# Compressible fluids interacting with plates – regularity and weak-strong
uniqueness

Srđan Trifunović Department of Mathematics and Informatics, University of Novi
Sad, Trg D. Obradovića 4, 21000 Novi Sad, Serbia, email<EMAIL_ADDRESS>
###### Abstract

In this paper, we study a nonlinear interaction problem between compressible
viscous fluids and plates. For this problem, we introduce relative entropy and
relative energy inequality for the finite energy weak solutions (FEWS). First,
we prove that for all FEWS, the structure displacement enjoys improved
regularity by utilizing the dissipation effects of the fluid onto the
structure and that all FEWS satisfy the relative energy inequality. Then, we
show that all FEWS enjoy the weak-strong uniqueness property, thus extending
the classical result for compressible Navier-Stokes system to this fluid-
structure interaction problem.

Keywords and phrases: fluid-structure interaction, compressible viscous fluid,
elastic plate, weak-strong uniqueness

AMS Mathematical Subject classification (2020): 76N10 (Primary), 35A02, 35R37,
74F10 (Secondary)

##  1 Introduction

The well-posedness theory for the nonlinear interaction problems between
fluids and thin elastic structures (plates or shells) has seen quite a
development in the last two decades. Such problems arise from various physical
phenomena and they are quite challenging for various reasons. For example, the
fluid domain changes in time and it depends on the displacement of the elastic
structure - an unknown in the system, it is a mixed-type coupled physical
system, usually there are both hyperbolic and parabolic equations in the
system etc. Majority of such results use the incompressible Navier-Stokes
system to model the fluid. We mention [5, 14, 17, 24, 32, 33, 34, 37] for the
existence of weak solutions, and [2, 18, 19, 25] for the existence of strong
solutions. On the other hand, for the compressible Navier-Stokes system, there
are only a handful of results which appeared in the last few years. First such
result was due to Breit and Schwarzacher in [4], where they proved the
existence of a weak solution for a shell interacting with a compressible
viscous fluid. Later, Mitra proved the existence of a unique local regular
solution in 2D, where the beam is viscoelastic in [29]. Maity and Takahashi
obtained a strong solution in $L^{p}-L^{q}$ framework for the interaction of
heat-conducting fluid with a viscoelastic plate, and later Maity, Roy and
Takahashi obtained a unique local regular solution in 3D [27], where the
structure is modeled by a wave equation and doesn’t have any viscosity.
Trifunović and Wang obtained the existence of a weak solution in 3D in [38],
where the structure is modeled by a nonlinear thermoelastic (or just elastic)
plate, by constructing a time-continuous splitting approximation scheme that
decouples the fluid and the structure.

While weak solutions are in general global, or in the case of fluid-structure
interaction exist up to a collision or self-collision of elastic structures,
they are very often not unique and have a low regularity that comes from the
energy of the problem. On the other hand, strong (or more regular) solutions
are usually unique and posses higher regularity, but are often local in time.
Thus, a natural question arises - when do weak and strong solutions coincide
if eminating from the same initial data, or in other words, when do weak
solutions enjoy so called weak-strong uniqueness property? Note that this only
gives a partial answer to the uniqueness problem, as it usually doesn’t hold
globally in time and requires a more regular initial data in order to ensure
the existence of a strong solution in the first place. Some systems, like the
incompressible Navier-Stokes equations, actually enjoy a better type of
uniqueness properties. In particular, finite energy weak solutions in 2D are
unique, while in 3D, a different type of a weaker uniquneness, so called
Prodi-Serrin, is proven to hold. This means that a weak solution that has the
integrability $\mathbf{u}\in L_{t}^{a}L_{x}^{b}$, for
$a\in(2,\infty),b\in(3,\infty)$ such that $\frac{2}{a}+\frac{3}{b}=1$, is
unique in the class of weak solutions. On the other hand, the compressible
Navier-Stokes system is not as fortunate, as only the results concerning the
weak-strong uniqueness are available so far (see [12, 11, 15]), at least
without any further simplifications.

The uniqueness of fluid-structure interaction problems has been studied mostly
for the interaction of rigid bodies and incompressible viscous fluids. The
following results are mentioned: the uniqueness in 2D case with no-slip
condition [16], the uniqueness and the energy identity in 2D with slip
condition [3], the weak-strong uniqueness with slip condition [6] and the
Serri-Prodin uniqueness with slip or no-slip [31]. The weak-strong uniqueness
for compressible viscous fluid on moving domains, surrounding a rigid body and
filling a rigid body were studied in [26, 22, 13], respectively. In the
mentioned results, when comparing two solutions, the usual approach is to
transform coordinates of both solutions to be expressed on a common fixed
domain and then compare them there. One should note that to be even able to do
this in the first place, the rigid bodies corresponding to two different
solutions need to be a priori isomorphic, while the fluid cavities need to be
the same. Consequently, if one of the two solutions is more regular, then the
rigid body and the fluid cavity for both of the solutions will need to have
the same (higher) regularity. In other words, the regularities of the
boundaries are a priori fixed and do not play a role in the analysis. However,
when one deals with the interaction of fluids and elastic plates (or shells),
the regularity of fluid domains directly depends on the regularity of the
plate displacements, and as such can be different for two different solutions.
Uniqueness of such problems was first studied by Guidoboni, Guidorzi, and
Padula in [20], where a 1D beam interacts with a 2D incompressible fluid. For
this problem, they proved the continuous time dependence of solutions on
initial data, and consequently the uniqueness of such weak solutions. Very
recently, Schwarzacher and Sroczinski in [35] studied the interaction of
incompressible viscous fluids and elastic plates, in 2D or 3D. Denoting the
fluid velocities as $\mathbf{u}_{i}$ and the plate displacements as $w_{i}$,
they proved that two finite energy weak solutions $(\mathbf{u}_{1},w_{1})$ and
$(\mathbf{u}_{2},w_{2})$ coincide if $\partial_{t}\mathbf{u}_{2}\in
L_{t}^{2}W_{x}^{-1,2}$ in 2D, or $\mathbf{u}_{2}\in L_{t}^{p}W_{x}^{1,q}$ and
$\partial_{t}\mathbf{u}_{2}\in L_{t}^{2}W_{x}^{-1,r}$ in 3D, for any
$p>2,q>3$. The authors of this paper argue that the condition on
$\partial_{t}\mathbf{u}_{2}$ was forgotten in [20] and it needs to be
additionally imposed. This is because when $(\mathbf{u}_{2},w_{2})$ is
compared with $(\mathbf{u}_{1},w_{1})$, it is mapped onto a domain
corresponding to the latter and this mapping (constructed in [20, section 6]
and also used in [35]) is chosen to preserve the divergence-free condition in
order to avoid the appearance of the pressure, but it doesn’t ensure that
$\partial_{t}\mathbf{u}_{2}$ is in the right dual space of divergence-free
functions corresponding to the domain of the solution
$(\mathbf{u}_{1},w_{1})$, so this is why this condition needs to be imposed
additionally.

In this paper, we study the finite energy weak solutions (FEWS) to the
interaction problem of compressible viscous fluids and elastic or viscoelastic
plates. The notions of relative entropy and relative energy inequality are
introduced. First, by utilizing the dissipation effects of the fluid onto the
structure, we prove that the additional regularity $\Delta w\in
L_{t}^{2}H_{x}^{s}$ can be obtained, for $s=s(\gamma,d,\alpha)>0$, where $w$
is the displacement of the plate, $\gamma$ is the adiabatic constant, $d=2,3$
is the dimension of the fluid and $\alpha$ is the viscoelasticity constant.
This is done by following the ideas from [32], where similar (but stronger)
estimates were obtained for the interaction of a incompressible viscous fluid
and a nonlinear Koiter shell. Afterwards, it is proved that all FEWS satisfy
the relative energy inequality. Finally, we prove that all FEWS enjoy the
weak-strong uniqueness property. The proof uses the strategy from [12] (see
also [11, 15]), and relies on using the relative entropy, which measures the
distance between two solutions, and the relative energy inequality, which if
tested by sufficiently regular solution of the same problem, gives the desired
result by means of the Gronwall lemma. Since the fluid domains of two
solutions are not the same in general, the coordinates of both solutions are
transformed to the ones of a common fixed domain. The time and space
derivatives are then replaced by the corresponding transformed derivatives
which depend on the domain transformation, and thus on the displacement of the
plate. This drastically increases the difficulty of the analysis. To close
these estimates, additional integrability is needed - $\nabla w\in
L_{t}^{2}L_{x}^{\infty}$, which is always ensured by the mentioned regularity
$\Delta w\in L_{t}^{2}H_{x}^{s}$ and the imbedding of Sobolev spaces (for any
$s>0$). In the last part of the paper, the obtained results are extended to
fluid-structure interaction models for which the structure is more nonlinear.
More precisely, we first study the case where the structure is modeled by a
semilinear plate model which generalizes Kirchhoff, von Karman and Berger
plates (see Appendix for more details). Then, this equation is then replaced
by a semilinear thermoelastic system and finally by a quasilinear
thermoelastic system, thus furthering the frontiers of the approach presented
here.

This paper is organized as follows. In section 2, the model we will study is
introduced, then the notions of weak and strong solution for this problem are
given, the relative entropy and relative energy inequality are introduced, and
the main results are presented. In sections 3 and 4, the main results are
proved. Finally in section 5, the main results are extended to interaction
problems including certain nonlinear elastic/thermoelastic plates.

##  2 The model

Here we deal with a compressible viscous fluid interacting with an elastic or
viscoelastic plate, where the dimensions of the fluid and the plate are $d$
and $d-1$, respectively, for $d=2,3$. The vertical plate displacement is
described by a scalar function $w:\Gamma\to\mathbb{R}$, where
$\Gamma\subset\mathbb{R}$ is a connected bounded domain with a Lipschitz
boundary. The fluid fills the domain defined as

$\displaystyle\Omega^{w}(t):=\\{(X,z):X\in\Gamma,-1<z<w(t,X)\\},\quad
t\in[0,T].$

We will denote the graph of $w$ as
$\Gamma^{w}(t)=\\{(X,z):X\in\Gamma,z=w(t,X)\\}$, while the entire rigid part
of the boundary $\partial\Omega^{w}(t)$ will be denoted as
$\Sigma:=(\Gamma\times\\{-1\\})\cup\partial\Gamma\times\\{-1,0\\}$. Finally,
by a slight abuse of notation, for a time interval $I$ and a time dependent
set $B(t)$, we will write

$\displaystyle I\times B(t):=\bigcup_{t\in I}\\{t\\}\times B(t),$

and the time-space cyllinders corresponding to our system will be denoted as

$\displaystyle
Q_{T}^{w}:=(0,T)\times\Omega^{w}(t),\quad\Gamma_{T}^{w}:=(0,T)\times\Gamma^{w}(t),\quad\Gamma_{T}:=(0,T)\times\Gamma.$

The governing equations for our coupled fluid-structure interaction problem
read:

The (visco)elastic plate equation on $\Gamma_{T}$:

$\partial_{t}^{2}w+\Delta^{2}w-\alpha\partial_{t}\Delta
w=-S^{w}\mathbf{f}_{fl}\cdot\mathbf{e_{3}};$ (1)

The compressible Navier-Stokes equations on $Q_{T}^{w}$:

$\displaystyle\partial_{t}(\rho\mathbf{u})+\nabla\cdot(\rho\mathbf{u}\otimes\mathbf{u})$
$\displaystyle=$ $\displaystyle-\nabla p(\rho)+\mathbb{S}(\nabla\mathbf{u}),$
(2) $\displaystyle\partial_{t}\rho+\nabla\cdot(\rho\mathbf{u})$
$\displaystyle=$ $\displaystyle 0;$ (3)

The fluid-structure coupling (kinematic and dynamic, resp.) on $\Gamma_{T}$:

$\displaystyle\partial_{t}w(t,X)\mathbf{e_{3}}$ $\displaystyle=$
$\displaystyle\mathbf{u}(t,X,w(t,X)),$ (4) $\displaystyle\mathbf{f}_{fl}(t,X)$
$\displaystyle=$
$\displaystyle\big{[}(-p(\rho)I+\nabla\cdot\mathbb{S}(\nabla\mathbf{u}))\nu^{w}\big{]}(t,X,w(t,X));$
(5)

The boundary conditions (clamped and no-slip, resp.):

$\displaystyle w(t,x)=\partial_{\nu}w(t,x)$ $\displaystyle=$ $\displaystyle
0,\quad\text{ on }(0,T)\times\partial\Gamma,$ (6) $\displaystyle\mathbf{u}$
$\displaystyle=$ $\displaystyle 0,\quad\text{ on}~{}~{}(0,T)\times\Sigma;$ (7)

The initial data:

$\displaystyle\rho(0,\cdot)=\rho_{0},\quad(\rho\mathbf{u})(0,\cdot)=(\rho\mathbf{u})_{0},\quad
w(0,\cdot)=w_{0},\quad\partial_{t}w(0,\cdot)=v_{0}.$ (8)

Here, $\alpha\geq 0$ is the viscoelastic coefficient, $S^{w}(t,X)$ is the
Jacobian of the transformation from the Eulerian to the Lagrangian coordinates
of the plate

$\displaystyle S^{w}(t,X)=\sqrt{1+|\nabla w|^{2}},$

$\mathbf{e}_{3}=(0,0,1)$ (or $\mathbf{e}_{3}=(0,1)$ if $d=2$), $\nu^{w}$ is
the unit outward normal vector on $\Gamma^{w}$, $p$ is the pressure given by
$p(\rho)=\rho^{\gamma}$, where111The lower bounds for $\gamma$ are imposed to
ensure the existence of a weak solution (see Remark $\ref{thatremark}(3)$ for
more details).

  1. $\gamma>1$, if $d=2$ and $\alpha\geq 0$,

  2. $\gamma>\frac{12}{7}$, if $d=3$, and $\alpha=0$,

  3. $\gamma>\frac{3}{2}$, if $d=3$ and $\alpha>0$,

$\mathbb{S}(\nabla\mathbf{u})$ is the Newton stress tensor given by

$\displaystyle\mathbb{S}(\nabla\mathbf{u}):=\mu\nabla\mathbf{u}+(\mu+\lambda)(\nabla\cdot\mathbf{u})I,$

with $\mu>0$ and $\lambda+\frac{2}{3}\mu>0$, and $\nu$ is the normal vector on
$\partial\Gamma$.

###  2.1 Weak solution

First, for any $1\leq p,q\leq\infty$, we define the following functional
spaces for variable domains

$\displaystyle L^{p}(0,T;L^{q}(\Omega^{w}(t))):=$ $\displaystyle\\{f\in
L^{1}((0,T)\times\Omega^{w}(t)):$ $\displaystyle f(t,\cdot)\in
L^{q}(\Omega^{w}(t))\text{ for a.a
}t\in(0,T),~{}||f(t,\cdot)||_{L^{q}(\Omega^{w}(t))}\in L^{p}(0,T)\\},$

and

$\displaystyle L^{p}(0,T;W^{1,q}(\Omega^{w}(t))):=\\{f\in
L^{p}(0,T;L^{q}(\Omega^{w}(t))):\nabla f\in
L^{p}(0,T;L^{q}(\Omega^{w}(t)))\\},$ $\displaystyle
L^{p}(0,T;W^{2,q}(\Omega^{w}(t))):=\\{f\in
L^{p}(0,T;W^{1,q}(\Omega^{w}(t))):\nabla^{2}f\in
L^{p}(0,T;L^{q}(\Omega^{w}(t)))\\}.$

Now, to introduce the weak formulation to the problem
$\eqref{structureeqs}-\eqref{initialdata}$, the following spaces that come
from the energy inequality $\eqref{energy}$ are defined:

  1. the structure displacement space222Here, $\alpha H^{1}(\Gamma_{T}):=\\{f\in L^{2}(\Gamma_{T}):\alpha f\in H^{1}(\Gamma_{T})\\}$.

