# Global weak solutions to a 3D/3D fluid-structure interaction problem
including possible contacts

Malte Kampschulte, Boris Muha, Srđan Trifunović

###### Abstract

In this paper, we study an interaction problem between a $3D$ compressible
viscous fluid and a $3D$ nonlinear viscoelastic solid fully immersed in the
fluid, coupled together on the interface surface. The solid is allowed to have
self-contact or contact with the rigid boundary of the fluid container. For
this problem, a global weak solution with defect measure is constructed by
using a multi-layered approximation scheme which decouples the body and the
fluid by penalizing the fluid velocity and allowing the fluid to pass through
the body, while the body is supplemented with a contact-penalization term. The
resulting defect measure is a consequence of pressure concentrations that can
appear where the fluid meets the (generally irregular) points of self-contact
of the solid. Moreover, we study some geometrical properties of the fluid-
structure interface and the contact surface. In particular, we prove a lower
bound on area of the interface.

Keywords and phrases: fluid-structure interaction, compressible viscous fluid,
second-grade viscoelasticity

AMS Mathematical Subject classification (2020): 76F10 (Primary), 76N06, 74B20
, 74M15 (Secondary)

##  1 Introduction

###  1.1 Problem definition

We consider the motion of $3D$ viscoelastic solid fully immersed into a
barotropic compressible fluid contained in a container with rigid walls. Let
$\Omega_{S}$, $\Omega\subset{\mathbb{R}}^{3}$ be open, bounded, connected,
$C^{1,\alpha}$ sets, $\alpha>0$, representing the reference position of the
viscoelastic solid and the container, respectively. The unknowns of the system
are the following:

  1. The elastic solid deformation ${\boldsymbol{\eta}}:(0,T)\times{\Omega_{S}}\to\Omega$;

  2. The fluid velocity $\mathbf{u}:(0,T)\times\Omega_{F}^{{\boldsymbol{\eta}}}(t)\to\mathbb{R}^{3}$;

  3. The fluid density $\rho:(0,T)\times\Omega_{F}^{{\boldsymbol{\eta}}}(t)\to\mathbb{R}$;

  4. The elastic solid at time $t$ $\Omega^{\boldsymbol{\eta}}_{S}(t):=\text{int}({\boldsymbol{\eta}}(t,\overline{\Omega_{S}}))$;

  5. The fluid domain at time $t$ $\Omega_{F}^{{\boldsymbol{\eta}}}(t):=\text{int}(\Omega\setminus\Omega_{S}^{\boldsymbol{\eta}}(t))$.

Notice that such a non-standard definition for fluid and solid domains (see
Figure 1) is used because we want to include the interior points of self-
contact into the structure domain, rather than into the fluid domain. This
happens when ${\boldsymbol{\eta}}(t)$ is not injective on
$\partial\Omega_{S}$. We assume $\Omega_{S}^{\boldsymbol{\eta}}(0)\neq\Omega$,
i.e. the solid does not completely fill the container at the initial time. We
will prove that, as a consequence of conservation of mass and boundedness of
energy, that the same is true at any time. Moreover, for time-dependent sets,
by slight abuse of notation, we will write $I\times S(t):=\cup_{t\in
I}\\{t\\}\times S(t)$, and some time-space domains which will be frequently
used will be denoted as:

$\displaystyle Q_{S,T}:=(0,T)\times{\Omega_{S}},\quad
Q_{S,T}^{\boldsymbol{\eta}}:=(0,T)\times\Omega_{S}^{{\boldsymbol{\eta}}}(t),\quad
Q_{F,T}^{{\boldsymbol{\eta}}}:=(0,T)\times\Omega_{F}^{{\boldsymbol{\eta}}}(t),\quad
Q_{T}:=(0,T)\times\Omega.$

Figure 1: Lagrangian coordinates (left) and physical coordinates (right) at a
time $t$.

####  1.1.1 Elastodynamics

The motion of the solid is given by nonlinear elastodynamics equations:

$\displaystyle\partial_{t}^{2}{{\boldsymbol{\eta}}}=\nabla\cdot\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\quad{\rm
in}\quad Q_{S,T},$ (1.1)

where $\boldsymbol{\sigma}$ is the first Piola-Kirchhoff stress tensor. We
require that deformation satisfies the Ciralet-Nečas condition introduced in
[12], i.e. it is orientation preserving and injective on the interior of
$\Omega_{S}$. Therefore we define the following function space for the
deformations:

$\displaystyle\mathcal{E}^{q}:=\left\\{{\boldsymbol{\eta}}\in
W^{2,q}(\Omega_{S}):|{\boldsymbol{\eta}}(\Omega_{S})|=\int_{\Omega_{s}}\det\nabla{\boldsymbol{\eta}},\quad\inf_{\Omega_{S}}\det\nabla{\boldsymbol{\eta}}>0,\quad\overline{\Omega_{S}^{\boldsymbol{\eta}}}\subseteq\overline{\Omega}\right\\},$

for $q>3$. We also prescribe the following boundary condition when the solid
is in contact with the rigid boundary, which states that the solid does not
penetrate or slip on the rigid boundary:

$\displaystyle\partial_{t}{\boldsymbol{\eta}}(t,x)=0,\quad{\rm for}\quad
t\in(0,T)\text{ and }x\in\partial\Omega_{S}\text{ such that
}{\boldsymbol{\eta}}(t,x)\in\partial\Omega.$ (1.2)

Here we consider nonlinearly elastic material of second order, i.e. a
generalized standard material (see e.g. [26], [32, Sec. 2.5]) with
constitutive law:

$\displaystyle\nabla\cdot\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}}):=-DE({\boldsymbol{\eta}})-D_{2}R({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}}),\quad\text{
in }\mathcal{D}^{\prime}(Q_{S,T}),$

where $E$ is the energy functional, $R$ is the dissipation functional, $D$ is
the Fréchet derivative and $D_{2}$ is the Fréchet derivative with respect to
the second argument. Since we are working with a second-gradient material, we
must impose another boundary condition on $\nabla^{2}{\boldsymbol{\eta}}$.
This condition is chosen so that the energy in our coupled system is conserved
properly. Since it is cumbersome to write it explicitly, we can define it
implicitly by imposing that

$\displaystyle\int_{\Omega_{S}}\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}}):\nabla\boldsymbol{\xi}=\langle
DE({\boldsymbol{\eta}}),\boldsymbol{\xi}\rangle+\langle
D_{2}R({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}}),\boldsymbol{\xi}\rangle,\quad\text{
for all }\boldsymbol{\xi}\in C^{\infty}(\Omega_{S}).$ (1.3)

Finally, we assume that $E$ and $R$ satisfy the following conditions:

  * (E1)

$E:W^{2,q}(\Omega_{S})\to\mathbb{R}\cup\\{+\infty\\}$ is bounded from below,
weakly lower-semicontinuous and coercive;

  * (E2)

For every $E_{0}>0$ there is a $\varepsilon_{0}>0$ such that
$\det\nabla{\boldsymbol{\eta}}(x)>\varepsilon_{0}$ for all $x\in\Omega_{S}$
and all $\boldsymbol{{\boldsymbol{\eta}}}$ with
$E(\boldsymbol{{\boldsymbol{\eta}}})<E_{0}$;

  * (E3)

Whenever it is finite, $E$ is Fréchet-differentiable with a derivative
$DE:\\{{\boldsymbol{\eta}}\in
W^{2,q}(\Omega_{S}):E({\boldsymbol{\eta}})<\infty\\}\to[W^{2,q}(\Omega_{S})]^{\prime}$
which is continuous with respect to weak topology;

  * (E4)

There is a Minty-type property: If
${\boldsymbol{\eta}}_{k}\rightharpoonup{\boldsymbol{\eta}}$ in
$W^{2,q}(\Omega_{S})$ and $E({\boldsymbol{\eta}}_{k})$ is bounded, then
$\liminf_{k\to\infty}\langle
DE({\boldsymbol{\eta}}_{k})-DE({\boldsymbol{\eta}}),\psi({\boldsymbol{\eta}}_{k}-{\boldsymbol{\eta}})\rangle\geq
0$, where $\psi\in C_{c}(\Omega_{S};[0,1])$ is an arbitrary cut-off function.
Furthermore, if also $\limsup_{k\to\infty}\langle
DE({\boldsymbol{\eta}}_{k})-DE({\boldsymbol{\eta}}),\psi({\boldsymbol{\eta}}_{k}-{\boldsymbol{\eta}})\rangle\leq
0$ for all such $\psi$, then this implies
${\boldsymbol{\eta}}_{k}\to{\boldsymbol{\eta}}$ in $W^{2,q}(\Omega_{S})$;

  * (R1)

$R:W^{2,q}(\Omega_{S})\times W^{1,2}(\Omega_{S})\to\mathbb{R}$ is weakly lower
semicontinuous, 2-homogeneous in the second argument, i.e.
$R(\boldsymbol{{\boldsymbol{\eta}}},\lambda\boldsymbol{b})=\lambda^{2}R(\boldsymbol{{\boldsymbol{\eta}}},\boldsymbol{b})$
for all $\lambda\in\mathbb{R}$ and continuously Fréchet-differentiable with
respect to the second argument;

  * (R2)

There exists a (possibly energy dependent) Korn-inequality: For any fixed
$E_{0}>0$ and all ${\boldsymbol{\eta}}\in\mathcal{E}^{q}$ with
$E({\boldsymbol{\eta}})\leq E_{0}$, we have $\|b\|_{W^{1,2}}\leq
c(E_{0})(\|b\|_{L^{2}}+R({\boldsymbol{\eta}},b))$ for all $b\in
W^{1,2}(\Omega_{S})$.

A prototypical example of such an energy-dissipation pair is of the form

$\displaystyle R({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})$
$\displaystyle:=\int_{\Omega_{S}}|\partial_{t}\nabla{\boldsymbol{\eta}}^{\tau}\nabla{\boldsymbol{\eta}}+\nabla{\boldsymbol{\eta}}^{\tau}\partial_{t}\nabla{\boldsymbol{\eta}}|^{2},$
$\displaystyle E({\boldsymbol{\eta}})$
$\displaystyle:=\begin{cases}\int_{\Omega_{S}}\mathcal{C}(\nabla{\boldsymbol{\eta}}^{\tau}\nabla{\boldsymbol{\eta}}-I)+\frac{1}{\det(\nabla{\boldsymbol{\eta}})^{a}}+\frac{1}{q}|\nabla^{2}{\boldsymbol{\eta}}|^{q},&\text{
if }\det\nabla{\boldsymbol{\eta}}>0\text{ a.e. in }\Omega_{S},\\\
\infty,&\text{ otherwise},\end{cases}$

where $\mathcal{C}$ is a positive definite quadratic tensor and
$a>\frac{3q}{q-3}$. For a much more detailed discussion, we refer to Sections
1.4 and 2.3 of [2].

####  1.1.2 Fluid equations

The fluid flow is governed by the compressible Navier-Stokes equations:

$\displaystyle\partial_{t}\rho+\nabla\cdot(\rho\mathbf{u})$ $\displaystyle=$
$\displaystyle 0\quad{\rm in}\quad Q_{F,T}^{{\boldsymbol{\eta}}},$ (1.4)
$\displaystyle\partial_{t}(\rho\mathbf{u})+\nabla\cdot(\rho\mathbf{u}\otimes\mathbf{u})$
$\displaystyle=$ $\displaystyle\nabla\cdot\mathbb{T}({\bf u},p)\quad{\rm
in}\quad Q_{F,T}^{{\boldsymbol{\eta}}},$ (1.5) $\displaystyle{\bf u}$
$\displaystyle=$ $\displaystyle 0\quad{\rm
on}\;(0,T)\times(\partial\Omega\setminus\partial\Omega_{S}^{\boldsymbol{\eta}}(t)),$
(1.6)

where $\mathbb{T}({\bf u},p)=\mathbb{S}({\bf u})-p\mathbb{I}$ is the fluid
Cauchy stress tensor. Here $\mathbb{S}$ is the viscous stress tensor whose
constitutive relation is given by standard Newton rheological law

$\displaystyle\mathbb{S}(\nabla\mathbf{u}):=\mu\big{(}\nabla\mathbf{u}+\nabla^{\tau}\mathbf{u}-\frac{2}{3}\nabla\cdot\mathbf{u}\big{)}+\zeta\nabla\cdot\mathbf{u},\quad\mu,\zeta>0,$

where $\mu$ is the shear viscosity coefficient and $\zeta$ the bulk viscosity
coefficient. We assume following state equation for the pressure

$\displaystyle p=p(\rho)=\rho^{\gamma},$ (1.7)

where $\gamma$ is the adiabatic exponent.

####  1.1.3 Coupling conditions

Since we consider a coupled problem, we need to prescribe two types of
coupling conditions for two types of interactions. They read as follows:

Fluid-structure coupling on the Lagrangian fluid-structure interface
$Q_{\boldsymbol{\eta}}^{I}$:

Kinematic coupling condition:
$\displaystyle~{}\partial_{t}{\boldsymbol{\eta}}$
$\displaystyle=\mathbf{u}\circ{\boldsymbol{\eta}},$ (1.8) Dynamic coupling
condition:
$\displaystyle~{}\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\mathbf{n}$
$\displaystyle=\det\nabla{\boldsymbol{\eta}}\left[\mathbb{T}({\bf
u},p)\circ{\boldsymbol{\eta}}\right]\nabla{\boldsymbol{\eta}}^{-\tau}\mathbf{n},$
(1.9)

where $Q_{\boldsymbol{\eta}}^{I}\subseteq(0,T)\times\partial\Omega_{S}$ is
defined in Section 2 and $\mathbf{n}$ is the unit outer normal vector on
$\partial\Omega_{S}$;

Structure-structure coupling on the set of self-contact

Kinematic coupling condition:
$\displaystyle~{}\partial_{t}{\boldsymbol{\eta}}(t,x)$
$\displaystyle=\partial_{t}{\boldsymbol{\eta}}(t,y),\quad$ (1.10) Dynamic
coupling condition:
$\displaystyle~{}\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})(t,x)\mathbf{n}(x)dS(x)$
$\displaystyle=-\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})(t,y)\mathbf{n}(y)dS(y),\quad$
(1.11)

for all $t\in(0,T)$ and $x\neq y\in\partial\Omega_{S}$ such that
${\boldsymbol{\eta}}(t,x)={\boldsymbol{\eta}}(t,y)$, where
$dS:\partial\Omega\to\mathbb{R}^{+}$ is the surface element.

Notice that since we prescribe coupling conditions on the reference domain, we
needed to apply the Piola transform to the Cauchy stress tensor of the fluid
to get the first Piola-Kirchhoff stress tensor in (1.9). Next, we prescribe
the inital data:

$\displaystyle{{\boldsymbol{\eta}}}(0,\cdot)={{\boldsymbol{\eta}}}_{0},\quad\partial_{t}{{\boldsymbol{\eta}}}(0,\cdot)=\mathbf{v}_{0},\quad~{}\rho(0,\cdot)=\rho_{0},\quad(\rho\mathbf{u})(0,\cdot)=(\rho\mathbf{u})_{0},$
(1.12)

and assume that it satifies the following compatibility conditions:

$\displaystyle\rho_{0}\in
L^{\gamma}(\Omega^{{\boldsymbol{\eta}}_{0}}),\quad\rho_{0}\geq
0,\quad\rho_{0}\not\equiv
0,\quad\rho_{0}|_{\mathbb{R}^{3}\setminus\Omega^{{\boldsymbol{\eta}}_{0}}}=0,$
$\displaystyle\rho_{0}>0\text{ in
}\\{x\in\Omega_{F}^{{\boldsymbol{\eta}}_{0}}:(\rho\mathbf{u})_{0}(x)>0\\},\quad\frac{(\rho\mathbf{u})_{0}^{2}}{\rho_{0}}\in
L^{1}(\Omega_{F}^{{\boldsymbol{\eta}}_{0}}).$ (1.13)

Finally, let us point out that a smooth solution of the problem (1.1)-(1.12)
satisfies the following energy identity in $(0,T]$:

$\displaystyle\frac{d}{dt}\int_{\Omega_{F}^{{\boldsymbol{\eta}}}}\Big{(}\frac{1}{2}\rho|\mathbf{u}|^{2}+\frac{\rho^{\gamma}}{\gamma-1}\Big{)}+\int_{\Omega_{F}^{{\boldsymbol{\eta}}}}\mathbb{S}(\nabla\mathbf{u}):\nabla\mathbf{u}$
$\displaystyle+\frac{d}{dt}\Big{(}\int_{{\Omega_{S}}}\frac{1}{2}|\partial_{t}{{\boldsymbol{\eta}}}|^{2}+E(\nabla{\boldsymbol{\eta}})\Big{)}+2R({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})=0.$
(1.14)

This identity is formally obtained at the end of Section 4.1. The goal of this
paper is to study global weak solutions with a pressure defect measure to
problem (1.1)-(1.12) which satisfy the corresponding energy inequality.

###  1.2 Main results and literature review

The main result of the paper is the following existence theorem:

######  Theorem 1.1.

Let $\Omega$, $\Omega_{S}$ be open bounded sets of class $C^{1,\alpha}$,
$\alpha>0$, $T>0$ and let structure energy and dissipation functionals $E$ and
$R$ satisfy conditions (E1)-(E4) and (R1)-(R2) respectively. Moreover, let
${\boldsymbol{\eta}}_{0}\in\mathcal{E}^{q}$ for $q>3$, ${\bf v}_{0}\in
L^{2}(\Omega_{S})$, $\rho_{0}\in
L^{\gamma}(\Omega_{F}^{{\boldsymbol{\eta}}_{0}})$ for $\gamma>12/7$ and
$(\rho\mathbf{u})_{0}\in
L^{\frac{2\gamma}{\gamma+1}}(\Omega_{F}^{{\boldsymbol{\eta}}_{0}})$ are such
that compatibility conditions $\eqref{IC_Compatibility}$ are satisfied and
$\Omega^{\boldsymbol{\eta}}_{S}(0)\neq\Omega$. Then there exists a global weak
solution with defect measure to the problem (1.1)-(1.12) on $(0,T)$ in the
sense of Definition 2.1.

One of the main difficulties of the considered problem is that the fluid-solid
interface can be quite irregular. Namely, while the space $\mathcal{E}^{q}$ is
embedded in $C^{1,\alpha}(\Omega_{S})$ and the surface deformation
${\boldsymbol{\eta}}|_{\partial\Omega_{S}}$ is a $C^{1,\alpha}$ local
diffeomorphism, it is in general not injective. Therefore one cannot avoid
irregularities in the fluid domain such as cusps or even surfaces that are not
representable by a union of overlapping graphs, which gives rise to defect
measures that correspond to concentrations of the fluid pressure. Please see
Section 3.1 for more details about possible irregularities of the fluid
domain. In this sense, our result is optimal and one cannot expect more
regular solutions without extra assumptions. However, we can prove that the
interface corresponding to a weak solution has certain nice topological
properties, see Theorem 3.2. In particular we prove the following result,
obtained there in Claim 4:

######  Theorem 1.2.

Let $({\boldsymbol{\eta}},\rho,{\bf u})$ be a weak solution to problem
(1.1)-(1.12) in the sense of Definition 2.1. Then there exits a constant
$A_{0}>0$, depending only on the initial energy and the domains
$\Omega,\Omega_{S}$, such that area of the fluid-solid interface is bounded
below by $A_{0}$.

The main novelty of these results is that we consider global solutions to a
$3D$/$3D$ fluid-structure interaction (FSI) problem that allow possible self-
contact of the solid and solid-rigid boundary contact. To the best of our
knowledge, this is the first result where such global solutions are
constructed and where contact conditions are formulated in the context of
fluid-structure interaction.

The problem of describing contact is one of the most challenging problems in
the analysis of the FSI problems and majority of results in the area do not
allow contact, i.e. analyse the problem only before the contact occurs. So far
there are only a few results that consider global-in-time solution in the
context of FSI. In the simplest case when the structure is rigid, the
existence of a global-in-time weak solution was proved for compressible fluid
and no-slip kinematic coupling conditions by Feireisl [14] and for the
incompressible fluid with slip kinematic coupling conditions by Chemetov and
Nečasova [11]. An analogous global existence result for a FSI system
consisting of $2D$ incompressible viscous fluid and a $1D$ elastic beam was
proved in [9]. In the context of strong solutions, there are several small
data global-time existence results. In [35] such a result was proved for the
incompressible Navier-Stokes equations coupled with a damped wave equation,
while in [28, 29] analogous results were proved for an FSI problem between
$3D$ fluid and $3D$ linearly elastic solid, but with some additional interface
or structure damping terms.

Another approach to proving global-in-time existence results for FSI problems
is to show that in certain situation the contact will not occur in finite time
– the so-called “no contact paradox”. This phenomenon is by now well
understood in the case where solid is rigid, see e.g. [27, 19]. Recently,
significant progress was made by Grandmont and Hillairet [24] who proved the
existence of a global-in-time strong solution to a 2D FSI problem involving a
viscoelastic beam, where they also showed that the contact does not occur in
finite time. We emphasize that this result essentially uses the global strong
solution with large initial data and therefore cannot be directly applied to
$3D$ case.

In contrast to this, in solid mechanics without intervening fluids, contact
has been long established as something that will occur and many methods to
deal with it have been established, from classic results, e.g. [46, 1, 12],
going all the way to most recent studies of hard dynamic self contact such as
[30, 10]. As we have the fluid acting as a kind of physical buffer to the
contact, our approximation relies on penalization by a soft potential,
something that has been well studied in particular in the context of numerics
(see e.g. [31]).

