we would like to test the shell equation
with $\eta$. In order to do that we we seek a corresponding extension
$\mathcal{E}(\eta)$ for the fluid equation such that
$\text{div}\mathcal{E}(\eta)=0$ and
$\text{tr}\mathcal{E}(\eta)=\eta\mathbf{e}_{3}$. Finding such a function
$\mathcal{E}(\eta)$ is not immediate. But in [27, Proposition 3.3], such a
construction was performed, along with corresponding estimates as long as the
compatibility conditions with the solenoidality is satisfied. We include the
properties of the operator $\mathcal{E}(\eta)$ in Proposition 4.7, that can be
found in the appendix. Roughly speaking $\mathcal{E}(\eta)\sim\eta$ in the
sense that obeys the same estimates up to a constant depending on
$\kappa,c_{0},\Omega$. As mentioned the compatibility condition with the
solenoidality of the extension has to be satisfied. For that we fix a smooth
bump function

(2.7) $\psi\in C_{0}^{\infty}(\omega;[0,1])\ \text{with}\int_{\omega}\psi
dx^{\prime}=1$

and by recalling the definition of $m$ from Remark 1.20 we find that the
following couple

$\left(\mathcal{F}_{\eta}\left(\mathcal{M}_{\eta}\left(\eta\right)\right),\mathcal{M}_{\eta}\left(\eta\right)\right)=\left(\mathcal{F_{\eta}}\left(\eta-m\psi\right),\eta-m\psi\right)$

is a test function for the weak formulation given in Definition 2.12. As we
show below, this test-function does provide the following estimate

(2.8) $2\int_{0}^{T}K\left(\eta\right)dt\leq\frac{1}{2}\sup_{0\leq t\leq
T}E\left(t\right)+c\left(C_{0}+C_{0}^{2}\right).$

Indeed, using the test-function in Definition 2.12 we find

$\displaystyle 2\int_{0}^{T}K\left(\eta\right)dt$
$\displaystyle=\int_{0}^{T}\left\langle
K^{\prime}\left(\eta\right),\eta\right\rangle dt\leq\int_{0}^{T}\left\langle
K^{\prime}\left(\eta\right),\eta-\mathcal{M}_{\eta}\left(\eta\right)\right\rangle
dt+\int_{0}^{T}\int_{\omega}\partial_{t}\eta\partial_{t}\left(\mathcal{M}_{\eta}\left(\eta\right)\right)dx^{\prime}dt$
$\displaystyle\quad+\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\mathbf{u}\cdot\partial_{t}\mathcal{F}_{\eta}\left(\mathcal{M}_{\eta}\left(\eta\right)\right)+\left(\nabla\mathbf{u}+\mathbf{u}\otimes\mathbf{u}\right):\nabla\mathcal{F}_{\eta}\left(\mathcal{M}\left(\eta\right)\right)dxdt$
$\displaystyle\quad+\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\mathbf{f}\cdot\mathcal{F}_{\eta}\left(\mathcal{M}\left(\eta\right)\right)dxdt+\int_{0}^{T}\int_{\omega}g\left(\mathcal{M}_{\eta}\left(\eta\right)\right)dx^{\prime}dt=:\sum_{k=1}^{7}I_{k}.$

Recalling now the properties of the divergence-free extension operator
$\mathcal{F}_{\eta}$ from Proposition 4.7 we employ Hölder’s inequalilty, the
Sobolev embedding, the $H^{2}$-coercivity of $K$ (1.9) and the estimate (2.5)
to get

$\displaystyle I_{1}$ $\displaystyle\leq
m\left\|K^{\prime}\left(\eta\right)\right\|_{L_{t}^{2}L_{x}^{2}}$
$\displaystyle I_{2}$
$\displaystyle\leq\left\|\partial_{t}\eta\right\|_{L_{t}^{2}L_{x}^{2}}^{2}\leq
cC_{0}$ $\displaystyle I_{3}+I_{4}+I_{5}$
$\displaystyle\leq\left(\left\|\mathbf{u}\right\|_{L_{t}^{2}W_{x}^{1,2}}+\left\|\mathbf{u}\right\|_{L_{t}^{2}W_{x}^{1,2}}^{2}\right)\left\|\mathcal{F}_{\eta}\left(\mathcal{M}_{\eta}\right)\right\|_{W_{t}^{1,2}L_{x}^{2}\cap
L_{t}^{\infty}W_{x}^{1,2}}$ $\displaystyle\leq
c(\sqrt{C_{0}}+C_{0})\left\|\mathcal{M}_{\eta}\right\|_{W_{t}^{1,\infty}L_{x}^{2}\cap
L_{t}^{\infty}W_{x}^{2,2}}$ $\displaystyle\leq
c(\sqrt{C_{0}}+C_{0})\sup_{t}\sqrt{E\left(t\right)}+c(C_{0}^{2}+C_{0})$
$\displaystyle I_{6}+I_{7}$
$\displaystyle\leq\left(\left\|\mathbf{f}\right\|_{L_{t}^{2}L_{x}^{2}}+\left\|g\right\|_{L_{t}^{2}L_{x}^{2}}\right)\left(\left\|\mathcal{F}_{\eta}\left(\mathcal{M}_{\eta}\right)\right\|_{L_{t}^{2}L_{x}^{2}}+\left\|\mathcal{M}_{\eta}\right\|_{L_{t}^{2}L_{x}^{2}}\right)$
$\displaystyle\leq c\sqrt{C_{0}}\sup_{t}\sqrt{E\left(t\right)}+cC_{0}$

All in all, using Young’s inequality we conclude that

$\sum_{k=1}^{7}I_{k}\leq\frac{1}{2}\sup_{t}E\left(t\right)+c(C_{0}+C_{0}^{2}).$

Hence we get (2.8) which together with (2.6) implies

(2.9) $\sup_{0\leq t\leq T}E\left(t\right)\leq c(C_{0}+C_{0}^{2}).$

###  2.2. Spaces of time-periodic functions

In the spirit of the formal estimates from above it is natural to consier the
following function spaces. For $T>0$ and a general (real) Banach space $X$ we
define

$C_{\text{per}}^{\infty}\left(0,T;X\right):=\left\\{\varphi:\mathbb{R}\to X:\
\varphi\text{\ is}\ C^{\infty}\ \text{and
}\varphi\left(t+T,\cdot\right)=\varphi\left(t,\cdot\right)\right\\}$

and then one obtains for $k\in\mathbb{Z}$ and $1\leq p\leq\infty$ the usual
periodic spaces

$L^{p}_{\text{per}}\left(0,T;X\right):=\overline{\left\\{f\in
C_{\text{per}}^{\infty}\left(0,T;X\right)\right\\}}^{\left\|\cdot\right\|_{p}}\quad
W^{k,p}_{\text{per}}\left(0,T;X\right):=\overline{\left\\{f\in
C_{\text{per}}^{\infty}\left(0,T;X\right)\right\\}}^{\left\|\cdot\right\|_{k,p}}.$

Further we consider $C_{w,\text{per}}(I,X)$ as the periodic weakly continuous
space.333We say that $x\in C_{w,\text{per}}(I,X)$ if for all $x^{*}\in
C_{\text{per}}(\overline{I},X^{*})$, $t\mapsto\langle x(t),x^{*}(t)\rangle$ is
continuous in $\overline{I}$; this concept can naturally be extended to time-
changing domains. We denote $I:=(0,T)$ the time interval, where $T$ is the
time-period.

In the following we will use spaces that change in time. Such spaces have been
investigated carefully in the previous literature (see for instance [22]) and
hence the concept of time-periodic spaces can be extended in a straight
forward manner. We have:

(2.10) $\displaystyle V_{\eta}\left(t\right):=$
$\displaystyle\left\\{\mathbf{u}\in
H^{1}\left(\Omega_{\eta\left(t\right)}\right):\text{div}\mathbf{u}=0\text{ in
}\Omega_{\eta(t)},\,\mathbf{u}=0\text{ on }\Gamma\right\\}$ $\displaystyle
V_{F,\text{per}}:=$ $\displaystyle\left\\{\mathbf{u}\in
C_{w,\text{per}}\left(I;L^{2}\left(\Omega_{\eta}\right)\cap
L_{\text{per}}^{2}\left(I;V_{\eta}\left(t\right)\right)\right)\right\\}$
$\displaystyle V_{K,\text{per}}:=$ $\displaystyle
C_{w,\text{per}}\left(\mathbb{R};H_{0}^{2}\left(\omega\right)\right)\cap
W_{\text{per}}^{1,\infty}\left(\mathbb{R};L^{2}\left(\omega\right)\right)$
$\displaystyle V_{S,\text{per}}:=$
$\displaystyle\left\\{\left(\mathbf{u},\eta\right)\in V_{F,\text{per}}\times
V_{K,\text{per}};\mathbf{u}\left(t,\phi_{\eta}\left(t,\cdot\right)\right)=\partial_{t}\eta\left(t,\cdot\right)\mathbf{e}_{3}\left(\eta\left(t,\cdot\right)\right)\text{
in }\omega\right\\}$ $\displaystyle V_{T,\text{per}}:=$
$\displaystyle\left\\{\left(\mathbf{q},\xi\right)\in
C^{1}_{\text{per}}(I;V_{\eta}(t))\times
V_{K,\text{per}}:\left(I\times\Omega_{\eta}\right),\mathbf{q}\left(t,\phi_{\eta}\left(t,\cdot\right)\right)=\xi\left(t,\cdot\right)\mathbf{e}_{3}\left(\eta\left(t,\cdot\right)\right)\right\\}.$

Note that $V_{S,\text{per}}$ is the space of time-periodic solutions and
$V_{T,\text{per}}$ is the space of time-periodic test-functions. In particular
$(\mathbf{u},\eta)\in V_{S,\text{per}}$ implies that $\mathbf{u}\equiv 0$ over
$(0,T)\times\Gamma$ as well as $\left(\mathbf{q},\xi\right)\in
V_{T,\text{per}}$ implies that $\mathbf{q}\equiv 0$ over $(0,T)\times\Gamma$.