$\displaystyle\mathcal{W}_{S}(0,T):=W^{1,\infty}(0,T;L^{2}(\Gamma))\cap
L^{\infty}(0,T;H_{0}^{2}(\Gamma))\cap\alpha H^{1}(\Gamma_{T}),$

  2. the space for the fluid density

$\displaystyle\mathcal{W}_{D}(0,T):=L^{\infty}(0,T;L^{\gamma}(\Omega^{w}(t))).$

  3. the fluid velocity space

$\displaystyle\mathcal{W}_{F}(0,T):=L^{\infty}(0,T;L^{2}(\Omega^{w}(t)))\cap
L^{2}(0,T;H^{1}(\Omega^{w}(t))),$

  4. and the coupled fluid-structure space

$\displaystyle\mathcal{W}_{FS}(0,T)=\\{(\mathbf{u},w)\in\mathcal{W}_{F}(0,T)\times\mathcal{W}_{S}(0,T):\gamma_{|\Gamma^{w}(t)}\mathbf{u}=\partial_{t}w\mathbf{e}_{3}\text{
and }\gamma_{|\Sigma}\mathbf{u}=0\text{ for a.a. }t\in(0,T)\\}.$

Here, $\gamma_{|\Gamma^{w}(t)}$ is the Lagrangian trace operator on
$\Gamma^{w}(t)$, which is a continuous linear operator from
$H^{1}(\Omega^{w}(t))$ to $H^{s}(\Gamma)$, for any $s<\frac{1}{2}$ (see [30]).

The weak solution of the problem $\eqref{structureeqs}-\eqref{initialdata}$ is
defined as:333See [38] for the derivation of this weak form for smooth
solutions.

######  Definition 2.1.

We say that
$(\rho,\mathbf{u},w)\in\mathcal{W}_{D}(0,T)\times\mathcal{W}_{FS}(0,T)$ is a
weak solution of the problem $\eqref{structureeqs}$-$\eqref{initialdata}$ if:

  1. 1.

The initial data $\rho_{0},(\rho\mathbf{u})_{0},w_{0},v_{0},\in
L^{\gamma}(\Omega^{w_{0}})\times
L^{\frac{2\gamma}{\gamma+1}}(\Omega^{w_{0}})\times H_{0}^{2}(\Gamma)\times
L^{2}(\Gamma)$ and is assumed to satisfy the following compatibility
conditions:

$\displaystyle\begin{aligned} \rho_{0}>0,&\text{ on
}\\{(\rho\mathbf{u})_{0}\neq 0\\},\\\
\frac{(\rho\mathbf{u})_{0}^{2}}{\rho_{0}}&\in L^{1}(\Omega^{w_{0}}),\\\
\partial_{\nu}w_{0}=w_{0}=0,&\text{ on }\partial\Gamma,\\\ w_{0}>-1,&\text{ on
}\Gamma.\end{aligned}$

  2. 2.

$\rho\in C_{w}(0,T;L^{\gamma}(\Omega^{w}(t)))$ and the continuity
equation444Here, the density $\rho$ is extended by $0$ to $\mathbb{R}^{3}$.

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{3}}\big{(}\rho\partial_{t}\varphi+\rho\mathbf{u}\cdot\nabla\varphi\big{)}=\int_{\mathbb{R}^{3}}\rho(T)\varphi(T)-\int_{\mathbb{R}^{3}}\rho_{0}\varphi(0),$
(9)

holds for all $\varphi\in C^{\infty}([0,T]\times\mathbb{R}^{3})$.

  3. 3.

$\rho\mathbf{u}\in C_{w}(0,T;L^{\frac{2\gamma}{\gamma+1}}(\Omega^{w}(t)))$ and
the coupled momentum equation

$\displaystyle\int_{Q_{T}^{w}}\rho\mathbf{u}\cdot\partial_{t}\mathbf{q}+\int_{Q_{T}^{w}}(\rho\mathbf{u}\otimes\mathbf{u}):\nabla\mathbf{q}+\int_{Q_{T}^{w}}\rho^{\gamma}(\nabla\cdot\mathbf{q})-\int_{Q_{T}^{w}}\mathbb{S}(\nabla\mathbf{u}):\nabla\mathbf{q}$
$\displaystyle+\int_{\Gamma_{T}}\partial_{t}w\partial_{t}\psi-\int_{\Gamma_{T}}\Delta
w\Delta\psi-\alpha\int_{\Gamma_{T}}\partial_{t}\nabla w\cdot\nabla\psi$
$\displaystyle=\int_{\Omega^{w}(T)}\rho(T)\mathbf{u}(T)\cdot\mathbf{q}(T)-\int_{\Omega^{w_{0}}}(\rho\mathbf{u})_{0}\cdot\mathbf{q}(0)+\int_{\Gamma}\partial_{t}w(T)\psi(T)-\int_{\Gamma}v_{0}\psi(0),\quad\quad\quad\quad\quad$
(10)

holds for all $\mathbf{q}\in C^{\infty}([0,T]\times\overline{\Omega^{w}(t)})$
and $\psi\in C_{0}^{\infty}([0,T]\times\Gamma)$ such that
$\mathbf{q}_{|\Sigma}=0$ and $\mathbf{q}(t,X,w(t,X))=\psi(t,X)\mathbf{e}_{3}$
on $\Gamma_{T}$.

If, in addition $(\rho,\mathbf{u},w)$ satisfies the following energy
inequality

$\displaystyle\frac{1}{2}\displaystyle\int_{\Omega^{w}(t)}(\rho|\mathbf{u}|^{2})(t)+\frac{1}{\gamma-1}\displaystyle\int_{\Omega^{w}(t)}\rho^{\gamma}(t)+\int_{0}^{t}\displaystyle\int_{\Omega^{w}(t)}\mathbb{S}(\nabla\mathbf{u}):\nabla\mathbf{u}$
$\displaystyle\quad+\frac{1}{2}\int_{\Gamma}|\partial_{t}w(t)|^{2}+\frac{1}{2}\int_{\Gamma}|\Delta
w(t)|^{2}+\alpha\int_{0}^{t}\displaystyle\int_{\Gamma}|\partial_{t}\nabla
w|^{2}$
$\displaystyle\leq\frac{1}{2}\displaystyle\int_{\Omega^{w_{0}}}\frac{(\rho\mathbf{u})_{0}^{2}}{\rho_{0}}+\frac{1}{\gamma-1}\displaystyle\int_{\Omega^{w_{0}}}\rho_{0}^{\gamma}+\frac{1}{2}\int_{\Gamma}|v_{0}|^{2}+\frac{1}{2}\int_{\Gamma}|\Delta
w_{0}|^{2},$ (11)

then we call $(\rho,\mathbf{u},w)$ a finite energy weak solution.

######  Theorem 2.1.

([38, Theorem 2.1]) Let $\alpha=0$ and $\gamma>\frac{12}{7}$ or $\alpha>0$ and
$\gamma>\frac{3}{2}$. Then there exists a finite energy weak solution
$(\rho,\mathbf{u},w)\in\mathcal{W}_{D}(0,T)\times\mathcal{W}_{FS}(0,T)$ in the
sense of the Definition 2.1. The lifespan $T>0$ is either any $T<T^{*}$, where
$T^{*}$ is the moment when the collision occurs
$\min\limits_{X\in\Gamma}w(T^{*},X)+1=0$, or $T=\infty$ if no collision
occurs.

######  Remark 2.1.

(1) In theorem above, the structure is modeled by a nonlinear thermoelastic
plate model, but the same result holds for the linear elastic plate model
given in $\eqref{structureeqs}$. In section 5, the nonlinear
elastic/thermoelastic plate models are considered.
(2) In [4], a weak solution in the sense very similar to the one in Definition
2.1 was constructed, for $\alpha=0$ and $\gamma>\frac{12}{7}$. There however,
the geometry is different - the elastic structure is a shell which deforms in
the normal direction of the reference domain boundary.
(3) One of the main difficulties in studying the existence of such weak
solutions is the weak $L^{1}$ convergence of the pressure. Since the structure
regularity that comes from the energy is not strong enough to ensure that the
fluid cavity is Lipschitz (and it also changes in time), the standard approach
based on Bogovskii operator fails in this framework. The alternative here
consists in proving the additional integrability of the pressure inside the
domain in the standard way and that the $L^{1}$ norm of the pressure doesn’t
concentrate near the boundary, thus giving the equiintegrability sufficient to
pass to the weak limit. The latter proof is very involved and it is based on
constructing a test function which sort of represents the distance with
respect to the boundary. As a consequence, this construction in [4] and [38]
varies greatly due to different geometries, as the latter has corners at
$\partial\Gamma\times\\{-1\\}$ and $\partial\Gamma\times\\{0\\}$. If
$\alpha=0$, in order to close the estimates in the mentioned proof, one needs
the condition $\gamma>12/7$ to compensate for the low integrability of the
structure velocity $\partial_{t}w\in L_{t}^{2}L_{x}^{p}$, for any $p<4$, which
comes from the trace regularity. If however $\alpha>0$, then one can use the
regularity $\partial_{t}w\in L_{t}^{2}H_{x}^{1}$, so $\gamma>3/2$ suffices.
This means that the same results as in [4, 38] hold for $\alpha>0$ and
$\gamma>3/2$, even though they weren’t explicitly considered.

###  2.2 Strong solution

Here, we refer to known results concerning strong and regular solutions to the
problem $\eqref{structureeqs}-\eqref{initialdata}$:

Reference, constants | Regularity of initial data | Regularity of solutions
---|---|---
[29, Theorem 1.1] | $\rho_{0}\circ A_{w_{0}}\in H^{2}$ | $\rho\circ A_{w}\in C_{t}^{0}H_{x}^{2}\cap C_{t}^{1}H_{x}^{1}$
$d=2$ | $\mathbf{u}_{0}\circ A_{w_{0}}\in H^{3}$ | $\mathbf{u}\circ A_{w}\in L_{t}^{2}H_{x}^{3}\cap C_{t}^{0}H_{x}^{\frac{5}{2}}\cap H_{t}^{1}H_{x}^{2}\cap C_{t}^{1}H_{x}^{1}\cap H_{t}^{2}L_{x}^{2}$
$\gamma\geq 1$ | $w_{0}=0$ | $w\in C_{t}^{0}H_{x}^{\frac{9}{2}}\cap H_{t}^{1}H_{x}^{4}\cap C_{t}^{1}H_{x}^{3}\cap H_{t}^{2}H_{x}^{2}$
$\alpha>0$ | $v_{0}\in H^{3}$ | $\cap C_{t}^{2}H_{x}^{1}\cap H_{t}^{3}L_{x}^{2}$
[27, Theorem 1.1] | $\rho_{0}\circ A_{w_{0}}\in H^{3}$ | $\rho\circ A_{w}\in H_{t}^{1}H_{x}^{3}\cap W_{t}^{1,\infty}H_{x}^{2}$
$d=2,3$ | $\mathbf{u}_{0}\circ A_{w_{0}}\in H^{3}$ | $\mathbf{u}\circ A_{w}\in L_{t}^{2}H_{x}^{4}\cap H_{t}^{1}H_{x}^{2}\cap H_{t}^{2}L_{x}^{2}$
$\gamma\geq 1$ | $w_{0}\in H^{4}$ | $w\in L_{t}^{\infty}H_{x}^{4}\cap W_{t}^{1,\infty}H_{x}^{3}\cap H_{t}^{2}H_{x}^{2}\cap H_{t}^{3}L_{x}^{2}$
$\alpha\geq 0$ | $v_{0}\in H^{3}$ |
[28, Theorem 1.1] | $\rho_{0}\circ A_{w_{0}}\in W^{1,q}$ | $\rho\circ A_{w}\in W_{t}^{p}L_{x}^{q}\cap L_{t}^{p}W_{x}^{q}$
$d=2,3$ | $\mathbf{u}_{0}\circ A_{w_{0}}\in B_{q,p}^{2(2-1/p)}$ | $\mathbf{u}\circ A_{w}\in L_{t}^{p}W_{x}^{2,q}\cap W_{t}^{1,p}L_{x}^{q}$
$\gamma\geq 1$ | $w_{0}\in B_{q,p}^{2(2-1/p)}$ | $w\in L_{t}^{p}W_{x}^{4,q}\cap W_{t}^{2,p}L_{x}^{q}$
$\alpha>0$ | $v_{0}\in B_{q,p}^{2(2-1/p)}$ | $p\in(1,\infty),q\in(d,\infty),\frac{1}{p}+\frac{1}{2q}\neq\frac{1}{4-d}$
Table 1: List of strong and regular solutions to
$\eqref{structureeqs}-\eqref{initialdata}$. The regularity of
$\rho,\mathbf{u}$ is given in ALE coordinates, by means of the mapping $A_{w}$
from $\eqref{ALE}$, and the compatibility conditions for the initial data are
omitted as they are quite natural and can be found in the respective
references.

######  Remark 2.2.

(1) In the above table, $B_{q,p}^{2(2-1/p)}$ stands for Besov spaces (see
[1]).
(2) In [27, 29], the plate domain $\Gamma$ is periodic, i.e.
$\Gamma=(R/L_{1}\mathbb{Z})\times(R/L_{2}\mathbb{Z})$ and
$\Gamma=(R/L_{1}\mathbb{Z})$ , respectively, for some $L_{1},L_{2}>0$, and
$\rho,\mathbf{u},w$ are periodic in horizontal coordinates as well. However,
in [28], the reference fluid domain is smooth and contains a flat part which
is deformable and constitutes a viscoelastic plate.
(3) In [28], the fluid conducts heat and it is governed by the Navier-Stokes-
Fourier system, but the same result holds for the barotropic case used in this
paper, as it was pointed out in [28, Remark 1.3(5)].
(4) In [27], the structure is actually governed by a wave equation.
Nevertheless, the same result can be obtained for the plate equation, both for
$\alpha=0$ and $\alpha>0$, as these cases have more regularity. However, in
[28, 29], $\alpha>0$ and the same results cannot be obtained in the same way
if $\alpha=0$. This is because viscoelasticity ensures maximal regularity [28]
and the analiticity [29] properties, which play a key role in the respective
approaches.

###  2.3 Relative entropy and relative energy inequality

Relative energy (or entropy) inequality in the context of compressible Navier-
Stokes system was first introduced by Germain in [15], and it was used as a
tool in proving the weak-strong uniqueness property for a sub-class of finite
energy weak solutions which have have a $L_{t}^{p}L_{x}^{q}$-integrable
gradient of density (for certain $p,q>1$). Note that the latter condition is
quite unnatural and existence of weak solutions within this class is non-
existant in the literature. Later, Feireisl, Novotný and Sun in [11]
introduced suitable weak solutions - a sub-class of finite energy weak
solutions that satisfy the relative energy inequality, then proved their
existence and showed the weak-strong uniqueness within this class, thus
removing the additional regularity on density imposed in [15]. Finally,
Feireisl, Novotny and Jin in [12] introduced the relative entropy in the class
of finite energy weak solutions and gave a definitive answer to the weak-
strong uniqueness question, proving that all finite energy weak solutions are
in fact suitable weak solutions and thus have the weak-strong uniqueness
property. Here, we extend the notions of relative entropy and relative energy
inequality to our fluid-structure interaction model.