The second significant novelty of our result is that we consider a nonlinear
$3D$ viscoelastic solid. The vast majority of papers that analyse FSI problems
involving $3D$ elastic solids work with local-in-time regular solutions, see
e.g. [13, 33, 43, 4]. For the analysis of the static case we refer reader to
[22, 17]. The standard elastic energy (e.g. Saint Venant–Kirchhoff) does not
provide enough regularity for even defining weak solution of Leray-Hopf type.
Therefore in order to work with weak solution one needs to use an elasticity
model with better regularity properties. To the best of our knowledge there
are two approaches in the literature. The first one is to consider an elastic
interface with mass, e.g. [41, 18]. In this paper we adapt the second
approach, i.e. we use a second order nonlinear viscoelastic model for the
solid. For such a model the existence a weak solution to FSI problem with
incompressible and compressible fluid has been obtained in [2, 6]. We note
that viscoelastic regularization is needed in our approach in order to deal
with the nonlinearities that arise when studying large deformations and moving
boundaries. However, viscoelastic terms are not merely a mathematical
regularization, but rather they are derived from the physical characteristics
of viscoelastic solids such as biological materials and polymers.

We finish this brief literature review by mentioning that well-posedness
theory for weak solutions to FSI problems with simpler models for solid is by
now well-developed: existence theory when the solid is rigid e.g. [25, 44],
uniqueness theory when the solid is rigid e.g. [21, 5, 38], existence theory
when the solid is described by a lower dimensional model (plate/shell/beam
etc.) e.g. [23, 40, 7, 39, 49], uniqueness theory when the solid is described
by a lowerdimensional model e.g. [45, 48]. Finally, we refer the interested
reader to a nice review of the current state-of-the-art in FSI [8], and [3]
for a discussion of applications of FSI in biomedicine.

###  1.3 The strategy of the proof and discussion of some resulting
implications

The weak solution is obtained as a limit of approximate solutions which are
constructed as follows. The problem is decoupled by allowing the fluid to pass
through the elastic solid and extending the fluid equations to the entire
domain $\Omega$, while the fluid and solid velocities are penalized by each
other on the solid domain. Moreover, the elastic solid is supplemented with a
contact-penalization term which prevents contact with the rigid boundary
$\partial\Omega$ and self-contact. The initial fluid density is zero on the
solid domain. Finally, some regularizations are added to the problem to ensure
that the scheme and the convergences work properly. Let us point out the main
advantages in this construction. First, the fluid and structure sub-problems
are solved separately, sending data to one another through a time-stepping
scheme. The fluid equations are solved by using a Galerkin-based approach on a
fixed domain rather than on a moving domain, while the structure equations are
in contrast solved by time-discretizing and using the variational techniques.
This is possible only because of the decoupling and it cannot be overstated
how much it simplifies the analysis. Moreover, the flexibility of this
construction paves the road for future extensions, where more complex fluid or
structure models can be used.

The second part of the proof are the convergences. Here, the kinematic
coupling is recovered due to existence of $H_{x}^{1}$ global velocity field
(which is a consequence of viscosity in both fluid and solid) and the fluid
cannot pass through the elastic solid anymore, so the initial zero density of
the fluid in the solid domain is proved to stay this way at all times. As the
contact-penalization vanishes, the structure can have contact with the rigid
boundary and with itself, while overlap is still prevented, thus giving us a
global solution. The proof of this part is mostly standard, as the methods
used for compressible viscous fluids and second-gradient elastic solids can be
used here as well. The key difference is the convergence of pressure. The
appearance of contact leads to the formation of an irregular boundary in the
fluid domain. At the point of contact, the fluid domain is bounded by two
graphs that are tangent to each other. As a result, the boundary of the fluid
domain will have cusps, and it is no longer Lipschitz, which means that it is
not differentiable at some points. This irregularity can result in
concentrations of pressure, which correspond to the defect measure in the
definition of our weak solution. Note that such behaviour is to be expected
because fluids are in general strongly affected by the shape of the domain and
the theory for compressible viscous fluids developed so far gives us no
information about the pressure near such irregular boundaries. The only result
that touches this issue was obtained by Kukučka in [34], where pressure at
cusps of $W^{1,s}$-regularity for $s\geq 2$ can be controlled provided
$\gamma\geq\frac{3s}{2s-3}$. Note that this result cannot be used in our
problem due to the reasons presented above. To conclude, the defect measure in
the weak solution definition allows us to capture concentrations of pressure
that appear because of the irregularities of the fluid domain due to the
contact. Therefore, it seems that the weak solution with defect measure is the
optimal framework to study our problem.

At the end, let us point out that one might consider the incompressible fluid
model instead, in order to avoid having to deal with defect measures directly.
However incompressible fluids do not allow us to study the properties of the
pressure nearly as well as the compressible case does, as it is only recovered
as a Lagrange multiplier instead of a quantity given by constitution law
(1.7). Proving existence of solutions would require the test functions
$\boldsymbol{\xi}$ to be divergence-free on the fluid domain and thus even
further dependent on the solution. This was done without contact in [2, Lemma
A.6] by constructing approximate test functions which are additionally
divergence-free inside a $\varepsilon$-neighbourhood of the interface and
could likely be extended to contact as well. However the resulting pressure
will still be a distribution and thus not an improvement on the solutions
obtained for the compressible case.

The paper is organized as follows. In Section 2, the concept of weak solution
is introduced. In Section 3, some topological properties of fluid and
structure domains for weak solutions are studied. Section 4, it is proved that
smooth solution are also weak solutions, and that weak solutions which are
smooth satisfy the original problem pointwise. In Section 5, the approximate
solutions are constructed via decoupling time-stepping penalization
approximation scheme. In Section 6, a first limit passage is performed in
order to obtain a solution to a fully coupled problem without contact.
Finally, in Section 7, the weak solution to the original problem is obtained
by removing the contact-penalization, thus allowing the contact to happen.

##  2 Weak solution

First, note that the elements of the set $\mathcal{E}^{q}$ of possible
deformations can still have self-contact or contact with the rigid boundary.
Therefore, it is useful to characterize the physical boundary for such
deformations:

$\displaystyle
C_{\boldsymbol{\eta}}:=\\{x\in\partial\Omega_{S}:{\boldsymbol{\eta}}(x)\in\partial\Omega\\},$
$\displaystyle I_{\boldsymbol{\eta}}:=\\{x\in\partial\Omega_{S}\setminus
C_{\boldsymbol{\eta}}:{\boldsymbol{\eta}}\text{ is injective at }x\\},$
$\displaystyle N_{\boldsymbol{\eta}}:=\\{x\in\partial\Omega_{S}\setminus
C({\boldsymbol{\eta}}):{\boldsymbol{\eta}}\text{ is not injective at }x\\}.$

Here injective at $x$ means: $(\forall
y\in\overline{\Omega_{S}}\setminus\\{x\\})\;{\boldsymbol{\eta}}(x)\neq{\boldsymbol{\eta}}(y)$.
Notice that since ${\boldsymbol{\eta}}\in\mathcal{E}^{q}$ and therefore
satisfies the Ciarlet-Nečas condition, injectivity at $x$ is equivalent to:
$(\forall
y\in\partial\Omega_{S}\setminus\\{x\\})\;{\boldsymbol{\eta}}(x)\neq{\boldsymbol{\eta}}(y)$.
By definition we have $\partial\Omega_{S}=C_{\boldsymbol{\eta}}\cup
I_{\boldsymbol{\eta}}\cup N_{\boldsymbol{\eta}}$ and
$C_{\boldsymbol{\eta}}\cap I_{\boldsymbol{\eta}}=C_{\boldsymbol{\eta}}\cap
N_{\boldsymbol{\eta}}=I_{\boldsymbol{\eta}}\cap
N_{\boldsymbol{\eta}}=\emptyset$.

Figure 2: Sets
${\boldsymbol{\eta}}(C_{\boldsymbol{\eta}}),{\boldsymbol{\eta}}(I_{\boldsymbol{\eta}})$
and ${\boldsymbol{\eta}}(N_{\boldsymbol{\eta}})$ for a given deformation
${\boldsymbol{\eta}}\in\mathcal{E}^{q}$.

Next, let us denote the time-dependent sets as

$\displaystyle C_{\boldsymbol{\eta}}(t):=C_{{\boldsymbol{\eta}}(t)},\qquad
I_{\boldsymbol{\eta}}(t):=I_{{\boldsymbol{\eta}}(t)},\qquad
N_{\boldsymbol{\eta}}(t):=N_{{\boldsymbol{\eta}}(t)}$

and the time-space cylinders

$\displaystyle Q_{\boldsymbol{\eta}}^{C}:=[0,T]\times
C_{\boldsymbol{\eta}}(t),\qquad Q_{\boldsymbol{\eta}}^{I}:=[0,T]\times
I_{\boldsymbol{\eta}}(t),\qquad Q_{\boldsymbol{\eta}}^{N}:=[0,T]\times
N_{\boldsymbol{\eta}}(t).$

We will study topological properties of these sets in Section 3.

To define weak solutions we need to introduce appropriate functional
framework. For a given ${\boldsymbol{\eta}}$, let us introduce the functional
spaces defined on time-dependent domains:

$\displaystyle
L^{p}(0,T;L^{q}(\Omega_{F}^{\boldsymbol{\eta}}(t))):=\left\\{f\in
L^{1}(Q_{F,T}^{\boldsymbol{\eta}}):\left(\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}|f(t,x)|^{q}dx\right)^{1/q}\in
L^{p}(0,T)\right\\}$

and

$\displaystyle L^{p}(0,T;W^{1,q}(\Omega_{F}^{\boldsymbol{\eta}}(t))):=\\{f\in
L^{1}(Q_{F,T}^{\boldsymbol{\eta}}):f,\nabla f\in
L^{p}(0,T;L^{q}(\Omega_{F}^{\boldsymbol{\eta}}(t)))\\}.$

Now, let us introduce the concept of weak solution we will study (see Section
4.1 for its formal derivation):

######  Definition 2.1.

Let $\Omega^{\boldsymbol{\eta}}_{S}(0)\neq\Omega$,
${\boldsymbol{\eta}}_{0}\in\mathcal{E}^{q}$ for $q>3$, ${\bf v}_{0}\in
L^{2}(\Omega_{S})$, $\rho_{0}\in
L^{\gamma}(\Omega_{F}^{{\boldsymbol{\eta}}_{0}})$ and $(\rho\mathbf{u})_{0}\in
L^{\frac{2\gamma}{\gamma+1}}(\Omega_{F}^{{\boldsymbol{\eta}}_{0}})$ such that
compatibility conditions $\eqref{IC_Compatibility}$ are satisfied. We say that
$({{\boldsymbol{\eta}}},\rho,\mathbf{u})$ is a weak solution to the problem
(1.1)-(1.12) with defect measure $\pi\in
L^{\infty}(0,T;\mathcal{M}^{+}(\partial N_{\boldsymbol{\eta}}(t)))$ if:

  1. 1.

${\boldsymbol{\eta}}\in L^{\infty}(0,T;\mathcal{E}^{q})$,
$\partial_{t}{\boldsymbol{\eta}}\in L^{2}(0,T;H^{1}(\Omega_{S}))$,
$\mathbf{u}\in L^{2}(0,T;H^{1}(\Omega_{F}^{\boldsymbol{\eta}}(t)))$,
$\rho\in L^{\infty}(0,T;L^{\gamma}(\Omega_{F}^{\boldsymbol{\eta}}(t)))\cap
L^{\gamma+\theta}\big{(}0,T;L_{loc}^{\gamma+\theta}(\Omega_{F}^{\boldsymbol{\eta}}(t))\big{)}$,
for $\theta=\frac{2}{3}\gamma-1$,
$\rho\mathbf{u}\in
L^{\infty}(0,T;L^{\frac{2\gamma}{\gamma+1}}(\Omega_{F}^{\boldsymbol{\eta}}(t)))$,
$\rho|\mathbf{u}|^{2}\in
L^{\infty}(0,T;L^{1}(\Omega_{F}^{\boldsymbol{\eta}}(t)))$;

  2. 2.

The kinematic coupling
$\partial_{t}{\boldsymbol{\eta}}=\mathbf{u}\circ{\boldsymbol{\eta}}$ holds in
the sense of traces on $Q_{\boldsymbol{\eta}}^{I}$;

  3. 3.

The structure velocities match at points of contact in the sense of traces,
i.e.

$\displaystyle\partial_{t}{\boldsymbol{\eta}}(t,x)=\partial_{t}{\boldsymbol{\eta}}(t,y),\quad\text{for
a.a. }t\in[0,T]\text{ and }x,y\in N_{\boldsymbol{\eta}}(t)\text{ such that
}{\boldsymbol{\eta}}(t,x)={\boldsymbol{\eta}}(t,y);$

  4. 4.

The renormalized continuity equation is satisfied in the weak sense

$\displaystyle\int_{Q_{F,T}^{\boldsymbol{\eta}}}\rho
B(\rho)(\partial_{t}\varphi+\mathbf{u}\cdot\nabla\varphi)-\int_{Q_{F,T}^{\boldsymbol{\eta}}}b(\rho)(\nabla\cdot\mathbf{u})\varphi=\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{0}}}\rho_{0}B(\rho_{0})\varphi(0,\cdot)$
(2.1)

for all $\varphi\in C_{0}^{\infty}([0,T)\times\mathbb{R}^{3})$ and any $b\in
L^{\infty}(0,\infty)\cap C[0,\infty)$ such that $b(0)=0$ with
$B(\rho)=B(1)+\int_{1}^{\rho}\frac{b(z)}{z^{2}}dz$.

  5. 5.

The coupled momentum equation

$\displaystyle\int_{Q_{F,T}^{{\boldsymbol{\eta}}}}\rho\mathbf{u}\cdot\partial_{t}\boldsymbol{\xi}+\int_{Q_{F,T}^{{\boldsymbol{\eta}}}}(\rho\mathbf{u}\otimes\mathbf{u}):\nabla\boldsymbol{\xi}+\int_{Q_{F,T}^{{\boldsymbol{\eta}}}}p(\rho)(\nabla\cdot\boldsymbol{\xi})+\int_{0}^{T}\langle\pi,\nabla\cdot\boldsymbol{\xi}\rangle_{[\mathcal{M}^{+},C](\partial
I_{\boldsymbol{\eta}}(t))}$
$\displaystyle\quad-\int_{Q_{F,T}^{{\boldsymbol{\eta}}}}\mathbb{S}(\nabla\mathbf{u}):\nabla\boldsymbol{\xi}+\int_{Q_{S,T}}\partial_{t}{{\boldsymbol{\eta}}}\cdot\partial_{t}\boldsymbol{\phi}-\int_{0}^{T}\langle
DE({\boldsymbol{\eta}}),\boldsymbol{\phi}\rangle-\int_{0}^{T}\langle
D_{2}R({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}}),\boldsymbol{\phi}\rangle$
$\displaystyle=\int_{\Omega^{{\boldsymbol{\eta}}_{0}}}(\rho\mathbf{u})_{0}\cdot\boldsymbol{\xi}(0,\cdot)+\int_{\Omega_{S}}\mathbf{v}_{0}\cdot\boldsymbol{\phi}(0,\cdot)$
(2.2)

holds for all $\boldsymbol{\xi}\in C_{c}^{\infty}([0,T)\times\Omega)$, where
$\boldsymbol{\phi}=\boldsymbol{\xi}\circ{\boldsymbol{\eta}}$ on111Note that
$\boldsymbol{\phi}$ inherits the regularity from ${\boldsymbol{\eta}}$ so
$\boldsymbol{\phi}\in L^{2}(0,T;W^{2,q}(\Omega_{S}))\cap
H^{1}(0,T;H^{1}(\Omega_{S}))$ and therefore all the integral terms are well-
defined. See also [2, Prop. A.4]. $Q_{S,T}$.

  6. 6.

Energy inequality

$\displaystyle\int_{\Omega_{F}^{{\boldsymbol{\eta}}}(t)}\Big{(}\frac{1}{2}\rho|\mathbf{u}|^{2}+\frac{\rho^{\gamma}}{\gamma-1}\Big{)}(t)+\frac{1}{\gamma-1}\int_{\partial
I_{\boldsymbol{\eta}}(t)}d\pi(t)+\int_{0}^{t}\int_{\Omega_{F}^{{\boldsymbol{\eta}}}(\tau)}(\mathbb{S}(\nabla\mathbf{u}):\nabla\mathbf{u})(\tau)~{}d\tau$
$\displaystyle\quad+\frac{1}{2}\int_{\Omega_{S}}|\partial_{t}{{\boldsymbol{\eta}}}|^{2}(t)+E({\boldsymbol{\eta}}(t))+2\int_{0}^{t}R({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})(\tau)~{}d\tau$
$\displaystyle\leq\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{0}}}\Big{(}\frac{1}{2\rho_{0}}|(\rho\mathbf{u})_{0}|^{2}+\frac{\rho_{0}^{\gamma}}{\gamma-1}\Big{)}+\int_{\Omega_{S}}\frac{1}{2}|\mathbf{v}_{0}|^{2}+E({\boldsymbol{\eta}}_{0})$
(2.3)

holds for all $t\in(0,T]$.

######  Remark 2.1.

(1) If there is no contact, i.e. $Q_{\boldsymbol{\eta}}^{C}\cup
Q_{\boldsymbol{\eta}}^{N}=\emptyset$, then the weak solution enjoys better
properties and coincides with the one in [6].
(2) Note that the defect measure $\pi$ is supported on
$(0,T)\times\mathcal{M}^{+}(\partial N_{\boldsymbol{\eta}}(t))$. It is a
consequence of lack of uniform integrability of pressure near the domain
irregularities caused by self-contact. While contact can appear with the rigid
boundary $\partial\Omega$ which also breaks uniform integrability, the test
functions are compactly supported and are thus zero on $\partial\Omega$, so
the possible measures supported on $\partial\Omega$ are irrelevant. The
physical interpretation of this is that there is no boundary condition on
$\partial\Omega$ for the fluid stress, while on the fluid-structure interface
there is the fluid-structure dynamic coupling $\eqref{dync}$ which needs to be
preserved in the test functions. Let us point out that if the solution is
smooth, both this condition and the structure-structure dynamic coupling
$\eqref{dynca}$ can be recovered in the pointwise sense, and the defect
measure $\pi$ vanishes. This is shown in Section 4.1.
(3) Let us point out that there is a possibility of vacuum appearing in weak
solutions to our problem. There are a few downsides to it. First, the
viscosity coefficients are constant which implies that even in vacuum there
can be fluid velocity which satisfies the following “ghost elliptic equation”
$\nabla\cdot\mathbb{S}(\nabla\mathbf{u})=0$. In our framework, this means that
the vacuum can produce stress which acts onto the structure. One way to avoid
all these issues is to use a model which has a density dependent viscosity
coefficients $\mu=\mu(\rho)$ and $\lambda=\lambda(\rho)$ which vanish inside
vacuum (we refer to [20, Chapter 30] for detailed coverage of this topic).
However, the analysis of this fluid model has not been done on bounded domains
because no boundary conditions conserve both energy and so called “BD entropy”
- a quantity which provides additional information that ensures the
convergence of pressure and stress. Note that in this case, the strong
convergence of density based on the convergence of viscous effective flux (for
the constant viscosity case) is out of reach, so the improved regularity
coming from the BD entropy compensates for this deficiency.

######  Remark 2.2.

The coupling of test functions between an Eulerian and a Lagrangian test-
function, as we have done here in the weak solution leads to some consequences
for the expected class of solutions. Indeed, at any pair $(x,y)$ of pre-images
of a contact point, i.e. with ${\boldsymbol{\eta}}(x)={\boldsymbol{\eta}}(y)$,
the corresponding values $\boldsymbol{\phi}(x),\boldsymbol{\phi}(y)$ of the
Lagrangian test-function must coincide with that of the Eulerian test function
$\boldsymbol{\xi}({\boldsymbol{\eta}}(x))$ and thus with each other. As a
result, at that point, it seems like it is not possible to discern any
information about the relative behaviour of the different parts, in particular
the forces expressed by the momentum balance.

Indeed this has been a well-known problem in the case of rigid bodies immersed
in a fluid, where this uncertainty can lead to non-uniqueness of solutions
[47]. However in our case, we are dealing with an elastic body. For a rigid
body, the same argument as above immediately fixes the test-function on the
whole body. In contrast, for an elastic body we have the freedom to choose the
test-function differently everywhere else, which gives us full information
about all the stresses and forces within the solid. And while those could
potentially form concentrations at the boundary, the interface itself has zero
mass and thus carries no momentum. That means that knowing the solid momentum
in the bulk is indeed sufficient to determine also the behaviour of the
interface. In particular, while the velocities will coincide at the contact
itself, this still allows for separation.

##  3 Topological properties of fluid and structure domains and their
interface for deformations with allowed contact

In order to study some topological and geometrical properties of the
interface, first we prove a Lemma that gives us some nice properties of
functions from $\mathcal{E}^{q}$.

######  Lemma 3.1.

Let ${\boldsymbol{\eta}}\in\mathcal{E}^{q}$. Then, one has:

  1. 1.