Further, if the geometry is decoupled from the solid deformation we shall use
the analogous notations $V_{T,\text{per}}^{\delta},V_{S,\text{per}}^{\delta}$
whenever we deal with functions $(u,\eta)$ for which

$\mathbf{u}\left(t,\phi_{\delta(t)}\right)=\partial_{t}\eta\mathbf{e}_{3}$

for a given function $\delta\in C\left(I\times\omega\right),\
\left\|\delta\right\|_{L^{\infty}\left(I\times\omega\right)}\leq\kappa$
defines a given time-changing domain.

######  Remark 2.11.

We have $V_{K,\text{per}}\hookrightarrow
C^{0,1-\theta}\left(\overline{I};C^{0,2\theta-1}\left(\omega\right)\right)$,
and therefore the displacement of the shell is Hölder continuous in time.
Indeed, using the embeddings $H^{2}\left(\omega\right)\hookrightarrow
H^{2\theta}\left(\omega\right)\hookrightarrow
C^{0,2\theta-1}\left(\omega\right)$ for $\theta\in(1/2,1)$ (since
$\omega\in\mathbb{R}^{2}$). Consequently we find

$\displaystyle\left\|\eta\left(t\right)-\eta\left(s\right)\right\|_{C^{0,2\theta-1}\left(\omega\right)}$
$\displaystyle\leq
c{\left\|{\eta\left(t\right)-\eta\left(s\right)}\right\|}_{H^{2\theta}(\omega)}\leq
c\left\|\eta\left(t\right)-\eta\left(s\right)\right\|_{H^{2}\left(\omega\right)}^{\theta}\left\|\eta\left(t\right)-\eta\left(s\right)\right\|_{L^{2}\left(\omega\right)}^{1-\theta}$
$\displaystyle\leq
c\left\|\eta\right\|_{L^{\infty}\left(I;H^{2}\left(\omega\right)\right)}^{\theta}\left\|\eta\right\|_{W^{1,\infty}\left(I;L^{2}\left(\omega\right)\right)}\left|t-s\right|^{1-\theta}.$

In particular, the condition $\eta(0,\cdot)=\eta(T,\cdot)$ holds in a strong
sense, in contrast with the conditions
$\mathbf{u}(0,\cdot)=\mathbf{u}(T,\cdot)$ and
$\partial_{t}\eta(0,\cdot)=\partial_{t}\eta(T,\cdot)$ which can only occur in
a weak sense (continuous in time and taking values in suitable functions
spaces endowed with a weak-topology), see Definition 2.12.

###  2.3. The definition of weak time-periodic solutions

Based on the a-priori estimates we can introduce the following notion of
_periodic solution_. Let $(q,\xi)\in V_{T,\text{per}}^{\eta}$ be a test
function which is smooth in time and space. First we notice that

$\displaystyle\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\partial_{t}\mathbf{u}\cdot\mathbf{q}dxdt=$
$\displaystyle\int_{0}^{T}\left(\frac{d}{dt}\int_{\Omega_{\eta\left(t\right)}}\mathbf{u}\cdot\mathbf{q}dx-\int_{\Omega_{\eta\left(t\right)}}\mathbf{u}\cdot\partial_{t}\mathbf{q}dx-\int_{\omega}\left(\partial_{t}\eta\right)^{2}\xi
dx^{\prime}\right)dt$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\mathbf{u}\cdot\partial_{t}\mathbf{q}dxdt-\int_{0}^{T}\int_{\omega}\left(\partial_{t}\eta\right)^{2}\xi
dx^{\prime}dt$

where we have used Reynolds’ transport Theorem 4.1 and the fact that
$\text{tr}_{\eta}\mathbf{u}=\partial_{t}\eta\mathbf{e}_{3}$ and also
$\text{tr}_{\eta}\mathbf{q}=\xi\mathbf{e}_{3}$. On the other hand if we look
at the convective term we see that

$\displaystyle\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}\cdot\mathbf{q}dxdt=$
$\displaystyle\sum_{i=1}^{3}\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\partial_{i}\left(\mathbf{u}^{i}\mathbf{u}^{j}\right)\mathbf{u}^{j}-\partial_{i}\mathbf{u}^{i}\mathbf{u}^{j}\mathbf{q}^{j}dxdt$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\int_{0}^{T}\int_{\partial\Omega_{\eta\left(t\right)}}\mathbf{u}^{i}\mathbf{u}^{j}\mathbf{q}^{j}\nu^{i}dx^{\prime}dt-\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\left(\mathbf{u}^{i}\mathbf{u}^{j}\right)\partial_{i}\mathbf{q}^{j}dxdt$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\omega}\left(\partial_{t}\eta\right)^{2}\xi
dx^{\prime}dt-\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\left(\mathbf{u}\cdot\nabla\right)\mathbf{q}\cdot\mathbf{u}dxdt$

where we have used the coupling of $\mathbf{u}$ and $\partial_{t}\eta$ , the
$\text{div}\mathbf{u}=0$ condition and Stokes’ Theorem. We also make use of
the Korn’s identity from Lemma 4.3 and get

$\int_{\Omega_{\eta\left(t\right)}}\nabla\mathbf{u}:\nabla\mathbf{q}dx=2\int_{\Omega_{\eta\left(t\right)}}D\left(\mathbf{u}\right):D\left(\mathbf{q}\right)dx.$

All in all we obtain the following definition.

######  Definition 2.12.

A couple $(\mathbf{u},\eta)\in V_{S,\text{per}}$ with
$\left\|\eta\right\|_{L^{\infty}\left(I\times\omega\right)}<\kappa$ is a _weak
time-periodic solution_ to (1.19) if

(2.13)
$\displaystyle\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}-\mathbf{u}\cdot\partial_{t}\mathbf{q}+\nabla\mathbf{u}:\nabla\mathbf{q}-\left(\mathbf{u}\cdot\nabla\right)\mathbf{q}\cdot\mathbf{u}dxdt+\int_{0}^{T}\left\langle
K^{\prime}\left(\eta\right),\xi\right\rangle-\int_{\omega}\partial_{t}\eta\partial_{t}\xi
dx^{\prime}dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Omega_{\eta\left(t\right)}}\mathbf{f}\cdot\mathbf{q}dxdt+\int_{\omega}g\xi
dx^{\prime}dt\quad\forall\left(\mathbf{q},\xi\right)\in
V_{T,\text{per}}^{\eta}.$

##  3\. Proof of the main result

###  3.1. The decoupled system

In the following we assume that $\delta\in C\left(I\times\omega\right),\
\left\|\delta\right\|_{L^{\infty}\left(I\times\omega\right)}<\kappa$ is
prescribing a periodic time-changing domain. This means in particular that
$\delta(0,\cdot)=\delta(T,\cdot)$ and
$\Omega_{\delta(t)}=\phi_{\delta(t)}(\Omega)$. Next we will regularize the
space variables of the prescribed geometry $\delta$, and therefore replace
$\delta$ by $\mathcal{R}_{\varepsilon}\delta$, where $\varepsilon$ is the
mollification parameter. We extend $\delta$ to $(-\infty,T]\times\omega$ by
$\delta(t,\cdot)=\delta(0,\cdot)$ for $t<0$.

######  Lemma 3.1 ([23],[4]).

There exists an operator

$\mathcal{R}_{\varepsilon}:C\left(\left[0,T\right]\times\omega\right)\mapsto
C^{4}\left(\left[0,T\right]\times\omega\right)$

such that,

  1. (a)

$\mathcal{R}_{\varepsilon}\delta\to\delta$ uniformly as $\varepsilon\to 0$;

  2. (b)

$\mathcal{R}_{\varepsilon}:L^{2}\left(0,T;H_{0}^{2}\left(\omega\right)\right)\mapsto
L^{2}\left(0,T;H_{0}^{2}\left(\omega\right)\right)$ and
$\mathcal{R}_{\varepsilon}\delta\to\delta$ in
$L^{2}\left(0,T;H_{0}^{2}\left(\omega\right)\right)$ as $\varepsilon\to 0$;

  3. (c)

If $\partial_{t}\delta\in L^{p}\left(\left(0,T\right)\times\omega\right)$ then
$\partial_{t}\mathcal{R}_{\varepsilon}\delta=\mathcal{R}_{\varepsilon}\partial_{t}\delta\to\partial_{t}\delta$
in $L^{p}\left(\left(0,T\right)\times\omega\right)$ as $\varepsilon\to 0$;

  4. (d)

If $\delta\in C^{\gamma}\left(\left(0,T\right)\times\omega\right)$ for some
$\gamma\in(0,1)$ then $\mathcal{R}_{\varepsilon}\delta\to\delta$ in
$C^{\gamma}\left(\left(0,T\right)\times\omega\right)$ as $\varepsilon\to 0$;

  5. (e)

$\left\|\mathcal{R}_{\varepsilon}\delta\right\|_{L^{\infty}\left(\left(0,T\right)\times\omega\right)}\leq\left\|\delta\right\|_{L^{\infty}\left(\left(0,T\right)\times\omega\right)}$.