Let $(\rho_{1},\mathbf{u}_{1},w_{1})$ be a finite energy weak solution in the
sense of Definition 2.1. For any $\rho_{2},\mathbf{u}_{2}\in
C^{\infty}([0,T]\times\overline{\Omega^{w_{1}}(t)})$ and $w_{2}\in
C_{0}^{\infty}([0,T]\times\Gamma)$ such that ${\mathbf{u}_{2}}_{|\Sigma}=0$
and $\mathbf{u}_{2}(t,X,w_{2}(t,X))=\partial_{t}w_{2}(t,X)\mathbf{e}_{3}$ on
$\Gamma_{T}$, we define the relative entropy with respect to
$(\rho_{2},\mathbf{u}_{2},w_{2})$ as

$\displaystyle\mathcal{E}\Big{(}(\rho_{1},\mathbf{u}_{1},w_{1})\Big{|}(\rho_{2},\mathbf{u}_{2},w_{2})\Big{)}(t)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega^{w_{1}}(t)}(\rho_{1}|\mathbf{u}_{1}-\mathbf{u}_{2}|^{2})(t)+\frac{1}{\gamma-1}\int_{\Omega^{w_{1}}(t)}\big{(}\rho_{1}^{\gamma}-\gamma\rho_{2}^{\gamma-1}(\rho_{1}-\rho_{2})-\rho_{2}^{\gamma}\big{)}(t)$
$\displaystyle\quad+\frac{1}{2}\int_{\Gamma}|\partial_{t}w_{1}-\partial_{t}w_{2}|^{2}(t)+\frac{1}{2}\int_{\Gamma}|\Delta
w_{1}-\Delta w_{2}|^{2}(t),$

and the relative energy inequality with respect to
$(\rho_{2},\mathbf{u}_{2},w_{2})$ as

$\displaystyle\mathcal{E}\Big{(}(\rho_{1},\mathbf{u}_{1},w_{1})\Big{|}(\rho_{2},\mathbf{u}_{2},w_{2})\Big{)}(t)+\int_{0}^{t}\int_{\Omega^{w_{1}}(t)}\mathbb{S}(\nabla\mathbf{u}_{1}-\nabla\mathbf{u}_{2}):(\nabla\mathbf{u}_{1}-\nabla\mathbf{u}_{2})+\alpha\int_{0}^{t}\int_{\Gamma}|\partial_{t}\nabla
w_{1}-\partial_{t}\nabla w_{2}|^{2}$
$\displaystyle\leq\mathcal{E}\Big{(}(\rho_{1},\mathbf{u}_{1},w_{1})\Big{|}(\rho_{2},\mathbf{u}_{2},w_{2})\Big{)}(0)+\int_{0}^{t}\mathcal{R}\Big{(}\rho_{1},\mathbf{u}_{1},w_{1},\rho_{2},\mathbf{u}_{2},w_{2}\Big{)},$
(12)

where the remainder term reads

$\displaystyle\mathcal{R}\Big{(}\rho_{1},\mathbf{u}_{1},w_{1},\rho_{2},\mathbf{u}_{2},w_{2}\Big{)}:=\int_{\Omega^{w_{1}}(t)}\mathbb{S}(\nabla\mathbf{u}_{2}):(\nabla\mathbf{u}_{2}-\nabla\mathbf{u}_{1})+\int_{\Omega^{w_{1}}(t)}\rho_{1}(\partial_{t}\mathbf{u}_{2}+\mathbf{u}_{1}\cdot\nabla\mathbf{u}_{2})\cdot(\mathbf{u}_{2}-\mathbf{u}_{1})$
$\displaystyle\quad+\frac{\gamma}{\gamma-1}\int_{\Omega^{w_{1}}(t)}\Big{[}(\rho_{2}\mathbf{u}_{2}-\rho_{1}\mathbf{u}_{1})\cdot\nabla(\rho_{2}^{\gamma-1})+(\rho_{1}-\rho_{2})\partial_{t}(\rho_{2}^{\gamma-1})\Big{]}+\int_{\Omega^{w_{1}}(t)}(\rho_{2}^{\gamma}-\rho_{1}^{\gamma})(\nabla\cdot\mathbf{u}_{2})$
$\displaystyle\quad+\int_{\Gamma^{w_{1}}(t)}(\partial_{t}w_{2}-\partial_{t}w_{1})\rho_{2}^{\gamma}\nu^{w^{1}}\cdot\mathbf{e}_{3}-\int_{\Gamma}(\partial_{t}w_{2}-\partial_{t}w_{1})\partial_{t}^{2}w_{2}-\int_{\Gamma}(\partial_{t}w_{2}-\partial_{t}w_{1})\Delta^{2}w_{2}$
$\displaystyle\quad+\alpha\int_{\Gamma}\partial_{t}\Delta
w_{2}(\partial_{t}w_{1}-\partial_{t}w_{2}).$

It is straightforward to see that by choosing
$\rho_{2}=0,\mathbf{u}_{2}=0,w_{2}=0$, the relative energy inequality reduces
to the energy inequality $\eqref{energy}$. Also, if one chooses
$w_{1}=w_{2}=0$, then relative entropy and relative energy inequality take the
form of the ones defined in [12].

###  2.4 Main results

First, the following characterization of finite energy weak solutions is
given:

######  Theorem 2.2.

(Main result I) Let $(\rho,\mathbf{u},w)$ be a finite energy weak solution in
the sense of Definition 2.1 defined on $(0,T)$ such that
$\min_{(0,T)\times\Omega^{w}(t)}1+w\geq c>0$. Then, one has the following:

  1. (1)

$\Delta w\in L^{2}(0,T;H^{s}(\Gamma))$, where

    1. 2D:

$\alpha=0$, $\gamma>1$ and
$0<s<\min\big{\\{}\frac{1}{2}-\frac{1}{2\gamma},\frac{1}{4}\big{\\}}$;
$\alpha>0$, $\gamma>1$ and
$0<s<\min\big{\\{}\frac{3}{4}-\frac{1}{2\gamma},\frac{1}{2}\big{\\}}$;

    2. 3D:

$\alpha=0$, $\gamma>\frac{12}{7}$ and
$0<s<\min\big{\\{}\frac{7}{12}-\frac{1}{\gamma},\frac{1}{4}\big{\\}}$;
$\alpha>0$: $\gamma>\frac{3}{2}$ and
$0<s<\min\big{\\{}\frac{2}{3}-\frac{1}{\gamma},\frac{1}{2}\big{\\}}$.

Consequently, $\nabla w\in L^{2}(0,T;L^{\infty}(\Gamma))$ in all the above
cases;

  2. (2)

$(\rho,\mathbf{u},w)$ satisfies the relative energy inequality
$\eqref{relen}$.

The second result of this paper states that all finite energy weak solutions
enjoy the weak-strong uniqueness property in the following sense:

######  Theorem 2.3.

(Main result II) Let $(\rho_{1},\mathbf{u}_{1},w_{1})$ be a finite energy weak
solution in the sense of Definition 2.1 defined on $(0,T_{1})$ such that

$\displaystyle\min\limits_{(0,T_{1})\times\Omega^{w_{1}}(t)}1+w_{1}\geq
c_{1}>0$

and let $(\rho_{2},\mathbf{u}_{2},w_{2})$ be a strong solution of the problem
$\eqref{structureeqs}$-$\eqref{initialdata}$ such that

$\displaystyle\rho_{2}\in L^{2}(0,T_{2};W^{1,q}(\Omega^{w_{2}}(t)))\cap
H^{1}(0,T_{2};L^{q}(\Omega^{w_{2}}(t))),$ $\displaystyle\mathbf{u}_{2}\in
L^{2}(0,T_{2};W^{2,q}(\Omega^{w_{2}}(t)))\cap
H^{1}(0,T_{2};L^{q}(\Omega^{w_{2}}(t))),$ $\displaystyle w_{2}\in
L^{2}(0,T_{2};H^{4}(\Gamma))\cap H^{2}(0,T_{2};L^{2}(\Gamma)),$

and

$\displaystyle
0<\inf\limits_{(0,T_{2})\times\Omega^{w_{2}}(t)}\rho_{2}\leq\sup\limits_{(0,T_{2})\times\Omega^{w_{2}}(t)}\rho_{2}<\infty,\qquad\min\limits_{(0,T_{2})\times\Omega^{w_{2}}(t)}1+w_{2}\geq
c_{2}>0.$

If

$\displaystyle\rho_{1}(0,\cdot)=\rho_{2}(0,\cdot),\quad\mathbf{u}_{1}(0,\cdot)=\mathbf{u}_{2}(0,\cdot),\quad
w_{1}(0,\cdot)=w_{2}(0,\cdot),\quad v_{1}(0,\cdot)=v_{2}(0,\cdot),$

and

  1. 2D:

$\alpha\geq 0$, $\gamma>1$ and $q>2$;

  2. 3D:

$\alpha=0$, $\gamma\in\big{(}\frac{12}{7},2\big{]}$, $\nabla\mathbf{u}_{2}\in
L^{\infty}((0,T_{2})\times\Omega^{w_{2}}(t))$ and
$q>\frac{4\gamma}{3\gamma-4}$;
$\alpha=0$, $\gamma>2$ and $q>4$;
$\alpha>0$, $\gamma>\frac{3}{2}$ and
$q>\max\\{3,\frac{6\gamma}{5\gamma-6}\\}$,

then

$\displaystyle(\rho_{1},\mathbf{u}_{1},w_{1})\equiv(\rho_{2},\mathbf{u}_{2},w_{2}),\quad\text{a.e.
in }(0,T_{2})\times\Omega^{w_{2}}(t).$

The following extension of main results is obtained in section 5:

######  Corollary 2.4.

Theorems 2.3 and 2.2 hold in the case where the linear plate model
$\eqref{structureeqs}$ is replaced by certain nonlinear plate models.

######  Remark 2.3.

The purpose of the regularity estimate in Theorem $\ref{main2}(1)$ is
threefold. First, it gives us additional properties for finite-energy weak
solutions which can be used on their own in further applications. Second, it
ensures that the domain $\Omega^{w}(t)$ is always Lipschitz (with values in
$L_{t}^{2}$), which allows us to use the divergence theorem and Raynolds
transport theorem in the proof of Theorem $\ref{main2}(2)$. Finally, it is
used in the proof of Theorem 2.3 to control 7th and 10th terms on the right-
hand side of inequality $\eqref{estimate}$.

##  3 Proof of Theorem 2.2

The following proof will be carried in two parts.

###  3.1 Part 1: the regularity estimates - Theorem $\ref{main2}(1)$

We start with introducing fractional Sobolev and Nikolskii functional spaces,
respectively (see [1] and [36] for more details):

######  Definition 3.1.

For $\alpha\in(0,1)$, $q\in(1,\infty)$, we say that
$g\in\mathcal{W}^{\alpha,q}(\Gamma)$ if its norm satisfies

$\displaystyle||g||_{\mathcal{W}^{\alpha,q}(\Gamma)}^{q}:=\Big{(}\int_{\Gamma}\int_{\Gamma}\frac{|g(x)-g(y)|^{q}}{|x-y|^{n+\alpha
q}}dxdy\Big{)}^{\frac{1}{q}}+\Big{(}\int_{\Gamma}|g(x)|^{q}dx\Big{)}^{\frac{1}{q}}<\infty,$

and that $g\in\mathcal{N}^{\alpha,q}(\Gamma)$ if its norm satisfies

$\displaystyle||g||_{\mathcal{N}^{\alpha,q}(\Gamma)}^{q}:=\sup\limits_{i\in\\{1,...,d\\}}\sup\limits_{h>0}\Big{(}\int_{\Gamma_{h}}\Big{|}\frac{g(x+h\mathbf{e}_{i})-g(x)}{h^{\alpha}}\Big{|}^{q}dx\Big{)}^{\frac{1}{q}}+\Big{(}\int_{\Gamma}|g(x)|^{q}dx\Big{)}^{\frac{1}{q}}<\infty,$

where $\mathbf{e}_{i}$ its the $i$-th unit vector and
$\Gamma_{h}=\\{x\in\Gamma:\text{dist}(x,\partial\Gamma)>h\\}$. Given any
$0<\alpha<\beta<1$ and $g\in\mathcal{N}^{\beta,q}(\Gamma)$, the following
imbedding is true

$\displaystyle||g||_{L^{\frac{nq}{n-\alpha q}}(\Gamma)}\leq
C_{1}||g||_{\mathcal{W}^{\alpha,q}(\Gamma)}\leq
C_{2}||g||_{\mathcal{N}^{\beta,q}(\Gamma)}\leq
C_{3}||g||_{\mathcal{W}^{\beta,q}(\Gamma)},\quad\text{ for }\alpha q<n,$

or

$\displaystyle||g||_{C^{\alpha-\frac{n}{q}}(\Gamma)}\leq
C_{1}||g||_{\mathcal{W}^{\alpha,q}(\Gamma)}\leq
C_{2}||g||_{\mathcal{N}^{\beta,q}(\Gamma)}\leq
C_{3}||g||_{\mathcal{W}^{\alpha,q}(\Gamma)},\quad\text{ for }\alpha q>n.$

Finally, for $s,h>0$ and a function $q:\Gamma\to\mathbb{R}$, we introduce the
quotient

$\displaystyle
D_{h,\mathbf{e}}^{s}(q)(y):=\frac{q(y+h\mathbf{e})-q(y)}{h^{s}},~{}~{}\text{for
any (unit) vector }\mathbf{e}\in\mathbb{R}^{2}.$

Since the direction of the unit vector will not be relevant in most cases, we
will usually omit it and just write
$D_{h}^{s}(q)(y):=D_{h,\mathbf{e}}^{s}(q)(y)$ instead, for any unit vector
$\mathbf{e}$.

The following notation will be useful in the upcoming analysis:

######  Definition 3.2.

For a given $1<b\leq\infty$ and a domain $E$, denote by

$\displaystyle L^{b^{-}}(E):=\bigcap_{p<b}L^{p}(E),\quad\quad
W^{{a^{-}},b}(E):=\bigcap_{s<a}W^{s,p}(E),$ $\displaystyle
W^{a,b^{-}}(E):=\bigcap_{p<b}W^{s,p}(E),\quad\quad
W^{a^{-},b^{-}}(E):=\bigcap_{s<a,p<b}W^{s,p}(E).$

We will write for any $1<b\leq\infty$,

$\displaystyle||f||_{L^{b^{-}}(E)}\leq D,$

for a constant $D\geq 0$, if for all $1\leq b^{\prime}<b$, there exists a
constant $C(b^{\prime})$ such that

$\displaystyle||f||_{L^{b^{\prime}}(E)}\leq C(b^{\prime})D.$

and similarly for the Sobolev spaces $W^{a^{-},b}(E)$, $W^{a,b^{-}}(E)$ and
$W^{a^{-},b^{-}}(E)$.

######  Remark 3.1.

(1) Note that $||f||_{L^{b^{-}}(E)}$ is just a notation, not a proper norm;
(2) Wherever we have $||f||_{L^{b^{-}}(E)}$ appearing in an estimate, it can
be replaced with $||f||_{L^{p}(E)}$ followed by "for $p$ close enough to $b$".
By using this notation, we avoid writing this below the majority of estimates,
especially in the case when there is more than one being used. Moreover, when
$b^{-}$ appears in the estimates (instead of somewhere afterwards), it makes
lengthy estimates easier to follow.

######  Remark 3.2.

(1) The following proof is the adaptation to the compressible case of [32,
Theorem 1.2], where similar estimates were obtained in the context of
incompressible viscous fluid interacting with a quasilinear elastic Koiter
shell;
(2) We will only prove the estimate for the 3D/2D case, since the estimates in
the 2D/1D case can be obtained more easily.

We separately observe two cases for the domain $\Gamma$, first the periodic
boundary domain and then a bounded Lipschitz domain.