If $x,y\in C_{\boldsymbol{\eta}}$ and
${\boldsymbol{\eta}}(x)={\boldsymbol{\eta}}(y)$, then $x=y$, i.e.
${\boldsymbol{\eta}}$ is injective on $C_{\boldsymbol{\eta}}$;

  2. 2.

If $x,y,z\in N_{\boldsymbol{\eta}}$ and
${\boldsymbol{\eta}}(x)={\boldsymbol{\eta}}(y)={\boldsymbol{\eta}}(z)$, then
at least one of $x=y$, $x=z$ and $y=z$ is true;

  3. 3.

$\partial\Omega_{S}^{\boldsymbol{\eta}}={\boldsymbol{\eta}}(I_{\boldsymbol{\eta}})\cup{\boldsymbol{\eta}}(C_{\boldsymbol{\eta}})\cup{\boldsymbol{\eta}}(\partial
N_{\boldsymbol{\eta}})$;

  4. 4.

$\partial\Omega_{F}^{\boldsymbol{\eta}}={\boldsymbol{\eta}}(\overline{I_{\boldsymbol{\eta}}})\cup\partial\Omega\setminus{\boldsymbol{\eta}}(\operatorname{Int}(C_{\boldsymbol{\eta}}))$;

  5. 5.

The fluid-structure interface equals
${\boldsymbol{\eta}}(\overline{I_{\boldsymbol{\eta}}})$, i.e.
$\partial\Omega_{S}^{\boldsymbol{\eta}}\cap\partial\Omega_{F}^{\boldsymbol{\eta}}={\boldsymbol{\eta}}(\overline{I_{\boldsymbol{\eta}}})$;

  6. 6.

Let ${\boldsymbol{\eta}}_{n}\to{\boldsymbol{\eta}}\;{\rm
in}\;\mathcal{E}^{q}$. Then
$|I_{\boldsymbol{\eta}}|\leq\liminf\limits_{n\to\infty}|I_{{\boldsymbol{\eta}}_{n}}|$.

###### Proof.

Claim 1. Assume the opposite, i.e. there exists $x,y\in
C_{\boldsymbol{\eta}}$, $x\neq y$ such that
${\boldsymbol{\eta}}(x)={\boldsymbol{\eta}}(y)\in\partial\Omega.$ Since
${\boldsymbol{\eta}}\in\mathcal{E}^{q}$ we have
$\det\nabla{\boldsymbol{\eta}}>c$ and therefore there exists $\delta>0$ such
that the restrictions ${\boldsymbol{\eta}}_{|B_{\delta}(x)}$ and
${\boldsymbol{\eta}}_{|B_{\delta}(y)}$ are each bijective. Moreover, since
$\Omega$ is a $C^{1,\alpha}$ domain, there exists $l>0$ such that
$B_{l}({\boldsymbol{\eta}}(x))\cap\partial\Omega$ can be represented as a
graph of a function $\phi:V_{l}\to\mathbb{R}$, where $V_{l}$ is an open and
connected set in $\mathbb{R}^{2}$, so that $\Omega$ lies below the graph of
$\phi$. Now we consider sets $\partial{\boldsymbol{\eta}}(B_{\delta}(x))$ and
$\partial{\boldsymbol{\eta}}(B_{\delta}(y))$. By assumption both sets contain
point ${\boldsymbol{\eta}}(x)$ and since ${\boldsymbol{\eta}}\in
C^{1}(\Omega_{S},{\mathbb{R}}^{3})$ tangents at that point coincide with the
tangent to $\partial\Omega$. Therefore, by taking $\delta$ and $l$ smaller if
needed, both
$B_{l}({\boldsymbol{\eta}}(x))\cap\partial{\boldsymbol{\eta}}(B_{\delta}(x))$
and
$B_{l}({\boldsymbol{\eta}}(x))\cap\partial{\boldsymbol{\eta}}(B_{\delta}(y))$
can be represented as graphs of functions
$\phi_{1},\phi_{2}:V_{l}\to\mathbb{R}$, respectively, in the same coordinate
system such that $\phi(x_{0})=\phi_{1}(x_{0})=\phi_{2}(x_{0})$, for some
$x_{0}\in V_{l}$. Furthermore, since
$\Omega_{S}^{\boldsymbol{\eta}}\subseteq\Omega$ we have
$\phi_{1},\phi_{2}\leq\phi$ on $V_{l}$. However, this means that the area that
lies below the function $\phi_{m}:=\min\\{\phi_{1},\phi_{2}\\}$ belongs to
${\boldsymbol{\eta}}(B_{\varepsilon}(x))\cap{\boldsymbol{\eta}}(B_{\varepsilon}(y))$,
which is a contradiction with ${\boldsymbol{\eta}}\in\mathcal{E}^{q}$ (more
precisely it is in contradiction with the Ciralet-Nečas condition).

Claim 2. This claim is similar to the first one (see also [12, Theorem 2]).
Assume the opposite, i.e. there exists $x,y,z\in N_{\boldsymbol{\eta}}$ such
that $x\neq y\neq z\neq x$ and
${\boldsymbol{\eta}}(x)={\boldsymbol{\eta}}(y)={\boldsymbol{\eta}}(z)$. Again
we can choose $\delta>0$ such that the corresponding restriction of
${\boldsymbol{\eta}}$ to $B_{\delta}(x)$, $B_{\delta}(y)$ and $B_{\delta}(z)$
are bijective. By reasoning in the same way as in the proof of Claim 1, we
conclude that $\partial{\boldsymbol{\eta}}(B_{\delta}(x))$,
$\partial{\boldsymbol{\eta}}(B_{\delta}(y))$ and
$\partial{\boldsymbol{\eta}}(B_{\delta}(z))$ that can be represented as graphs
of functions in some neighbourhood of ${\boldsymbol{\eta}}(x)$ in the same
coordinate system (notice that in this case $\Omega_{S}^{\boldsymbol{\eta}}$
can lie on both sides of the graphs). However, this is again in contradiction
with the Ciralet-Nečas condition.

Claim 3. First note that
${\boldsymbol{\eta}}(C_{\boldsymbol{\eta}})\cup{\boldsymbol{\eta}}(I_{\boldsymbol{\eta}})\subseteq\partial\Omega_{S}^{\boldsymbol{\eta}}\subseteq{\boldsymbol{\eta}}(\partial\Omega_{S})={\boldsymbol{\eta}}(I_{\boldsymbol{\eta}})\cup{\boldsymbol{\eta}}(C_{\boldsymbol{\eta}})\cup{\boldsymbol{\eta}}(N_{\boldsymbol{\eta}})$.
Therefore it is enough to prove
$\partial\Omega_{S}^{\boldsymbol{\eta}}\setminus{\boldsymbol{\eta}}(I_{\boldsymbol{\eta}}\cup
C_{\boldsymbol{\eta}})={\boldsymbol{\eta}}(\partial N_{\boldsymbol{\eta}}).$

Let
$x\in\partial\Omega_{S}^{\boldsymbol{\eta}}\setminus{\boldsymbol{\eta}}(I_{\boldsymbol{\eta}}\cup
C_{\boldsymbol{\eta}})$. Then there exists $y\in N_{\boldsymbol{\eta}}$ such
that ${\boldsymbol{\eta}}(y)=x.$ Moreover, by definition of
$N_{\boldsymbol{\eta}}$ and Claim 2, there exists unique $z\in
N_{\boldsymbol{\eta}}$ such that
${\boldsymbol{\eta}}(y)={\boldsymbol{\eta}}(z)$. We will prove that at least
one of $y,z$ is in $\partial N_{\boldsymbol{\eta}}$. Assume the opposite, i.e.
$y,z\in{\rm int}N_{\boldsymbol{\eta}}$. Then there exist open sets $O_{1}$ and
$O_{2}$ in $N_{\boldsymbol{\eta}}$ such that $y\in O_{1}$ and $z\in O_{2}$ and
${\boldsymbol{\eta}}(O_{1})\subseteq{\boldsymbol{\eta}}(O_{2})$. In this case,
we can represent ${\boldsymbol{\eta}}(O_{1})\cap
B_{\delta}({\boldsymbol{\eta}}(x))$ as a graph of a function
$\phi:V_{\delta}\to\mathbb{R}$, for $\delta>0$ small enough, so that above and
below is $\Omega_{S}^{\boldsymbol{\eta}}$. This means that
${\boldsymbol{\eta}}(O_{1})\cap
B_{\delta}({\boldsymbol{\eta}}(x))\subseteq\text{int}({\boldsymbol{\eta}}(\overline{\Omega_{S}}))=\Omega_{S}^{\boldsymbol{\eta}}$,
which is a contradiction with $x\in\partial\Omega_{S}^{\boldsymbol{\eta}}$.
Therefore $x\in{\boldsymbol{\eta}}(\partial N_{\boldsymbol{\eta}})$ and the
first inclusion is proved.

Let $x\in{\boldsymbol{\eta}}(\partial N_{\boldsymbol{\eta}})$ and let
$z\in\partial N_{\boldsymbol{\eta}}$ be such that ${\boldsymbol{\eta}}(z)=x$.
Then there is a sequence $z_{n}\in I_{\boldsymbol{\eta}}$ such that $z_{n}\to
z$ (since otherwise $z_{n}\in C_{\boldsymbol{\eta}}$ would imply that $z\in
C_{\boldsymbol{\eta}}$, due to closedness of $C_{\boldsymbol{\eta}}$ which is
proved in Theorem 3.2 Claim 3). Then
${\boldsymbol{\eta}}(z_{n})\in\partial\Omega_{S}^{\boldsymbol{\eta}}$ and
${\boldsymbol{\eta}}(z_{n})\to{\boldsymbol{\eta}}(z)=x$. Since
$\partial\Omega_{S}^{\boldsymbol{\eta}}$ is closed we have
$x\in\partial\Omega_{S}^{\boldsymbol{\eta}}$. Therefore we have proved the
second inclusion and thus the claim.

Claim 4. First we prove inclusion
${\boldsymbol{\eta}}(\overline{I_{\boldsymbol{\eta}}})\cup(\partial\Omega\setminus{\boldsymbol{\eta}}(\text{int}(C_{\boldsymbol{\eta}})))\subseteq\partial\Omega_{F}^{\boldsymbol{\eta}}$.
Since ${\boldsymbol{\eta}}$ is an open mapping (because
${\boldsymbol{\eta}}\in\mathcal{E}^{q}$) and
$\partial\Omega_{F}^{\boldsymbol{\eta}}$ is a closed set it is enough to prove
${\boldsymbol{\eta}}(I_{\boldsymbol{\eta}})\cup(\partial\Omega\setminus{\boldsymbol{\eta}}(C_{\boldsymbol{\eta}}))\subseteq\partial\Omega_{F}^{\boldsymbol{\eta}}$.
Let
$x\in{\boldsymbol{\eta}}(I_{\boldsymbol{\eta}})\cup(\partial\Omega\setminus{\boldsymbol{\eta}}(C_{\boldsymbol{\eta}}))$
and denote $y={\boldsymbol{\eta}}(x)$. Then, for every
$\delta<\text{dist}({\boldsymbol{\eta}}(x),\partial\Omega)$, one has
$B_{\delta}({\boldsymbol{\eta}}(x))\setminus\Omega_{S}^{\boldsymbol{\eta}}\neq\emptyset$
and $B_{\delta}({\boldsymbol{\eta}}(x))\cap\Omega^{c}=\emptyset$, which then
implies that
$(B_{\delta}({\boldsymbol{\eta}}(x))\setminus\Omega_{S}^{\boldsymbol{\eta}})\subseteq\overline{\Omega_{F}^{\boldsymbol{\eta}}}$
(since
$\overline{\Omega}=\overline{\Omega_{F}^{\boldsymbol{\eta}}}\cup\overline{\Omega_{S}^{\boldsymbol{\eta}}}$).
Therefore, in this case
${\boldsymbol{\eta}}(x)\in\partial\Omega_{F}^{\boldsymbol{\eta}}$. If on the
other hand
$y\in\partial\Omega\setminus{\boldsymbol{\eta}}(C_{\boldsymbol{\eta}})$, then
for every $\delta<\text{dist}(y,\partial\Omega_{S}^{\boldsymbol{\eta}})$, one
has $B_{\delta}(y)\cap\Omega_{S}^{\boldsymbol{\eta}}=\emptyset$ and
$B_{\delta}(y)\cap\Omega\neq\emptyset$, so
$B_{\delta}(y)\cap\Omega\subseteq\overline{\Omega_{F}^{\boldsymbol{\eta}}}$
and consequently $y\in\partial\Omega_{F}^{\boldsymbol{\eta}}$.

To prove other inclusion first we notice that
$\partial\Omega_{F}^{\boldsymbol{\eta}}\subseteq\partial\Omega\cup{\boldsymbol{\eta}}(\partial\Omega_{S})$.
Let $x\in\partial\Omega_{F}^{\boldsymbol{\eta}}$. If $x\in\partial\Omega$, by
definition of $C_{\boldsymbol{\eta}}$, one has $x\notin{\rm
Int}C_{{\boldsymbol{\eta}}}$. Similarly, by definition of
$N_{\boldsymbol{\eta}}$ it immediately follows $x\notin{\rm
Int}N_{\boldsymbol{\eta}}$. Therefore if $x\notin\partial\Omega$, we have
$x\in\overline{I_{\boldsymbol{\eta}}}$ which proves the second inclusion.

Claim 5. Is a direct consequence of previous two claims.

Claim 6. First we prove $|\Omega_{S}\setminus
C_{\boldsymbol{\eta}}|\leq\limsup_{n}|\Omega_{S}\setminus
C_{{\boldsymbol{\eta}}_{n}}|$. Let $\chi:=\chi|_{\Omega_{S}\setminus
C_{\boldsymbol{\eta}}}$ and $\chi_{n}:=\chi|_{\Omega_{S}\setminus
C({\boldsymbol{\eta}}_{n})}$ be characteristic functions of sets
$\Omega_{S}\setminus C({\boldsymbol{\eta}}_{n})$ and $\Omega_{S}\setminus
C({\boldsymbol{\eta}}_{n})$, respectively.

Let $X\in{\rm int}(\Omega_{S}\setminus C({\boldsymbol{\eta}}_{n}))$. By
definition of $C({\boldsymbol{\eta}})$ it means that
$d({\boldsymbol{\eta}}(x),\partial\Omega)>0.$ Morever, by uniform convergence
of ${\boldsymbol{\eta}}_{n}$ there exists $n_{x}\in{\mathbb{N}}$ such that
$d({\boldsymbol{\eta}}_{n}(x),\partial\Omega_{S})>0$, $n\geq n_{x}$, i.e.
$1=\chi(x)=\chi_{n}(x)$, $n\geq n_{x}$. Therefore, we proved:

$(1-\chi)\leq\liminf\limits_{n\to\infty}(1-\chi_{n})\quad{\rm
a.\;e.\;in}\quad\Omega_{S}.$

Now, by Fatou’s Lemma we have

$|\Omega\setminus
C({\boldsymbol{\eta}})|=\int_{\partial\Omega_{S}}(1-\chi)\leq\int_{\partial\Omega_{S}}\liminf\limits_{n\to\infty}(1-\chi_{n})\leq\liminf\limits_{n\to\infty}\int_{\partial\Omega_{S}}(1-\chi_{n})=\liminf\limits_{n\to\infty}|\Omega\setminus
C_{{\boldsymbol{\eta}}_{n}}|.$

By working additionally with characteristic function for $\Omega_{S}\setminus
N_{\boldsymbol{\eta}}$ and $\Omega_{S}\setminus N_{\boldsymbol{\eta}}$, we
analogously prove $|\Omega_{S}\setminus
N_{\boldsymbol{\eta}}|\leq\limsup_{n}|\Omega_{S}\setminus
N_{{\boldsymbol{\eta}}_{n}}|$, so

$\displaystyle|I_{\boldsymbol{\eta}}|$
$\displaystyle=|\Omega\setminus\left(C_{\boldsymbol{\eta}}\cup
N_{\boldsymbol{\eta}}\right)|\leq\liminf\limits_{n\to\infty}|\Omega\setminus\left(C_{{\boldsymbol{\eta}}_{n}}\cup
N_{{\boldsymbol{\eta}}_{n}}\right)|=\liminf\limits_{n\to\infty}|I_{{\boldsymbol{\eta}}_{n}}|.\qed$

Now we are in the position to prove a theorem that gives us some geometrical
properties of the fluid and structure domains and their interface, associated
to a weak solution.

######  Theorem 3.2.

Let $({\boldsymbol{\eta}},\rho,\mathbf{u})$ satisfy the energy inequality
$\eqref{enineq}$ and let $\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\rho(t)=m>0$
for all $t\in[0,T]$. Then, one has:

  1. 1.

$|\Omega_{F}^{\boldsymbol{\eta}}(t)|\geq c(E_{0},m,\gamma)>0$ and
$|\Omega_{S}^{\boldsymbol{\eta}}(t)|\geq c(E_{0})>0$, for all $t\in[0,T]$;

  2. 2.

$|\partial\Omega_{F}^{\boldsymbol{\eta}}(t)|\geq c(E_{0},m,\gamma)>0$ and
$|\partial\Omega_{S}^{\boldsymbol{\eta}}(t)|\geq c(E_{0})>0$, for all
$t\in[0,T]$;

  3. 3.

$Q_{\boldsymbol{\eta}}^{I}$ is open and
$Q_{\boldsymbol{\eta}}^{C},Q_{\boldsymbol{\eta}}^{N}$ are closed in
$[0,T]\times\partial\Omega_{S}$;

  4. 4.

There exists a constant $c=c(E_{0})$ independent of ${\boldsymbol{\eta}}$ such
that we have the following lower bound on the area of the fluid-structure
interface:
$|\partial\Omega_{F}^{\boldsymbol{\eta}}(t)\cap\partial\Omega_{S}^{\boldsymbol{\eta}}(t)|\geq
c(E_{0}),\quad t\in[0,T].$

###### Proof.

Claim 1. Since mass of the fluid is fixed, one has

$\displaystyle
m=||\rho||_{L^{1}(\Omega_{F}^{\boldsymbol{\eta}}(t))}\leq||\rho||_{L^{\gamma}(\Omega_{F}^{\boldsymbol{\eta}}(t))}||1||_{L^{\frac{\gamma}{\gamma-1}}(\Omega_{F}^{\boldsymbol{\eta}}(t))}\leq
E_{0}^{\frac{1}{\gamma}}|\Omega_{F}^{\boldsymbol{\eta}}|^{\frac{\gamma-1}{\gamma}},$

so

$\displaystyle|\Omega_{F}^{\boldsymbol{\eta}}(t)|\geq\frac{m^{\frac{\gamma}{\gamma-1}}}{E_{0}^{\frac{1}{\gamma-1}}},$

for all $t\in[0,T]$. Note that $E_{0}>0$ since $m>0$. Now, since
$\det\nabla{\boldsymbol{\eta}}\geq c$ on $[0,T]\times\Omega_{S}$, one has
$|\Omega_{S}^{\boldsymbol{\eta}}(t)|=\int_{\Omega_{S}}\det\nabla{\boldsymbol{\eta}}(t)\geq
c|\Omega_{S}|>0$.

Claim 2. First, since $\text{int}(I_{\boldsymbol{\eta}}(t))$ and
$\partial\Omega\setminus C_{\boldsymbol{\eta}}(t)$ have a well-defined surface
element, by Lemma 3.1 Claim 3 and continuity, one has that
$\partial\Omega_{F}^{\boldsymbol{\eta}}(t)$ has a well-defined surface element
everywhere and therefore it is a measurable surface for all $t\in[0,T]$ (and
similarly for the surface $\partial\Omega_{S}^{\boldsymbol{\eta}}(t)$). From
the isoperimetric inequality we have

$\displaystyle|\partial\Omega_{F}^{\boldsymbol{\eta}}(t)|\geq
3|\Omega_{F}^{\boldsymbol{\eta}}(t)|^{\frac{2}{3}}\left(\frac{4}{3}\pi\right)^{\frac{1}{3}}\geq
c(E_{0},m,\gamma),$

and similarly for $|\partial\Omega_{S}^{\boldsymbol{\eta}}(t)|$.