We will also regularize functions $\mathbf{v}\in
L^{2}((0,T)\times\mathbb{R}^{3})$ by convolution with standard mollifying
kernels. Thus, if $\psi_{\varepsilon}$ denote the standard time-space
mollification kernels, we set

$\mathcal{R}_{\varepsilon}v\left(t,x\right):=\int_{\mathbb{R}^{3+1}}\psi_{k}\left(t-s,x-y\right)\chi_{\left(0,T\right)\times\Omega_{\mathcal{R}_{\varepsilon}\delta}}\left(s,y\right)\mathbf{v}\left(s,y\right)dsdy.$

Observe, that the convective term can be linearized in such a way that the
energy estimates are preserved via Reynolds’ Transport Theorem. Indeed,
observing that

$\int_{\Omega_{\delta\left(t\right)}}\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}\cdot\mathbf{q}dx+\int_{\Omega_{\delta\left(t\right)}}\left(\mathbf{u}\cdot\nabla\right)\mathbf{q}\cdot\mathbf{u}dx=\int_{\omega}\left(\partial_{t}\eta\right)^{2}\xi
dx^{\prime}$

we will rewrite (in order to linearize) the convective term as follows:

(3.2)
$\displaystyle\int_{0}^{T}\int_{\Omega_{\delta\left(t\right)}}\left(\mathbf{u}\cdot\nabla\right)\mathbf{q}\cdot\mathbf{u}dxdt=$
$\displaystyle\frac{1}{2}\int_{0}^{T}\int_{\Omega_{\delta\left(t\right)}}\left(\mathbf{u}\cdot\nabla\right)\mathbf{q}\cdot\mathbf{u}dx-\frac{1}{2}\int_{\Omega_{\delta\left(t\right)}}\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}\cdot\mathbf{q}dx$
$\displaystyle+\frac{1}{2}\int_{\omega}\left(\partial_{t}\eta\right)^{2}\xi
dx^{\prime}dt.$

We now use the above to define decoupled time-periodic weak solutions. For
that we use the regularizing operators from Lemma 3.1. Within this section we
will suppress the $\varepsilon$ from the notations and will denote
$\mathcal{R}\delta:=\mathcal{R}_{\varepsilon}\delta$. At this level we assume
$\varepsilon$ is sufficiently small and fixed. We introduce the following:

######  Definition 3.3.

Given $\left(\mathbf{v},\delta\right)\in
L^{\infty}\left(I\times\omega\right)\times
L^{2}\left(I\times\mathbb{R}^{3}\right)$ with
$\delta(0,\cdot)=\delta(T,\cdot)$ and
$\left\|\delta\right\|_{L^{\infty}\left(I\times\omega\right)}<\kappa$, we say
a couple $(\mathbf{u},\eta)\in V_{S,\text{per}}^{\mathcal{R}\delta}$ is a
time-periodic weak solution the _decoupled_ , _linear_ and _egularized_
problem if:444Note that the trace operator
$\text{tr}:W^{1,2}\left(\Omega_{\mathcal{R}\delta\left(t\right)}\right)\mapsto
W^{1/2,2}\left(\omega\right)\hookrightarrow L^{2}\left(\omega\right)$ is well
defined since by construction the operator $\mathcal{R}_{\varepsilon}$ enjoys
the estimate
$\left\|\mathcal{R}\delta\right\|_{L^{\infty}\left(I\times\omega\right)}\leq\left\|\delta\right\|_{L^{\infty}\left(I\times\omega\right)}<\kappa$.

(3.4)
$\begin{split}\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}-\mathbf{u}\cdot\partial_{t}\mathbf{q}+\nabla\mathbf{u}:\nabla\mathbf{q}dx+\frac{1}{2}\left(\mathcal{R}\mathbf{u}\cdot\nabla\right)\mathbf{u}\cdot\mathbf{q}-\frac{1}{2}\left(\mathcal{R}\mathbf{v}\cdot\nabla\right)\mathbf{q}\cdot\mathbf{u}dxdt&-\\\
-\int_{0}^{T}\int_{\omega}\frac{1}{2}\partial_{t}\eta\partial_{t}\mathcal{R}\delta\xi+\partial_{t}\eta\partial_{t}\xi
dx^{\prime}dt+\int_{0}^{T}\left\langle
K^{\prime}\left(\eta\right),\xi\right\rangle dt&=\\\
\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\mathbf{f}\cdot\mathbf{q}dx+\int_{\omega}g\xi
dx^{\prime}dt\quad\forall(\mathbf{q},\xi)\in
V_{T,\text{per}}^{\mathcal{R}\delta}.\end{split}$

The main result of this subsection is the following:

######  Proposition 3.5.

Assume (in addition to the previous discussion) that

$\left(\delta,\mathbf{v}\right)\in\left(L^{\infty}\left(I;H_{0}^{2}\left(\omega\right)\right)\cap
W^{1,\infty}\left(I;L^{2}\left(\omega\right)\right)\right)\times
L^{2}\left(I;W^{1,2}\left(I;\mathbb{R}^{3}\right)\right)$

with

(3.6)
$\left\|\delta\right\|_{L^{\infty}\left(I;H_{0}^{2}\left(\omega\right)\right)}+\left\|\delta\right\|_{W^{1,\infty}\left(I;L^{2}\left(\omega\right)\right)}\leq
M_{1}\ \text{and}\
\left\|\mathbf{v}\right\|_{L^{2}\left(I;W^{1,2}\left(I;\mathbb{R}^{3}\right)\right)}\leq
M_{2}$

where $M_{1}$ is sufficiently small so that ensures
$\left\|\delta\right\|_{L^{\infty}\left(I\times\omega\right)}<\kappa$ (see
(3.39) for the precise choice). Let $m\in\mathbb{R}$. Then there exists a
time-periodic weak solution $(\mathbf{u},\eta)$ for the decoupled and
regularized problem with data $(\mathbf{v},\delta)$ such that

(3.7) $\int_{\omega}\eta\left(t,\cdot\right)dx^{\prime}=m\ \quad\forall t\in
I.$

and which satisfies

(3.8)
$\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\nabla\mathbf{u}\right|^{2}dxdt\leq\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\mathbf{f}\cdot\mathbf{u}dx+\int_{0}^{T}\int_{\omega}g\partial_{t}\eta
dx^{\prime}dt.$

and, after denoting

$E\left(t\right):=\int_{\mathcal{R}\delta\left(t\right)}\left|\mathbf{u}\right|^{2}dx+\int_{\omega}\left|\partial_{t}\eta\right|^{2}dx^{\prime}+K\left(\eta(t)\right),\
t\in\left[0,T\right]$

that 555Recall the definition of $C(\mathbf{f},g)$ from (2.5).

(3.9)
$\sup_{t\in\left[0,T\right]}E(t)+\int_{0}^{T}\int_{\mathcal{R}\delta\left(t\right)}\left|\nabla\mathbf{u}\right|^{2}dxdt\leq
c\left(\Omega,\kappa,c_{0},T\right)\cdot\left(C\left(\mathbf{f},g\right)^{2}+C\left(\mathbf{f},g\right)+m^{2}\right)$

The proof of the proposition is split into several parts.

###  3.2. The construction of a decoupled, regularized, periodic solution

We consider a sequence $(\mathbf{f}_{n},,g_{n})$ of smooth functions such that

$\left(\mathbf{f}_{n},g_{n}\right)\xrightarrow{n\to\infty}\left(\mathbf{f},g\right)\
\text{in}\
L_{\text{loc}}^{2}\left(I\times\mathbb{R}^{3};\mathbb{R}^{3}\right)\times
L_{\text{loc}}^{2}\left(\mathbb{R}\times\omega;\mathbb{R}\right)$

and

(3.10)
$\int_{0}^{T}\int_{\mathbb{R}^{3}}\left|\mathbf{f}_{n}\right|^{2}dxdt+\int_{0}^{T}\int_{\omega}\left|g_{n}\right|^{2}dx^{\prime}dt\leq\int_{0}^{T}\int_{\mathbb{R}^{3}}\left|\mathbf{f}\right|^{2}dxdt+\int_{0}^{T}\int_{\omega}\left|g\right|^{2}dx^{\prime}dt=:C(\mathbf{f},g).$

Next let $(\hat{Y}_{k})_{k\geq 1}$ be a basis of $\left\\{Y\in
H_{0}^{2}\left(\omega\right):\ \int_{\omega}Ydx^{\prime}=0\right\\}$ which is
orthogonal in $L^{2}(\omega)$. Proceeding in the same manner as in [9] we
extend each $\hat{Y}_{k}$ to a divergence-free function
$\mathbf{U}_{k}:\Omega\mapsto\mathbb{R}^{3}$ by solving the Stokes problem

$\begin{cases}-\Delta\mathbf{U}_{k}+\nabla P_{k}=0&\text{in}\ \Omega\\\
\text{div}\ \mathbf{U}_{k}=0&\text{in}\ \Omega\\\
\mathbf{U}_{k}=\hat{Y}_{k}\mathbf{e}_{3}&\text{on}\ \partial\Omega\end{cases}$

Then we use the Piola transform from Lemma 4.5 and define

(3.11)
$\left(\mathbf{Y}_{k}\left(t,\cdot\right),Y_{k}\left(t,\cdot\right)\right)=\left(\mathcal{J}_{\mathcal{R\delta}\left(t\right)}\mathbf{U}_{k},\hat{Y}_{k}\right)\in
V_{T,\text{per}}^{\mathcal{R}\delta}$

We consider also $(\hat{Z}_{k})_{k\geq 1}$ an $L^{2}$-orthonormal basis of the
space $H^{1}_{0,\text{div}}$ which we extend in the same manner to

(3.12)
$\left(\mathbf{Z}_{k}\left(t,\cdot\right),Z_{k}\left(t,\cdot\right)\right)=\left(\mathcal{J}_{\mathcal{R\delta}\left(t\right)}Z_{k},0\right)\in
V_{T,\text{per}}^{\mathcal{R}\delta}.$

Let us finally define

(3.13)
$\left(\mathbf{X}_{k}\left(t,\cdot\right),X_{k}\left(t,\cdot\right)\right):=\begin{cases}\left(\mathbf{Y}_{(k+1)/2}\left(t,\cdot\right),Y_{(k+1)/2}\left(t,\cdot\right)\right)&k\
\text{odd}\\\ \left(\mathbf{Z}_{k/2}\left(t,\cdot\right),0\right)&k\
\text{even}\end{cases}\quad\text{for all }k\in\mathbb{N}$

######  Remark 3.14.

With reasons very similar to the ones from [23, p. 234] it can be proved that
the space

$\text{span}\left\\{\varphi\mathbf{X}_{k},\varphi X_{k}:\varphi\in
C\left[0,T\right]\cap C^{1}\left(0,T\right),\
\varphi\left(0\right)=\varphi\left(T\right)\right\\}$

is dense in the space of test functions
$V_{T,\text{per}}^{\mathcal{R}\delta}$.