####  3.1.1 The periodic boundary case

Here, we assume that $\Gamma=(R/L_{1}\mathbb{Z})\times(R/L_{2}\mathbb{Z})$ for
$L_{1},L_{2}>0$. Let $\mathbf{R}_{w}$ be an extension operator defined as

$\displaystyle\mathbf{R}_{w}:f(t,X)\mapsto
f(t,X)\frac{z+1}{w(t,X)+1}\mathbf{e}_{3}.$

Fix $h>0$ and in the coupled momentum equation $\eqref{weaksolmom}$, choose
$(\mathbf{q},\psi)=(\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]},D_{-h}^{s}D_{h}^{s}[w])$
to obtain:

$\displaystyle||D_{h}^{s}\Delta w||_{L^{2}(\Gamma_{T})}^{2}$
$\displaystyle=-\displaystyle\int_{Q_{T}^{w}}(\rho\mathbf{u}\otimes\mathbf{u}):\nabla\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]}-\displaystyle\int_{Q_{T}^{w}}\rho\mathbf{u}\cdot\partial_{t}\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]}-\displaystyle\int_{Q_{T}^{w}}\rho^{\gamma}\big{(}\nabla\cdot\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]}\big{)}$
$\displaystyle\quad+\mu\displaystyle\int_{Q_{T}^{w}}\nabla\mathbf{u}:\nabla\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]}+(\mu+\lambda)\displaystyle\int_{Q_{T}^{w}}(\nabla\cdot\mathbf{u})(\nabla\cdot\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]})$
$\displaystyle\quad+||D_{h}^{s}\partial_{t}w||_{L^{2}(\Gamma_{T})}+\alpha\displaystyle\int_{\Gamma_{T}}\partial_{t}\nabla
w\cdot\nabla D_{-h}^{s}D_{h}^{s}[w]$
$\displaystyle\quad+\int_{\Omega^{w}(T)}\rho(T)\mathbf{u}(T)\cdot\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w(T)]\big{]}-\int_{\Omega^{w_{0}}}(\rho\mathbf{u})_{0}\cdot\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w_{0}]\big{]})$
$\displaystyle\quad+\int_{\Gamma}\partial_{t}w(T)D_{-h}^{s}D_{h}^{s}w(T)-\int_{\Gamma}v_{0}D_{-h}^{s}D_{h}^{s}w_{0},$
(13)

where we used the following commutative property

$\displaystyle-\int_{\Gamma}fD_{-h}^{s}D_{h}^{s}g=\int_{\Gamma}D_{h}^{s}fD_{h}^{s}g,\quad
f,g\in L^{2}(\Gamma).$

The proof will follow once we show that right-hand side of $\eqref{aaaa}$ can
be bounded by a constant independent of $h$. We start with

$\displaystyle\displaystyle\int_{Q_{T}^{w}}(\rho\mathbf{u}\otimes\mathbf{u}):\nabla\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]}$
$\displaystyle\leq
C||\rho||_{L^{\infty}(0,T;L^{\gamma}(\Omega^{w}(t)))}||\mathbf{u}||_{L^{2}(0,T;L^{6-}(\Omega^{w}(t)))}^{2}\left|\left|\nabla\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]}\right|\right|_{L^{\infty}(0,T;L^{p}(\Omega^{w}(t)))}$
$\displaystyle\leq C\Big{|}\Big{|}D_{-h}^{s}D_{h}^{s}[\nabla
w]\frac{z+1}{w+1}+D_{-h}^{s}D_{h}^{s}[w]\nabla\Big{(}\frac{z+1}{w+1}\Big{)}\Big{|}\Big{|}_{L^{\infty}(0,T;L^{p}(\Omega^{w}(t)))}$
$\displaystyle\leq C\left(||D_{-h}^{s}D_{h}^{s}[\nabla
w]||_{L^{\infty}(0,T;L^{p}(\Gamma))}+||D_{-h}^{s}D_{h}^{s}[w]||_{L^{\infty}(0,T;L^{p^{-}}(\Gamma))}\Big{|}\Big{|}\nabla\Big{(}\frac{z+1}{w+1}\Big{)}\Big{|}\Big{|}_{L^{\infty}(0,T;L^{\infty^{-}}(\Omega^{w}(t)))}\right)$
$\displaystyle\leq C\big{(}||\nabla
w||_{L^{\infty}(0,T;W^{2s,p}(\Gamma))}+||w||_{L^{\infty}(0,T;W^{2s,p^{-}}(\Gamma))}||\nabla
w||_{L^{\infty}(0,T;L^{\infty^{-}}(\Gamma))}\big{)}$ $\displaystyle\leq
C||w||_{L^{\infty}(0,T;H^{2}(\Gamma))}\leq C,$

where we have used the Hölder’s inequality to obtain the second line which
constraints $p>\frac{3\gamma}{2\gamma-3}$ and the imbedding
$H^{1}(\Gamma)\hookrightarrow W^{2s,p}(\Gamma)$ to obtain the last line which
constraints $s<\frac{1}{2}$ and $\frac{4}{2-(1-2s)2}=\frac{1}{s}\geq p$, so

$\displaystyle s<\frac{1}{2}~{}\text{ and }~{}\frac{1}{s}\geq
p>\frac{3\gamma}{2\gamma-3}\implies
0<s<\min\left\\{\frac{2}{3}-\frac{1}{\gamma},\frac{1}{2}\right\\}.$

Next,

$\displaystyle\displaystyle\int_{Q_{T}^{w}}\rho\mathbf{u}\cdot\partial_{t}\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]}\leq||\rho\mathbf{u}||_{L^{2}(0,T;L^{(\frac{6\gamma}{5\gamma-6})^{-}}(\Omega^{w}(t)))}\left|\left|\partial_{t}\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]}\right|\right|_{L^{2}(0,T;L^{p}(\Omega^{w}(t)))}$
$\displaystyle\leq
C\Big{|}\Big{|}D_{-h}^{s}D_{h}^{s}[\partial_{t}w]\frac{z+1}{w+1}-D_{-h}^{s}D_{h}^{s}[w]\frac{z+1}{(w+1)^{2}}\partial_{t}w\Big{|}\Big{|}_{L^{2}(0,T;L^{p}(\Omega^{w}(t)))}$
$\displaystyle\leq
C\big{(}||D_{-h}^{s}D_{h}^{s}[\partial_{t}w]||_{L^{2}(0,T;L^{p}(\Gamma))}+||D_{-h}^{s}D_{h}^{s}[w]\partial_{t}w||_{L^{2}(0,T;L^{p}(\Gamma))}\big{)}$
$\displaystyle\leq
C\big{(}||\partial_{t}w||_{L^{2}(0,T;W^{2s,p}(\Gamma))}+||D_{-h}^{s}D_{h}^{s}[w]||_{L^{\infty}(0,T;L^{\infty}(\Gamma))}||\partial_{t}w||_{L^{2}(0,T;L^{p}(\Gamma))}\big{)}$
$\displaystyle\leq
C(1+||D_{-h}^{s}D_{h}^{s}[w]||_{L^{\infty}(0,T;W^{1,q}(\Gamma))})$
$\displaystyle\leq C(1+||w||_{L^{\infty}(0,T;W^{1+2s,q}(\Gamma))})\leq
C(1+||w||_{L^{\infty}(0,T;H^{2}(\Gamma))})\leq C,$

where we have used the Hölder’s inequality in the first inequality which
constraints $p>\frac{6\gamma}{5\gamma-6}$, the imbedding in the line before
the last one which constraints $q>2$ (but close to $2$), the imbedding in the
last line which constraints $s<\frac{1}{2}$ so that
$H^{2}(\Gamma)\hookrightarrow W^{1+2s,q}(\Gamma)$ and

  1. 1.

Case $\alpha=0$: in order to bound

$\displaystyle||\partial_{t}w||_{L^{2}(0,T;W^{2s,p}(\Gamma))}\leq
C||\partial_{t}w||_{L^{2}(0,T;H^{(\frac{1}{2})^{-}}(\Gamma))}\leq C$

we need the following imbedding to hold
$H^{(\frac{1}{2})^{-}}(\Gamma)\hookrightarrow W^{2s,p}(\Gamma)$ which
constraints $s<\frac{1}{4}$, $p\leq\frac{4}{2-(1/2-2s)2}=\frac{4}{1+4s}$ and
$p<4$, so

$\displaystyle s<\frac{1}{4}~{}\text{ and }~{}\frac{4}{1+4s}\geq
p>\frac{6\gamma}{5\gamma-6}\implies
0<s<\min\Big{\\{}\frac{7}{12}-\frac{1}{\gamma},\frac{1}{4}\Big{\\}};$

  2. 2.

Case $\alpha>0$: in order to bound

$\displaystyle||\partial_{t}w||_{L^{2}(0,T;W^{2s,p}(\Gamma))}\leq
C||\partial_{t}w||_{L^{2}(0,T;H^{1}(\Gamma))}\leq C$

we need $H^{1}(\Gamma)\hookrightarrow W^{2s,p}(\Gamma)$ to hold which
constraints $s<\frac{1}{2}$, $p\leq\frac{4}{2-(1-2s)2}=\frac{1}{s}$ and
$p<\infty$, so

$\displaystyle s<\frac{1}{2}~{}\text{ and }~{}\frac{1}{s}\geq
p>\frac{6\gamma}{5\gamma-6}\implies
0<s<\min\Big{\\{}\frac{5}{6}-\frac{1}{\gamma},\frac{1}{2}\Big{\\}}.$

Next, notice that we can explicitly calculate on $Q_{T}^{w}$

$\displaystyle\nabla\cdot\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]}=D_{-h}^{s}D_{h}^{s}[w]\frac{1}{w+1},$

so one can bound

$\displaystyle\displaystyle\int_{Q_{T}^{w}}\rho^{\gamma}\big{(}\nabla\cdot\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w]\big{]}\big{)}\leq
C||\rho^{\gamma}||_{L^{\infty}(0,T;L^{1}(\Omega^{w}(t)))}\Big{|}\Big{|}D_{-h}^{s}D_{h}^{s}[w]\frac{1}{w+1}\Big{|}\Big{|}_{L^{1}(0,T;L^{\infty}(\Gamma))}$
$\displaystyle\leq
C||D_{-h}^{s}D_{h}^{s}[w]||_{L^{2}(0,T;W^{1,p}(\Gamma))}\leq
C||w||_{L^{2}(0,T;W^{1+2s,p}(\Gamma))}\leq
C||w||_{L^{2}(0,T;H^{2}(\Gamma))}\leq C,$

by the imbedding of Sobolev spaces $L^{\infty}(\Gamma)\hookrightarrow
W^{1,p}(\Gamma)$ and $H^{2}(\Gamma)\hookrightarrow W^{2s,p}(\Gamma)$, which
constraints $p>2$ (but arbitrarily close to $2$) and $s<\frac{1}{2}$. The
remaining terms can be bounded in a similar fashion as the previous ones, the
proof is finished.

####  3.1.2 Bounded domain case

In this case, we choose $\Gamma$ to be a bounded and Lipschitz domain. There
are three parts to this proof, first one are the interior estimates, second
one are the boundary estimates for the flat boundary case and the third one
are the boundary estimates for general boundary case. The detailed calculation
is omitted here and only the main steps are presented. An interested reader is
referred to [10, Section 6.3], where the same steps are taken to obtain the
improved regularity estimates for the second order elliptic non-homogeneous
equation.

Step 1 - interior estimates. Let $V\Subset\Gamma$ and fix $h>0$ such that
$h<\text{dist}(V,\partial\Gamma)$. Let $\phi\in C_{c}^{\infty}(\Gamma)$ be a
cut-off function such that $0\leq\phi\leq 1$ on $\Gamma$ and $\phi=1$ on $V$.
We choose
$(\mathbf{q},\psi)=(\mathbf{R}_{w}\big{[}D_{-h}^{s}D_{h}^{s}[w\phi]\big{]},D_{-h}^{s}D_{h}^{s}[w\phi])$
in the coupled momentum equation $\eqref{weaksolmom}$. We obtain an identity
almost the same to the one in $\eqref{aaaa}$ with some additional terms
involving the derivatives of $\phi$, so by estimating in the same way as in
the periodic domain case, we obtain that $||\phi D_{h}^{s}\Delta
w||_{L^{2}(\Gamma_{T})}^{2}\leq C(\text{dist}(V,\partial\Gamma))$. Since $V$
was arbitrary, this gives us $\Delta w\in L^{2}(0,T;H_{loc}^{s}(\Gamma))$,
where $s$ satisfies the conditions given in Theorem $\ref{main2}(1)$.

Step 2 - boundary estimates (flat boundary case). Here, we assume that that we
are dealing with a flat part of the boundary. More precisely, let w.l.o.g.
$(x_{0},y_{0})=X_{0}\in\partial\Gamma$ and

$\displaystyle B(X_{0},r)\cap\Gamma=\\{(x,y)\in B(X_{0},r):x\geq
x_{0}\\},\quad\text{for some }r>0.$

Fix $h<0$ small enough and let $\phi\in C_{c}(B(X_{0},r))$ be a cut-off
function such that $\phi=1$ on $B(X_{0},\frac{r}{2})$. We choose
$(\mathbf{q},\psi)=(\mathbf{R}_{w}\big{[}D_{h,-\mathbf{e}_{2}}^{s}D_{h,\mathbf{e}_{2}}^{s}[w\phi]\big{]},D_{h,-\mathbf{e}_{2}}^{s}D_{h,\mathbf{e}_{2}}^{s}[w\phi])$
in the coupled momentum equation $\eqref{weaksolmom}$ (note that the direction
of $\mathbf{e}_{2}$ which appears in the quotient is parallel with
$B(X_{0},r)\cap\partial\Gamma$). Therefore, we can once again close the
estimates in the same way as in the periodic domain case to obtain

$\displaystyle||D_{h,\mathbf{e}_{2}}^{s}[\Delta
w]||_{L^{2}(0,T;L^{2}(B(X_{0},\frac{r}{2})))}\leq C.$ (14)

It remains to prove that

$\displaystyle||D_{h,\mathbf{e}_{1}}^{s}[w]||_{L^{2}(0,T;L^{2}(B(X_{0},\frac{r}{2})))}\leq
C.$

For this purpose, we can extend $w$ to be defined on $(0,T)\times B(X_{0},r)$
and odd with respect to the boundary $x=x_{0}$

$\displaystyle\hat{w}(t,x,y):=\begin{cases}w(t,x,y),&\quad\text{for }x\geq
x_{0},\\\ -w(t,x_{0}+(x_{0}-x),y),&\quad\text{for }x<x_{0}.\end{cases}$

This gives us that the clamped boundary condition is satisfied for the
following test function (here w.l.o.g. we assume that $\phi$ is also odd with
respect to $\\{x=x_{0}\\}$)

$\displaystyle\partial_{\mathbf{e}_{1}}D_{h,-\mathbf{e}_{1}}^{s}D_{h,\mathbf{e}_{1}}^{s}[\hat{w}\phi]=D_{h,-\mathbf{e}_{1}}^{s}D_{h,\mathbf{e}_{1}}^{s}[\hat{w}\phi]=0,\quad\text{on
}B(X_{0},r)\cap\\{x=x_{0}\\},$

while outside of $B(X_{0},r)$ we have that $\hat{w}\phi=0$, so we can test
$\eqref{weaksolmom}$ with the pair
$(\mathbf{R}_{w}\big{[}D_{h,-\mathbf{e}_{1}}^{s}D_{h,\mathbf{e}_{1}}^{s}[\hat{w}\phi]\big{]},D_{h,-\mathbf{e}_{1}}^{s}D_{h,\mathbf{e}_{1}}^{s}[\hat{w}\phi])$
and once again obtain

$\displaystyle||D_{h,\mathbf{e}_{2}}^{s}[\Delta
w]||_{L^{2}(0,T;L^{2}(B(X_{0},\frac{r}{2})))}\leq C,$

which combined with $\eqref{ydir}$ finally implies

$\displaystyle||\Delta w||_{L^{2}(0,T;H^{s}(B(X_{0},\frac{r}{2})))}\leq C.$

Step 3 - boundary estimates (general boundary case). Here, we assume that the
part of the boundary we are dealing with is a sub-graph of a $C^{0,1}$
function. The idea is to “straighten out the boundary” by means of a local
coordinate transform. For that purpose, denote by $x^{\prime},y^{\prime}$
variables in new coordinate system and
$(\mathbf{e}_{1}^{\prime},\mathbf{e}_{2}^{\prime})$ the corresponding unit
basis vectors and for $X_{0}\in\partial\Gamma$ let

$\displaystyle\Phi:B(0,r_{1})\to B(X_{0},r_{2})$

be a $C^{0,1}$ local coordinate transform such that

$\displaystyle\Phi\big{(}B(0,r_{1})\cap\\{x^{\prime}>0\\}\big{)}=B(X_{0},r_{2})\cap\Gamma,\qquad\Phi\big{(}B(0,r_{1})\cap\\{x=0\\}\big{)}=B(X_{0},r_{2})\cap\partial\Gamma.$

We can express

$\displaystyle\int_{B(X_{0},r_{2})\cap\Gamma}\Delta
w\Delta\psi~{}dxdy=\int_{B(0,r_{1})\cap\\{x^{\prime}>0\\}}|\det\nabla\Phi|(\Delta
w)\circ\Phi(\Delta\psi)\circ\Phi~{}dx^{\prime}dy^{\prime}.$

Now, we can repeat the calculation from step 2 by choosing $\psi$ to satisfy
$(\Delta\psi)\circ\Phi=D_{h,-\mathbf{e}_{2}^{\prime}}^{s}D_{h,\mathbf{e}_{2}^{\prime}}^{s}\big{[}(\Delta
w)\circ\Phi\big{]}$ on $B(0,r_{1})$, or equivalently

$\displaystyle\psi=\Delta^{-1}\Big{[}\Phi^{-1}\big{(}D_{h,-\mathbf{e}_{2}^{\prime}}^{s}D_{h,\mathbf{e}_{2}^{\prime}}^{s}\big{[}(\Delta
w)\circ\Phi\big{]}\big{)}\Big{]},\quad\text{ on }B(X_{0},r_{2}),$

where $\Delta^{-1}$ is the inverse Laplace operator with zero Dirichlet
boundary data, while the fluid test function is chosen as
$\mathbf{q}=\mathbf{R}_{w}[\psi]$. With these test functions, the equation
$\eqref{weaksolmom}$ leads to the following estimate

$\displaystyle
c\int_{B(0,r_{1})\cap\\{x^{\prime}>0\\}}|D_{h,\mathbf{e}_{2}^{\prime}}^{s}[(\Delta(\phi
w))\circ\Phi]|^{2}~{}dx^{\prime}dy^{\prime}$
$\displaystyle\leq\int_{B(0,r_{1})\cap\\{x^{\prime}>0\\}}|\det\nabla\Phi||D_{h,\mathbf{e}_{2}^{\prime}}^{s}[(\Delta(\phi
w))\circ\Phi]|^{2}~{}dx^{\prime}dy^{\prime}\leq C$

and similarly in the direction $\mathbf{e}_{1}^{\prime}$

$\displaystyle\int_{B(0,r_{1})\cap\\{x^{\prime}>0\\}}|D_{h,\mathbf{e}_{1}^{\prime}}^{s}[(\Delta(\phi
w))\circ\Phi]|^{2}~{}dx^{\prime}dy^{\prime}\leq C.$

Thus, we conclude

$\displaystyle||\Delta(\phi
w)||_{L^{2}(0,T;H^{s}(B(X_{0},r_{2})\cap\Gamma))}\leq C||[\Delta(\phi
w)]\circ\Phi||_{L^{2}(0,T;H^{s}(B(0,r_{1})\cap\\{x^{\prime}>0\\}))}\leq C$

because $\Phi$ is a $C^{0,1}$ bijective map. Since $X_{0}$ was an arbitrary
point and boundary $\partial\Gamma$ can be covered by a finite number of balls
$B(X_{i},\frac{r_{1}}{2})$ with non-empty intersection such that
$X_{i}\in\partial\Gamma$ and $B(X_{i},r_{1})\cap\partial\Gamma$ is a sub-graph
of a $C^{0,1}$ function, the proof is finished.