Claim 3. Closedness of $Q^{C}_{\boldsymbol{\eta}}$ is a direct consequence of
continuity of ${\boldsymbol{\eta}}$. Let $(t_{n},x_{n})$ be a convergent
sequence in $Q^{C}_{\boldsymbol{\eta}}$ and $(t,x)$ its limit. Then we have

$\displaystyle
0=\lim\limits_{n\to\infty}\text{dist}({\boldsymbol{\eta}}(t_{n},x_{n}),\partial\Omega)=\text{dist}({\boldsymbol{\eta}}(\lim\limits_{n\to\infty}(t_{n},x_{n})),\partial\Omega)=\text{dist}({\boldsymbol{\eta}}(t,x),\partial\Omega),$

and therefore $(t,x)\in Q^{C}_{\boldsymbol{\eta}}.$

To prove that $Q^{N}_{\boldsymbol{\eta}}$ is closed, we take a convergent
sequence $(t_{n},x_{n})$ in $Q^{N}_{\boldsymbol{\eta}}$ and
$(t,x)=\lim\limits_{n\to\infty}(t_{n},x_{n})$. By Lemma 3.1, Claim 2), and the
definition of $Q^{N}_{\boldsymbol{\eta}}$ for every $n$ there exists a unique
$y_{n}\in N_{{\boldsymbol{\eta}}(t_{n})}$ such that $y_{n}\neq x_{n}$ and
${\boldsymbol{\eta}}(t_{n},x_{n})={\boldsymbol{\eta}}(t_{n},y_{n})$. Since
$y_{n}$ is a sequence in $\partial\Omega_{S}$ which is compact, there exists a
convergent subsequence which we will still denote by $y_{n}$. By the
continuity of ${\boldsymbol{\eta}}$, one concludes

$\displaystyle{\boldsymbol{\eta}}(t,x)={\boldsymbol{\eta}}\left(\lim\limits_{n\to\infty}(t_{n},x_{n})\right)={\boldsymbol{\eta}}\left(\lim\limits_{n\to\infty}(t_{n},y_{n})\right)={\boldsymbol{\eta}}(t,y)$

where $y:=\lim\limits_{n\to\infty}y_{n}$. Since ${\boldsymbol{\eta}}$ is
locally bijective for on balls of uniform radius $\varepsilon$, one must have
$|x_{n}-y_{n}|\geq\varepsilon$ for all $n\geq n_{0}\in\mathbb{N}$, so $x\neq
y$, which proves that $Q_{\boldsymbol{\eta}}^{N}$ is closed. Finally, for the
last set, since $Q_{\boldsymbol{\eta}}^{C}\cup Q_{\boldsymbol{\eta}}^{N}$ is
now closed, one gets that
$Q_{\boldsymbol{\eta}}^{T}=[0,T]\times\partial\Omega_{S}\setminus(Q_{\boldsymbol{\eta}}^{C}\cup
Q_{\boldsymbol{\eta}}^{N})$ is open.

Claim 4. Let
$X=\\{{\boldsymbol{\eta}}\in\mathcal{E}^{q}:\|{\boldsymbol{\eta}}\|_{W^{2,q}(\Omega_{S})}\leq
E_{0},~{}|\Omega_{S}^{\boldsymbol{\eta}}|\leq C(E_{0})<|\Omega|\\}$. First,
note that by Lemma 3.1, Claim 5 we have

$\partial\Omega_{F}^{\boldsymbol{\eta}}\cap\partial\Omega_{S}^{\boldsymbol{\eta}}={\boldsymbol{\eta}}(\overline{I_{\boldsymbol{\eta}}}).$

Moreover, there exists a constant $C_{0}$ such that

$|{\boldsymbol{\eta}}(I_{\boldsymbol{\eta}})|=\left|\int_{I_{\boldsymbol{\eta}}}\det\nabla{\boldsymbol{\eta}}|\nabla{\boldsymbol{\eta}}^{-\tau}{\bf
n}|\right|\geq C_{0}|I_{\boldsymbol{\eta}}|,\quad{\boldsymbol{\eta}}\in X.$

Therefore, since
$|{\boldsymbol{\eta}}(\overline{I_{\boldsymbol{\eta}}})|\geq|{\boldsymbol{\eta}}(I_{\boldsymbol{\eta}})|$,
it is enough to prove that there exists a constant $c=c(E_{0})$ such that

$|I_{\boldsymbol{\eta}}|\geq c(E_{0})>0,\quad\text{for all
}{\boldsymbol{\eta}}\in X.$

Assume the opposite, i.e. there exists a sequence ${\boldsymbol{\eta}}_{n}\in
X$ such that $|I_{{\boldsymbol{\eta}}_{n}}|\to 0.$ By Lemma 3.1, Claim 6 we
have:

$|I_{\boldsymbol{\eta}}|\leq\liminf\limits_{n\to\infty}|I_{{\boldsymbol{\eta}}_{n}}|=0.$

which is a contradiction, since $I_{\boldsymbol{\eta}}$ is an open set and it
must have a positive measure. ∎

###  3.1 Closed components of the the fluid domain: a surprising example

While we have all these conditions, the boundary of the fluid region can still
be highly irregular. Assuming sufficient regularity of the fixed reference
domain $\Omega_{S}$ and finite elastic energy, the surface deformation
${\boldsymbol{\eta}}|_{\partial\Omega_{S}}$ will be a regular (i.e. local
diffeomorphism) $C^{1,\alpha}$-map of a $C^{1,\alpha}$-domain. However it will
not be injective or even an immersion.

Here, in particular the latter is what makes the situtation interesting, but
difficult. While it can be already seen from simple two dimensional examples
(cmp. Figure 2), that this will lead to irregularities in the fluid domain,
this still leaves the question of how many of these irregularities can appear.
In other words, what is the cardinality and the measure of $\partial
I_{\boldsymbol{\eta}}\subset\partial\Omega_{S}$? The somewhat surprising
answer is that even in an otherwise benign situation, this can be a set of
full, i.e. $d-1=2$-dimensional measure.

To see this, let us first note, that locally the fluid domain always looks
like one of three possible configurations: the bulk, the area above a
$C^{1,\alpha}$-graph or the area between two such graphs. The former two are
well understood. Only in the latter configuration can these cusps appear. For
our example we can further simplify the situation by considering a problem
with a vertical symmetry, i.e. the functions defining the two graphs differ by
only a sign. Then the question reduces to the following: Given a smooth, non-
negative function $f:Q\subset\mathbb{R}^{2}\to\mathbb{R}$, what can (locally)
be said about the size of the set $\partial\\{x\in Q:f(x)>0\\}$?

It turns out, that the answer is, not much, as the following Cantor-type
construction shows. Take any fixed non-negative function
$g:C^{\infty}(\mathbb{R})$ with $\operatorname{supp}g=[-1,1]$. Then for a
sequence of vertical and horizontal scalings
$(a_{k})_{k\in\mathbb{N}},(b_{k})_{k\in\mathbb{N}}$, for each $k$, we add
$2^{k}$ copies of $x\mapsto a_{k}g(b_{k}x)$ to our function, shifted in such a
way that they sit precisely in the gaps of all the previous copies (see Figure
3). The final result of this construction will be our function $f$. As the
choice of $a_{k}$ does not influence the set $\partial\\{x\in Q:f(x)>0\\}$,
this can be done in a way such that $f$ is smooth and or arbitrary small norm.
If we now choose $b_{k}=\frac{1}{4^{k}}$, it is easy to show that
$\partial\\{x\in Q:f(x)>0\\}$ ends up being the well known “fat”-Cantor set,
which has non-zero $1$-dimensional measure. Similar constructions can be done
in higher dimensions, where it is even possible for the resulting fluid domain
to still be connected.

Nevertheless, one should keep in mind that such a situation is not necessarily
unnatural. If two mostly flat elastic solids (or parts of the same solid) are
pushed against each other in a fluid, then small inclusions of the fluid
between them are to be expected and there is no a-priori way to bound their
size or number. In fact, the solid itself could already have this type of
roughness in its relaxed configuration. It is thus impossible to exclude such
a behaviour in the generic setup that we study here.

Figure 3: A sketch of the construction given above. Vertical and horizontal
scaling of the individual domain components are exagarated for better
visibility.

##  4 Formal equivalence between weak and smooth solutions

In this section, the goal is to show that our concept of weak solution is
justified. This will be done in two steps. First we will show that all smooth
solutions $\eqref{elasticity}-\eqref{initialdata}$ are in fact weak solutions
in the sense of Definition 2.1. Then, we will show that a weak solution
$(\rho,\mathbf{u},{\boldsymbol{\eta}})$, which is smooth, with defect measure
$\pi$, satisfies the problem $\eqref{elasticity}-\eqref{initialdata}$
pointwise and that the defect measure vanishes.

###  4.1 Derivation of weak formulation for smooth solutions

Let $(\rho,\mathbf{u},{\boldsymbol{\eta}})$ be a smooth solution to the
problem $\eqref{elasticity}-\eqref{initialdata}$. The goal is to prove that
this is also a weak solution in the sense of Definition 2.1. Let us first
observe the fluid domain $\Omega_{F}^{\boldsymbol{\eta}}$. Due to Lemma 3.1
Claim 4, one has that
$\partial\Omega_{F}^{\boldsymbol{\eta}}={\boldsymbol{\eta}}(\overline{I_{\boldsymbol{\eta}}})\cup\partial\Omega\setminus{\boldsymbol{\eta}}(\text{Int}(C_{\boldsymbol{\eta}}))$.
Assuming that
$|{\boldsymbol{\eta}}(\overline{I_{\boldsymbol{\eta}}})|=|{\boldsymbol{\eta}}(I_{\boldsymbol{\eta}})$|
and
$|\partial\Omega\setminus{\boldsymbol{\eta}}(\text{Int}(C_{\boldsymbol{\eta}}))|=|\partial\Omega\setminus{\boldsymbol{\eta}}(C_{\boldsymbol{\eta}})|$,
one has that both ${\boldsymbol{\eta}}(I_{\boldsymbol{\eta}})$ and
$\partial\Omega\setminus{\boldsymbol{\eta}}(C_{\boldsymbol{\eta}})$ have a
well-defined trace and normal vector, so one can use the divergence theorem on
$\Omega_{F}^{\boldsymbol{\eta}}$

$\displaystyle\int_{\Omega_{F}^{\boldsymbol{\eta}}}\nabla\cdot\mathbf{f}=\int_{{\boldsymbol{\eta}}(I_{\boldsymbol{\eta}})}\mathbf{f}\cdot\mathbf{n}dS+\int_{\partial\Omega\setminus{\boldsymbol{\eta}}(C_{\boldsymbol{\eta}})}\mathbf{f}\cdot\mathbf{n}^{\boldsymbol{\eta}}dS^{\boldsymbol{\eta}},$

where $\mathbf{n}^{\boldsymbol{\eta}}$ and $dS^{\boldsymbol{\eta}}$ are the
normal vector and the surface element on
$\partial\Omega_{S}^{\boldsymbol{\eta}}$, respectively. On the other hand, for
any function $\mathbf{f}\in C^{\infty}([0,T]\times\Omega)$, one has the
Reynolds transport theorem on $\Omega_{S}^{\boldsymbol{\eta}}(t)$

$\displaystyle\frac{d}{dt}\int_{\Omega_{S}^{\boldsymbol{\eta}}(t)}\mathbf{f}=\int_{\Omega_{S}^{\boldsymbol{\eta}}(t)}\partial_{t}\mathbf{f}+\int_{\partial\Omega_{S}^{\boldsymbol{\eta}}(t)}\mathbf{f}(\partial_{t}{\boldsymbol{\eta}}\cdot\mathbf{n}^{\boldsymbol{\eta}})\circ{\boldsymbol{\eta}}^{-1}dS^{\boldsymbol{\eta}},$

so

$\displaystyle\frac{d}{dt}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\mathbf{f}=\frac{d}{dt}\int_{\Omega}\mathbf{f}-\frac{d}{dt}\int_{\Omega_{S}^{\boldsymbol{\eta}}(t)}\mathbf{f}$
$\displaystyle=\int_{\Omega}\partial_{t}\mathbf{f}-\int_{\Omega_{S}^{\boldsymbol{\eta}}(t)}\partial_{t}\mathbf{f}-\int_{\partial\Omega_{S}^{\boldsymbol{\eta}}(t)}\mathbf{f}(\partial_{t}{\boldsymbol{\eta}}\cdot\mathbf{n}^{\boldsymbol{\eta}})\circ{\boldsymbol{\eta}}^{-1}dS^{\boldsymbol{\eta}}$
$\displaystyle=\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\partial_{t}\mathbf{f}-\int_{\partial\Omega_{S}^{\boldsymbol{\eta}}(t)}\mathbf{f}(\partial_{t}{\boldsymbol{\eta}}\cdot\mathbf{n}^{\boldsymbol{\eta}})\circ{\boldsymbol{\eta}}^{-1}dS^{\boldsymbol{\eta}}.$

Finally, since
$\partial_{t}{\boldsymbol{\eta}}(t,x)=\partial_{t}{\boldsymbol{\eta}}(t,y)$
whenever ${\boldsymbol{\eta}}(t,x)={\boldsymbol{\eta}}(t,y)$, the last
integral vanishes on $N_{\boldsymbol{\eta}}(t)$, so one concludes that the
Reynolds transport theorem holds on $\Omega_{F}^{\boldsymbol{\eta}}(t)$ in the
following form

$\displaystyle\frac{d}{dt}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\mathbf{f}=\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\partial_{t}\mathbf{f}-\int_{{\boldsymbol{\eta}}(t,I_{\boldsymbol{\eta}}(t))}\mathbf{f}(\partial_{t}{\boldsymbol{\eta}}\cdot\mathbf{n}^{\boldsymbol{\eta}})\circ{\boldsymbol{\eta}}^{-1}dS^{\boldsymbol{\eta}}$
(4.1)

for all functions $\mathbf{f}$ defined on $(0,T)\times\Omega$ for which the
above terms are well-defined. We will assume that both $\rho$ and
$\rho\mathbf{u}$ can be represented as restrictions of functions defined on
$(0,T)\times\Omega$ which are regular enough so that
$\eqref{reynolds:omega:f}$ holds for them.

Multiplying the fluid momentum equation with a test function
$\boldsymbol{\xi}\in C_{0}^{\infty}([0,T)\times\overline{\Omega})$, and then
integrating over $(0,T)\times\Omega_{F}^{\boldsymbol{\eta}}(t)$ gives us

$\displaystyle\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\partial_{t}(\rho\mathbf{u})\cdot\boldsymbol{\xi}+\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}(\nabla\cdot(\rho\mathbf{u}\otimes\mathbf{u}))\cdot\boldsymbol{\xi}$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}(\nabla\cdot\mathbb{S}({\bf
u}))\cdot\boldsymbol{\xi}-\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\nabla\rho^{\gamma}\cdot\boldsymbol{\xi}.$

First, by $\eqref{reynolds:omega:f}$, one has

$\displaystyle\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\partial_{t}(\rho\mathbf{u})\cdot\boldsymbol{\xi}=\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\partial_{t}(\rho\mathbf{u}\cdot\boldsymbol{\xi})-\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\rho\mathbf{u}\cdot\partial_{t}\boldsymbol{\xi}$
$\displaystyle=\int_{0}^{T}\underbrace{\frac{d}{dt}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\rho\mathbf{u}\cdot\boldsymbol{\xi}}_{\mathclap{=-\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}(\rho\mathbf{u})_{0}\cdot\boldsymbol{\xi}(0,\cdot)}}+\int_{0}^{T}\int_{{\boldsymbol{\eta}}(t,I_{\boldsymbol{\eta}}(t))}\rho\mathbf{u}\cdot\boldsymbol{\xi}(\partial_{t}{\boldsymbol{\eta}}\cdot\mathbf{n})\circ{\boldsymbol{\eta}}^{-1}dS-\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\rho\mathbf{u}\cdot\partial_{t}\boldsymbol{\xi}.$

Next

$\displaystyle\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}(\nabla\cdot(\rho\mathbf{u}\otimes\mathbf{u}))\cdot\boldsymbol{\xi}$
$\displaystyle=\int_{0}^{T}\int_{{\boldsymbol{\eta}}(t,I_{\boldsymbol{\eta}}(t))}(\rho\mathbf{u}\cdot\boldsymbol{\xi})(\mathbf{u}\cdot\mathbf{n}^{\boldsymbol{\eta}})dS+\underbrace{\int_{0}^{T}\int_{\partial\Omega\setminus{\boldsymbol{\eta}}(t,C_{\boldsymbol{\eta}}(t))}(\rho\mathbf{u}\cdot\boldsymbol{\xi})(\mathbf{u}\cdot\mathbf{n})dS}_{=0}-\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}(\rho\mathbf{u}\otimes\mathbf{u}):\nabla\boldsymbol{\xi}$

since $\boldsymbol{\xi}=0$ on $\partial\Omega$, so combining the above
identites and using the kinematic fluid-structure coupling condition
$\eqref{kinc}$ to cancel the boundary terms, one obtains

$\displaystyle\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\partial_{t}(\rho\mathbf{u})\cdot\boldsymbol{\xi}+\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}(\nabla\cdot(\rho\mathbf{u}\otimes\mathbf{u}))\cdot\boldsymbol{\xi}$
$\displaystyle=-\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}(\rho\mathbf{u})_{0}\cdot\boldsymbol{\xi}(0,\cdot)-\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\rho\mathbf{u}\cdot\partial_{t}\boldsymbol{\xi}-\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}(\rho\mathbf{u}\otimes\mathbf{u}):\nabla\boldsymbol{\xi}.$
(4.3)

Next, one has

$\displaystyle\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}(\nabla\cdot\mathbb{S}({\bf
u}))\cdot\boldsymbol{\xi}-\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\nabla\rho^{\gamma}\cdot\boldsymbol{\xi}$
$\displaystyle=\int_{0}^{T}\int_{{\boldsymbol{\eta}}(t,I_{\boldsymbol{\eta}}(t))}(\mathbb{S}({\bf
u})\mathbf{n}^{\boldsymbol{\eta}})\cdot\boldsymbol{\xi}dS^{\boldsymbol{\eta}}+\underbrace{\int_{0}^{T}\int_{\partial\Omega\setminus{\boldsymbol{\eta}}(t,C_{\boldsymbol{\eta}}(t))}(\mathbb{S}({\bf
u})\mathbf{n})\cdot\boldsymbol{\xi}dS}_{=0}$
$\displaystyle\quad-\int_{0}^{T}\int_{{\boldsymbol{\eta}}(I_{\boldsymbol{\eta}}(t))}\rho^{\gamma}\mathbf{n}^{\boldsymbol{\eta}}\cdot\boldsymbol{\xi}dS^{\boldsymbol{\eta}}-\underbrace{\int_{0}^{T}\int_{\partial\Omega\setminus{\boldsymbol{\eta}}(C_{\boldsymbol{\eta}})}\rho^{\gamma}\mathbf{n}\cdot\boldsymbol{\xi}dS}_{=0}$
$\displaystyle\quad-\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\mathbb{S}(\nabla\mathbf{u}):\nabla\boldsymbol{\xi}+\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\rho^{\gamma}(\nabla\cdot\boldsymbol{\xi})$
$\displaystyle=\int_{0}^{T}\int_{I_{\boldsymbol{\eta}}(t)}\det\nabla{\boldsymbol{\eta}}\left(\left[\mathbb{T}({\bf
u},p)\circ{\boldsymbol{\eta}}\right]\nabla{\boldsymbol{\eta}}^{-\tau}\mathbf{n}\right)\cdot\boldsymbol{\xi}\circ{\boldsymbol{\eta}}dS-\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\mathbb{S}(\nabla\mathbf{u}):\nabla\boldsymbol{\xi}+\int_{0}^{T}\int_{\Omega_{F}^{\boldsymbol{\eta}}(t)}\rho^{\gamma}(\nabla\cdot\boldsymbol{\xi}),$
(4.4)

where the last term is obtained by a coordinate transformation by composing
with ${\boldsymbol{\eta}}^{-1}$. Next, multiplying the structure momentum
equation with $\boldsymbol{\phi}:=\boldsymbol{\xi}\circ{\boldsymbol{\eta}}$
and integrating over $(0,T)\times\Omega_{S}$ gives us

$\displaystyle\int_{0}^{T}\int_{\Omega_{S}}\partial_{t}^{2}{\boldsymbol{\eta}}\boldsymbol{\phi}-\int_{0}^{T}\int_{\Omega_{S}}\nabla\cdot\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\cdot\boldsymbol{\phi}$
$\displaystyle=-\int_{0}^{T}\int_{\Omega_{S}}\partial_{t}{\boldsymbol{\eta}}\cdot\partial_{t}\boldsymbol{\phi}+\int_{\Omega_{S}}\mathbf{v}_{0}\cdot\boldsymbol{\phi}(0,\cdot)-\int_{0}^{T}\int_{\partial\Omega_{S}}(\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\mathbf{n})\cdot\boldsymbol{\phi}+\underbrace{\int_{0}^{T}\int_{\Omega_{S}}\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}}):\nabla\boldsymbol{\phi}}_{=\int_{0}^{T}\langle
DE({\boldsymbol{\eta}}),\boldsymbol{\phi}\rangle+\int_{0}^{T}\langle
D_{2}R({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}}),\boldsymbol{\phi}\rangle}.$
(4.5)

by $\eqref{2nd:order:bnd}$. Noticing that

$\displaystyle\int_{\partial\Omega_{S}}(\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\mathbf{n})\cdot\boldsymbol{\phi}$
$\displaystyle=\int_{I_{\boldsymbol{\eta}}(t)}(\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\mathbf{n})\cdot\boldsymbol{\phi}dS+\int_{N_{\boldsymbol{\eta}}(t)}(\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\mathbf{n})\cdot\boldsymbol{\phi}dS+\underbrace{\int_{C_{\boldsymbol{\eta}}(t)}(\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\mathbf{n})\cdot\boldsymbol{\phi}dS}_{=0},$

where the second term on RHS also vanishes due to Lemma 3.1 Claim 2 and the
structure-structure coupling $\eqref{dynca}$, and that

$\displaystyle\int_{I_{\boldsymbol{\eta}}}(\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\mathbf{n})\cdot\boldsymbol{\phi}dS=\int_{I_{\boldsymbol{\eta}}}\det\nabla{\boldsymbol{\eta}}\left[\mathbb{T}({\bf
u},p)\circ{\boldsymbol{\eta}}\right]\nabla{\boldsymbol{\eta}}^{-\tau}\mathbf{n}dS,$

by the dynamic fluid-structure coupling $\eqref{dync}$, one can use the
identities $\eqref{fl:calc0}$, $\eqref{fl:calc}$ and $\eqref{str:calc}$ in
$\eqref{fl}$ to obtain the coupled momentum equation $\eqref{momeqweak}$.