We now make the following ansatz:

(3.15)
$\begin{split}\eta_{n}\left(t,x\right)&=\sum_{k=1}^{n}b_{n}^{k}\left(t\right)X_{k}+m\Psi,\
t\in\left[0,T\right],x\in\omega\\\
\mathbf{u}_{n}\left(t,x\right)&=\sum_{k=1}^{n}a_{n}^{k}\left(t\right)\mathbf{X}_{k}\left(t,x\right),\
t\in\left[0,T\right],\
x\in\Omega_{\mathcal{R}\delta\left(t\right)}.\end{split}$

and we impose the condition
$\text{tr}_{\delta}\mathbf{u}_{n}=\partial_{t}\eta_{n}\mathbf{e}_{3}$ which is

(3.16)
$a_{n}^{k}\left(t\right)=\left(b_{n}^{k}\right)^{\prime}\left(t\right)t\in\left[0,T\right],$

by (3.11). Due to the trace operator (see Lemma 4.6) we get

(3.17) $\int_{\omega}\left|\partial_{t}\eta_{n}\right|^{2}dx^{\prime}\leq
c_{1}\left(\Omega,k\right)\int_{\Omega_{\mathcal{R\delta}\left(t\right)}}\left|\mathbf{u}_{n}\right|^{2}dx\
\forall t\in\left[0,T\right].$

We now seek for

$\mathbf{b}_{n}:=\left({b}_{n}^{k}\right)_{k=1}^{n}:\left[0,T\right]\mapsto\mathbb{R}^{n}$

such that the following equation is satisfied:

(3.18) $\displaystyle\begin{aligned}
&\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\partial_{t}\mathbf{u}_{n}\cdot\mathbf{X}_{k}dx+\frac{1}{2}\int_{\omega}\partial_{t}\eta_{n}\partial_{t}\mathcal{R}\delta
Y_{k}dx^{\prime}+\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\nabla\mathbf{u}_{n}\cdot\nabla\mathbf{X}_{k}dx\\\
&\quad+\frac{1}{2}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left(\mathcal{R}\mathbf{v}\cdot\nabla\right)\mathbf{u}_{n}\cdot\mathbf{X}_{k}dx-\frac{1}{2}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left(\mathcal{R}\mathbf{v}\cdot\nabla\right)\mathbf{X}_{k}\cdot\mathbf{u}_{n}dx+\int_{\omega}\partial_{tt}\eta_{n}X_{k}dx^{\prime}+\left\langle
K^{\prime}\left(\eta_{n}\right),X_{k}\right\rangle\\\
&=\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\mathbf{f}_{n}\cdot\mathbf{X}_{k}dx+\int_{\omega}g_{n}X_{k}dx^{\prime}\quad\forall
t\in[0,T],\forall 1\leq k\leq n\end{aligned}$

Please note that (3.16) and (3.18) read as a second order linear ordinary
differential equation with the unknown $\mathbf{b}_{n}$. Now we prove that for
any given initial values
$\mathbf{b}_{n}\left(0\right),\mathbf{b}_{n}^{\prime}\left(0\right)$ this
differential equation has a unique solution: the coefficient of
$\left(\mathbf{b}_{n}\right)^{\prime\prime}$ is given by the mass matrix

$M\left(t\right):=\left(\int_{\Omega_{\delta\left(t\right)}}\mathbf{X}_{i}\cdot\mathbf{X}_{j}dx\right)_{1\leq
i,j\leq n}+I_{n}=:\tilde{M}\left(t\right)+I_{n}.$

Since for any $\xi\in\mathbb{R}^{n}$ we have that

(3.19) $\displaystyle M\left(t\right)\xi\cdot\xi$
$\displaystyle=\tilde{M}\left(t\right)\xi\cdot\xi+\text{diag}\left(1,0,1,0,\ldots\right)\xi\cdot\xi$
$\displaystyle\geq\sum_{i,j=1}^{n}\xi_{i}\xi_{j}\int_{\Omega_{\delta\left(t\right)}}\mathbf{X}_{i}\cdot\mathbf{X}_{j}dx$
$\displaystyle\geq\int_{\Omega_{\delta\left(t\right)}}\left(\sum_{i=1}^{n}\xi_{i}\mathbf{X}_{i}\right)\cdot\left(\sum_{j=1}^{n}\xi_{j}\mathbf{X}_{j}\right)dx$
$\displaystyle\geq 0$

it follows that $M\xi=0$ implies that $\sum_{i}\xi_{i}\mathbf{X}_{i}=0$ and
this in turn yields (after applying the inverse of the Piola transform
$\mathcal{J}_{\mathcal{R}\delta(t)}$) that $\xi=\mathbf{0}$ due to linear
independence of the vectors from (3.13) Hence it follows that the vector field
$\mathbf{b}_{n}$ exists in an interval $[0,T_{0}]$ for some $T_{0}>0$. The
energy of the discretized system will play an important role in what follows
and it is introduced via the formula

(3.20)
$E_{n}\left(t\right):=\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\mathbf{u}_{n}\left(t\right)\right|^{2}dx+\int_{\omega}\left|\partial_{t}\eta_{n}\left(t\right)\right|^{2}dx^{\prime}+K\left(\eta_{n}\left(t\right)\right),\
t\in\left[0,T\right].$

By multiplying (3.18) with $a_{n}^{k}$ and summing for $k=1,n$ we get

(3.21)
$\frac{d}{dt}E_{n}\left(t\right)+\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\nabla\mathbf{u}_{n}\right|^{2}dx=\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\mathbf{f}_{n}\cdot\mathbf{u}_{n}dx+\int_{\omega}g_{n}\eta_{n}dx^{\prime}\
\forall t\in\left[0,T\right].$

In particular this allows to extend $\mathbf{b}_{n}$ to the whole interval
$[0,T]$.

###  3.3. The key estimate

The following proposition is the main effort to obtain the existence of a
fixed point.

######  Proposition 3.22.

Suppose that

$E_{n}(T)\geq E_{n}(0)$

(or, more general, that (3.23) holds). Then there exists a constant

$R=c\left(\Omega,\kappa,c_{0},T\right)\cdot\left(C\left(\mathbf{f},g\right)^{2}+C\left(\mathbf{f},g\right)+m^{2}\right)>0$

for which

$\ E_{n}\left(T\right)\leq R.$

In particular

$E_{n}\left(0\right)\leq E_{n}\left(T\right)\leq R.$

###### Proof.

The assumption $E_{n}(T)\geq E_{n}(0)$ yields

(3.23)
$\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\mathbf{f}_{n}\cdot\mathbf{u}_{n}dxdt+\int_{0}^{T}\int_{\omega}g_{n}\partial_{t}\eta_{n}dx^{\prime}dt\geq\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\nabla\mathbf{u}_{n}\right|^{2}dxdt.$

From (3.23), Young’s inequality, Poincare’s inequality and the trace operator
(see Lemma 4.6) we get

(3.24)
$\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\nabla\mathbf{u}_{n}\right|^{2}+\left|\mathbf{u}_{n}\right|^{2}dx+\int_{\omega}\left|\partial_{t}\eta_{n}\right|^{2}dx^{\prime}dt\leq
c_{2}\left(\Omega,\kappa\right)\left(\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\mathbf{f}\right|^{2}dxdt+\int_{\omega}\left|g\right|^{2}dxdt\right).$

Recall the relation (3.18), and note that we can, for a function $\phi\in
C^{\infty}(0,T)$ with $\phi\leq 1$, $\phi\equiv 1$ on $[T/4,T]$ and
$\phi^{\prime}\leq\frac{c}{T}$, use the test function
$\phi^{3}\left(\mathbf{u}_{n},\partial_{t}\eta_{n}\right)$ and obtain

(3.25)
$\begin{split}\phi^{3}E_{n}\left(t\right)\leq&c\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\phi^{2}\phi^{\prime}\left|\mathbf{u}_{n}\right|^{2}+\phi^{3}\mathbf{f}_{n}\cdot\mathbf{u}_{n}dxdt+\\\
&c\int_{0}^{T}\int_{\omega}\phi^{3}g_{n}\partial_{t}\eta_{n}+\phi^{2}\left(\partial_{t}\eta_{n}\right)^{2}dx^{\prime}dt+c\int_{0}^{T}\phi^{2}\phi^{\prime}K\left(\eta_{n}\right)dt\\\
\leq&c\frac{c\left(\mathbf{f},g\right)}{T}+c\fint_{0}^{T/4}\phi^{2}K\left(\eta_{n}\right)dt\end{split}$

We aim to estimate uniformly the term $\fint_{0}^{T/4}E\left(t\right)dt$.