###  3.2 Part 2: relative energy inequality - Theorem $\ref{main2}(2)$

We start by choosing $(\mathbf{q},\psi)=(\mathbf{u}_{2},\partial_{t}w_{2})$ in
the coupled momentum equation $\eqref{weaksolmom}$ on555Note that the coupled
momentum equation $\eqref{weaksolmom}$ and the continuity equation
$\eqref{cont}$ hold on $(0,t)$, for any $t\in(0,T]$, due to the weak
continuity in time of $\rho_{1}$ and $\rho_{1}\mathbf{u}_{1}$ and
$\partial_{t}w_{1}$. $(0,t)$ to obtain

$\displaystyle\int_{\Omega^{w_{1}}(t)}(\rho_{1}\mathbf{u}_{1}\cdot\mathbf{u}_{2})(t)+\int_{\Gamma}(\partial_{t}w_{1}\partial_{t}w_{2})(t)$
$\displaystyle=\int_{\Omega^{w_{1}}(0)}(\rho_{1}\mathbf{u}_{1}\cdot\mathbf{u}_{2})(0)+\int_{\Gamma}(\partial_{t}w_{1}\partial_{t}w_{2})(0)+\int_{Q_{t}^{w_{1}}}\rho_{1}\mathbf{u}_{1}\cdot\partial_{t}\mathbf{u}_{2}$
$\displaystyle\quad+\int_{Q_{t}^{w_{1}}}(\rho_{1}\mathbf{u}_{1}\otimes\mathbf{u}_{1}):\nabla\mathbf{u}_{2}+\int_{Q_{t}^{w_{1}}}\rho_{1}^{\gamma}(\nabla\cdot\mathbf{u}_{2})-\underbrace{\int_{Q_{t}^{w_{1}}}\mathbb{S}(\nabla\mathbf{u}_{1}):\nabla\mathbf{u}_{2}}_{=\int_{Q_{t}^{w_{1}}}\mathbb{S}(\nabla\mathbf{u}_{2}):\nabla\mathbf{u}_{1}}$
$\displaystyle\quad+\int_{\Gamma_{t}}\partial_{t}w_{1}\partial_{t}^{2}w_{2}-\int_{\Gamma_{t}}\Delta
w_{1}\Delta\partial_{t}w_{2}-\alpha\displaystyle\int_{\Gamma_{t}}\partial_{t}\nabla
w_{1}\cdot\partial_{t}\nabla w_{2}.$ (15)

Next, in the continuity equation $\eqref{cont}$ on $(0,t)$, choose
$\varphi=\frac{1}{2}|\mathbf{u}_{2}|^{2}$ to obtain

$\displaystyle\frac{1}{2}\int_{\Omega^{w_{1}}(t)}(\rho_{1}|\mathbf{u}_{2}|^{2})(t)=\frac{1}{2}\int_{\Omega^{w_{1}}(0)}(\rho_{1}|\mathbf{u}_{2}|^{2})(0)+\int_{Q_{t}^{w_{1}}}\rho_{1}\partial_{t}\mathbf{u}_{2}\cdot\mathbf{u}_{2}+\int_{Q_{t}^{w_{1}}}\rho_{1}(\mathbf{u}_{1}\cdot\nabla\mathbf{u}_{2})\cdot\mathbf{u}_{2},\quad\quad$
(16)

and then choose $\varphi=\frac{\gamma}{\gamma-1}\rho_{2}^{\gamma-1}$

$\displaystyle\frac{\gamma}{\gamma-1}\int_{\Omega^{w_{1}}(t)}(\rho_{1}\rho_{2}^{\gamma-1})(t)$
$\displaystyle=\frac{\gamma}{\gamma-1}\int_{\Omega^{w_{1}}(0)}(\rho_{1}\rho_{2}^{\gamma-1})(0)+\frac{\gamma}{\gamma-1}\int_{Q_{t}^{w_{1}}}\rho_{1}\partial_{t}(\rho_{2}^{\gamma-1})+\frac{\gamma}{\gamma-1}\int_{Q_{t}^{w_{1}}}\rho_{1}\mathbf{u}_{1}\cdot\nabla(\rho_{2}^{\gamma-1}).$
(17)

Now, by Theorem $\ref{main2}(1)$, one has that $w\in
L^{2}(0,T;C^{0,1}(\Gamma))$ which means that $\Omega^{w}(t)$ is Lipschitz for
a.a. $t\in(0,T)$. This allows us to use the divergence theorem to obtain

$\displaystyle\frac{\gamma}{\gamma-1}\int_{Q_{t}^{w_{1}}}\rho_{2}\mathbf{u}_{2}\cdot\nabla(\rho_{2}^{\gamma-1})=\int_{Q_{t}^{w_{1}}}\mathbf{u}_{2}\cdot\nabla(\rho_{2}^{\gamma})=-\int_{Q_{t}^{w_{1}}}\rho_{2}^{\gamma}(\nabla\cdot\mathbf{u}_{2})+\int_{\Gamma_{t}^{w_{1}}}\rho_{2}^{\gamma}\partial_{t}w_{2}\nu^{w_{1}}\cdot\mathbf{e}_{3},$

and the Raynolds transport theorem to obtain

$\displaystyle\int_{0}^{t}\frac{d}{dt}\int_{\Omega^{w_{1}}(t)}\rho_{2}^{\gamma}=\frac{\gamma}{\gamma-1}\int_{Q_{t}^{w_{1}}}\rho_{2}\partial_{t}(\rho_{2}^{\gamma-1})+\int_{\Gamma_{t}^{w_{1}}}\rho_{2}^{\gamma}\partial_{t}w_{1}\nu^{w^{1}}\cdot\mathbf{e}_{3},$

which together imply

$\displaystyle\int_{\Omega^{w_{1}}(t)}\rho_{2}^{\gamma}(t)=\int_{\Omega^{w_{1}}(0)}\rho_{2}^{\gamma}(0)+\int_{\Gamma_{t}^{w_{1}}}\rho_{2}^{\gamma}(\partial_{t}w_{1}-\partial_{t}w_{2})\nu^{w_{1}}\cdot\mathbf{e}_{3}$
$\displaystyle+\frac{\gamma}{\gamma-1}\int_{Q_{t}^{w_{1}}}\Big{[}\rho_{2}\mathbf{U}_{2}\cdot\nabla(\rho_{2}^{\gamma-1})+\rho_{2}\partial_{t}(\rho_{2}^{\gamma-1})\Big{]}+\int_{Q_{t}^{w_{1}}}\rho_{2}^{\gamma}(\nabla\cdot\mathbf{u}_{2}).$
(18)

Finally, denoting $a\pm b=a+b-b$, the last three terms on the RHS of
$\eqref{ws1}$ can be transformed as

$\displaystyle\int_{\Gamma_{t}}\Delta w_{1}\partial_{t}\Delta
w_{2}\pm\frac{1}{2}\Big{(}\int_{\Gamma}|\Delta
w_{2}|^{2}\Big{)}\Big{|}_{0}^{t}=\int_{\Gamma_{t}}w_{1}\partial_{t}\Delta^{2}w_{2}-\frac{1}{2}\int_{0}^{t}\frac{d}{dt}\int_{\Gamma}|\Delta
w_{2}|^{2}+\frac{1}{2}\Big{(}\int_{\Gamma}|\Delta
w_{2}|^{2}\Big{)}\Big{|}_{0}^{t}$
$\displaystyle=-\int_{\Gamma_{t}}\partial_{t}w_{1}\Delta^{2}w_{2}+\Big{(}\int_{\Gamma}w_{1}\Delta^{2}w_{2}\Big{)}\Big{|}_{0}^{t}-\int_{\Gamma_{t}}\partial_{t}\Delta
w_{2}\Delta w_{2}+\frac{1}{2}\Big{(}\int_{\Gamma}|\Delta
w_{2}|^{2}\Big{)}\Big{|}_{0}^{t}$
$\displaystyle=\int_{\Gamma_{t}}(\partial_{t}w_{2}-\partial_{t}w_{1})\Delta^{2}w_{2}+\Big{(}\int_{\Gamma}\Delta
w_{1}\Delta w_{2}\Big{)}\Big{|}_{0}^{t}+\frac{1}{2}\Big{(}\int_{\Gamma}|\Delta
w_{2}|^{2}\Big{)}\Big{|}_{0}^{t},$

and

$\displaystyle-\int_{\Gamma_{t}}\partial_{t}w\partial_{t}^{2}w_{2}\pm\Big{(}\frac{1}{2}\int_{\Gamma}|\partial_{t}w_{2}|^{2}\Big{)}\Big{|}_{0}^{t}=\int_{\Gamma_{t}}(\partial_{t}w_{2}-\partial_{t}w_{1})\partial_{t}^{2}w_{2}+\Big{(}\frac{1}{2}\int_{\Gamma}|\partial_{t}w_{2}|^{2}\Big{)}\Big{|}_{0}^{t}~{},$
$\displaystyle-\displaystyle\int_{\Gamma_{t}}\partial_{t}\nabla
w_{1}\cdot\partial_{t}\nabla w_{2}\pm\int_{\Gamma_{t}}|\partial_{t}\nabla
w_{1}|^{2}=\int_{\Gamma}\partial_{t}\Delta
w_{2}(\partial_{t}w_{1}-\partial_{t}w_{2})+\int_{\Gamma_{t}}|\partial_{t}\nabla
w_{1}|^{2}~{},$

so we sum $\eqref{energy}-\eqref{ws1}+\eqref{ws2}-\eqref{ws3}-\eqref{ws5}$,
which then gives us $\eqref{relen}$.

##  4 Proof of Theorem 2.3

When working with free boundary problems, it is common to transform the
coordinate system in order to define the problem on a fixed domain. Also, in
this proof, we will be working with two solutions which are in general defined
on different domains, so transforming them onto a common domain will allow us
to compare them.

###  4.1 Part 1: change of coordinates

We start by introducing the fixed domain

$\displaystyle\Omega:=\Gamma\times(-1,0),$

and the family of so called arbitrary Lagrangian-Eulerian (ALE) mappings

$\displaystyle A_{w}(t):$ $\displaystyle\Omega\to\Omega^{w}(t)$ (19)
$\displaystyle(X,z)\mapsto(X,(z+1)w(t,X)+z).$

For an arbitrary (vector or scalar) function $\mathbf{f}$ defined on
$\Omega^{w}(t)$, corresponding to $A_{w}$, we denote the pull-back,
transformed gradient and transformed divergence of $\mathbf{f}$ on $\Omega$,
respectively, as

$\displaystyle\mathbf{f}^{w}:=\mathbf{f}\circ A_{w},$
$\displaystyle\nabla^{w}\mathbf{f}^{w}:=(\nabla\mathbf{f})^{w}=\nabla\mathbf{f}^{w}(\nabla
A_{w})^{-1}\circ A_{w},$
$\displaystyle\nabla^{w}\cdot\mathbf{f}^{w}:=\text{Tr}(\nabla^{w}\mathbf{f}^{w}).$

where

$\displaystyle(\nabla A_{w}^{-1})\circ A_{w}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\
-\frac{\displaystyle z+1}{\displaystyle w+1}\partial_{x}w&-\frac{\displaystyle
z+1}{\displaystyle w+1}\partial_{y}w&\frac{\displaystyle 1}{\displaystyle
w+1}\end{bmatrix}.$

The Jacobian and the ALE velocity of the mapping $A_{w}$, respectively, read

$\displaystyle J$ $\displaystyle:=$ $\displaystyle\text{det}\nabla A_{w}=1+w,$
$\displaystyle\mathbf{w}$ $\displaystyle:=$
$\displaystyle\frac{d}{dt}A_{w}=(z+1)\partial_{t}w\mathbf{e}_{3},$

and the time-space cylinder is denoted as

$\displaystyle Q_{T}:=(0,T)\times\Omega.$

Since we will work with pairs of solutions, the following notation will be
useful. For $i=1,2$ and a given (scalar or vector function) $\mathbf{f}$,
denote

$\displaystyle\nabla_{i}\mathbf{f}:=\nabla^{w_{i}}\mathbf{f},\quad\nabla_{i}\cdot\mathbf{f}:=\nabla^{w_{i}}\cdot\mathbf{f},\quad
A_{i}:=A_{w_{i}},\quad S^{i}:=S^{w^{i}},\quad\nu^{i}:=\nu^{w^{i}},$

and the pull-backs

$\displaystyle r_{i}$ $\displaystyle:=$
$\displaystyle\rho_{i}^{w_{i}}=\rho_{i}\circ A_{i},$
$\displaystyle\mathbf{U}_{i}$ $\displaystyle:=$
$\displaystyle\mathbf{u}_{i}^{w_{i}}=\mathbf{u}_{i}\circ A_{i}.$

Finally, the subscript for the Jacobian will be omitted

$\displaystyle J:=J_{1}=1+w_{1},$

since it will appear frequently.