Next, the renormalized continuity equation $\eqref{reconteqweak}$ in obtained
in a similar fashion, i.e. we multiply $\eqref{conteq}$ with $(B(\rho)+\rho
B^{\prime}(\rho))\varphi$, then integrate over
$(0,T)\times\Omega_{F}^{\boldsymbol{\eta}}(t)$ and use the integration by
parts and Reynolds transport theorem $\eqref{reynolds:omega:f}$.

Finally, the energy (in)equality is obtained by choosing

$\displaystyle\boldsymbol{\xi}=\begin{cases}\mathbf{u},&\quad\text{ on
}(0,T)\times\Omega_{F}^{\boldsymbol{\eta}}(t),\\\
\partial_{t}{\boldsymbol{\eta}}\circ{\boldsymbol{\eta}}^{-1},&\quad\text{ on
}(0,T)\times\Omega_{S}^{\boldsymbol{\eta}}(t),\end{cases}$

in $\eqref{momeqweak}$, then
$B(\rho)=\frac{1}{\gamma-1}\rho^{\gamma},~{}b(\rho)=\rho^{\gamma}$ and
$\varphi=1$ in $\eqref{reconteqweak}$, and $B(\rho)=1,~{}b(\rho)=0$ and
$\varphi=-\frac{1}{2}|\mathbf{u}|^{2}$ and then summing up these 3 identities.

###  4.2 Consistency

Let us now assume that $({{\boldsymbol{\eta}}},\rho,\mathbf{u})$ is a weak
solution in the sense of Definition $\eqref{weaksol}$ which is smooth and let
$\pi\in L^{\infty}(0,T;\mathcal{M}^{+}(\partial N_{\boldsymbol{\eta}}(t)))$ be
the corresponding defect measure. Now, by choosing test functions
$\boldsymbol{\xi}$ with compact supports in
$(0,T)\times\Omega_{F}^{\boldsymbol{\eta}}(t)$ and
$(0,T)\times\Omega_{S}^{\boldsymbol{\eta}}(t)$, one directly recovers the
momentum equations $\eqref{momeq}$ and $\eqref{elasticity}$. The fluid-
structure and structure-structure kinematic coupling conditions $\eqref{kinc}$
and $\eqref{kinca}$ are automatically satisfied by definition. By similar
calculation as above, one can conclude that

$\displaystyle\int_{0}^{T}\int_{I_{\boldsymbol{\eta}}(t)}\det\nabla{\boldsymbol{\eta}}\left(\left[\mathbb{T}({\bf
u},p)\circ{\boldsymbol{\eta}}\right]\nabla{\boldsymbol{\eta}}^{-\tau}\mathbf{n}\right)\cdot\boldsymbol{\phi}dS+\int_{0}^{T}\langle\pi,\nabla\cdot\boldsymbol{\xi}\rangle_{[\mathcal{M}^{+},C](\partial
I_{\boldsymbol{\eta}}(t))}$
$\displaystyle\quad-\int_{0}^{T}\int_{I_{\boldsymbol{\eta}}(t)}(\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\mathbf{n})\cdot\boldsymbol{\phi}-\int_{0}^{T}\int_{N_{\boldsymbol{\eta}}(t)}(\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\mathbf{n})\cdot\boldsymbol{\phi}=0$
(4.6)

for any $\boldsymbol{\xi}\in C_{0}^{\infty}([0,T)\times\overline{\Omega})$,
where $\boldsymbol{\phi}=\boldsymbol{\xi}\circ{\boldsymbol{\eta}}$. Now, for
every $t_{0}\in[0,T]$ and $x_{0}\in I_{\boldsymbol{\eta}}(t_{0})$, there is an
interval $I\ni t_{0}$ and a ball $B=B({\boldsymbol{\eta}}(t_{0},x_{0}),r)$,
for some $r>0$, such that for all $t\in I$, one has
$B\cap\partial\Omega_{F}^{\boldsymbol{\eta}}(t)\subset{\boldsymbol{\eta}}(t,I_{\boldsymbol{\eta}}(t))$.
Then, we define a sequence of smooth test functions $\psi_{n}$ supported on
$I\times B$ so that $\psi_{n}\to\delta_{(t_{0},x_{0})}$, so by testing
$\eqref{test:eq}$ with $\psi_{n}\boldsymbol{\xi}$ and letting $n\to\infty$,
one has that $\left(\det\nabla{\boldsymbol{\eta}}\left(\left[\mathbb{T}({\bf
u},p)\circ{\boldsymbol{\eta}}\right]\nabla{\boldsymbol{\eta}}^{-\tau}\mathbf{n}\right)\cdot\boldsymbol{\phi}\right)(t_{0},x_{0})=\left((\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\mathbf{n})\cdot\boldsymbol{\phi}\right)(t_{0},x_{0})$.
Since $(t_{0},x_{0})$ and $\boldsymbol{\xi}$ were arbitrary, one obtains that
the fluid-structure dynamic condition
$\det\nabla{\boldsymbol{\eta}}\left(\left[\mathbb{T}({\bf
u},p)\circ{\boldsymbol{\eta}}\right]\nabla{\boldsymbol{\eta}}^{-\tau}\mathbf{n}\right)=\boldsymbol{\sigma}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})\mathbf{n}$
holds on $(0,T)\times I_{\boldsymbol{\eta}}(t)$, and by continuity on
$[0,T]\times\overline{I_{\boldsymbol{\eta}}(t)}$. By a similar approach, one
can also conclude that the structure-structure dynamic condition
$\eqref{dynca}$ holds on $[0,T]\times N_{\boldsymbol{\eta}}(t)$ (here the key
point is that due to the coupling
$\boldsymbol{\phi}=\boldsymbol{\xi}\circ{\boldsymbol{\eta}}$ on $\Omega_{S}$,
one has that $\boldsymbol{\phi}(t,x)=\boldsymbol{\phi}(t,y)$ whenever
${\boldsymbol{\eta}}(t,x)={\boldsymbol{\eta}}(t,y)$), so one has that all
$({\boldsymbol{\eta}},\rho,\mathbf{u})$ satisfy
$\eqref{elasticity}-\eqref{initialdata}$ pointwise. This implies that we can
show that the energy identity (without the defect measure) holds

$\displaystyle\int_{\Omega_{F}^{{\boldsymbol{\eta}}}(t)}\Big{(}\frac{1}{2}\rho|\mathbf{u}|^{2}+\frac{\rho^{\gamma}}{\gamma-1}\Big{)}(t)+\int_{0}^{t}\int_{\Omega_{F}^{{\boldsymbol{\eta}}}(\tau)}(\mathbb{S}(\nabla\mathbf{u}):\nabla\mathbf{u})(\tau)~{}d\tau$
$\displaystyle\quad+\frac{1}{2}\int_{\Omega_{S}}|\partial_{t}{{\boldsymbol{\eta}}}|^{2}(t)+E({\boldsymbol{\eta}}(t))+\int_{0}^{t}2R({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})(\tau)~{}d\tau$
$\displaystyle=\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{0}}}\Big{(}\frac{1}{2\rho_{0}}|(\rho\mathbf{u})_{0}|^{2}+\frac{\rho_{0}^{\gamma}}{\gamma-1}\Big{)}+\int_{\Omega_{S}}\frac{1}{2}|\mathbf{v}_{0}|^{2}+E({\boldsymbol{\eta}}_{0}),$

for all $t\in(0,T]$, which compared to the energy inequality $\eqref{enineq}$,
by taking into consideration that $\pi\geq 0$, implies that $\pi\equiv 0$.

##  5 Construction of approximate solutions

The goal of this section is to obtain approximate solutions to our problem
which depend on given parameters $h,\varsigma,\varepsilon>0$, where $h=T/N$
for $N\in\mathbb{N}$. As these parameters vanish (in certain order), we will
prove that the approximations converge to a weak solution in sense of
Definition 2.1.

###  5.1 Time-stepping decoupling approximation scheme

Before we start, let us point out that for our scheme to work, we must avoid
self-contact or contact with the rigid boundary of the elastic body initially,
at least for $\varepsilon>0$, so the initial contact can be recovered as
$\varepsilon\to 0$. The key idea here is to use the bulk nature of the solid
and to “pull inside” using the reference configuration. For this we have to
construct a smooth diffeomorphism $\Phi:\Omega_{S}\to\Omega_{S}$ which mostly
is equal to the identity, but maps the contact set
$C({\boldsymbol{\eta}}_{0})\cup N({\boldsymbol{\eta}}_{0})$ to the interior of
$\Omega_{S}$. A simple example would be

$\displaystyle\Phi_{\varepsilon}(x):=x-\varepsilon\boldsymbol{\nu}(x)\psi(x),$

where $\boldsymbol{\nu}:\overline{\Omega_{S}}\to B_{1}(0)$ is a sufficiently
smooth continuation of the unit normal and
$\psi(x):\overline{\Omega_{S}}\to[0,1]$ is a cut-off equal to one on
$C({\boldsymbol{\eta}}_{0})\cup N({\boldsymbol{\eta}}_{0})$. Since we are
dealing with a fixed, smooth enough domain $\Omega_{S}$, for $\varepsilon$
small enough, then $\Phi_{\varepsilon}$ is injective and has a uniformly
positive Jacobian. With this, we can then define

$\displaystyle{\boldsymbol{\eta}}_{0,\varepsilon}(x):={\boldsymbol{\eta}}_{0}\circ\Phi_{\varepsilon}(x)$

which itself has the same regularity as ${\boldsymbol{\eta}}_{0}$, satisfies
$\det\nabla{\boldsymbol{\eta}}_{\varepsilon}=\det\nabla{\boldsymbol{\eta}}_{0}\det\nabla\Phi_{\varepsilon}>c>0$
for some $c$ and per construction
$C({\boldsymbol{\eta}}_{0,\varepsilon})=N({\boldsymbol{\eta}}_{0,\varepsilon})=\emptyset$.
Additionally we have
${\boldsymbol{\eta}}_{0,\varepsilon}\to{\boldsymbol{\eta}}_{0}$ in all
relevant norms. Note that this approximate data can then be further
regularized, e.g. by a standard convolution argument, without making it
inadmissible by causing overlap. For more details also compare Step 1 of the
proof of Theorem 2.4 in [10].

Next, we define the contact-penalization term given by

$\displaystyle
K_{\varepsilon}({\boldsymbol{\eta}}):=\int_{\Omega^{S}}\kappa_{\varepsilon}(\operatorname{dist}({\boldsymbol{\eta}},\partial\Omega))+\iint_{\Omega^{S}\times\Omega^{S}\setminus\\{|x-y|<\sqrt{\varepsilon}\\}}\kappa_{\varepsilon}(|{\boldsymbol{\eta}}(x)-{\boldsymbol{\eta}}(y)|)$

with $\kappa_{\varepsilon}(r)=\kappa(r/\varepsilon)$ and $\kappa$ such that
$\kappa(r):=+\infty$ for $r\leq 0$, $\kappa(r)=0$ for $r\geq 1$ and convex in
between with $\lim_{r\to 0}\kappa(r)=+\infty$ and $\lim_{r\to
1}\kappa^{\prime}(r)=0$. Also we assume that there is a constant $c$ such that
$|\kappa^{\prime}(r)r|\leq c(\kappa(r)+1)$ for all $r>0$. Note that the last
translates into

$\displaystyle|\kappa_{\varepsilon}^{\prime}(r)r|\leq
c\varepsilon(\kappa_{\varepsilon}(r)+1).$ (5.1)

A canonical example of such a function would be $\kappa(r):=\frac{1}{r}+r-2$
for $r\in(0,1)$ and $\kappa(r)=0$ for $r\geq 1$. Additionally, the following
regularisation of elastic and viscoelastic energy terms is introduced:

$\displaystyle
E_{\varepsilon}({\boldsymbol{\eta}}):=E({\boldsymbol{\eta}})+\varepsilon^{a_{0}}\left\|{\boldsymbol{\eta}}\right\|_{W^{k_{0},2}}^{2},\quad
R_{\varepsilon}({\boldsymbol{\eta}},\boldsymbol{b}):=R({\boldsymbol{\eta}}_{k},\boldsymbol{b})+\varepsilon\left\|\boldsymbol{b}\right\|_{W^{k_{0},2}}^{2},$

and the approximate stress tensor $\boldsymbol{\sigma}_{\varepsilon}$ is
defined as

$\displaystyle\nabla\cdot\boldsymbol{\sigma}_{\varepsilon}:=-DE_{\varepsilon}({\boldsymbol{\eta}})-D_{2}R_{\varepsilon}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}}),\quad\text{
in }\mathcal{D}^{\prime}(Q_{S,T})$

and the exponent $a_{0}$ is needed for convergence later. Finally, define the
time delay operator $\tau_{h}$ as

$\displaystyle\tau_{h}f(t):=f(t-h).$

The structure sub-problem (SSP):
By induction on $0\leq n\leq N-1$, assume that:

  * Case $n=0$: ${\boldsymbol{\eta}}^{0}(0,\cdot):={\boldsymbol{\eta}}_{0,\varepsilon}$, $\partial_{t}{\boldsymbol{\eta}}^{0}(0,\cdot):=\mathbf{v}_{0}$;

  * Case $n\geq 1$: the solution ${\boldsymbol{\eta}}^{n}$ of $(SSP)$ and the solution $(\rho^{n},\mathbf{u}^{n})$ of $(FSP)$ (defined below) are already obtained.

Find ${\boldsymbol{\eta}}^{n+1}$ so that:

  1. 1.

${\boldsymbol{\eta}}^{n+1}\in
L^{\infty}(nh,(n+1)h;W^{k_{0},2}(\Omega_{S})),~{}\partial_{t}{\boldsymbol{\eta}}^{n+1}\in
L^{2}(nh,(n+1)h;W^{k_{0},2}(\Omega_{S}))$;

  2. 2.

The structure equation

$\displaystyle\int_{nh}^{(n+1)h}\int_{\Omega_{S}}\frac{\partial_{t}{\boldsymbol{\eta}}^{n+1}-\tau_{h}\mathbf{U}^{n}}{h}\cdot\boldsymbol{\phi}+\int_{0}^{T}\langle
DE_{\varepsilon}({\boldsymbol{\eta}}^{n+1}),\boldsymbol{\phi}\rangle$
$\displaystyle\quad+\int_{0}^{T}\langle
D_{2}R_{\varepsilon}({\boldsymbol{\eta}}^{n+1},\partial_{t}{\boldsymbol{\eta}}^{n+1}),\boldsymbol{\phi}\rangle+\int_{nh}^{(n+1)h}\langle
DK_{\varepsilon}({\boldsymbol{\eta}}^{n+1}),\boldsymbol{\phi}\rangle=0$ (5.2)

holds for all $\boldsymbol{\phi}\in
C^{\infty}([nh,(n+1)h]\times\overline{\Omega_{S}})$;

  3. 3.

Structure energy inequality

$\displaystyle
E_{\varepsilon}({\boldsymbol{\eta}}^{n+1}(t))+K_{\varepsilon}({\boldsymbol{\eta}}^{n+1}(t))+2\int_{nh}^{t}R_{\varepsilon}({\boldsymbol{\eta}}^{n+1},\partial_{t}{\boldsymbol{\eta}}^{n+1})$
$\displaystyle\quad+\frac{1}{2h}\int_{nh}^{t}\int_{\Omega_{S}}\Big{(}|\partial_{t}{\boldsymbol{\eta}}^{n+1}-\tau_{h}\mathbf{U}^{n}|^{2}+|\partial_{t}{\boldsymbol{\eta}}^{n+1}|^{2}\Big{)}$
$\displaystyle\leq
E_{\varepsilon}({\boldsymbol{\eta}}^{n+1}(nh))+K_{\varepsilon}({\boldsymbol{\eta}}^{n+1}(nh))+\frac{1}{2h}\int_{nh}^{t}\int_{\Omega_{S}}|\tau_{h}\mathbf{U}^{n}|^{2}(nh)$
(5.3)

for all $t\in(nh,(n+1)h]$.

We extend the fluid domain to the entire $\Omega$ by extending the initial
data and the fluid dissipation. More precisely, we take sufficiently regular
$\rho_{0,\varepsilon}$ and $(\rho\mathbf{u})_{0,\varepsilon}$ such that

$\displaystyle\rho_{0,\varepsilon}\geq
0,~{}~{}\rho_{0,\varepsilon}|_{\mathbb{R}^{3}\setminus\Omega^{{\boldsymbol{\eta}}_{0,\varepsilon}}}=0,~{}~{}\int_{\Omega}\rho_{0,\varepsilon}^{\gamma}+\varepsilon\rho_{0,\varepsilon}^{\beta}\leq
c,$ $\displaystyle(\rho\mathbf{u})_{0,\varepsilon}=0\text{ wherever
}\rho_{0,\varepsilon}=0,$ $\displaystyle\rho_{0,\varepsilon}\to\rho_{0}\text{
in }L^{\gamma}(\Omega^{{\boldsymbol{\eta}}_{0}})\text{ and
}(\rho\mathbf{u})_{0,\varepsilon}\to(\rho\mathbf{u})_{0}\text{ in
}L^{\frac{2\gamma}{\gamma+1}}(\Omega^{{\boldsymbol{\eta}}_{0}}),\text{ as
}\varepsilon\to 0,$

where $\beta>0$ is a sufficiently large number, and for given $\varepsilon>0$
and ${\boldsymbol{\eta}}$, we define

$\displaystyle\mathbb{S}_{\varepsilon,{\boldsymbol{\eta}}}(\nabla\mathbf{u}):=\mathbb{S}(\nabla\mathbf{u})\times\begin{cases}1,\quad\text{on
}\Omega^{\boldsymbol{\eta}},\\\ \varepsilon,\quad\text{elsewhere}.\end{cases}$

We now introduce:

The fluid sub-problem (FSP):
By induction on $0\leq n\leq N-1$, assume that:

  * Case $n=0$: $\rho^{0}(0,\cdot):=\rho_{0,\varepsilon}(\cdot)$, $(\rho\mathbf{u})^{0}(0,\cdot):=(\rho\mathbf{u})_{0,\varepsilon}(\cdot)$;

  * Case $n\geq 1$: the solution $(\rho^{n},\mathbf{u}^{n})$ of $(FSP)$ and the solution ${\boldsymbol{\eta}}^{n+1}$ of $(SSP)$ are already obtained.

Find $(\rho^{n+1},\mathbf{u}^{n+1})$ so that:

  1. 1.

$\rho^{n+1}\in L^{\infty}(nh,(n+1)h;L^{\beta}(\Omega))$, $\mathbf{u}^{n+1}\in
L^{2}(nh,(n+1)h;H_{0}^{1}(\Omega))$;

  2. 2.

$\rho^{n+1}(nh,\cdot)=\rho^{n}(nh,\cdot)$ and
$(\rho\mathbf{u})^{n+1}(nh,\cdot)=(\rho\mathbf{u})^{n}(nh,\cdot)$ in weakly
continuous sense in time;

  3. 3.

The following damped continuity equation

$\displaystyle\int_{nh}^{(n+1)h}\displaystyle\int_{\Omega}\rho^{n+1}(\partial_{t}\varphi+\mathbf{u}^{n+1}\cdot\nabla\varphi)-\int_{nh}^{(n+1)h}\displaystyle\int_{\Omega_{S}^{{\boldsymbol{\eta}}^{n+1}}}(\rho^{n+1})^{2}\varphi-\varsigma\int_{nh}^{(n+1)h}\displaystyle\int_{\Omega}\nabla\rho^{n+1}\cdot\nabla\varphi$
$\displaystyle=-\int_{nh}^{(n+1)h}\frac{d}{dt}\int_{\Omega}\rho^{n+1}\varphi$

holds for all $\varphi\in C^{\infty}([nh,(n+1)h]\times\mathbb{R}^{3})$;

  4. 4.

The following momentum equation

$\displaystyle\int_{nh}^{(n+1)h}\displaystyle\int_{\Omega}\rho^{n+1}\mathbf{u}^{n+1}\cdot\partial_{t}\boldsymbol{\xi}+\int_{nh}^{(n+1)h}\displaystyle\int_{\Omega}(\rho^{n+1}\mathbf{u}^{n+1}\otimes\mathbf{u}^{n+1}):\nabla\boldsymbol{\xi}$
$\displaystyle\quad+\int_{nh}^{(n+1)h}\displaystyle\int_{\Omega}\big{(}(\rho^{n+1})^{\gamma}+\varepsilon(\rho^{n+1})^{\beta}\big{)}\nabla\cdot\boldsymbol{\xi}-\int_{nh}^{(n+1)h}\displaystyle\int_{\Omega}\mathbb{S}_{\varepsilon,{\boldsymbol{\eta}}^{n+1}}(\nabla\mathbf{u}^{n+1}):\nabla\boldsymbol{\xi}$
$\displaystyle\quad-\frac{1}{2}\int_{nh}^{(n+1)h}\int_{\Omega_{S}^{{\boldsymbol{\eta}}^{n+1}}(t)}(\rho^{n+1})^{2}\mathbf{u}^{n+1}\cdot\boldsymbol{\xi}-\varsigma\int_{nh}^{(n+1)h}\int_{\Omega}(\nabla\rho^{n+1}\cdot\nabla\mathbf{u}^{n+1})\cdot\boldsymbol{\xi}$
$\displaystyle\quad-\int_{nh}^{(n+1)h}\displaystyle\int_{\Omega_{S}}\frac{\displaystyle\mathbf{U}^{n+1}-\partial_{t}{\boldsymbol{\eta}}^{n+1}}{\displaystyle
h}\cdot\boldsymbol{\phi}$
$\displaystyle=\int_{\Omega}\rho^{n+1}\mathbf{u}^{n+1}\cdot\boldsymbol{\xi}~{}\Big{|}_{nh}^{(n+1)h}$
(5.4)

holds for all $\boldsymbol{\xi}\in C^{\infty}([nh,(n+1)h]\times\Omega)$, where

$\displaystyle\mathbf{U}^{n+1}:=\mathbf{u}^{n+1}\circ{\boldsymbol{\eta}}^{n+1},\quad\boldsymbol{\phi}:=\boldsymbol{\xi}\circ{\boldsymbol{\eta}}^{n+1};$

  5. 5.