To this end, let us consider a function $\psi\in
C_{0}^{\infty}\left(0,T\right)$ such that $\psi=\phi$ on $[0,3T/4]$. Note that
we may assume that $\psi^{\prime}\leq\frac{c}{T}$. By multiplying the equation
(3.18) with $\psi^{2}b_{n}^{k}$ we are allowed to use the test function

(3.26)
$\psi^{2}\left(\mathbf{B}_{n},\mathcal{M}\left(\eta_{n}\right)\right):=\psi^{2}\left(\sum_{k=1}^{n}b_{n}^{k}\mathbf{X}_{k},\sum_{k=1}^{n}b_{n}^{k}X_{k}\right)\in
V_{T,}^{\mathcal{R}\delta},$

this is the discrete represent of the test-function related to ”$\eta-m$”. We
obtain

(3.27) $\displaystyle\int_{0}^{T}\psi^{2}K\left(\eta_{n}\right)dt\leq$
$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\psi^{2}\left(\left|\nabla\mathbf{u}_{n}:\nabla\mathbf{B}_{n}\right|+\left|\nabla\mathbf{B}_{n}:\mathcal{R}\mathbf{v}\otimes\mathbf{u}_{n}\right|+\left|\nabla\mathbf{u}_{n}:\mathbf{B}_{n}\otimes\mathcal{R}\mathbf{v}\right|\right)dxdt+$
$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\mathbf{u}_{n}\cdot\partial_{t}\left(\psi^{2}\mathbf{B}_{n}\right)\right|dxdt+\int_{0}^{T}\psi^{2}\int_{\omega}\left|\eta_{n}\partial_{t}\eta_{n}\partial_{t}\mathcal{R}\delta\right|+\left|\partial_{t}\eta_{n}\partial_{t}\left(\eta_{n}\phi\right)\right|dx^{\prime}dt+$
$\displaystyle+\int_{0}^{T}\psi^{2}\left\langle
K^{\prime}\left(\eta_{n}\right),m\Psi\right\rangle
dt+c\left(\mathbf{f},g\right)$
$\displaystyle=:\sum_{k=1}^{7}I_{k}+C\left(\mathbf{f},g\right)$

Now we need some properties of $\mathbf{B}_{n}$ (from (3.26)) which follow rom
(3.23) and the definition of $\mathbf{X}_{k}$ and $X_{k}$:

(3.28)
$\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\partial_{t}\mathbf{B}_{n}\right|^{2}+\left|\mathbf{B}_{n}\right|^{2}+\left|\nabla\mathbf{B}\right|^{2}dx\leq
C\left(\Omega,\kappa,c_{0}\right)\int_{0}^{T}K\left(\eta_{n}\left(t,\cdot\right)\right)dt+C(\mathbf{f},g).$

Let $\theta>0$; we use Young’s inequality and (3.23), (3.28) to get that

(3.29)
$I_{1}\leq\theta\int_{0}^{T}\int_{0}^{T}\psi^{2}K\left(\eta_{n}\left(t\right)\right)dt+\frac{c(C(\mathbf{f},g)))}{\theta}$

and for $\theta$ sufficiently small we can absorb the $K(\eta_{n})$ term in
the LHS. Next,

(3.30) $\displaystyle\sum_{k=2}^{6}I_{k}$
$\displaystyle\leq\left\|\psi^{2}\nabla\mathbf{B}_{n}\right\|_{L_{t}^{\infty}L_{x}^{3/2}}\left\|\mathcal{R}\mathbf{v}\right\|_{L_{t}^{2}L_{x}^{6}}\left\|\mathbf{u}_{n}\right\|_{L_{t}^{2}L_{x}^{6}}+\left\|\psi^{2}\nabla\mathbf{B}_{n}\right\|_{L_{t}^{\infty}L_{x}^{3}}\left\|\mathcal{R}\mathbf{v}\right\|_{L_{t}^{2}L_{x}^{6}}\left\|\nabla\mathbf{u}_{n}\right\|_{L_{t}^{2}L_{x}^{2}}$
$\displaystyle\quad+\left\|\partial_{t}\mathcal{R}\mathbf{\delta}\right\|_{L_{t}^{2}L_{x}^{2}}\left\|\partial_{t}\eta_{n}\right\|_{L_{t}^{2}L_{x}^{2}}\left\|\psi^{2}\eta_{n}\right\|_{L_{t}^{\infty}L_{x}^{\infty}}+cC(\mathbf{f},g)$
$\displaystyle\leq\theta\sup_{t\in\left[0,T\right]}\psi^{2}K\left(\eta_{n}\left(t\right)\right)+\frac{c((C(\mathbf{f},g))+(C(\mathbf{f},g))^{2})}{\theta}$
$\displaystyle\leq\theta\sup_{t\in\left[0,T\right]}\psi^{2}E_{n}\left(t\right)+\frac{c((C(\mathbf{f},g))+(C(\mathbf{f},g))^{2})}{\theta}$

and

(3.31) $\displaystyle
I_{7}\leq\theta\left\|\eta\right\|_{L_{t}^{2}H_{x}^{2}}^{2}+\frac{c}{\theta}m^{2}$

From (3.25), (3.29), (3.30) (3.31) and the fact that $\psi\leq\phi$ we
conclude that

(3.32) $\sup_{t\in\left[0,T\right]}\phi^{2}E_{n}\left(t\right)\leq
c\left(\Omega,\kappa,c_{0},T\right)\cdot\left(C\left(\mathbf{f},g\right)^{2}+C\left(\mathbf{f},g\right)+m^{2}\right)$

and in particular

(3.33) $E_{n}(T)\leq c.$

This now proves Proposition 3.22 by considering

(3.34)
$R:=c\left(\Omega,\kappa,c_{0},T\right)\cdot\left(C\left(\mathbf{f},g\right)^{2}+C\left(\mathbf{f},g\right)+m^{2}\right).$

###  3.4. End of the proof of Proposition 3.5

We aim to prove that the following mapping possesses a fixed-point:

(3.35) $F_{n}:\mathbb{R}^{2\times n}\mapsto\mathbb{R}^{2\times
n},F_{n}:\left(\mathbf{b}_{n}\left(0\right),\mathbf{b}_{n}^{\prime}\left(0\right)\right)\mapsto\left(\mathbf{b}_{n}\left(T\right),\mathbf{b}_{n}^{\prime}\left(T\right)\right).$

This mapping associates to any set of initial data the solution
$\mathbf{b}_{n}$ of (3.18) evaluated at $t=T$. We will apply Schäffer’s Fixed
Point Theorem. The mapping $F_{n}$ is _continuous_ since a solution of the
above linear ODE depends continuously on the initial data. The _compactness_
is automatically ensured because we work in a finite dimensional Banach space
$\mathbb{R}^{2\times n}$.

In order to use Schäffer’s fixed-point Theorem 4.4 it suffices to prove that
the set

(3.36)
$S:=\left\\{\left(\mathbf{b}_{n}\left(0\right),\mathbf{b}_{n}^{\prime}\left(0\right)\right):\left(\mathbf{b}_{n}\left(0\right),\mathbf{b}_{n}^{\prime}\left(0\right)\right)=\lambda\left(\mathbf{b}_{n}\left(T\right),\mathbf{b}_{n}^{\prime}\left(T\right)\right)\
\text{for some}\ \lambda\in\left[0,1\right]\right\\}$

is uniformly bounded with respect to $\lambda$. Let us consider
$\left(b_{0},b_{1}\right)\in S$. We can exclude the trivial case $\lambda=0$
and assume $0<\lambda\leq 1$. Then it follows that

$E_{n}\left(T\right)=\frac{1}{\lambda^{2}}E_{n}\left(0\right)\geq
E_{n}\left(0\right).$

But now Proposition 3.22 provides a uniform bound on $S$ and a fixed-point is
obtained. The bounds on $S$ are in particular bounds on the fixed-point and
imply

(3.37)
$\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\mathbf{u}_{n}\right|^{2}+\left|\nabla\mathbf{u}_{n}\right|^{2}dxdt+\int_{\omega}\left|\partial_{t}\eta_{n}\right|^{2}dx^{\prime}dt\leq
c\left(\Omega,\kappa\right)C\left(\mathbf{f},g\right).$

Further the energy equality is satisfied for the fixed-point value:

(3.38)
$\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\nabla\mathbf{u}_{n}\right|^{2}dxdt=\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\mathbf{f}_{n}\cdot\mathbf{u}_{n}dx+\int_{\omega}g_{n}\partial_{t}\eta_{n}dx^{\prime}dt.$

Hence we are left to pass to the limit with $n\to\infty$. From (3.6) since
$\left\|\delta\right\|_{L^{\infty}\left(I\times\omega\right)}\leq
c\left(\omega\right)M_{1}$ we will choose

(3.39) $M_{1}\leq\frac{3\kappa}{4c\left(\omega\right)}$

while for $M_{2}$, since we want the mapping $\mathbf{v}\mapsto\mathbf{u}$ to
admit a fixed point, we choose $M_{2}$ so that the bounds on $\mathbf{v,u}$
are uniform w.r.t. $\varepsilon$ . In accordance with (3.37) we choose