####  4.1.1 The relative entropy and relative energy inequality on the fixed
reference domain

Let $(\rho_{1},\mathbf{u}_{1},w_{1})$ be a finite energy weak solution in the
sense of Definition 2.1 and let $r_{2},\mathbf{U}_{2}\in
C^{\infty}([0,T]\times\overline{\Omega})$ and $w_{2}\in
C_{0}^{\infty}(\Gamma_{T})$ such that ${\mathbf{U}_{2}}_{|\Sigma}=0$ and
${\mathbf{U}_{2}}_{|\Gamma\times\\{0\\}}=\partial_{t}w_{2}\mathbf{e}_{3}$.
For666Here, the weak solution on the fixed reference domain is not defined
explicitly since it will not be required in the proof, but this definition can
be found in [38, section 2.3]. $(r_{1},\mathbf{U}_{1},w_{1})=(\rho_{1}\circ
A_{1},\mathbf{u}_{1}\circ A_{1},w_{1})$, the relative entropy on the fixed
reference domain with respect to $(r_{2},\mathbf{U}_{2},w_{2})$ will be also
denoted by $\mathcal{E}$ and it reads as

$\displaystyle\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}(t)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}(Jr_{1}|\mathbf{U}_{1}-\mathbf{U}_{2}|^{2})(t)+\frac{1}{\gamma-1}\int_{\Omega}J\big{(}r_{1}^{\gamma}-\gamma
r_{2}^{\gamma-1}(r_{1}-r_{2})-r_{2}^{\gamma}\big{)}(t)$
$\displaystyle\quad+\frac{1}{2}\int_{\Gamma}|\partial_{t}w_{1}-\partial_{t}w_{2}|^{2}(t)+\frac{1}{2}\int_{\Gamma}|\Delta
w_{1}-\Delta w_{2}|^{2}(t).$

By using Theorem $\ref{main2}(2)$ and the identities

$\displaystyle\int_{\Gamma^{w_{1}}(t)}(\partial_{t}w_{2}-\partial_{t}w_{1})\rho_{2}^{\gamma}I\nu^{1}\cdot\mathbf{e}_{3}=\int_{\Gamma}(\partial_{t}w_{2}-\partial_{t}w_{1})r_{2}^{\gamma}\underbrace{S_{1}I\nu^{1}\cdot\mathbf{e}_{3}}_{=1},$
$\displaystyle\frac{\displaystyle d}{\displaystyle dt}(\mathbf{q}\circ
A_{i})=(\partial_{t}\mathbf{q})\circ
A_{i}+\mathbf{w}_{i}\cdot\nabla_{i}(\mathbf{q}\circ A_{i}),\quad\text{
on}\quad\Omega,$ (20)

where $\mathbf{q}$ is a sufficiently regular function defined on
$Q_{T}^{w_{i}}$, one can easily see that the triplet
$(r_{1},\mathbf{U}_{1},w_{1})$ satisfies the relative energy inequality on the
fixed reference domain with respect to $(r_{2},\mathbf{U}_{2},w_{2})$ of the
form

$\displaystyle\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}(t)+\int_{Q_{t}}J\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2}):(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})+\alpha\int_{0}^{t}\int_{\Gamma}|\partial_{t}\nabla
w_{1}-\partial_{t}\nabla w_{2}|^{2}$
$\displaystyle\leq\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}(0)+\int_{0}^{t}\mathcal{R}\Big{(}r_{1},\mathbf{U}_{1},w,r_{2},\mathbf{U}_{2},w_{2}\Big{)}$
(21)

where the remainder term reads

$\displaystyle\mathcal{R}\Big{(}r_{1},\mathbf{U}_{1},w_{1},r_{2},\mathbf{U}_{2},w_{2}\Big{)}:=\int_{\Omega}J\mathbb{S}(\nabla_{1}\mathbf{U}_{2}):(\nabla_{1}\mathbf{U}_{2}-\nabla_{1}\mathbf{U}_{1})+\int_{\Omega}Jr_{1}(\partial_{t}\mathbf{U}_{2}+\mathbf{U}_{1}\cdot\nabla_{1}\mathbf{U}_{2})\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\quad-\int_{\Omega}Jr_{1}\big{[}\mathbf{w}_{1}\cdot\nabla_{1}\mathbf{U}_{2}\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})+\frac{\gamma}{\gamma-1}\Big{[}\int_{\Omega}J(r_{2}\mathbf{U}_{2}-r_{1}\mathbf{U}_{1})\cdot\nabla_{1}(r_{2}^{\gamma-1})$
$\displaystyle\quad+J(r_{1}-r_{2})\big{(}\partial_{t}(r_{2}^{\gamma-1})-\mathbf{w}_{1}\cdot\nabla_{1}(r_{2}^{\gamma-1})\big{)}\Big{]}+\int_{\Omega}J(r_{2}^{\gamma}-r_{1}^{\gamma})(\nabla_{1}\cdot\mathbf{U}_{2})$
$\displaystyle\quad+\int_{\Gamma}(\partial_{t}w_{2}-\partial_{t}w_{1})r_{2}^{\gamma}-\int_{\Gamma}(\partial_{t}w_{2}-\partial_{t}w_{1})\partial_{t}^{2}w_{2}-\int_{\Gamma}(\partial_{t}w_{2}-\partial_{t}w_{1})\Delta^{2}w_{2}+\alpha\int_{\Gamma}\partial_{t}\Delta
w_{2}(\partial_{t}w_{1}-\partial_{t}w_{2}).$ (22)

###  4.2 Part 2: carrying out the proof on the fixed reference domain

Here, we compare (see figure 1) a finite energy weak solution
$(\rho_{1},\mathbf{u}_{1},w_{1})$ and a strong solution
$(\rho_{2},\mathbf{u}_{2},w_{2})$ belonging to the regularity class specified
in Theorem 2.3, which emanate from the same initial data.

Figure 1: The weak and the strong solution on both physical and fixed
reference domain coordinates.

The goal is to prove that

$\displaystyle\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}(t)\leq
C\int_{0}^{t}h(\tau)\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}(\tau)d\tau,\quad
h\in L^{1}(0,T),$

for a.a. $t\in(0,T_{2})$, then the desired result will follow immediately by
Gronwall’s lemma.

Step 1: transforming the remainder term
$\mathcal{R}\Big{(}r_{1},\mathbf{U}_{1},w_{1},r_{2},\mathbf{U}_{2},w_{2}\Big{)}$

First, one can see that $(r_{2},\mathbf{U}_{2},w_{2})$ satisfies the following
transformed problem in the strong sense

$\displaystyle\begin{cases}\partial_{t}^{2}w_{2}+\Delta^{2}w_{2}-\alpha\partial_{t}\Delta
w_{2}=-S^{2}\big{[}(\mathbb{S}(\nabla_{2}\mathbf{U}_{2})-r_{2}^{\gamma}I)\nu^{2}\big{]}\cdot\mathbf{e}_{3},\quad\text{
on }~{}\Gamma_{T_{2}},\\\\[7.11317pt]
\partial_{t}r_{2}-\mathbf{w}_{2}\cdot\nabla_{2}r_{2}+\nabla_{2}\cdot(r_{2}\mathbf{U}_{2})=0,\quad\text{
in }~{}Q_{T_{2}},\\\\[7.11317pt]
\partial_{t}\mathbf{U}_{2}-\mathbf{w}_{2}\cdot\nabla_{2}\mathbf{U}_{2}+\mathbf{U}_{2}\cdot\nabla_{2}\mathbf{U}_{2}=-\frac{1}{r_{2}}\nabla_{2}(r_{2}^{\gamma})+\frac{1}{r_{2}}\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2}),\quad\text{
in }~{}Q_{T_{2}},\end{cases}$ (23)

by using $\eqref{aleid}$, and that

$\displaystyle r_{2}\in L^{2}(0,T_{2};W^{1,q}(\Omega))\cap
H^{1}(0,T_{2};L^{q}(\Omega)),$ $\displaystyle\mathbf{U}_{2}\in
L^{2}(0,T_{2};W^{2,q}(\Omega))\cap H^{1}(0,T_{2};L^{q}(\Omega)),$
$\displaystyle w_{2}\in L^{2}(0,T_{2};H^{4}(\Gamma))\cap
H^{2}(0,T_{2};L^{2}(\Gamma)),$ $\displaystyle
0<\inf\limits_{Q_{T_{2}}}r_{2}\leq\sup\limits_{Q_{T_{2}}}r_{2}<\infty,$

with $q=q(d,\alpha,\gamma)$ being specified in Theorem 2.3. Using this
regularity, we test $\eqref{relenfixed}$ with $(r_{2},\mathbf{U}_{2},w_{2})$
(by the density argument), and then transform the remainder term given in
$\eqref{caprfixed}$ by $\eqref{strongfixed}$, as follows.

The 2nd term on RHS of $\eqref{caprfixed}$:

$\displaystyle\int_{Q_{t}}Jr_{1}(\partial_{t}\mathbf{U}_{2}+\mathbf{U}_{1}\cdot\nabla_{1}\mathbf{U}_{2})(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle=\int_{Q_{t}}Jr_{1}(\partial_{t}\mathbf{U}_{2}+\mathbf{U}_{2}\cdot\nabla_{2}\mathbf{U}_{2})(\mathbf{U}_{2}-\mathbf{U}_{1})+\int_{Q_{t}}Jr_{1}\big{[}\mathbf{U}_{2}\cdot(\nabla_{1}-\nabla_{2})\mathbf{U}_{2}\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\quad+\int_{Q_{t}}Jr_{1}\big{[}(\mathbf{U}_{1}-\mathbf{U}_{2})\cdot\nabla_{1}\mathbf{U}_{2}\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})\pm\int_{Q_{t}}Jr_{1}(\mathbf{w}_{2}\cdot\nabla_{2}\mathbf{U}_{2})(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\stackrel{{\scriptstyle\eqref{strongfixed}_{3}}}{{=}}-\int_{Q_{t}}J\frac{r_{1}}{r_{2}}\nabla_{2}(r_{2}^{\gamma})(\mathbf{U}_{2}-\mathbf{U}_{1})+\int_{Q_{t}}J\frac{1}{r_{2}}\big{[}\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\quad+\int_{Q_{t}}Jr_{1}\big{[}\mathbf{U}_{2}\cdot(\nabla_{1}-\nabla_{2})\mathbf{U}_{2}\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})+\int_{Q_{t}}Jr_{1}\big{[}(\mathbf{U}_{1}-\mathbf{U}_{2})\cdot\nabla_{1}\mathbf{U}_{2}\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\quad+\int_{Q_{t}}Jr_{1}(\mathbf{w}_{2}\cdot\nabla_{2}\mathbf{U}_{2})(\mathbf{U}_{2}-\mathbf{U}_{1}).$
(24)

The 2nd term on RHS of $\eqref{bound1}$:

$\displaystyle\int_{Q_{t}}J\frac{1}{r_{2}}r_{1}\big{[}\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle=\int_{Q_{t}}J\frac{1}{r_{2}}(r_{1}-r_{2})\big{[}\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})+\int_{Q_{t}}(J-J_{2})\big{[}\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\quad+\int_{Q_{t}}J_{2}\big{[}\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1}),$
(25)

where the last term can be transformed by partial integration on the physical
domain

$\displaystyle\int_{Q_{t}}J_{2}\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})(\mathbf{U}_{2}-\mathbf{U}_{1})=\int_{0}^{t}\int_{\Omega^{w_{2}}(t)}\nabla\cdot\mathbb{S}(\nabla\mathbf{u}_{2})(\mathbf{u}_{2}-\mathbf{U}_{1}\circ
A_{2}^{-1})$
$\displaystyle=-\int_{0}^{t}\int_{\Omega^{w_{2}}(t)}\mathbb{S}(\nabla\mathbf{U}_{2}):(\nabla\mathbf{u}_{2}-\nabla\mathbf{U}_{1}\circ
A_{2}^{-1})+\int_{0}^{t}\int_{\Gamma^{w_{2}}(t)}\big{[}(\mathbb{S}(\nabla\mathbf{u}_{2}))\nu^{2}\big{]}\cdot\mathbf{e}_{3}(\partial_{t}w_{2}-\partial_{t}w_{1})$
$\displaystyle=-\int_{Q_{t}}J_{2}\mathbb{S}(\nabla_{2}\mathbf{U}_{2}):(\nabla_{2}\mathbf{U}_{2}-\nabla_{2}\mathbf{U}_{1})+\int_{\Gamma_{t}}S_{2}\big{[}(\mathbb{S}(\nabla_{2}\mathbf{U}_{2}))\nu^{2}\big{]}\cdot\mathbf{e}_{3}(\partial_{t}w_{2}-\partial_{t}w_{1}).$
(26)

The 4th term on RHS of $\eqref{caprfixed}$ and the 1st term on RHS of
$\eqref{bound1}$:

$\displaystyle\frac{\gamma}{\gamma-1}\Big{[}\int_{Q_{t}}J(r_{2}\mathbf{U}_{2}-r_{1}\mathbf{U}_{1})\cdot\nabla_{1}(r_{2}^{\gamma-1})+J(r_{1}-r_{2})\big{[}\partial_{t}(r_{2}^{\gamma-1})-\mathbf{w}_{1}\cdot\nabla_{1}(r_{2}^{\gamma-1})\big{]}\Big{]}$
$\displaystyle-\int_{Q_{t}}\underbrace{J\frac{r_{1}}{r_{2}}\nabla_{2}(r_{2}^{\gamma})}_{=\frac{\gamma}{\gamma-1}Jr_{1}\nabla_{2}(r_{2}^{\gamma-1})}(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle=\frac{\gamma}{\gamma-1}\int_{Q_{t}}J(r_{2}-r_{1})\Big{[}\mathbf{U}_{2}\cdot\nabla_{1}(r_{2}^{\gamma-1})+\partial_{t}(r_{2}^{\gamma-1})-\mathbf{w}_{1}\cdot\nabla_{1}(r_{2}^{\gamma-1})\Big{]}$
$\displaystyle\quad+\int_{Q_{t}}r_{1}(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})(\mathbf{U}_{2}-\mathbf{U}_{1})\pm\frac{\gamma}{\gamma-1}\int_{Q_{t}}J(r_{2}-r_{1})\Big{[}\mathbf{U}_{2}\cdot\nabla_{2}(r_{2}^{\gamma-1})-\mathbf{w}_{2}\cdot\nabla_{2}(r_{2}^{\gamma-1})\Big{]}$
$\displaystyle=\frac{\gamma}{\gamma-1}\int_{Q_{t}}J(r_{2}-r_{1})\Big{[}\underbrace{\partial_{t}(r_{2}^{\gamma-1})+\mathbf{U}_{2}\cdot\nabla_{2}(r_{2}^{\gamma-1})-\mathbf{w}_{2}\cdot\nabla_{1}(r_{2}^{\gamma-1})}_{=-(\gamma-1)(\nabla_{2}\cdot\mathbf{U}_{2})r_{2}^{\gamma-1},~{}~{}\text{by
}\eqref{strongfixed}_{2}}\Big{]}$
$\displaystyle\quad+\frac{\gamma}{\gamma-1}\int_{Q_{t}}J(r_{2}-r_{1})\Big{[}\mathbf{U}_{2}\cdot(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})-(\mathbf{w}_{1}-\mathbf{w}_{2})\cdot\nabla_{1}(r_{2}^{\gamma-1})-\mathbf{w}_{2}\cdot(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})\Big{]}$
$\displaystyle\quad+\int_{Q_{t}}Jr_{1}(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})\cdot(\mathbf{U}_{2}-\mathbf{U}_{1}).$
(27)

The 5th term on RHS of $\eqref{caprfixed}$ and the 1st term on RHS of
$\eqref{bound3}$:

$\displaystyle\int_{Q_{t}}J(r_{2}^{\gamma}-r_{1}^{\gamma})(\nabla_{1}\cdot\mathbf{U}_{2})-\gamma\int_{Q_{t}}J(r_{2}-r_{1})(\nabla_{2}\cdot\mathbf{U}_{2})r_{2}^{\gamma-1}$
$\displaystyle=\int_{Q_{t}}J\Big{[}r_{2}^{\gamma}-\gamma
r_{2}^{\gamma-1}(r_{2}-r_{1})-r_{1}^{\gamma}\Big{]}(\nabla_{1}\cdot\mathbf{U}_{2})+\gamma\int_{Q_{t}}J(r_{2}-r_{1})r_{2}^{\gamma-1}(\nabla_{1}-\nabla_{2})\cdot\mathbf{U}_{2}.$
(28)

The plate equation
$\eqref{strongfixed}_{1}\times(\partial_{t}w_{2}-\partial_{t}w_{1})$:

$\displaystyle\int_{\Gamma_{t}}(\partial_{t}w_{2}-\partial_{t}w_{1})\partial_{t}^{2}w_{2}+\int_{\Gamma_{t}}(\partial_{t}w_{2}-\partial_{t}w_{1})\Delta^{2}w_{2}-\alpha\int_{\Gamma_{t}}\partial_{t}\Delta
w_{2}(\partial_{t}w_{2}-\partial_{t}w_{1})$
$\displaystyle=\int_{\Gamma_{t}}(\partial_{t}w_{2}-\partial_{t}w_{1})r_{2}^{\gamma}\underbrace{S^{2}I\nu^{2}\cdot\mathbf{e}_{3}}_{=1}-\int_{\Gamma_{t}}(\partial_{t}w_{2}-\partial_{t}w_{1})S_{2}\big{[}(\mathbb{S}(\nabla_{2}\mathbf{U}_{2}))\nu^{2}\big{]}\cdot\mathbf{e}_{3}.\quad$
(29)