The fluid energy inequality

$\displaystyle\int_{\Omega}\Big{(}\frac{1}{2}\rho^{n+1}|\mathbf{u}^{n+1}|^{2}+\frac{(\rho^{n+1})^{\gamma}}{\gamma-1}+\varepsilon\frac{(\rho^{n+1})^{\beta}}{\beta-1}\Big{)}(t)+\int_{nh}^{t}\int_{\Omega}\mathbb{S}_{\varepsilon,{\boldsymbol{\eta}}^{n+1}}(\nabla\mathbf{u}^{n+1}):\nabla\mathbf{u}$
$\displaystyle\quad+\int_{nh}^{t}\int_{\Omega_{S}^{{\boldsymbol{\eta}}^{n+1}}(t)}\left(\frac{1}{\gamma-1}(\rho^{n+1})^{\gamma+1}+\frac{\varepsilon}{\beta-1}(\rho^{n+1})^{\beta+1}\right)$
$\displaystyle\quad+\varsigma\int_{nh}^{t}|\nabla\rho^{n+1}|^{2}\left(\gamma(\rho^{n+1})^{\gamma-2}+\varepsilon\beta(\rho^{n+1})^{\beta-2}\right)$
$\displaystyle\quad+\frac{1}{2h}\int_{nh}^{t}\int_{\Omega_{S}}\Big{(}|\mathbf{U}^{n+1}-\partial_{t}{\boldsymbol{\eta}}^{n+1}|^{2}+|\mathbf{U}^{n+1}|^{2}\Big{)}$
$\displaystyle\leq\int_{\Omega}\Big{(}\rho^{n}|\mathbf{u}^{n}|^{2}+\frac{(\rho^{n})^{\gamma}}{\gamma-1}\Big{)}(nh)+\frac{1}{2h}\int_{\Omega_{S}}|\partial_{t}{\boldsymbol{\eta}}^{n+1}|^{2}(nh)$
(5.5)

holds for all $t\in(nh,(n+1)h]$.

###  5.2 Discussion about the scheme

The fluid equations are extended to the entire domain $\Omega$ and the fluid
is allowed to pass through the fluid-structure interface and occupy the
structure domain. This way, we obtain a problem on a fixed domain. This idea
combines decoupling, penalization and domain extension and was first done in
[37] in the context of interaction between a heat-conducting fluid and a
thermoelastic shell. The penalization and domain extension approach for
compressible viscous fluids on moving domains without the structure comes from
[15]. Next, the mentioned decoupling method in fluid-structure interaction
which splits the coupled momentum equation in the structure acceleration term
was first done in [40], where the fluid is incompressible and the elastic
solid is a beam. The constructed scheme was stationary both for fluid and
structure, and they are solved through a time-marching scheme. Later, in [49],
this scheme was adjusted to the case where the structure is a nonlinear plate,
and a hybrid scheme was constructed which is stationary for the fluid and
time-continuous for the plate. This way, the energy for the plate (which is
not discretizable in time) is properly conserved for the approximate solution.
Finally, in [50], the interaction between a compressible viscous fluid and a
nonlinear thermoelastic plate is studied, and the decoupling scheme is fully
time-continuous, which allowed the standard theory for solving both sub-
problems to be used.

The solutions to the solid equations in turn are constructed using a method
related to that employed in [2] where the effects of inertia are initially
dealt with on a different time-scale than those resulting from energy and
dissipation. This allows us to begin by treating the problem as a gradient
flow with forces. For this type of problems we can then employ the variational
method of minimizing movements. In contrast to other approaches relying on
linearization or finite dimensional approximations, this allows us to directly
work with nonlinearities, both in the equation as well as in that the space of
admissible deformations which for this kind of problem is generally not closed
under any form of linear interpolation.

We deviate from [2] in two key aspects though. Firstly, in favor of the more
simple approach described above, we choose to not construct an approximation
to the fluid equations using the same method (compare [6] to see how this can
be done). Secondly, while the approach can in fact deal with “hard” collisions
of solids (see [10]), we instead employ a soft penalization potential to avoid
collisions in our approximation, as is well established, in particular in
numerics (see e.g. [31] for a modern approach). This allows us to have regular
approximate fluid domains, which makes working with them easier. To the best
of our knowledge this is the first time this type of approach has been applied
to the fluid-structure interactions.

Let us now point out the roles of the added terms in the equations.

  1. 1.

The contact-penalization $K_{\varepsilon}$ is added to ensure that the
deformations satisfy the Ciarlet-Nečas condition;

  2. 2.

The approximate energy functionals $E_{\varepsilon}$ and $R_{\varepsilon}$
provide stronger regularity which ensures that we can use
$\partial_{t}{\boldsymbol{\eta}}$ as a test function in the structure sub-
problem and derive the energy inequality;

  3. 3.

The term
$\smash{\chi_{\Omega_{S}^{{\boldsymbol{\eta}}^{n+1}}}(\rho^{n+1})^{2}}$ in the
continuity equation ensures that pressure is bounded in $L_{t,x}^{p}$ for
$p>1$ on the structure domain
$\smash{\Omega_{S}^{{\boldsymbol{\eta}}^{n+1}}}$, since we cannot obtain the
improved pressure estimates inside the structure domain uniformly w.r.t.
parameter $h$, due to structure terms. Then, the term
$\smash{-\frac{1}{2}\chi_{\Omega_{S}^{{\boldsymbol{\eta}}^{n+1}}}(\rho^{n+1})^{2}\mathbf{u}^{n+1}}$
in the momentum equation is added to correct the energy. Both of these terms
will converge to zero weakly as $h,\varsigma\to 0$, due to Lemma 6.2. Since
the solution to the fluid sub-problem is obtained as a limit of a system in a
finite basis, the term $-\varsigma\Delta\rho^{n+1}$ is added (or rather kept)
to ensure that we can identify the mentioned nonlinear terms, while once again
$-\varsigma\nabla\rho^{n+1}\cdot\nabla\mathbf{u}^{n+1}$ is added to the fluid
momentum equation to correct the energy;

  4. 4.

The artificial pressure $\varepsilon(\rho^{n+1})^{\beta}$ is standard in the
theory of compressible Navier-Stokes equations. It is added to ensure that
effective viscous flux converges properly as artificial density damping
vanishes $-\varsigma\Delta\rho\to 0$;

  5. 5.

The terms
$\chi_{\Omega_{S}}\frac{\mathbf{U}^{n+1}-\partial_{t}{\boldsymbol{\eta}}^{n+1}}{h}$
and $\frac{\partial_{t}{\boldsymbol{\eta}}^{n+1}-\tau_{h}\mathbf{U}^{n}}{h}$
allow us to decouple the coupled momentum equation $\eqref{momeqweak}$ and
thus decouple the problem into fluid and structure sub-problems. Moreover,
they act as a two-sided penalization, and force the limiting functions
$\partial_{t}{\boldsymbol{\eta}}$ and $\mathbf{u}\circ{\boldsymbol{\eta}}$ to
become equal on the solid domain as $\varsigma,h\to 0$, which will ensure the
kinematic coupling condition. The coupled momentum equation will be recovered
by summing up $\eqref{momeqSSP}$ and $\eqref{momeqFSP}$ with a common test
function, so the mentioned terms, when summed up, will give
$\frac{\mathbf{U}^{n+1}-\tau_{h}\mathbf{U}^{n}}{h}$, a term which converges to
$\partial_{t}\mathbf{U}=\partial_{t}^{2}{\boldsymbol{\eta}}$ – the
acceleration of the solid, as $h\to 0$.

###  5.3 Solving the fluid sub-problem

The solution of the fluid sub-problem can be obtained as in standard theory of
compressible viscous fluids. The idea is to span the fluid velocity in a
finite Galerkin basis on $\Omega$. The solution of such problem is obtained by
Schauder’s fixed point theorem, while the estimates are obtained by testing
the continuity equation with
$\frac{1}{\gamma-1}\rho^{\gamma}+\varepsilon\frac{1}{\beta-1}\rho^{\beta-1}$
and $-\frac{1}{2}|\mathbf{u}|^{2}$ and the momentum equation with $\mathbf{u}$
and summing these 3 identities together. Afterwards, the limit in the number
of basis functions is passed to, so the solution is obtained. For more
details, the reader is referred to [42, Section 7].

###  5.4 Solving the structure sub-problem

We introduce a time-discretized version. First, let us split the time interval
$[nh,(n+1)h]$ into $M$ time intervals of length $\Delta t$, that is
$[nh,nh+\Delta t],[nh+\Delta t,nh+2\Delta t],...,[nh+(M-1)\Delta t,(n+1)h]$.
Finally, we discretize the function $\mathbf{U}^{n}$ in time and denote it
with

$\displaystyle\mathbf{U}_{k}^{n}:=\frac{1}{\Delta t}\int_{nh+k\Delta
t}^{nh+(k+1)\Delta t}\mathbf{U}^{n}dt,\quad k=0,1,2,...,M-1.$

For $1\leq k\leq M$, we define the problem as

$\displaystyle\frac{1}{h}\int_{\Omega_{S}}\left(\frac{{\boldsymbol{\eta}}_{k}^{n+1}-{\boldsymbol{\eta}}_{k-1}^{n+1}}{\Delta
t}-\mathbf{U}_{k}^{n}\right)\cdot\boldsymbol{\phi}+\langle
DE_{\varepsilon}({\boldsymbol{\eta}}_{k}^{n+1}),\boldsymbol{\phi}\rangle+\langle
D_{2}R_{\varepsilon}({\boldsymbol{\eta}}_{k}^{n+1},\partial_{t}{\boldsymbol{\eta}}_{k}^{n+1}),\boldsymbol{\phi}\rangle+\langle
DK_{\varepsilon}({\boldsymbol{\eta}}_{k}^{n+1}),\boldsymbol{\phi}\rangle=0$

for all $\boldsymbol{\phi}\in C^{\infty}(\Omega_{S})$, where
${\boldsymbol{\eta}}_{0}^{n+1}:={\boldsymbol{\eta}}^{n}(nh)$.

The approach we use for this part of the proof is quite similar to [2, Theorem
3.5]. However for the sake of completeness and because of the additional
collision potential $K_{\varepsilon}$ we will still carry out the details. The
idea is that if $\boldsymbol{U}_{k}^{n}$ is treated as given data, then at
this point the problem has the structure of a gradient flow with forces. It
can thus be approached by a minimizing movements scheme. As $n$ will be fixed
in this section and apart from initial and right hand side data, all
quantities have the same index $n+1$, we will drop this superscript everywhere
where it is not necessary.

Starting with ${\boldsymbol{\eta}}_{0}^{n+1}:={\boldsymbol{\eta}}^{n}(nh)$ we
begin by considering the iterative problem of minimizing

$\displaystyle{\boldsymbol{\eta}}\mapsto
E_{\varepsilon}({\boldsymbol{\eta}})+K_{\varepsilon}({\boldsymbol{\eta}})+\Delta
tR_{\varepsilon}\left({\boldsymbol{\eta}}_{k},\frac{\boldsymbol{{\boldsymbol{\eta}}}-\boldsymbol{{\boldsymbol{\eta}}}_{k}}{\Delta
t}\right)+\frac{\Delta
t}{2h}\left\|\frac{\boldsymbol{{\boldsymbol{\eta}}}-\boldsymbol{{\boldsymbol{\eta}}}_{k}}{\Delta
t}-\mathbf{U}_{k}\right\|_{L^{2}(\Omega_{S})}^{2}$

among the set $\mathcal{E}^{q}\cap W^{k_{0},2}(\Omega_{S})$ to obtain
${\boldsymbol{\eta}}_{k+1}$. Here ${\boldsymbol{\eta}}_{k}$ is the solution
obtained in the previous $\Delta t$-step.

We first note that this problem has a minimizer as a consequence of the direct
method. Specifically, since all terms are non-negative, the growth condition
on $E$ gives us coercivity in $W^{k_{0},2}$ and by our assumptions or
respectively their specific form, all terms are weakly lower-semicontinuous in
${\boldsymbol{\eta}}$ with respect to that space. Thus any minimizing sequence
has a converging subsequence and a limit, which then has to be a minimizer
(cmp. also [2, Prop. 2.13] for details). However one should keep in mind that
such a minimizer is not necessarily unique.

Let us now denote the resulting piecewise constant and piecewise affine
approximations by

$\displaystyle\overline{\boldsymbol{\eta}}_{\Delta t}(t)$
$\displaystyle:={\boldsymbol{\eta}}_{k+1},$ for $\displaystyle t\in[nh+k\Delta
t,nh+(k+1)\Delta t),$ $\displaystyle\underline{{\boldsymbol{\eta}}}_{\Delta
t}(t)$ $\displaystyle:={\boldsymbol{\eta}}_{k},$ for $\displaystyle
t\in[nh+k\Delta t,nh+(k+1)\Delta t),$
$\displaystyle\hat{\boldsymbol{\eta}}_{\Delta t}(t)$
$\displaystyle:=\frac{t-nh-k\Delta t}{\Delta
t}{\boldsymbol{\eta}}_{k+1}+\frac{nh-(k+1)\Delta t-t}{\Delta
t}{\boldsymbol{\eta}}_{k},$ for $\displaystyle t\in[nh+k\Delta
t,nh+(k+1)\Delta t).$

Additionally since we know that ${\boldsymbol{\eta}}_{k+1}$ is a minimizer, we
can compare it with the choice ${\boldsymbol{\eta}}_{k}$ in the minimization
to obtain the initial estimate:

$\displaystyle
E_{\varepsilon}({\boldsymbol{\eta}}_{k+1})+K_{\varepsilon}({\boldsymbol{\eta}}_{k+1})+2\Delta
tR_{\varepsilon}\left({\boldsymbol{\eta}}_{k},\frac{\boldsymbol{{\boldsymbol{\eta}}}_{k+1}-\boldsymbol{{\boldsymbol{\eta}}}_{k}}{\Delta
t}\right)+\frac{\Delta
t}{2h}\left\|\frac{\boldsymbol{{\boldsymbol{\eta}}}_{k+1}-\boldsymbol{{\boldsymbol{\eta}}}_{k}}{\Delta
t}-\mathbf{U}_{k}\right\|^{2}$ $\displaystyle\leq
E_{\varepsilon}({\boldsymbol{\eta}}_{k})+K_{\varepsilon}(\boldsymbol{{\boldsymbol{\eta}}}_{k})+\frac{\Delta
t}{2h}\left\|\mathbf{U}_{k}\right\|^{2}.$

Summing up this estimate over $k$ and replacing the ${\boldsymbol{\eta}}_{k}$
and the difference quotients by the respective approximations then results in

$\displaystyle E_{\varepsilon}(\overline{\boldsymbol{\eta}}_{\Delta
t}(t))+K_{\varepsilon}(\overline{\boldsymbol{\eta}}_{\Delta
t}(t))+2\int_{nh}^{t}R_{\varepsilon}\left(\underline{{\boldsymbol{\eta}}}_{\Delta
t}(t),\partial_{t}\hat{\boldsymbol{\eta}}_{\Delta
t}(t)\right)+\frac{1}{2h}\left\|\partial_{t}\hat{\boldsymbol{\eta}}_{\Delta
t}(t)-\underline{\mathbf{U}}_{\Delta t}(t)\right\|^{2}dt$ $\displaystyle\leq
E_{\varepsilon}({\boldsymbol{\eta}}_{0}^{n+1})+K_{\varepsilon}(\boldsymbol{{\boldsymbol{\eta}}}_{0}^{n+1})+\int_{nh}^{t}\frac{1}{2h}\left\|\underline{\mathbf{U}}_{\Delta
t}(t)\right\|^{2}dt$

for all $t\in[nh,(n+1)h]$ where $\underline{\mathbf{U}}_{\Delta
t}(t):=\mathbf{U}_{k}$ for $t\in[nh+k{\Delta t},nh+(k+1){\Delta t})$ similar
to before.

Per Jensen’s inequality we additionally have
$\int_{nh}^{t}\left\|\underline{\mathbf{U}}_{\Delta
t}(t)\right\|^{2}dt\leq\int_{nh}^{t}\left\|\mathbf{U}^{n}(t)\right\|^{2}dt$.
So in particular this gives us ${\Delta t}$-independent a-priori estimates in
the form of

$\displaystyle\sup_{t\in[nh,(n+1)h]}E_{\varepsilon}(\overline{\boldsymbol{\eta}}_{\Delta
t}(t))\leq C,$
$\displaystyle\quad\sup_{t\in[nh,(n+1)h]}K_{\varepsilon}(\overline{\boldsymbol{\eta}}_{\Delta
t}(t))\leq C,$
$\displaystyle\quad\int_{nh}^{(n+1)h}R_{\varepsilon}\left(\underline{{\boldsymbol{\eta}}}_{\Delta
t}(t),\partial_{t}\hat{\boldsymbol{\eta}}_{\Delta t}(t)\right)dt\leq C,$
$\displaystyle\quad\int_{nh}^{(n+1)h}\frac{1}{2h}\left\|\partial_{t}\hat{\boldsymbol{\eta}}_{\Delta
t}(t)\right\|^{2}dt\leq C.$

Using the coercivity of $E_{\varepsilon}$, the fact that
$\overline{\boldsymbol{\eta}}_{\Delta t}$ and
$\underline{{\boldsymbol{\eta}}}_{\Delta t}$ only differ by a small shift in
time and that $\hat{\boldsymbol{\eta}}_{\Delta t}$ is their linear
interpolation we then get that

$\displaystyle\overline{\boldsymbol{\eta}}_{\Delta
t},\underline{{\boldsymbol{\eta}}}_{\Delta t},\hat{\boldsymbol{\eta}}_{\Delta
t}$ $\displaystyle\text{ are bounded in
}L^{\infty}(nh,(n+1)h;W^{k_{0},2}(\Omega;\mathbb{R}^{n})),$
$\displaystyle\hat{\boldsymbol{\eta}}_{\Delta t}$ $\displaystyle\text{ is
bounded in }W^{1,2}(nh,(n+1)h;W^{k_{0},2}(\Omega;\mathbb{R}^{n})).$

Using the classical Aubin-Lions Lemma and the Banach-Alaoglu theorem, this
allows us to take a subsequence and a limit
$\boldsymbol{{\boldsymbol{\eta}}}^{n}$ such that

$\displaystyle\hat{\boldsymbol{\eta}}_{\Delta
t}\to\boldsymbol{{\boldsymbol{\eta}}}^{n}\text{ in
}C^{0}([nh,(n+1)h];W^{k_{0},2}(\Omega;\mathbb{R}^{n}))\cap
W^{1,2}(nh,(n+1)h;W^{k_{0},2}(\Omega;\mathbb{R}^{n})),$

as $\Delta t\to 0$. Additionally the closeness of the three interpolations
then also implies

$\displaystyle\overline{\boldsymbol{\eta}}_{\Delta
t},\underline{{\boldsymbol{\eta}}}_{\Delta
t}\rightharpoonup^{*}\boldsymbol{{\boldsymbol{\eta}}}^{n}$
$\displaystyle\text{ in
}L^{\infty}(nh,(n+1)h;W^{k_{0},2}(\Omega;\mathbb{R}^{n})),\quad\text{ as
}\Delta t\to 0.$

Next we go back to the initial minimization and note that as a minimizer
${\boldsymbol{\eta}}_{k+1}$ fulfills the Euler-Lagrange equation

$\displaystyle\langle
DE_{\varepsilon}({\boldsymbol{\eta}}_{k+1}),\boldsymbol{\phi}\rangle+\langle
DK_{\varepsilon}({\boldsymbol{\eta}}_{k+1}),\boldsymbol{\phi}\rangle+\left\langle
D_{2}R_{\varepsilon}\left({\boldsymbol{\eta}}_{k},\frac{{\boldsymbol{\eta}}_{k+1}-{\boldsymbol{\eta}}_{k}}{\Delta
t}\right),\boldsymbol{\phi}\right\rangle+\frac{1}{h}\int_{\Omega_{S}}\left(\frac{{\boldsymbol{\eta}}_{k+1}-{\boldsymbol{\eta}}_{k}}{\Delta
t}-\mathbf{U}_{k}\right)\cdot\boldsymbol{\phi}=0$

for any $\boldsymbol{\phi}\in W^{k_{0},2}(\Omega_{S})$, which in terms of the
previous approximations can be restated as