(3.40) $M_{2}\leq c\left(c_{0},\Omega,\kappa\right)C(\mathbf{f},g).$

Now let $t\in[0,T]$ be a fixed time and let us integrate (3.21) on $[0,t]$. By
using (3.24) and (3.34) we get

$\displaystyle E_{n}\left(t\right)$
$\displaystyle=E_{n}\left(0\right)+\int_{0}^{t}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\mathbf{f}_{n}\cdot\mathbf{u}_{n}dxdt+\int_{0}^{t}\int_{\omega}g_{n}\partial_{t}\eta_{n}dx^{\prime}ds$
$\displaystyle\leq
R+\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\mathbf{f}_{n}\cdot\mathbf{u}_{n}\right|dxdt+\int_{0}^{T}\int_{\omega}\left|g_{n}\partial_{t}\eta_{n}\right|dx^{\prime}ds$
$\displaystyle\leq R+C\left(\omega,\kappa\right)C(\mathbf{f},g)$

which then reads as

(3.41) $\sup_{t\in\left[0,T\right]}E_{n}(t)\leq
C\left(\Omega,\kappa,c_{0},T\right)\cdot\Big{(}\left(C\left(\mathbf{f},g\right)\right)^{2}+C\left(\mathbf{f},g\right)+m^{2}\Big{)}.$

Now from (3.41) it follows that there is $(\mathbf{u},\eta)$ for which

(3.42) $\begin{split}\eta_{n}\to\eta&\text{\ weakly * \ in}\
L^{\infty}\left(0,T;H_{0}^{2}\left(\omega\right)\right)\\\
\partial_{t}\eta_{n}\to\partial_{t}\eta&\text{\ weakly * \ in}\
L^{\infty}\left(0,T;L^{2}\left(\omega\right)\right)\\\
\mathbf{u}_{n}\to\mathbf{u}&\text{\ weakly * \ in}\
V_{F,\text{per}}^{\mathcal{R}\delta}.\end{split}$

By weak lower semicontinuity we have

(3.43)
$\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\left|\nabla\mathbf{u}\right|^{2}dxdt\leq
c(c_{0},\kappa,\Omega)\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\mathbf{f}\cdot\mathbf{u}dx+\int_{0}^{T}\int_{\omega}g\partial_{t}\eta
dx^{\prime}dt.$

Using (3.42) and the Remark 2.11 it follows that

$\eta(0,\cdot)=\eta(T,\cdot).$

Note that from (3.42) and
$\text{tr}_{\mathcal{R}\delta}\mathbf{u}_{n}=\partial_{t}\eta_{n}\mathbf{e}_{3}$
by letting $n\to\infty$ we obtain
$\text{tr}_{\mathcal{R}\delta}\mathbf{u}=\partial_{t}\eta\mathbf{e}_{3}.$ To
end the proof, we multiply (3.18) by $\varphi\in C^{1}\left(0,T\right)\cap
C^{0}[0,T],\ \varphi\left(0\right)=\varphi\left(T\right)$ and integrate by
parts on $[0,T]$. We get

(3.44) $\displaystyle\begin{aligned}
&\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}-\mathbf{u}_{n}\cdot\partial_{t}\left(\varphi\left(t\right)\mathbf{X}_{k}\right)dx+\nabla\mathbf{u}_{n}:\nabla\left(\varphi\left(t\right)\mathbf{X}_{k}\right)dxdt\\\
&\quad+\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\frac{1}{2}\left(\mathcal{R}\mathbf{v}\cdot\nabla\right)\mathbf{u}_{n}\cdot\left(\varphi\left(t\right)\mathbf{X}_{k}\right)-\frac{1}{2}\left(\mathcal{R}\mathbf{v}\cdot\nabla\right)\left(\varphi\left(t\right)\mathbf{X}_{k}\right)\cdot\mathbf{u}_{n}dxdt\\\
&\quad-\int_{0}^{T}\int_{\omega}\frac{1}{2}\partial_{t}\eta_{n}\left(\varphi\left(t\right)X_{k}\right)\partial_{t}\mathcal{R}\delta\
dx^{\prime}dt+\partial_{t}\eta_{n}\partial_{t}\left(\varphi\left(t\right)X_{k}\right)dx^{\prime}+\int_{0}^{T}\left\langle
K^{\prime}\left(\eta_{n}\right),X_{k}\right\rangle dt\\\
&=\int_{0}^{T}\int_{\Omega_{\mathcal{R}\delta\left(t\right)}}\mathbf{f}_{n}\cdot\left(\varphi\left(t\right)\mathbf{X}_{k}\right)dxdt+\int_{0}^{T}\int_{\omega}g_{n}\varphi\left(t\right)X_{k}dx^{\prime}dt.\end{aligned}$

for all $k\in\\{1,...,n\\}$ We can now take an arbitrary linear combination of
the above and let $n\to\infty$ in (3.44); note that the weak convergences from
(3.42) are enough, since the problem is linear. Recall Remark 3.14 and the
proof is complete. ∎

###  3.5. $\varepsilon$-regularized periodic solutions

This part follows the already developed existence theory for the respective
Cauchy problem [9, 25, 23, 27]. Indeed, since the a-priori estimates for the
here constructed time-periodic system are the same as for the Cauchy problem,
the existence of solutions follows by the analysis provided there. For the
convenience of the reader we repeat here the main steps that have to be
performed in order to produce a weak solution. For the technical details we do
however refer to the references given above.

During this subsection we regain the coupling between fluid and shell, that is
we prove that the mapping $(\delta,\mathbf{v})\mapsto(\eta,\mathbf{u})$ has a
fixed point. To this end we use Theorem 4.10 which can be found in the
Appendix. However, note that the motion of the shell $\eta$ will still depend
on $\varepsilon$, since we still need its regularity ensured by the operators
introduced in Lemma 3.1. The last step will therefore consist of letting
$\varepsilon\to 0$ however this procedure takes place in the next section
only.

######  Definition 3.45.

We say that a couple
$(\mathbf{u},\eta)=(\mathbf{u}_{\varepsilon},\eta_{\varepsilon})\in
V_{S,\text{per}}^{\mathcal{R}_{\varepsilon}\eta}$ is a weak time-periodic
solution of the $\varepsilon$-regularized problem in the interval $I=(0,T)$ if
$\left\|\eta\right\|_{L^{\infty}\left(I\times\omega\right)}<\kappa$, if

(3.46)
$\begin{split}\int_{0}^{T}\int_{\Omega_{\mathcal{R}_{\varepsilon}\eta\left(t\right)}}-\mathbf{u}\cdot\partial_{t}\mathbf{q}+\nabla\mathbf{u}:\nabla\mathbf{q}+\frac{1}{2}\left(\mathcal{R}_{\varepsilon}\mathbf{u}\cdot\nabla\right)\mathbf{u}\cdot\mathbf{q}-\frac{1}{2}\left(\mathcal{R}_{\varepsilon}\mathbf{u}\cdot\nabla\right)\mathbf{q}\cdot\mathbf{u}dxdt&-\\\
\int_{0}^{T}\int_{\omega}\frac{1}{2}\partial_{t}\eta\partial_{t}\mathcal{R}_{\varepsilon}\eta\xi+\partial_{t}\eta\partial_{t}\xi
dx^{\prime}+\left\langle
K_{\varepsilon}^{\prime}\left(\eta\right),\xi\right\rangle dt&=\\\
\int_{0}^{T}\int_{\Omega_{\mathcal{R}_{\varepsilon}\eta\left(t\right)}}\mathbf{f}\cdot\mathbf{q}dx+\int_{\omega}g\xi
dx^{\prime}dt&\forall\left(\mathbf{u},\xi\right)\in
V_{T,\text{per}}^{\mathcal{R}_{\varepsilon}\eta}\end{split}.$

Then we aim to prove that

######  Proposition 3.47.

Let $m\in\mathbb{R}$. There exists a constant

(3.48) $C_{0}=C_{0}(c_{0},\Omega,\kappa,T)$

such that if

(3.49) $m^{2}+C\left(\mathbf{f},g\right)^{2}\leq C_{0}$

then for all sufficiently small $\varepsilon>0$, there exists at least one
weak time periodic solution
$\left(\mathbf{u},\eta\right)=\left(\mathbf{u}_{\varepsilon},\eta_{\varepsilon}\right)$
for the $\varepsilon$-regularized problem such that

(3.50) $\int_{\omega}\eta\left(t,\cdot\right)dx^{\prime}=m\ \quad\forall t\in
I$

Furthermore, by denoting (as before)

$E\left(t\right):=\int_{\Omega_{\mathcal{R}_{\varepsilon}\eta\left(t\right)}}\left|\mathbf{u}\right|^{2}dx+\int_{\omega}\left|\partial_{t}\eta\right|^{2}dx^{\prime}+K\left(\eta(t)\right),\
t\in\left[0,T\right]$

it holds that

(3.51) $\text{ess}\sup_{t\in
I}E\left(t\right)+\int_{0}^{T}\int_{\Omega_{\mathcal{R}_{\varepsilon}\eta}}\left|\nabla\mathbf{u}\right|^{2}dxdt\leq
C(\Omega,\kappa,c_{0},T)\cdot\Big{(}\left(C\left(\mathbf{f},g\right)\right)^{2}+C\left(\mathbf{f},g\right)+m^{2}\Big{)}$

and

(3.52)
$\int_{0}^{T}\int_{\Omega_{\mathcal{R}_{\varepsilon}\eta\left(t\right)}}\left|\nabla\mathbf{u}\right|^{2}dxdt\leq\int_{0}^{T}\int_{\Omega_{\mathcal{R}_{\varepsilon}\eta\left(t\right)}}\mathbf{f}\cdot\mathbf{u}dxdt+\int_{\omega}g\cdot\partial_{t}\eta
dx^{\prime}dt.$