Now, the remainder term given in $\eqref{caprfixed}$ is transformed by using
the identities
$\eqref{bound1},\eqref{bound2},\eqref{partial},\eqref{bound3}$,$\eqref{bound5}$
and $\eqref{bound4}$ (in that order), which then by $\eqref{relenfixed}$ gives
us:

$\displaystyle\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}(t)+\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2}$
$\displaystyle\leq\int_{Q_{t}}J\mathbb{S}(\nabla_{1}\mathbf{U}_{2}):(\nabla_{1}\mathbf{U}_{2}-\nabla_{1}\mathbf{U}_{1})-\int_{Q_{t}}J_{2}\mathbb{S}(\nabla_{2}\mathbf{U}_{2}):(\nabla_{2}\mathbf{U}_{2}-\nabla_{2}\mathbf{U}_{1})$
$\displaystyle\quad\int_{Q_{t}}J\frac{1}{r_{2}}(r_{1}-r_{2})\big{[}\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})+\int_{Q_{t}}(J-J_{2})\big{[}\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\quad+\frac{\gamma}{\gamma-1}\int_{Q_{t}}J(r_{2}-r_{1})\Big{[}\mathbf{U}_{2}\cdot(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})-(\mathbf{w}_{1}-\mathbf{w}_{2})\cdot\nabla_{1}(r_{2}^{\gamma-1})-\mathbf{w}_{2}\cdot(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})\Big{]}$
$\displaystyle\quad+\int_{Q_{t}}Jr_{1}\big{[}\mathbf{U}_{2}\cdot(\nabla_{1}-\nabla_{2})\mathbf{U}_{2}\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})+\int_{Q_{t}}Jr_{1}\big{[}(\mathbf{U}_{1}-\mathbf{U}_{2})\cdot\nabla_{1}\mathbf{U}_{2}\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\quad+\int_{Q_{t}}Jr_{1}(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})+\gamma\int_{Q_{t}}J(r_{2}-r_{1})r_{2}^{\gamma-1}(\nabla_{1}-\nabla_{2})\cdot\mathbf{U}_{2}$
$\displaystyle\quad+\int_{Q_{t}}J\big{[}r_{2}^{\gamma}-\gamma
r_{2}^{\gamma-1}(r_{2}-r_{1})-r_{1}^{\gamma}\big{]}(\nabla_{1}\cdot\mathbf{U}_{2})+\int_{Q_{t}}Jr_{1}\Big{[}\mathbf{w}_{1}\cdot\nabla_{1}\mathbf{U}_{2}-\mathbf{w}_{2}\cdot\nabla_{2}\mathbf{U}_{2}\Big{]}\cdot(\mathbf{U}_{1}-\mathbf{U}_{2}).$
(30)

The aim is to bound the right-hand side by

$\displaystyle
C(\delta)\int_{0}^{t}h\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2},$
(31)

for some $h\in L^{1}(0,T)$ and a small $\delta$. This will be done in step 3.

Step 2: studying the difference $(r_{1}-r_{2})$

Denote

$\displaystyle f(x,y):=x^{\gamma}-\gamma y^{\gamma-1}(x-y)-y^{\gamma}.\quad
x,y\geq 0.$

It is easy to see that for any $C>0$, $f(x,C)$ is convex in $x$ and
$f(C,C)=\partial_{x}f(C,C)=0$. This means that for a fixed $C>0$, one can
choose $c=c(C)>0$ so that

$\displaystyle f(x,C)\geq c|x-C|^{2\land\gamma},\quad
x\in\Big{[}\frac{C}{2},2C\Big{]}.$

where $2\land\gamma=\min\\{2,\gamma\\}$. Since
$0<c_{r_{2}}=\inf\limits_{Q_{T_{2}}}r_{2}\leq\sup\limits_{Q_{T_{2}}}r_{2}=C_{r_{2}}$,
there is a constant $c=c(c_{r_{2}},C_{r_{2}},\gamma)$ for which

$\displaystyle f(x,r_{2})$ $\displaystyle\geq$ $\displaystyle
c|x-r_{2}|^{2\land\gamma},\quad\text{for }\frac{r_{2}}{2}\leq x\leq 2r_{2},$
(32) $\displaystyle f(x,r_{2})$ $\displaystyle\geq$ $\displaystyle
c(1+x^{2\land\gamma}),\quad\text{for
}x\in\mathbb{R}_{0}^{+}\setminus\Big{[}\frac{r_{2}}{2},2r_{2}\Big{]}.$ (33)

Now, for $0\leq r_{1}(t)\leq\frac{r_{2}(t)}{2}$, one has

$\displaystyle||r_{1}(t)-r_{2}(t)||_{L^{2\land\gamma}\big{(}\big{\\{}0\leq
r_{1}(t)\leq\frac{r_{2}(t)}{2}\big{\\}}\big{)}}^{2}\leq\int_{\big{\\{}0\leq
r_{1}(t)\leq\frac{r_{2}(t)}{2}\big{\\}}}c$
$\displaystyle\leq\int_{\big{\\{}0\leq
r_{1}(t)\leq\frac{r_{2}(t)}{2}\big{\\}}}c(1+r_{1}^{\gamma}(t))\leq
C\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})|(r_{2},\mathbf{U}_{2},w_{2})\Big{)}(t),$

from $\eqref{f2}$, then for $\frac{r_{2}(t)}{2}\leq r_{1}(t)\leq 2r_{2}(t)$

$\displaystyle||r_{1}(t)-r_{2}(t)||_{L^{2\land\gamma}\big{(}\big{\\{}\frac{r_{2}(t)}{2}\leq
r_{1}(t)\leq 2r_{2}(t)\big{\\}}\big{)}}^{2}\leq
c||r_{1}(t)-r_{2}(t)||_{L^{2\land\gamma}\big{(}\big{\\{}\frac{r_{2}(t)}{2}\leq
r_{1}(t)\leq 2r_{2}(t)\big{\\}}\big{)}}^{2\land\gamma}$ $\displaystyle\leq
c\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}(t),$

from $\eqref{f1}$, and finally for $r_{1}(t)\geq 2r_{2}(t)$

$\displaystyle||r_{1}(t)-r_{2}(t)||_{L^{2\land\gamma}(\\{r_{1}(t)\geq
2r_{2}(t)\\})}^{2}=\Big{|}\Big{|}\Big{(}1-\frac{r_{2}(t)}{r_{1}(t)}\Big{)}r_{1}(t)\Big{|}\Big{|}_{L^{2\land\gamma}(\\{r_{1}(t)\geq
2r_{2}(t)\\})}^{2}\leq||r_{1}(t)||_{L^{2\land\gamma}(\\{r_{1}(t)\geq
2r_{2}(t)\\})}^{2}$ $\displaystyle\leq C\Big{(}\int_{\\{r_{1}(t)\geq
2r_{2}(t)\\}}1+r_{1}^{2\land\gamma}\Big{)}^{\frac{2}{2\land\gamma}}\leq
C\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}^{\frac{2}{2\land\gamma}}(t)$
$\displaystyle\leq
C\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}(t),$

from $\eqref{f2}$, since
$\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}(t)\leq
C$, so by combining previous three inequalities, one obtains

$\displaystyle||r_{1}(t)-r_{2}(t)||_{L^{2\land\gamma}(\Omega)}^{2}\leq
C\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}(t).$
(34)

Step 3: closing the estimates

Here, we will prove the right-hand side of inequality $\eqref{estimate}$ can
be controlled by $\eqref{provethis}$, as follows.

The 1st and the 2nd term on RHS of $\eqref{estimate}$:

$\displaystyle\int_{Q_{t}}J\mathbb{S}(\nabla_{1}\mathbf{U}_{2}):(\nabla_{1}\mathbf{U}_{2}-\nabla_{1}\mathbf{U}_{1})-\int_{Q_{t}}J_{2}\mathbb{S}(\nabla_{2}\mathbf{U}_{2}):(\nabla_{2}\mathbf{U}_{2}-\nabla_{2}\mathbf{U}_{1})$
$\displaystyle=\int_{Q_{t}}(J-J_{2})\mathbb{S}(\nabla_{1}\mathbf{U}_{2}):(\nabla_{1}\mathbf{U}_{2}-\nabla_{1}\mathbf{U}_{1})+\int_{Q_{t}}J_{2}\mathbb{S}((\nabla_{1}-\nabla_{2})\mathbf{U}_{2}):(\nabla_{1}\mathbf{U}_{2}-\nabla_{1}\mathbf{U}_{1})$
$\displaystyle\quad+\int_{Q_{t}}J_{2}\mathbb{S}(\nabla_{2}\mathbf{U}_{2}):(\nabla_{1}-\nabla_{2})(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\leq
C\int_{0}^{t}||w_{1}-w_{2}||_{L^{\infty}(\Gamma)}||\nabla\mathbf{U}_{2}||_{L^{p}(\Omega)}||\nabla
A_{1}||_{L^{\infty^{-}}(\Omega)}||\nabla\mathbf{U}_{1}-\nabla\mathbf{U}_{2}||_{L^{2^{-}}(\Omega)}||\nabla
A_{1}||_{L^{\infty^{-}}(\Omega)}$
$\displaystyle\quad+C\int_{0}^{t}||A_{1}-A_{2}||_{W^{1,\infty^{-}}(\Omega)}||\nabla\mathbf{U}_{2}||_{L^{p}(\Omega)}||\nabla
A_{1}||_{L^{\infty^{-}}(\Omega)}||\nabla\mathbf{U}_{1}-\nabla\mathbf{U}_{2}||_{L^{2^{-}}(\Omega)}||\nabla
A_{1}||_{L^{\infty^{-}}(\Omega)}$
$\displaystyle\quad+C\int_{0}^{t}||\nabla\mathbf{U}_{2}||_{L^{p}(\Omega)}||\nabla
A_{2}||_{L^{\infty}(\Omega)}||A_{1}-A_{2}||_{W^{1,\infty^{-}}(\Omega)}||\nabla\mathbf{U}_{1}-\nabla\mathbf{U}_{2}||_{L^{2^{-}}(\Omega)}$
$\displaystyle\leq C\int_{0}^{t}||\Delta w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}||\mathbf{U}_{1}-\mathbf{U}_{2}||_{W^{1,2^{-}}(\Omega)}$
$\displaystyle\leq
C(\delta)\int_{0}^{t}\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2}|^{2},$

for any $p\in(2,3)$, since $J,J_{2}\leq C$ and $\nabla\mathbf{U}_{2}\in
L^{\infty}(0,T;L^{p}(\Omega))$, where we also used

$\displaystyle||\mathbf{U}_{1}-\mathbf{U}_{2}||_{W^{1,2^{-}}(\Omega)}^{2}\leq
C||\nabla
A_{1}||_{L^{\infty^{-}}(\Omega)}\int_{\Omega}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2}$
$\displaystyle\leq C||\nabla
A_{1}||_{H^{1}(\Omega)}^{2}\int_{\Omega}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2}\leq
C\underbrace{||\Delta w_{1}||_{L^{2}(\Omega)}^{2}}_{\leq
C}\int_{\Omega}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2}$

and

$\displaystyle||(\nabla_{1}-\nabla_{2})f||_{L^{p^{-}}(\Omega)}\leq
C||A_{1}-A_{2}||_{W^{1,\infty^{-}}(\Omega)}||\nabla f||_{L^{p}(\Omega)}\leq
C||\Delta w_{1}-\Delta w_{2}||_{L^{2}(\Gamma)}||\nabla f||_{L^{p}(\Omega)},$

for any $f\in W^{1,p}(\Omega)$, by the imbedding of Sobolev spaces.

The 3rd term on RHS of $\eqref{estimate}$:

$\displaystyle\int_{Q_{t}}J\frac{1}{r_{2}}(r_{1}-r_{2})\big{[}\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\leq
C\int_{0}^{t}||r_{1}-r_{2}||_{L^{2\land\gamma}(\Omega)}\underbrace{||\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})||_{L^{q}(\Omega)}}_{:=h_{1}\in
L^{2}(0,T)}||\mathbf{U}_{2}-\mathbf{U}_{1}||_{L^{6^{-}}(\Omega)}$
$\displaystyle\leq
C(\delta)\int_{0}^{t}h_{1}^{2}||r_{1}-r_{2}||_{L^{2\land\gamma}(\Omega)}^{2}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2}$
$\displaystyle\leq
C(\delta)\int_{0}^{t}h_{1}^{2}\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2},$

for $q>\max\big{\\{}3,\frac{6\gamma}{5\gamma-6}\big{\\}}$, by using
$\eqref{twolandgamma}$ and the following inequalities

$\displaystyle||\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})||_{L^{q}(\Omega)}\leq
C\Big{(}||\nabla^{2}\mathbf{U}_{2}||_{L^{q}(\Omega)}||(\nabla
w_{2})^{2}||_{L^{\infty}(\Gamma)}+||\nabla\mathbf{U}_{2}||_{L^{\infty}(\Omega)}||\Delta
w_{2}||_{L^{\infty}(\Gamma)}\Big{)},$
$\displaystyle||\mathbf{U}_{1}-\mathbf{U}_{2}||_{L^{6^{-}}(\Omega)}^{2}\leq
C||\mathbf{U}_{1}-\mathbf{U}_{2}||_{W^{1,2^{-}}(\Omega)}^{2}\leq
C\underbrace{||\nabla A_{1}||_{L^{\infty^{-}}(\Omega)}^{2}}_{\leq
C}\int_{\Omega}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2}.$

The 4th term on RHS of $\eqref{estimate}$:

$\displaystyle\int_{Q_{t}}(J-J_{2})\nabla_{2}\cdot\mathbb{S}(\nabla_{2}\mathbf{U}_{2})(\mathbf{U}_{2}-\mathbf{U}_{1})\leq\int_{0}^{t}||w_{1}-w_{2}||_{L^{\infty}(\Gamma)}h_{1}||\mathbf{U}_{1}-\mathbf{U}_{2}||_{L^{6^{-}}(\Omega)}$
$\displaystyle\leq
C(\delta)\int_{0}^{t}h_{1}^{2}\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2}.$

The 5th term on RHS of $\eqref{estimate}$ \- case $\alpha=0,\gamma>12/7$:

$\displaystyle\int_{Q_{t}}J(r_{2}-r_{1})\Big{[}\mathbf{U}_{2}\cdot(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})-(\mathbf{w}_{1}-\mathbf{w}_{2})\cdot\nabla_{1}(r_{2}^{\gamma-1})-\mathbf{w}_{2}\cdot(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})\Big{]}$
$\displaystyle\leq
C\int_{0}^{t}||r_{1}-r_{2}||_{L^{2\land\gamma}(\Omega)}\Big{[}||\Delta
w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}\underbrace{||\mathbf{U}_{2}\cdot\nabla(r_{2}^{\gamma-1})||_{L^{q}(\Omega)}}_{:=h_{2}\in
L^{2}(0,T)}+||\partial_{t}w_{1}-\partial
w_{2}||_{L^{4^{-}}(\Gamma)}\underbrace{||\nabla(r_{2}^{\gamma-1})||_{L^{q}(\Omega)}}_{:=h_{3}\in
L^{2}(0,T)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad~{}+||\Delta
w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}\underbrace{||\mathbf{w}_{2}\cdot\nabla(r_{2}^{\gamma-1})||_{L^{q}(\Omega)}}_{:=h_{4}\in
L^{2}(0,T)}\Big{]}$ $\displaystyle\leq
C(\delta)\int_{0}^{t}(h_{2}^{2}+h_{3}^{2}+h_{4}^{2})||r_{1}-r_{2}||_{L^{2\land\gamma}(\Omega)}^{2}+C\int_{0}^{t}||\Delta
w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}^{2}+\delta\int_{0}^{t}||\partial_{t}w_{1}-\partial
w_{2}||_{H^{({\frac{1}{2}})^{-}}(\Gamma)}^{2}$ $\displaystyle\leq
C(\delta)\int_{0}^{t}(h_{2}^{2}+h_{3}^{2}+h_{4}^{2}+1)\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})|(r_{2},\mathbf{U}_{2},w_{2})\Big{)}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2},$

for any $q>\max\\{4,\frac{4\gamma}{3\gamma-4}\\}$, where we used

$\displaystyle||\partial_{t}w_{1}-\partial_{t}w_{2}||_{L^{4^{-}}(\Gamma)}^{2}\leq
C||\partial_{t}w_{1}-\partial_{t}w_{2}||_{W^{(\frac{1}{2})^{-},2^{-}}(\Gamma)}^{2}\leq
C||\mathbf{U}_{1}-\mathbf{U}_{2}||_{W^{1,2^{-}}(\Omega)}^{2}$
$\displaystyle\leq
C\int_{\Omega}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2},$
(35)

by imbedding and trace inequality.