$\displaystyle\langle DE_{\varepsilon}(\overline{\boldsymbol{\eta}}_{\Delta
t}(t)),\boldsymbol{\phi}\rangle+\langle
DK_{\varepsilon}(\overline{\boldsymbol{\eta}}_{\Delta
t}(t)),\boldsymbol{\phi}\rangle+\left\langle
D_{2}R_{\varepsilon}\left(\underline{{\boldsymbol{\eta}}}_{\Delta
t},\partial_{t}\hat{\boldsymbol{\eta}}_{\Delta
t}(t)\right),\boldsymbol{\phi}\right\rangle+\frac{1}{h}\int_{\Omega_{S}}\left(\partial_{t}\hat{\boldsymbol{\eta}}_{\Delta
t}(t)-\underline{\mathbf{U}}_{\Delta t}(t)\right)\cdot\boldsymbol{\phi}=0$

for almost all $t\in[nh,(n+1)h]$. Integrating in time and taking the limit
${\Delta t}\to 0$ then results in

$\displaystyle\int_{nh}^{(n+1)h}\langle
DE_{\varepsilon}({\boldsymbol{\eta}}^{n+1}(t)),\boldsymbol{\phi}\rangle+\langle
DK_{\varepsilon}({\boldsymbol{\eta}}^{n+1}(t)),\boldsymbol{\phi}\rangle$
$\displaystyle\quad+\left\langle
D_{2}R_{\varepsilon}\left({\boldsymbol{\eta}}^{n+1},\partial_{t}{\boldsymbol{\eta}}^{n+1}(t)\right),\boldsymbol{\phi}\right\rangle+\frac{1}{h}\int_{\Omega_{S}}\left(\partial_{t}{\boldsymbol{\eta}}^{n+1}(t)-\mathbf{U}^{n}(t)\right)\cdot\boldsymbol{\phi}\,dt=0.$

Finally we derive an energy inequality for this equation. For this we test
with $\boldsymbol{\phi}=\partial_{t}{\boldsymbol{\eta}}^{n+1}$ to obtain via
the chain-rule

$\displaystyle 0$ $\displaystyle=\int_{nh}^{(n+1)h}\langle
DE_{\varepsilon}({\boldsymbol{\eta}}^{n+1}(t)),\partial_{t}{\boldsymbol{\eta}}^{n+1}\rangle+\langle
DK_{\varepsilon}({\boldsymbol{\eta}}^{n+1}(t)),\partial_{t}{\boldsymbol{\eta}}^{n+1}\rangle$
$\displaystyle\quad+\left\langle
D_{2}R_{\varepsilon}\left({\boldsymbol{\eta}}^{n+1},\partial_{t}{\boldsymbol{\eta}}^{n+1}(t)\right),\partial_{t}{\boldsymbol{\eta}}^{n+1}\right\rangle+\frac{1}{h}\int_{\Omega_{S}}\left({\boldsymbol{\eta}}^{n+1}(t)-\mathbf{U}^{n}(t)\right)\cdot\partial_{t}{\boldsymbol{\eta}}^{n+1}\,dt$
$\displaystyle=E_{\varepsilon}({\boldsymbol{\eta}}^{n+1}((n+1)h))-E_{\varepsilon}({\boldsymbol{\eta}}^{n+1}(nh))+K_{\varepsilon}({\boldsymbol{\eta}}^{n+1}((n+1)h))-K_{\varepsilon}({\boldsymbol{\eta}}^{n+1}(nh))$
$\displaystyle\quad+\int_{nh}^{(n+1)h}2R_{\varepsilon}\left({\boldsymbol{\eta}}^{n+1},\partial_{t}{\boldsymbol{\eta}}^{n+1}(t)\right)+\frac{1}{h}\int_{\Omega_{S}}\left({\boldsymbol{\eta}}^{n+1}(t)-\mathbf{U}^{n}(t)\right)\cdot\partial_{t}{\boldsymbol{\eta}}^{n+1}\,dt,$

so by using the identity $2a(a-b)=a^{2}+(a-b)^{2}-b^{2}$, one obtains the
(in)equality $\eqref{SSPen}$.

###  5.5 Coupled back problem and uniform estimates

Here, we sum up equations $\eqref{momeqSSP}$ and $\eqref{momeqFSP}$ with a
common test function $\boldsymbol{\phi}$, then the energy inequalities
$\eqref{SSPen}$ and $\eqref{FSPen}$ and finally sum over $n=0,...,N-1$.
Denoting $f(t):=f^{n}(t)$ for $t\in[(n-1)h,nh)$, we obtain that the
constructed approximate solutions $(\rho,\mathbf{u},{\boldsymbol{\eta}})$
satisfy:

######  Definition 5.1.

We say that $(\rho,\mathbf{u},{\boldsymbol{\eta}})$ is a weak solution to the
regularized problem on an extended domain with decoupled velocities if:

  1. 1.

The following damped continuity equation is satisfied in the weak sense

$\displaystyle\int_{0}^{T}\int_{\Omega}\rho(\partial_{t}\varphi+\mathbf{u}\cdot\nabla\varphi)-\int_{0}^{T}\displaystyle\int_{\Omega_{S}^{\boldsymbol{\eta}}(t)}\rho^{2}\varphi-\varsigma\int_{0}^{T}\nabla\rho\cdot\nabla\varphi=\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{0}}}\rho_{0}\varphi(0,\cdot)$
(5.6)

for all $\varphi\in C_{0}^{\infty}([0,T)\times\mathbb{R}^{3})$;

  2. 2.

The coupled momentum equation

$\displaystyle\int_{Q_{T}}\rho\mathbf{u}\cdot\partial_{t}\boldsymbol{\xi}+\int_{Q_{T}}(\rho\mathbf{u}\otimes\mathbf{u}):\nabla\boldsymbol{\xi}+\int_{Q_{T}}(\rho^{\gamma}+\varepsilon\rho^{\beta})\nabla\cdot\boldsymbol{\xi}-\int_{Q_{T}}\mathbb{S}_{\varepsilon,{\boldsymbol{\eta}}}(\nabla\mathbf{u}):\nabla\boldsymbol{\xi}$
$\displaystyle\quad-\frac{1}{2}\int_{0}^{T}\int_{\Omega_{S}^{\boldsymbol{\eta}}(t)}\rho^{2}\mathbf{u}\cdot\boldsymbol{\xi}-\varsigma\int_{0}^{T}\int_{\Omega}(\nabla\rho\cdot\nabla\mathbf{u})\cdot\boldsymbol{\xi}-\int_{Q_{S,T}}\frac{\partial_{t}{\boldsymbol{\eta}}-\tau_{h}\partial_{t}{{\boldsymbol{\eta}}}}{h}\cdot\boldsymbol{\phi}$
$\displaystyle\quad-\int_{0}^{T}\langle
DE_{\varepsilon}({\boldsymbol{\eta}}),\boldsymbol{\phi}\rangle-\int_{0}^{T}\langle
D_{2}R_{\varepsilon}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}}),\boldsymbol{\phi}\rangle-\int_{0}^{T}\langle
DK_{\varepsilon}({\boldsymbol{\eta}}),\boldsymbol{\phi}\rangle$
$\displaystyle=\int_{\Omega}(\rho\mathbf{u})_{0,\varepsilon}\cdot\boldsymbol{\xi}(0,\cdot)+\int_{\Omega_{S}}\mathbf{v}_{0}\cdot\boldsymbol{\phi}(0,\cdot)$
(5.7)

holds for all $\boldsymbol{\xi}\in
C_{0}^{\infty}([0,T)\times\overline{\Omega})$, where
$\boldsymbol{\phi}=\boldsymbol{\xi}\circ{\boldsymbol{\eta}}$ on $Q_{S,T}$;

  3. 3.

The following energy inequality

$\displaystyle\int_{\Omega}\Big{(}\frac{1}{2}\rho|\mathbf{u}|^{2}+\frac{\rho^{\gamma}}{\gamma-1}\Big{)}(t)+\int_{0}^{t}\int_{\Omega}\mathbb{S}_{\varepsilon,{\boldsymbol{\eta}}}(\nabla\mathbf{u}):\nabla\mathbf{u}$
$\displaystyle\quad+\int_{0}^{T}\int_{\Omega_{S}^{\boldsymbol{\eta}}(t)}\left(\frac{1}{\gamma-1}\rho^{\gamma+1}+\frac{\varepsilon}{\beta-1}\rho^{\beta+1}\right)+\varsigma\int_{0}^{T}|\nabla\rho|^{2}\left(\gamma\rho^{\gamma-2}+\varepsilon\beta\rho^{\beta-2}\right)$
$\displaystyle\quad+E_{\varepsilon}({\boldsymbol{\eta}})(t)+\int_{0}^{t}2R_{\varepsilon}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})+K_{\varepsilon}({\boldsymbol{\eta}})(t)$
$\displaystyle\quad+\frac{1}{2h}\int_{nh}^{t}\int_{\Omega_{S}}|\mathbf{U}|^{2}+\frac{1}{2h}\int_{0}^{t}\int_{\Omega_{S}}\Big{(}|\mathbf{U}-\partial_{t}{\boldsymbol{\eta}}|^{2}+|\partial_{t}{\boldsymbol{\eta}}-\tau_{h}\mathbf{U}|^{2}\Big{)}$
$\displaystyle\leq\int_{\Omega}\Big{(}\rho_{0}|\mathbf{u}_{0}|^{2}+\frac{\rho_{0}^{\gamma}}{\gamma-1}\Big{)}+\frac{1}{2h}\int_{\Omega_{S}}|\mathbf{v}_{0}|^{2}+E_{\varepsilon}({\boldsymbol{\eta}}_{0})$

for all $0\leq n\leq N-1$ and $t\in[nh,(n+1)h)$.

Figure 4: At this approximate level, the fluid is allowed to pass through the
elastic solid, which is emphasized with a dashed line. After the limit $h\to
0$, the kinematic coupling is achieved and the fluid cannot pass through the
interface.

######  Corollary 5.1.

The above weak solution satisfies the following estimates:

$\displaystyle\sqrt{\varsigma}||\nabla\rho||_{L^{2}((0,T)\times\Omega)}\leq
C(\varepsilon),$ (5.8)
$\displaystyle||\mathbf{u}||_{L^{2}(0,T;H^{1}(\Omega))}\leq C(\varepsilon),$
(5.9) $\displaystyle||\mathbf{u}||_{L^{2}(0,T;L^{6}(\Omega))}\leq
C(\varepsilon),$
$\displaystyle||\rho\mathbf{u}||_{L^{\infty}(0,T;L^{\frac{2\gamma}{\gamma+1}}(\Omega))}\leq
C(\varepsilon),$
$\displaystyle||\rho\mathbf{u}||_{L^{2}(0,T;L^{\frac{6\gamma}{\gamma+6}}(\Omega))}\leq
C(\varepsilon),$
$\displaystyle||\rho\mathbf{u}\otimes\mathbf{u}||_{L^{2}(0,T;L^{\frac{6\gamma}{4\gamma+3}}(\Omega))}\leq
C(\varepsilon),$
$\displaystyle||{\boldsymbol{\eta}}||_{L^{\infty}(0,T;W^{k_{0},2}(\Omega_{S}))}+||\partial_{t}{\boldsymbol{\eta}}||_{L^{2}(0,T;W^{k_{0},2}(\Omega_{S}))}+\sup\limits_{t\in(0,T)}K_{\varepsilon}(t)\leq
C(\varepsilon).$

##  6 The interaction limit $h\to 0$ and the density dissipation limit
$\varsigma\to 0$

The solution obtained in the previous section for a given
$\varsigma,h,\varepsilon>0$ in the sense of Definition 5.1 is denoted as
$({\boldsymbol{\eta}}_{h},\rho_{h},\mathbf{u}_{h})$. The goal is to pass to
the limit $\varsigma,h\to 0$ (we can choose w.l.o.g. that $\varsigma=h$) and
to prove that the limiting functions $({\boldsymbol{\eta}},\rho,\mathbf{u})$
satisfy the following problem:

######  Definition 6.1.

We say that $({{\boldsymbol{\eta}}},\rho,\mathbf{u})$ is a weak solution of a
regularized problem on an extended domain if

$\displaystyle{\boldsymbol{\eta}}\in L^{\infty}(0,T;\mathcal{E}^{q}\cap
W^{k_{0},2}),~{}\partial_{t}{\boldsymbol{\eta}}\in
L^{2}(0,T;W^{k_{0},2}(\Omega_{S})),$ $\displaystyle\mathbf{u}\in
L^{2}(0,T;H^{1}(\Omega)),~{}\rho\in L^{\infty}(0,T;L^{\gamma}(\Omega)),$

and

  1. 1.

The fluid and solid velocities match on the solid domain, i.e.
$\partial_{t}{\boldsymbol{\eta}}=\mathbf{u}\circ{\boldsymbol{\eta}}$ on
$Q_{S,T}$;

  2. 2.

The renormalized continuity equation

$\displaystyle\int_{Q_{F,T}^{{\boldsymbol{\eta}}}}\rho
B(\rho)(\partial_{t}\varphi+\mathbf{u}\cdot\nabla\varphi)$ (6.1)
$\displaystyle=\int_{Q_{F,T}^{{\boldsymbol{\eta}}}}b(\rho)(\nabla\cdot\mathbf{u})\varphi+\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{0,\varepsilon}}}\rho_{0,\varepsilon}B(\rho_{0,\varepsilon})\varphi(0,\cdot),$

holds for all $\varphi\in C^{\infty}([0,T)\times\mathbb{R}^{3})$ and any $b\in
L^{\infty}(0,\infty)\cap C[0,\infty)$ such that $b(0)=0$ with
$B(x)=B(1)+\int_{1}^{x}\frac{b(z)}{z^{2}}dz$;

  3. 3.

The coupled momentum equation

$\displaystyle\int_{Q_{F,T}^{{\boldsymbol{\eta}}}}\rho\mathbf{u}\cdot\partial_{t}\boldsymbol{\xi}+\int_{Q_{F,T}^{{\boldsymbol{\eta}}}}(\rho\mathbf{u}\otimes\mathbf{u}):\nabla\boldsymbol{\xi}+\int_{Q_{F,T}^{{\boldsymbol{\eta}}}}(\rho^{\gamma}+\varepsilon\rho^{\beta})\nabla\cdot\boldsymbol{\xi}-\int_{Q_{F,T}^{{\boldsymbol{\eta}}}}\mathbb{S}_{\varepsilon,{\boldsymbol{\eta}}}(\nabla\mathbf{u}):\nabla\boldsymbol{\xi}$
$\displaystyle\quad+\int_{Q_{S,T}}\partial_{t}{{\boldsymbol{\eta}}}\cdot\partial_{t}\boldsymbol{\phi}-\int_{0}^{T}\langle
DE_{\varepsilon}({\boldsymbol{\eta}}),\boldsymbol{\phi}\rangle-\int_{0}^{T}\langle
D_{2}R_{\varepsilon}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}}),\boldsymbol{\phi}\rangle-\int_{0}^{T}\langle
DK_{\varepsilon}({\boldsymbol{\eta}}),\boldsymbol{\phi}\rangle$
$\displaystyle=\int_{\Omega^{{\boldsymbol{\eta}}_{0}}}(\rho\mathbf{u})_{0,\varepsilon}\cdot\boldsymbol{\xi}(0,\cdot)+\int_{\Omega_{S}}\mathbf{v}_{0}\cdot\boldsymbol{\phi}(0,\cdot)$
(6.2)

holds for all $\boldsymbol{\xi}\in
C_{0}^{\infty}([0,T)\times\overline{\Omega})$, where
$\boldsymbol{\phi}=\boldsymbol{\xi}\circ{\boldsymbol{\eta}}$ on $Q_{S,T}$;

  4. 4.

Energy inequality

$\displaystyle\int_{\Omega_{F}^{{\boldsymbol{\eta}}}(t)}\Big{(}\frac{1}{2}\rho|\mathbf{u}|^{2}+\frac{\rho^{\gamma}}{\gamma-1}+\varepsilon\frac{\rho^{\beta}}{\beta-1}\Big{)}(t)+\int_{0}^{t}\int_{\Omega}\left(\mathbb{S}_{\varepsilon,{\boldsymbol{\eta}}}(\nabla\mathbf{u}):\nabla\mathbf{u}\right)(\tau)~{}d\tau$
$\displaystyle\quad+\frac{1}{2}\int_{\Omega_{S}}|\partial_{t}{{\boldsymbol{\eta}}}|^{2}(t)+E_{\varepsilon}({\boldsymbol{\eta}}(t))+\int_{0}^{t}2R_{\varepsilon}({\boldsymbol{\eta}},\partial_{t}{\boldsymbol{\eta}})+K_{\varepsilon}({\boldsymbol{\eta}})(t)$
$\displaystyle\leq\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{0}}}\Big{(}\frac{1}{2\rho_{0,\varepsilon}}|(\rho\mathbf{u})_{0,\varepsilon}|^{2}+\frac{\rho_{0,\varepsilon}^{\gamma}}{\gamma-1}\Big{)}+\frac{1}{2}\int_{\Omega_{S}}|\mathbf{v}_{0}|^{2}+E_{\varepsilon}({\boldsymbol{\eta}}_{0,\varepsilon}),$

holds for all $t\in(0,T]$.

######  Corollary 6.1.

The above weak solution satisfies additional estimates independent from
$\varepsilon$:

$\displaystyle||\rho\mathbf{u}||_{L^{\infty}(0,T;L^{\frac{2\gamma}{\gamma+1}}(\Omega_{F}^{\boldsymbol{\eta}}(t)))}\leq
C,$
$\displaystyle||\rho\mathbf{u}||_{L^{2}(0,T;L^{\frac{6\gamma}{\gamma+6}}(\Omega_{F}^{\boldsymbol{\eta}}(t)))}\leq
C,$
$\displaystyle||\rho\mathbf{u}\otimes\mathbf{u}||_{L^{2}(0,T;L^{\frac{6\gamma}{4\gamma+3}}(\Omega_{F}^{\boldsymbol{\eta}}(t)))}\leq
C,$
$\displaystyle||{\boldsymbol{\eta}}||_{L^{\infty}(0,T;W^{2,q}(\Omega_{S}))}+||\partial_{t}{\boldsymbol{\eta}}||_{L^{2}(0,T;H^{1}(\Omega_{S}))}\leq
C.$

###  6.1 Limit in the continuity equation and vanishing density inside the
structure domain

Based on the uniform estimates, we have

$\displaystyle\mathbf{u}_{h}\rightharpoonup\mathbf{u},\qquad\text{in
}L^{2}(0,T;H^{1}(\Omega)),$ $\displaystyle\rho_{h}\to\rho,\qquad\text{in
}L^{\infty}(0,T;L^{\gamma}(\Omega)).$

Moreover, noticing that the coupled momentum equation $\eqref{coupledmomapp}$
can be tested by any $\boldsymbol{\phi}\in
L^{2}(0,T;W_{0}^{k_{0},2}(\Omega_{S}))$, where $\boldsymbol{\xi}=0$ on
$[0,T]\times\overline{\Omega_{F}^{\boldsymbol{\eta}}}(t)$, one can deduce that
$\partial_{tt}{\boldsymbol{\eta}}_{h}$ is uniformly bounded in
$L^{2}(0,T;W^{-k_{0},2}(\Omega_{S}))$, so one can use the Aubin-Lions lemma to
deduce that

$\displaystyle\partial_{t}{\boldsymbol{\eta}}_{h}\to\partial_{t}{\boldsymbol{\eta}},\qquad$
$\displaystyle\text{in }L^{2}(0,T;L^{2}(\Omega_{S})),$ (6.3)

which combined with the bound
$\frac{1}{2h}\int_{0}^{T}\int_{\Omega_{S}^{{\boldsymbol{\eta}}_{h}}(t)}|\mathbf{u}_{h}-\partial_{t}{\boldsymbol{\eta}}_{h}\circ{\boldsymbol{\eta}}_{h}^{-1}|^{2}\leq
C$ implies

$\displaystyle\mathbf{u}_{h}\to\partial_{t}{\boldsymbol{\eta}}\circ{\boldsymbol{\eta}}^{-1},\qquad\text{in
}L^{2}((0,T)\times\Omega_{F}^{\boldsymbol{\eta}}(t)).$

By using $||\mathbf{u}_{h}||_{L^{2}(0,T;L^{6}(\Omega))}\leq C(\varepsilon)$
and
$||\partial_{t}{\boldsymbol{\eta}}_{h}\circ{\boldsymbol{\eta}}_{h}^{-1}||_{L^{2}(0,T;L^{6}(\Omega_{S}^{{\boldsymbol{\eta}}_{h}}(t)))}\leq
C(\varepsilon)$, one then has

$\displaystyle\mathbf{u}_{h}\to\partial_{t}{\boldsymbol{\eta}}\circ{\boldsymbol{\eta}}^{-1},\qquad\text{in
}L^{2}(0,T;L^{p}(\Omega_{F}^{\boldsymbol{\eta}}(t))),\quad\text{ for any
}p<6,$ (6.4)

which gives us

$\displaystyle\rho_{h}\mathbf{u}_{h}\to\rho\partial_{t}{\boldsymbol{\eta}}\circ{\boldsymbol{\eta}}^{-1},\qquad\text{in
}L^{2}(0,T;L^{p}(\Omega_{S}^{\boldsymbol{\eta}}(t))),\quad\text{ for any
}p<\frac{6\gamma}{\gamma+6}.$ (6.5)

From the coupled momentum equation, one can also deduce that

$\displaystyle||\partial_{t}(\rho_{h}\mathbf{u}_{h})||_{L^{2}(0,T;W^{-k_{0},2}(\Omega_{F}^{\boldsymbol{\eta}}(t)))}\leq
C(\varepsilon),\qquad\text{for some }p>1,$ (6.6)

and since $\rho_{h}\mathbf{u}_{h}$ is bounded in
$L^{2}(0,T;L^{\frac{6\gamma}{4\gamma+3}}(\Omega))$ and
$L^{\frac{6\gamma}{4\gamma+3}}(\Omega)\hookrightarrow\hookrightarrow
H^{-s}(\Omega)$ for some $s<1$, we deduce by the Aubin-Lions lemma

$\displaystyle\rho_{h}\mathbf{u}_{h}\to\rho\mathbf{u}\qquad\text{in
}L^{2}(0,T;H^{-1}(\Omega_{F}^{\boldsymbol{\eta}}(t))).$ (6.7)

Finally, due to the estimate
$\sqrt{\varsigma}||\nabla\rho_{h}||_{L^{2}((0,T)\times\Omega)}\leq C$, one has
that $\varsigma\nabla\rho\rightharpoonup 0$. This means that the limiting
functions $\rho,\mathbf{u}$ satisfy the continuity equation in the weak form.
Note that at this point, there is still a term in the continuity equation
$\overline{{\rho}^{2}}$, which is the weak limit of $\rho_{h}^{2}$ in
$(0,T)\times\Omega_{S}(t)$. Moreover, this equation holds on the entire domain
$\Omega$. In order to show that $\rho$ satisfies $\eqref{reconteqweakh}$, let
us prove that the density vanishes inside the structure domain:

######  Lemma 6.2.

We have

$\displaystyle\rho=0\text{ and }\overline{{\rho}^{2}}=0,\qquad\text{a.e. in
}(0,T)\times\Omega_{S}(t),$

where $\overline{{\rho}^{2}}$ is the weak limit of $\rho_{h}^{2}$ in
$(0,T)\times\Omega_{S}(t)$.