###### Proof.

Let us consider the space

$Z:=L_{\text{per}}^{2}\left(\mathbb{R};L^{2}\left(\mathbb{R}^{3}\right)\right)\times
C_{\text{per}}^{0}\left(\mathbb{R};C^{0}\left(\overline{\omega}\right)\right)\
$

and its convex subset

$D:=\left\\{\left(\mathbf{v},\delta\right)\in
Z:\left\|\delta\right\|_{L_{t}^{\infty}L_{x}^{\infty}}\leq
M_{1},\left\|\mathbf{v}\right\|_{L_{t}^{2}L_{x}^{2}}\leq M_{2},\
\int_{\omega}\delta\left(t,\cdot\right)d\equiv m\right\\}$

where $M_{1}$ and $M_{2}$ are given by (3.39) and (3.40). We define

$F:Z\mapsto\mathcal{P}(Z),\quad
F\left(\mathbf{v},\delta\right):=\left\\{\left(\mathbf{u},\eta\right):\left(\mathbf{u},\eta\right)\in
V_{T,\text{per}}^{\mathcal{R}\delta}\text{ is a solution for}\ \eqref{eq:eps-
regularized}\right\\}.$

Here we extend for almost every $t\in I$ the function $\mathbf{u}(t,\cdot)$ to
$\mathbb{R}^{3}$ by zero. We would like to have

$F:D\subset Z\mapsto\mathcal{P}(D)$

and therefore from (3.9), (3.39), (3.40) we need to impose the smallness
condition (3.50). Further we check the assumptions of Theorem 4.10:

  1. (i)

For all $(\mathbf{v},\delta)\in D$ the set $F(\mathbf{v},\delta)$ is _non-
empty_ (due to Proposition 3.5), _convex_ (since the problem is linearized).
The fact that $F(D)\subset D$ follows by imposing the _smallness_ condition
(3.49) on the forces. More precisely, this due to the estimate (3.41).

  2. (ii)

$F(D)$ is _relatively compact_ in $Z$: here we need to employ a compactness
result, for example the one from [23, Proposition 3.8]. Please observe, that
this can indeed be applied since after establishing the Proposition 3.22 we
are able to select a set of initial data from a uniformly bounded (by $R$, see
(3.34)) set. Note that for sufficiently small $\varepsilon$ the regularizing
operators ensure according to Lemma 3.1 that

$\mathcal{R}:\left\\{\delta\in
C\left(\overline{I}\times\omega\right):\delta\left(0,\cdot\right)=\delta\left(T,\cdot\right)\right\\}\mapsto
C^{3}\left(\overline{I}\times\omega\right)\hookrightarrow\hookrightarrow
C^{2}\left(\overline{I}\times\omega\right).$

  3. (iii)

$F$ has _closed graph_ since if $(\mathbf{v}_{n},\
\delta_{n})\to(\mathbf{v},\delta)$ and
$\left(\mathbf{v}_{n},\delta_{n}\right)\in
F\left(\mathbf{v}_{n},\delta_{n}\right)$ with
$\left(\mathbf{u}_{n},\eta_{n}\right)\to\left(\mathbf{u},\eta\right)$, then
$(\eta,\mathbf{u})\in F(\delta,\mathbf{v})$. The latter is an easy exercise
especially since the problem is linearized. To prove that
$\eta(0,\cdot)=\eta(T,\cdot)$ it suffices to note that
$\eta_{n}(0,\cdot)=\eta_{n}(T,\cdot)$ and we get strong convergence for every
time due to the Remark 2.11.

Therefore the assumptions of the Theorem 4.10 are fulfilled and we can
guarantee the existence of a pair $(\eta,\mathbf{u})\in D$ for which
$(\eta,\mathbf{u})\in F(\eta,\mathbf{u})$. Once the fixed-point is
established, the estimates (3.51) and (3.52) follow from (3.8) and (3.9). ∎

######  Remark 3.53.

We would like to point out that the same reasoning would have worked at the
discrete (Galerkin) level, by replacing $\delta$ with $\delta_{n}$\- a
discrete but already regularized version. We refer to [5, Section 4.1] where
this was successfully performed.

###  3.6. $\varepsilon\to 0$ limit and the proof of Theorem 1.22

###### Proof.

Our proof follows the same lines as in [23] and [27].
Recalling the definition of $\varepsilon$-regularized periodic solutions from
(3.46), we know that for all sufficiently small $\varepsilon>0$ it holds that:

(3.54)
$\displaystyle\int_{0}^{T}\int_{\Omega_{\mathcal{R}_{\varepsilon}\eta_{\varepsilon}\left(t\right)}}-\mathbf{u}_{\varepsilon}\cdot\partial_{t}\mathbf{q}_{\varepsilon}dx+\nabla\mathbf{u}_{\varepsilon}:\nabla\mathbf{q}_{\varepsilon}+\frac{1}{2}\left(\mathcal{R}_{\varepsilon}\mathbf{u}_{\varepsilon}\cdot\nabla\right)\mathbf{u}_{\varepsilon}\cdot\mathbf{q}_{\varepsilon}-\frac{1}{2}\int_{\Omega_{\mathcal{R}_{\varepsilon}\eta\left(t\right)}}\left(\mathcal{R}_{\varepsilon}\mathbf{u}_{\varepsilon}\cdot\nabla\right)\mathbf{q}\cdot\mathbf{u}_{\varepsilon}dxdt+$
$\displaystyle\int_{0}^{T}\int_{\omega}-\partial_{t}\eta_{\varepsilon}\partial_{t}\xi_{\varepsilon}dx^{\prime}+\left\langle
K_{\varepsilon}^{\prime}\left(\eta_{\varepsilon}\right),\xi_{\varepsilon}\right\rangle
dt=\int_{0}^{T}\int_{\Omega_{\mathcal{R}_{\varepsilon}\eta\left(t\right)}}\mathbf{f}\cdot\mathbf{q}_{\varepsilon}dx+\int_{\omega}g\xi_{\varepsilon}dx^{\prime}dt\quad\forall\left(\mathbf{q}_{\varepsilon},\xi_{\varepsilon}\right)\in
V_{T,\text{per}}^{\mathcal{R}_{\varepsilon}\eta_{\varepsilon}}$

We obtain the existence of a pair $(\mathbf{u},\eta,)$ for which

(3.55)
$\displaystyle\eta_{\varepsilon},\mathcal{R}_{\varepsilon}\eta_{\varepsilon}\to\eta$
$\displaystyle\quad\text{uniformly and weakly}^{*}\ \text{in}\
L^{\infty}\left(I;H_{0}^{2}\left(\omega\right)\right)$
$\displaystyle\partial_{t}\eta_{\varepsilon},\mathcal{R}_{\varepsilon}\partial_{t}\eta_{\varepsilon}\to\partial_{t}\eta$
$\displaystyle\quad\text{weakly}^{*}\ \text{in}\
L^{\infty}\left(I;L^{2}\left(\omega\right)\right)$
$\displaystyle\mathbf{u}_{\varepsilon}\to\mathbf{u}$
$\displaystyle\quad\text{weakly}^{*}\ \text{in}\
L^{\infty}\left(I;L^{2}\left(\mathbb{R}^{3}\right)\right)$
$\displaystyle\nabla\mathbf{u}_{\varepsilon}\to\nabla\mathbf{u}$
$\displaystyle\quad\text{weakly in}\
L^{2}\left(I;L^{2}\left(\mathbb{R}^{3}\right)\right)$

for some subsequences which we choose not to relabel in order to keep the
notation light. Note that this suffices to ensure that
$\eta(0,\cdot)=\eta(T,\cdot)$ in $\omega$ and
$\left\|\eta\right\|_{L^{\infty}\left(I\times\omega\right)}<\kappa$. This time
however our problem is _nonlinear_ and therefore strong convergence is needed
in order to pass to the limit in (3.54). Following now [9, Section 4] or [27,
Lemma 6.3] we get that

(3.56)
$\left(\mathbf{u}_{\varepsilon},\partial_{t}\eta_{\varepsilon}\right)\to\left(\mathbf{u},\partial_{t}\eta\right)\quad\text{in}\
L^{2}\left(I;L^{2}\left(\mathbb{R}^{3}\right)\right)\times
L^{2}\left(I;L^{2}\left(\omega\right)\right).$

In particular, by interpolating $L^{4}$ between $L^{2}$ and $L^{5}$ we easiliy
obtain that

(3.57)
$\mathbf{u}_{\varepsilon},\mathcal{R}_{\varepsilon}\mathbf{u}_{\varepsilon}\to\mathbf{u}\quad\text{in}\
L^{2}\left(I;L^{4}\left(\mathbb{R}^{3}\right)\right).$

This now proves the Theorem 1.22. ∎

## Acknowledgments

C. M. and S. S. thank the support of the Primus research programme
PRIMUS/19/SCI/01 and of the program GJ19-11707Y of the Czech National Grant
Agency (GAČR). They also thank the SVV program SVV-2020-260583 of the Faculty
of Mathematics and Physics of Charles University.

Moreover, S.S. thanks for the support of the University Centre UNCE/SCI/023 of
Charles University.

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##  4\. Appendix

###  4.1. Reynolds’ transport theorem

We recall the classical Reynolds’ Transport Theorem, which in our context (see
Subsection 1.2) reads as follows

######  Theorem 4.1.

It holds that

$\frac{d}{dt}\int_{\Omega_{\eta\left(t\right)}}g\left(t,x\right)dx=\int_{\Omega_{\eta\left(t\right)}}\partial_{t}g\left(t,x\right)dx+\int_{\partial\Omega_{\eta\left(t\right)}}\partial_{t}\eta\circ\psi_{\eta}^{-1}\mathbf{e}_{3}\cdot\nu_{\eta}gdx^{\prime}$

whenever all the terms that appear are well-defined.

The latter result will be used intensively during this work.

###  4.2. Poincaré’s inequality

######  Theorem 4.2.

[24, Theorem 13.15] If $\Omega\subset\mathbb{R}^{N}$ is an open set that lies
between two hyperplanes which are at distance $d$, $m\in\mathbb{N}$ and $1\leq
p<\infty$, then there is a constant $c=c(m,N,p)>0$ such that

$\int_{\Omega}\left|\nabla^{k}\mathbf{u}\right|^{p}dx\leq
c\frac{d^{p\left(m-k\right)}}{p\left(m-k\right)}\int_{\Omega}\left|\nabla^{m}\mathbf{u}\right|^{p}dx\quad\forall\mathbf{u}\in
W_{0}^{m,p}(\Omega),0\leq k\leq m-1.$

###  4.3. Korn’s identity

######  Lemma 4.3.

[23, Lemma A.5] In the context of the introduction it holds that

$\int_{\Omega_{\eta\left(t\right)}}\left(\nabla\mathbf{u}\right)^{T}:\nabla\mathbf{\xi}dx=0$

for all functions $\mathbf{\xi}$ with $\text{tr}_{\eta}\mathbf{\xi}=b\nu$ for
some scalar function $b$; in particular for $\mathbf{\xi}=\mathbf{u}$.

###  4.4. The Schäffer’s/ Leray-Schauder’s fixed point Theorem

######  Theorem 4.4.

[11, Theorem 4] Let $X$ be a Banach space and $A:X\mapsto X$ be a continuous
and compact mapping such that the set

$\left\\{x\in X:x=\lambda Ax\ \text{for some}\
\lambda\in\left[0,1\right]\right\\}$

is bounded. Then $A$ possesses a fixed point.

###  4.5. The Piola mapping

We follow [23, Remark 2.5] and state the following

######  Lemma 4.5.

For any $\delta\in C^{2}(\omega)$ with
$\left\|\delta\right\|_{L^{\infty}\left(\omega\right)}<\kappa$ and
$\varphi:\omega\mapsto\mathbb{R}^{3}$ we define the _Piola transform_ of
$\varphi$ under $\psi_{\delta}$ by

$\mathcal{J}_{\delta}\varphi:=\left(d\psi_{\delta}\left(\det
d\psi_{\delta}\right)^{-1}\varphi\right)\circ\psi_{\delta}^{-1}.$

Then $\mathcal{J}_{\delta}$ defines an isomorphism between the Lebesgue and
Sobolev spaces on $\Omega$ and the corresponding ones on $\Omega_{\delta}$
which moreover preserves the zero boundary values and the divergence-free
condition.

###### Proof.

It can be shown that
$\partial_{y}\mathcal{J}_{\delta}\varphi\circ\psi_{\delta}=\left(\det
d\psi_{\delta}\right)^{-1}\partial_{i}\varphi\ $ and thus
$\text{div}\mathcal{J}_{\delta}\varphi\circ\psi_{\delta}=\left(\det
d\psi_{\delta}\right)^{-1}\text{div}\varphi$. ∎

###  4.6. The trace operator

In order to rigorously justify the traces of functions defined on
$\Omega_{\eta(t)}$ we use

######  Lemma 4.6.

[23, Corollary 2.9] If $1<p\leq\infty$ and $\eta\in H^{2}(\omega)$ with
$\left\|\eta\right\|_{L^{\infty}\left(\omega\right)}<\kappa$, then for any
$r\in(1,p)$ the mapping

$\text{tr}_{\eta}:W^{1,p}\left(\Omega_{\eta}\right)\mapsto
W^{1-\frac{1}{r},r}\left(\omega\right),\quad\text{tr}_{\eta}\left(v\right):=\left(v\circ\psi_{\eta}\right)_{|\omega}$

is well defined and continuous, with continuity constant depending on
$\Omega,r,p$ and a bound for $\left\|\eta\right\|_{H^{2}\left(\omega\right)}$
and $\tau(\eta)$.

###  4.7. A divergence-free extension operator

######  Proposition 4.7.

[27, Proposition 3.3] Let $\eta\in L^{\infty}(I;W^{1,2}(\omega))$ be with
$\left\|\eta\right\|_{L^{\infty}\left(I\times\omega\right)}<\alpha<\kappa$ and
denote
$S_{\alpha}:=\left\\{x:\text{dist}\left(x,\partial\Omega\right)<\alpha\right\\}$.
Then there exists linear and bounded operators

$\mathcal{M}_{\eta}:L^{1}\left(\omega\right)\mapsto\mathbb{R},\quad\mathcal{F}_{\eta}:\left\\{\xi\in
L^{1}\left(I;W^{1,1}\left(\omega\right)\right):\int_{\omega}\xi
dx^{\prime}=0\right\\}\mapsto
L^{1}\left(I;W_{\text{div}}^{1,1}\left(\Omega\cup S_{\alpha}\right)\right)$

such that for all $\xi\in L^{\infty}(I;W^{2,2}(\omega))\cap
W^{1,\infty}(I;L^{2}(\omega))$ we have that

(4.8)
$\displaystyle\mathcal{F}_{\eta}\left(\mathcal{M}_{\eta}\left(\xi\right)\right)\in
L^{\infty}\left(I;L^{2}\left(\Omega_{\eta}\right)\right)\cap
L^{2}\left(I;W_{\text{div}}^{1,2}\left(\Omega_{\eta}\right)\right)$
$\displaystyle\mathcal{M_{\eta}}\left(\xi\right)\in
L^{\infty}\left(I;W^{2,2}\left(\omega\right)\right)\cap
W^{1,\infty}\left(I;L^{2}\left(\omega\right)\right)$
$\displaystyle\text{tr}_{\eta}\mathcal{F}_{\eta}\left(\mathcal{M_{\eta}}\left(\xi\right)\right)=\mathcal{M_{\eta}}\left(\xi\right)\mathbf{\nu}$
$\displaystyle\mathcal{F}_{\eta}\left(\mathcal{M}\left(\xi\right)\right)=0\quad\text{on}\
I\times\Omega\setminus S_{\alpha}$

and furthermore for all $p\in(1,\infty)$ and $q\in(1,\infty]$ we have

$\left\|\mathcal{F}_{\eta}\left(\mathcal{M}\left(\xi\right)\right)\right\|_{L^{q}\left(I;W^{1,p}\left(\Omega\cup
S_{\alpha}\right)\right)}\lesssim\left\|\xi\right\|_{L^{q}\left(I;W^{1,p}\left(\omega\right)\right)}+\left\|\xi\nabla\eta\right\|_{L^{q}\left(I;L^{p}\left(\omega\right)\right)}$

and

$\left\|\partial_{t}\mathcal{F}_{\eta}\left(\mathcal{M}_{\eta}\left(\xi\right)\right)\right\|_{L^{q}\left(I;L^{p}\left(\Omega\cup
S_{\alpha}\right)\right)}\lesssim\left\|\partial_{t}\xi\right\|_{L^{q}\left(I;L^{p}\left(\omega\right)\right)}+\left\|\xi\partial_{t}\eta\right\|_{L^{q}\left(I;L^{p}\left(\omega\right)\right)}.$

The continuity constant depends only on $\Omega$, $p$, and $\kappa$.

###  4.8.  Sobolev embedding on moving domains

######  Lemma 4.9.

[23, Corollary 2.10] If $1<p<3$ and $\eta\in H^{2}(\omega)$ with
$\left\|\eta\right\|_{L^{\infty}\left(\omega\right)}<\kappa$ then

$W^{1,p}\left(\Omega_{\eta}\right)\hookrightarrow
L^{p^{*}}\left(\Omega_{\eta}\right)\ \text{with}\ p^{*}:=\frac{3p}{3-p}$

and the embedding constant depends on $\Omega$, $p$ and a bound for
$\left\|\eta\right\|_{H^{2}\left(\omega\right)}$ and $\tau(\eta)$.

###  4.9. The Kakutani-Glicksberg-Fan fixed point theorem

We will need the following set-valued fixed-point result which can be found in
[18] or in [19, Chapter 2, Section 5.8]

######  Theorem 4.10 (Kakutani-Glicksberg-Fan).

Let $C$ be a convex subset of a normed vector space $Z$ and let
$F:C\to\mathcal{P}(C)$ be a set-valued mapping which has closed graph.
Moreover, let $F(C)$ be contained in a compact subset of $C$, and let $F(z)$
be non-empty, convex, and compact for all $z\in C$. Then $F$ possesses a fixed
point, that is there is $c_{0}\in C$ with $c_{0}\in F(c_{0})$.

We say a set-valued mapping $F:C\mapsto\mathcal{P}(C)$ has _closed graph_
provided that the set $\left\\{\left(x,y\right):y\in F\left(x\right)\right\\}$
is closed in $X\times Y$ with the product topology or that, equivalently, for
any sequences $x_{n}\to x$ and $y_{n}\to y$ with $y_{n}\in F(x_{n})$ for any
$\geq 1$ it follows that $y\in F(x)$.