###### Proof.

Let $\tau\in(0,T)$. We choose $B(\rho)=1$ and $\varphi\in
H^{1}(0,\tau;H_{0}^{1}(\Omega_{S}^{\boldsymbol{\eta}}(t)))$ in the continuity
equation to obtain

$\displaystyle\int_{0}^{\tau}\int_{\Omega_{S}^{\boldsymbol{\eta}}(t)}\rho(\partial_{t}\varphi+\mathbf{u}\cdot\nabla\varphi)=\int_{\Omega_{S}^{\boldsymbol{\eta}}(\tau)}(\rho\varphi)(\tau)-\underbrace{\int_{\Omega_{S}^{{\boldsymbol{\eta}}_{0,\varepsilon}}(0)}\rho_{0,\varepsilon}\varphi}_{=0}+\int_{0}^{\tau}\int_{\Omega_{S}^{\boldsymbol{\eta}}(t)}\overline{\rho^{2}}\varphi$

since $\rho_{0,\varepsilon}=0$ on $\Omega_{S}(0)=0$, by construction. Now,
recalling that
$\mathbf{u}=\partial_{t}{\boldsymbol{\eta}}\circ{\boldsymbol{\eta}}^{-1}$ on
$(0,T)\times\Omega_{S}(t)$, one has

$\displaystyle(\partial_{t}\varphi+\mathbf{u}\cdot\nabla\varphi)(t,{\boldsymbol{\eta}}(t,x))=\frac{d}{dt}\varphi(t,{\boldsymbol{\eta}}(t,x))$

so one can compose both integrals with ${\boldsymbol{\eta}}$

$\displaystyle\int_{0}^{t}\int_{\Omega_{S}}\det\nabla{\boldsymbol{\eta}}\tilde{\rho}\partial_{t}\tilde{\varphi}=\int_{\Omega_{S}}(\det\nabla{\boldsymbol{\eta}}\tilde{\rho}\tilde{\varphi})(t)+\int_{0}^{t}\int_{\Omega_{S}}(\det\nabla{\boldsymbol{\eta}}\tilde{\overline{{\rho}^{2}}}\tilde{\varphi})$

where $\tilde{\rho}=\rho\circ{\boldsymbol{\eta}}$,
$\tilde{\varphi}=\varphi\circ{\boldsymbol{\eta}}$ and
$\tilde{\overline{{\rho}^{2}}}=\overline{{\rho}^{2}}\circ{\boldsymbol{\eta}}$.
Now, for any $\tilde{\varphi}\in H^{1}(\Omega_{S})$ independent of time (note
that then $\varphi=\tilde{\varphi}\circ{\boldsymbol{\eta}}^{-1}$ is a suitable
test function), one has

$\displaystyle\int_{\Omega_{S}}(\det\nabla{\boldsymbol{\eta}}\tilde{\rho}\tilde{\varphi})(\tau)+\int_{0}^{t}\int_{\Omega_{S}}(\det\nabla{\boldsymbol{\eta}}\tilde{\overline{{\rho}^{2}}}\tilde{\varphi})=0.$

We can now easily construct a sequence of functions $\tilde{\varphi}$ which
converge to $1_{|\Omega_{S}}$, and since
$\tilde{\rho},\tilde{\overline{{\rho}^{2}}}\geq 0$ and
$\det\nabla{\boldsymbol{\eta}}>0$, this implies that $\tilde{\rho}=0$ and
$\tilde{\overline{{\rho}^{2}}}=0$ on $\Omega_{S}$ and the conclusion follows.
∎

Finally, let us point out that since $\rho\in
L^{\infty}(0,T;L^{\beta}(\Omega_{F}^{\boldsymbol{\eta}}(t)))$, it means that
$\rho$ is square integrable so one can use [42, Lemma 6.9] to deduce that the
limiting functions also satisfy the renormalized continuity equation
$\eqref{reconteqweakh}$.

###  6.2 Limit in the coupled momentum equation

First, by using $\eqref{fl:mom:conv}$ and the weak convergence of
$\mathbf{u}_{h}$ in $L^{2}(0,T;H^{1}(\Omega_{F}^{\boldsymbol{\eta}}(t)))$, one
deduces that

$\displaystyle\rho_{h}\mathbf{u}_{h}\otimes\mathbf{u}_{h}\rightharpoonup\rho\mathbf{u}\otimes\mathbf{u},\qquad\text{in
}L^{1}((0,T)\times\Omega_{F}^{\boldsymbol{\eta}}(t)),$

while combining $\eqref{str:mom:conv}$ and $\eqref{str:mom:conv2}$ implies

$\displaystyle\rho_{h}\mathbf{u}_{h}\otimes\mathbf{u}_{h}\rightharpoonup\rho\partial_{t}{\boldsymbol{\eta}}\circ{\boldsymbol{\eta}}^{-1}\otimes\partial_{t}{\boldsymbol{\eta}}\circ{\boldsymbol{\eta}}^{-1}=\rho\mathbf{u}\otimes\mathbf{u},\qquad\text{in
}L^{1}((0,T)\times\Omega_{S}^{\boldsymbol{\eta}}(t)),$

so

$\displaystyle\rho_{h}\mathbf{u}_{h}\otimes\mathbf{u}_{h}\rightharpoonup\rho\mathbf{u}\otimes\mathbf{u},\qquad\text{in
}L^{1}((0,T)\times\Omega).$

Next, due to the uniform estimate on $\mathbf{u}_{h}$ in
$L^{2}(0,T;H^{1}(\Omega))$ and the strong convergence of
${\boldsymbol{\eta}}_{h}$ in $C([0,T]\times\Omega_{S})$, one deduces

$\displaystyle\mathbb{S}_{\varepsilon,{\boldsymbol{\eta}}_{h}}(\nabla\mathbf{u}_{h})\to\mathbb{S}_{\varepsilon,{\boldsymbol{\eta}}}(\nabla\mathbf{u}),\qquad\text{in
}L^{2}(0,T;L^{2}(\Omega)).$

Finally, due to $\eqref{time:der:est}$ and uniform boundedness of
$\rho_{h}\mathbf{u}_{h}$ in
$L^{\infty}(0,T;L^{\frac{2\gamma}{\gamma+1}}(\Omega))$, one deduces

$\displaystyle\rho_{h}\mathbf{u}_{h}\to\rho\mathbf{u},\qquad\text{in
}L^{\infty}(0,T;L^{\frac{2\gamma}{\gamma+1}}(\Omega_{F}^{\boldsymbol{\eta}}(t))).$

To show the convergence of the structure terms, first note that the test
function $\boldsymbol{\phi}$ in $\eqref{coupledmomapp}$ depends on
${\boldsymbol{\eta}}_{h}$, so the derivatives of $\boldsymbol{\phi}$ depend on
the derivatives of ${\boldsymbol{\eta}}_{h}$. The following convergences hold:

$\displaystyle\partial_{t}{\boldsymbol{\eta}}_{h}\rightharpoonup\partial_{t}{\boldsymbol{\eta}},$
$\displaystyle\text{in }L^{2}(0,T;W^{k_{0},2}(\Omega_{S})),$
$\displaystyle{\boldsymbol{\eta}}_{h}\rightharpoonup{\boldsymbol{\eta}},$
$\displaystyle\text{in }L^{\infty}(0,T;W^{k_{0},2}(\Omega_{S}))$
$\displaystyle{\boldsymbol{\eta}}_{h}\to{\boldsymbol{\eta}},$
$\displaystyle\text{in }L^{2}(0,T;W^{k_{0},2}(\Omega_{S})),$

where the last convergence is due to the Minty property and it follows in the
same way as in [2, Proposition 2.23]. This combined with
$\eqref{eta:t:strong}$ allows us to pass to the limit all of the structure
terms.

Finally, it remains to show the convergence of pressure.

####  6.2.1 Convergence of pressure

In order to prove that $\rho_{h}^{\gamma}+\varepsilon\rho_{h}^{\beta}$ has a
weak limit $\overline{p}$ in
$L^{1}(0,T\times\Omega_{S}^{\boldsymbol{\eta}}(t))$, we utilize the bound
$\varepsilon\int_{0}^{T}\int_{\Omega_{S}^{\boldsymbol{\eta}}(t)}(\rho_{h}^{\gamma+1}+\varepsilon\rho_{h}^{\beta+1})\leq
C$. Due to Lemma 6.2, one has that $\overline{p}=0$ on
$(0,T)\times\Omega_{S}^{\boldsymbol{\eta}}(t)$. so it remains to prove the
weak convergence of $\rho_{h}^{\gamma}+\varepsilon\rho_{h}^{\beta}$ on the
fluid domain $(0,T)\times\Omega_{F}^{\boldsymbol{\eta}}(t)$:

######  Lemma 6.3.

Let $I$ be an interval and $S\Subset\Omega_{F}^{\boldsymbol{\eta}}(t)$ for all
$t\in I$ be a set with regular boundary. Then, there is an $h_{S}>0$ such that
for all $0<h<h_{S}$, one has the following improved pressure estimates:

$\displaystyle\int_{I\times
S}\rho_{h}^{\gamma+1}+\varepsilon\rho_{h}^{\beta+1}\leq C(S),$

where $C(S)$ depends on the initial energy and blows up as
$\text{dist}(S,\partial\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t))$ goes to zero.

###### Proof.

The proof is done by testing $\eqref{momeqweakh}$ with
$\boldsymbol{\xi}:=\varphi\nabla\Delta_{S^{\prime}}^{-1}[\rho_{h}]$, where
$S\subset S^{\prime}\Subset\Omega_{F}^{\boldsymbol{\eta}}(t)$ for all $t\in I$
is a set with regular boundary, $\varphi$ is a smooth cut-off function which
is equal to $1$ on $S$ and $0$ outside of $S^{\prime}$, and $\Delta_{S}^{-1}$
is the inverse Laplace operator on $S$ with Dirichlet data. At this point,
this proof is standard and can be found in [7, Lemma 6.3]. ∎

######  Lemma 6.4.

There is a $h_{0}>0$ such that for every $\varepsilon^{\prime}>0$, there is a
$\delta^{\prime}>0$, such that

$\displaystyle\int_{0}^{T}\int_{\\{d(x,\partial\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t))<\delta^{\prime}\\}}\rho_{h}^{\gamma}+\varepsilon\rho_{h}^{\beta}\leq
C(\varepsilon)\varepsilon^{\prime},$

for all $\varepsilon\leq\varepsilon_{0}$.

###### Proof.

The proof follows the approach from [34], so only the main steps are
presented. Let $h>0$ and $t_{0}\in[0,T]$ be fixed. First note that, due to the
regularity of ${\boldsymbol{\eta}}_{h}$ and $\partial\Omega$, for all
$t\in(t_{0}-s,t_{0}+s)\cap(0,T)$ where $s>0$, the boundary
$\partial\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t)$ can be represented as a
union of graphs of functions $\phi_{i}:(t_{0}-s,t_{0}+s)\times
V_{i}\to\mathbb{R}$ in local coordinate systems
$(x_{i}^{1},x_{i}^{2},x_{i}^{3})$, where $1\leq i\leq m$, so that the fluid
domain lies below all the graphs. In other words, for every
$(t,x)\in(t_{0}-s,t_{0}+s)\times(\partial\Omega_{S}\cup\partial\Omega)$, there
is an $1\leq i\leq m$ and $y\in V_{i}$ such that
${\boldsymbol{\eta}}_{h}(t,x)=\mathbf{s}_{i}((y,0)+\phi_{i}(t)(y)\mathbf{e}_{3}^{i})$
(and similarly for every $(t,x)\in(t_{0}-s,t_{0}+s)\times\partial\Omega$),
where $\mathbf{e}_{3}^{i}$ is the unit vector in $x_{i}^{3}$ direction in the
corresponding local coordinate system, and
$\mathbf{s}_{i}(\mathbf{v})=\mathbf{r}_{i}+A_{i}\mathbf{v}$ is a combination
of translation $\mathbf{r}_{i}$ and rotation $A_{i}$. Note that we can choose
a uniform $h_{0}>0$ such that these graphs are well-defined on
$(t_{0}-s,t_{0}+s)$ for all $0<h<h_{0}$. We define

$\displaystyle
v_{i}(t,x_{i}^{1},x_{i}^{2},x_{i}^{3}):=\begin{cases}(x_{i}^{3}-\phi_{i}(t,x_{1},x_{2}))^{\lambda},&\quad
x_{i}^{3}\leq\phi_{i}(t,x_{1},x_{2}),\\\ 0,&\quad\text{elsewhere},\end{cases}$

for $\frac{2\gamma-3}{6\gamma}<\lambda<1$, and

$\displaystyle\boldsymbol{\xi}_{h}(t,x):=\sum_{i=1}^{m}\psi_{i}(x)v_{i}(t,\mathbf{s}_{i}^{-1}(x))\mathbf{s}_{i}(\mathbf{e}_{3}^{i}).$

Here $\\{\psi_{i}\\}_{1\leq i\leq m+2}$ is a partition of unity, subordinate
to sets $\Omega_{1},...,\Omega_{m+2}$ such that
$\cup_{i=1}^{m+2}\Omega_{i}\supseteq\Omega$, for all $t\in(t_{0}-s,t_{0}+s)$
the sets $\Omega_{1},...,\Omega_{m}$ contain the graphs of $m$ functions
$\phi_{1}(t),...,\phi_{m}(t)$,
$\Omega_{m+1}\subset\text{int}\cap_{t\in(t_{0}-s,t_{0}+s)}\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t)$
and
$\Omega_{m+2}\subset\text{int}\cap_{t\in(t_{0}-s,t_{0}+s)}\Omega_{S}^{{\boldsymbol{\eta}}_{h}}(t)$.
Additionally, each of the sets $\Omega_{i}$ for $1\leq i\leq m$ can only have
a non-empty intersection with $\Omega_{m+1},\Omega_{m+2}$ and $\Omega_{j}$
such that $\phi_{i}$ and $\phi_{j}$ are neighbouring graphs.

Note that $\boldsymbol{\xi}_{h}$ vanishes outside of
$\Omega_{F}^{{\boldsymbol{\eta}}_{h}}$. Moreover, $\boldsymbol{\xi}_{h}\in
L^{\infty}(0,T;W^{1,q}(\Omega))\cap H^{1}(0,T;L^{4}(\Omega))$ for any
$q<\frac{1}{1-\lambda}$, due to the regularity of
${{\boldsymbol{\eta}}_{h}}_{|\partial\Omega_{S}^{{\boldsymbol{\eta}}_{h}}}$
and the fact that it behaves as a distance function to power $\lambda$ near
the boundary. Due to the specific lower bound for $\lambda$, we also have that
$\boldsymbol{\xi}_{h}\in
L^{\infty}(0,T;W^{1,\frac{6\gamma}{4\gamma+3}}(\Omega))$.

Finally, note also that we can extend this construction to the entire interval
$[0,T]$ by covering $[0,T]$ with a finite number of time intervals of length
$2s$ and using the partition of unity in time $\\{\chi_{i}\\}_{1\leq i\leq k}$
(here the key point is that $s$ can be chosen uniformly due to the regularity
of ${\boldsymbol{\eta}}_{h}$). Let the new function defined on $[0,T]$ be also
denoted by
$\boldsymbol{\xi}_{h}(t,x)=\sum_{j=1}^{k}\sum_{i=1}^{m}\chi_{j}(t)\psi_{i}^{j}(x)v_{i}^{j}(t,(\mathbf{s}_{i}^{j})^{-1}(x))\mathbf{s}_{i}^{j}(\mathbf{e}_{3}^{i,j})$.
Now, by testing $\eqref{momeqweakh}$ with $\boldsymbol{\xi}_{h}$, one obtains

$\displaystyle\int_{0}^{T}\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t)}(\rho_{h}^{\gamma}+\varepsilon\rho_{h}^{\beta})\nabla\cdot\boldsymbol{\xi}_{h}$
$\displaystyle=-\int_{0}^{T}\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t)}\rho_{h}\mathbf{u}_{h}\cdot\partial_{t}\boldsymbol{\xi}_{h}-\int_{0}^{T}\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t)}(\rho_{h}\mathbf{u}_{h}\otimes\mathbf{u}_{h}):\nabla\boldsymbol{\xi}_{h}$
$\displaystyle\quad+\int_{0}^{T}\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t)}\mathbb{S}_{\varepsilon,{\boldsymbol{\eta}}_{h}}(\nabla\mathbf{u}_{h}):\nabla\boldsymbol{\xi}_{h}-\frac{1}{2}\int_{0}^{T}\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t)}\rho_{h}\mathbf{u}_{h}\cdot\boldsymbol{\xi}_{h}+\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t)}\rho_{h}\mathbf{u}_{h}\cdot\boldsymbol{\xi}_{h}\Big{|}_{0}^{T}.$

Due to the estimates on $\rho_{h},\mathbf{u}_{h}$ and $\boldsymbol{\xi}_{h}$
(note that the term
$\rho_{h}\mathbf{u}_{h}\cdot\partial_{t}\boldsymbol{\xi}_{h}$ is uniformly
bounded due to $\gamma>12/7$ even w.r.t. $\varepsilon$), the right-hand side
is uniformly bounded for a fixed $\varepsilon$, so

$\displaystyle\int_{0}^{T}\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t)}(\rho_{h}^{\gamma}+\varepsilon\rho_{h}^{\beta})\nabla\cdot\boldsymbol{\xi}_{h}\leq
C(\varepsilon).$

The $\varepsilon$ here appears due to the penalization $K_{\varepsilon}$,
which directly affects the shape of the fluid domain (i.e. how close to
contact and self-contact can the elastic body be) and therefore the
construction of $\boldsymbol{\xi}_{h}$ and its bounds. This implies

$\displaystyle\int_{0}^{T}\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t)}\underbrace{(\rho_{h}^{\gamma}+\varepsilon\rho_{h}^{\beta})\sum_{j=1}^{k}\sum_{i=1}^{m}\chi_{j}(t)\psi_{i}^{j}(x)\nabla\cdot\big{(}v_{i}^{j}(t,(\mathbf{s}_{i}^{j})^{-1}(x))\mathbf{s}_{i}^{j}(\mathbf{e}_{3}^{i,j})}_{\geq
0}\big{)}$ $\displaystyle\leq
C(\varepsilon)-\int_{0}^{T}\int_{\Omega_{F}^{{\boldsymbol{\eta}}_{h}}(t)}(\rho_{h}^{\gamma}+\varepsilon\rho_{h}^{\beta})\sum_{j=1}^{k}\sum_{i=1}^{m}\chi_{j}(t)\nabla\psi_{i}^{j}(x)\cdot
v_{i}^{j}(t,(\mathbf{s}_{i}^{j})^{-1}(x))\mathbf{s}_{i}^{j}(\mathbf{e}_{3}^{i,j})$
$\displaystyle\leq C(\varepsilon).$

The key point here is that
$\nabla\cdot(v_{i}^{j}(t,(\mathbf{s}_{i}^{j})^{-1}(x))\mathbf{s}_{i}^{j}(\mathbf{e}_{3}^{i,j}))=\lambda
d_{i}^{j}(t,x)^{\lambda-1}$, where $d_{i}^{j}(t,x)$ is the distance of the
point $x$ and the boundary $\partial\Omega_{S}^{{\boldsymbol{\eta}}_{h}}(t)$
in the $\mathbf{s}_{i}^{j}(\mathbf{e}_{3}^{i,j})$-direction. These divergences
blow up as one approaches the boundary, so one has that for every
$\varepsilon^{\prime}>0$, there is a $\delta^{\prime}>0$ such that