The 5th term on RHS of $\eqref{estimate}$ \- case $\alpha>0,\gamma>3/2$:

$\displaystyle\int_{Q_{t}}J(r_{2}-r_{1})\Big{[}\mathbf{U}_{2}\cdot(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})-(\mathbf{w}_{1}-\mathbf{w}_{2})\cdot\nabla_{1}(r_{2}^{\gamma-1})-\mathbf{w}_{2}\cdot(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})\Big{]}$
$\displaystyle\leq
C\int_{0}^{t}||r_{1}-r_{2}||_{L^{2\land\gamma}(\Omega)}\Big{[}||\Delta
w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}\underbrace{||\mathbf{U}_{2}\cdot\nabla(r_{2}^{\gamma-1})||_{L^{q}(\Omega)}}_{:=h_{2}\in
L^{2}(0,T)}+||\partial_{t}w_{1}-\partial
w_{2}||_{L^{\infty^{-}}(\Gamma)}\underbrace{||\nabla(r_{2}^{\gamma-1})||_{L^{q}(\Omega)}}_{:=h_{3}\in
L^{2}(0,T)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad~{}+||\Delta
w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}\underbrace{||\mathbf{w}_{2}\cdot\nabla(r_{2}^{\gamma-1})||_{L^{q}(\Omega)}}_{:=h_{4}\in
L^{2}(0,T)}\Big{]}$ $\displaystyle\leq
C(\delta)\int_{0}^{t}(h_{2}^{2}+h_{3}^{2}+h_{4}^{2}+1)\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}+\delta\alpha\int_{0}^{t}||\partial_{t}\nabla
w_{1}-\partial_{t}\nabla w_{2}||_{L^{2}(\Gamma)}^{2},$

for any $q>3$.

The 6th term on RHS of $\eqref{estimate}$:

$\displaystyle\int_{Q_{t}}Jr_{1}\big{[}\mathbf{U}_{2}\cdot(\nabla_{1}-\nabla_{2})\mathbf{U}_{2}\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\leq
C\int_{0}^{t}||r_{1}||_{L^{\gamma}(\Omega)}||\mathbf{U}_{2}||_{L^{\infty}(\Omega)}||\Delta
w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}\underbrace{||\nabla\mathbf{U}_{2}||_{L^{\infty}(\Omega)}}_{=h_{5}\in
L^{2}(0,T)}||\mathbf{U}_{1}-\mathbf{U}_{2}||_{L^{6^{-}}(\Omega)}$
$\displaystyle\leq
C(\delta)\int_{0}^{t}h_{5}^{2}\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2}.$

The 7th term on RHS of $\eqref{estimate}$:

$\displaystyle\int_{Q_{t}}Jr_{1}\big{[}(\mathbf{U}_{1}-\mathbf{U}_{2})\cdot\nabla_{1}\mathbf{U}_{2}\big{]}\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})$
$\displaystyle\leq\int_{0}^{t}\Big{[}\underbrace{||\nabla
w_{1}||_{L^{\infty}(\Gamma)}||\nabla\mathbf{U}_{2}||_{L^{\infty}(\Omega)}}_{:=h_{6}\in
L^{1}(0,T)}\int_{\Omega}Jr_{1}|\mathbf{U}_{2}-\mathbf{U}_{1}|^{2}\Big{]}\leq
C\int_{0}^{t}h_{6}\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)},$

since $\nabla w_{1}\in L^{2}(0,T;L^{\infty}(\Gamma))$, by Theorem
$\ref{main2}(1)$.

The 8th term on RHS of $\eqref{estimate}$:

$\displaystyle\int_{Q_{t}}Jr_{1}(\nabla_{1}-\nabla_{2})(r_{2}^{\gamma-1})\cdot(\mathbf{U}_{2}-\mathbf{U}_{1})\leq
C\int_{0}^{t}\underbrace{||r_{1}||_{L^{\gamma}(\Omega)}||\nabla
r_{2}||_{L^{p}(\Omega)}}_{:=h_{7}\in L^{2}(0,T)}||\Delta w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}||\mathbf{U}_{1}-\mathbf{U}_{2}||_{L^{6^{-}}(\Omega)}$
$\displaystyle\leq
C(\delta)\int_{0}^{t}h_{7}^{2}\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2},$

for any $p>\frac{6\gamma}{5\gamma-6}$.

The 9th term on RHS of $\eqref{estimate}$:

$\displaystyle\int_{Q_{t}}J(r_{2}-r_{1})r_{2}^{\gamma-1}(\nabla_{1}-\nabla_{2})\cdot\mathbf{U}_{2}\leq
C\int_{0}^{t}\underbrace{||\nabla\mathbf{U}_{2}||_{L^{\infty}(\Omega)}}_{:=h_{8}\in
L^{2}(0,T)}||r_{1}-r_{2}||_{L^{2\land\gamma}(\Omega)}||\Delta w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}$ $\displaystyle\leq
C\int_{0}^{t}h_{8}\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}.$

The 10th term on RHS of $\eqref{estimate}$:

$\displaystyle\int_{Q_{t}}J\big{[}r_{2}^{\gamma}-\gamma
r_{2}^{\gamma-1}(r_{2}-r_{1})-r_{1}^{\gamma}\big{]}(\nabla_{1}\cdot\mathbf{U}_{2})$
$\displaystyle\leq\int_{0}^{t}\Big{[}\underbrace{||\nabla
w_{1}||_{L^{\infty}(\Gamma)}||\nabla\mathbf{U}_{2}||_{L^{\infty}(\Omega)}}_{:=h_{9}\in
L^{1}(0,T)}\int_{\Omega}J\big{(}r_{2}^{\gamma}-\gamma
r_{2}^{\gamma-1}(r_{2}-r_{1})-r_{1}^{\gamma}\big{)}\Big{]}\leq
C\int_{0}^{t}h_{9}\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}.$

The 11th term on RHS of $\eqref{estimate}$ \- case $\alpha=0$,
$\gamma\in(\frac{12}{7},2]$:

Assuming additionally that $\nabla\mathbf{U}_{2}\in L^{\infty}(Q_{T})$, one
can bound

$\displaystyle\int_{Q_{t}}Jr_{1}\Big{[}\mathbf{w}_{1}\cdot\nabla_{1}\mathbf{U}_{2}-\mathbf{w}_{2}\cdot\nabla_{2}\mathbf{U}_{2}\Big{]}\cdot(\mathbf{U}_{1}-\mathbf{U}_{2})$
$\displaystyle\int_{Q_{t}}J(r_{1}-r_{2})\Big{[}\mathbf{w}_{1}\cdot\nabla_{1}\mathbf{U}_{2}-\mathbf{w}_{2}\cdot\nabla_{2}\mathbf{U}_{2}\Big{]}\cdot(\mathbf{U}_{1}-\mathbf{U}_{2})+\int_{Q_{t}}Jr_{2}\Big{[}\mathbf{w}_{1}\cdot\nabla_{1}\mathbf{U}_{2}-\mathbf{w}_{2}\cdot\nabla_{2}\mathbf{U}_{2}\Big{]}\cdot(\mathbf{U}_{1}-\mathbf{U}_{2})$
$\displaystyle\leq
C\int_{0}^{t}||r_{1}-r_{2}||_{L^{\gamma}(\Omega)}||\nabla\mathbf{U}_{2}||_{L^{\infty}(\Omega)}\underbrace{\Big{[}||\partial_{t}w_{1}\nabla
w_{1}||_{L^{4^{-}}(\Gamma)}+||\partial_{t}w_{2}\nabla
w_{2}||_{L^{\infty}(\Gamma)}\Big{]}}_{:=h_{10}\in
L^{2}(0,T)}||\mathbf{U}_{1}-\mathbf{U}_{2}||_{L^{6^{-}}(\Omega)}$
$\displaystyle\quad+C\int_{0}^{t}||r_{2}||_{L^{\infty}(\Omega)}||\nabla\mathbf{U}_{2}||_{L^{\infty}(\Omega)}\Big{[}||\nabla
w_{1}||_{L^{\infty^{-}}(\Gamma)}||\partial_{t}w_{1}-\partial_{t}w_{2}||_{L^{2}(\Gamma)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+||\partial_{t}w_{2}||_{L^{\infty}(\Gamma)}||\Delta
w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}\Big{]}||\mathbf{U}_{1}-\mathbf{U}_{2}||_{L^{6^{-}}(\Omega)}$
$\displaystyle\leq
C(\delta)\int_{0}^{t}\Big{[}h_{10}^{2}||r_{1}-r_{2}||_{L^{\gamma}(\Omega)}^{2}+||\partial_{t}w_{1}-\partial_{t}w_{2}||_{L^{2}(\Gamma)}^{2}+||\Delta
w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}^{2}\Big{]}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2}$
$\displaystyle\leq
C(\delta)\int_{0}^{t}(h_{10}^{2}+1)\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2}.$

The 11th term on RHS of $\eqref{estimate}$ \- case $\alpha=0$, $\gamma>2$:

$\displaystyle\int_{Q_{t}}Jr_{1}\Big{[}\mathbf{w}_{1}\cdot\nabla_{1}\mathbf{U}_{2}-\mathbf{w}_{2}\cdot\nabla_{2}\mathbf{U}_{2}\Big{]}\cdot(\mathbf{U}_{1}-\mathbf{U}_{2})$
$\displaystyle\leq
C\int_{0}^{t}||\sqrt{r_{1}}||_{L^{2\gamma}(\Omega)}\underbrace{||\nabla\mathbf{U}_{2}||_{L^{\infty}(\Omega)}}_{:=h_{11}\in
L^{2}(0,T)}\Big{[}||\nabla
w_{1}||_{L^{\infty^{-}}(\Gamma)}||\partial_{t}w_{1}-\partial_{t}w_{2}||_{L^{4^{-}}(\Gamma)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+||\partial_{t}w_{2}||_{L^{\infty}(\Gamma)}||\Delta
w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}\Big{]}||\sqrt{r_{1}}(\mathbf{U}_{1}-\mathbf{U}_{2})||_{L^{2}(\Omega)}$
$\displaystyle\leq C\int_{0}^{t}||\Delta w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}^{2}+C(\delta)\int_{0}^{t}h_{11}^{2}\int_{\Omega}Jr_{1}|\mathbf{U}_{1}-\mathbf{U}_{2}|^{2}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2}$
$\displaystyle\leq
C(\delta)\int_{0}^{t}(h_{11}^{2}+1)\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2},$

by $\eqref{tracedis}$.

The 11th term on RHS of $\eqref{estimate}$ \- case $\alpha>0$, $\gamma>3/2$:

$\displaystyle\int_{Q_{t}}Jr_{1}\Big{[}\mathbf{w}_{1}\cdot\nabla_{1}\mathbf{U}_{2}-\mathbf{w}_{2}\cdot\nabla_{2}\mathbf{U}_{2}\Big{]}\cdot(\mathbf{U}_{1}-\mathbf{U}_{2})$
$\displaystyle\leq
C\int_{0}^{t}||\sqrt{r_{1}}||_{L^{3}(\Omega)}\underbrace{||\nabla\mathbf{U}_{2}||_{L^{\infty}(\Omega)}}_{=h_{12}\in
L^{2}(0,T)}\Big{[}||\nabla
w_{1}||_{L^{\infty^{-}}(\Gamma)}||\partial_{t}w_{1}-\partial_{t}w_{2}||_{L^{\infty^{-}}(\Gamma)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+||\partial_{t}w_{2}||_{L^{\infty}(\Gamma)}||\Delta
w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}\Big{]}||\sqrt{r_{1}}(\mathbf{U}_{1}-\mathbf{U}_{2})||_{L^{2}(\Omega)}$
$\displaystyle\leq C\int_{0}^{t}||\Delta w_{1}-\Delta
w_{2}||_{L^{2}(\Gamma)}^{2}+C(\delta)\int_{0}^{t}h_{12}^{2}\int_{\Omega}Jr_{1}|\mathbf{U}_{1}-\mathbf{U}_{2}|^{2}+\delta\alpha\int_{0}^{t}||\partial_{t}\nabla
w_{1}-\partial_{t}\nabla w_{2}||_{L^{2}(\Gamma)}^{2}$ $\displaystyle\leq
C\int_{0}^{t}(h_{12}^{2}+1)\mathcal{E}\Big{(}(r_{1},\mathbf{U}_{1},w_{1})\Big{|}(r_{2},\mathbf{U}_{2},w_{2})\Big{)}+\delta\alpha\int_{0}^{t}||\partial_{t}\nabla
w_{1}-\partial_{t}\nabla w_{2}||_{L^{2}(\Gamma)}^{2}.$

Combining previous inequalities, the proof is finished.

##  5 Some nonlinear plate models

Here, we study the interaction between compressible viscous fluids and
nonlinear plates. The goal is to prove that the main results of this paper can
be extended to such interaction problems. We note that the existence of weak
solutions for the following interaction problems were constructed in [38],
while the existence of strong solutions seem to be non-existent in the
literature.

###  5.1 A semilinear elastic plate

Here, the equation $\eqref{structureeqs}$ is replaced by

$\displaystyle\partial_{t}^{2}w+\Delta^{2}w-\alpha\partial_{t}\Delta
w+\mathcal{F}(w)=-S^{w}\mathbf{f}_{fl}\cdot\mathbf{e_{3}},$

where $\mathcal{F}$ satisfies the following assumptions:

  1. (A1)

Mapping $\mathcal{F}$ is locally Lipschitz from $H_{0}^{2-\epsilon}(\Gamma)$
into $H^{-2}(\Gamma)$ for some $\epsilon>0$, i.e.

$\displaystyle||\mathcal{F}(w_{1})-\mathcal{F}(w_{2})||_{H^{-2}(\Gamma)}\leq
C_{R}||w_{1}-w_{2}||_{H^{2-\epsilon}(\Gamma)},$

for a constant $C_{R}>0$, for any $||w_{i}||_{H^{2}(\Gamma)}\leq R$ ($i=1,2$).

  2. (A2)

Mapping $\mathcal{F}$ is locally Lipschitz from $H_{0}^{2}(\Gamma)$ into
$H^{-a}(\Gamma)$ for some $0\leq a<1/2$, i.e.

$\displaystyle||\mathcal{F}(w_{1})-\mathcal{F}(w_{2})||_{H^{-a}(\Gamma)}\leq
C_{R}||w_{1}-w_{2}||_{H^{2}(\Gamma)},$ (36)

for a constant $C_{R}>0$, for any $||w_{i}||_{H^{2}(\Gamma)}\leq R$ ($i=1,2$).

  3. (A3)

$\mathcal{F}(w)$ has a potential in $H_{0}^{2}(\Gamma)$, i.e. there exists a
Fréchet differentiable functional $\Pi(w)$ on $H_{0}^{2}(\Gamma)$ such that
$\Pi^{\prime}(w)=\mathcal{F}(w)$, and there are $0<\kappa<1/2$ and $C^{*}\geq
0$, such that the following inequality holds,

$\displaystyle\kappa||\Delta w||_{L^{2}(\Gamma)}^{2}+\Pi(w)+C^{*}\geq
0,~{}~{}\text{for all }w\in H_{0}^{2}(\Gamma).$

Moreover, $\Pi$ is bounded on bounded sets in $H_{0}^{2}(\Gamma)$.

######  Remark 5.1.

(1) This plate model is a generalization of Kirchhoff, von Karman and Berger
plates which are presented in Appendix.
(2) In [38], the condition $(A2)$ was not necessary for the existence proof
and was thus removed. However, in the proof that follows, it is needed.

Here, we give a sketch of the proofs of theorems 2.2 and 2.3 for this plate
model. Let $(\rho_{1},\mathbf{u}_{1},w_{1})$ be a finite energy weak solution.
In this case, the relative entropy stays the same, while in the relative
energy inequality takes the form

$\displaystyle\mathcal{E}\Big{(}(\rho_{1},\mathbf{u}_{1},w_{1})\Big{|}(\rho_{2},\mathbf{u}_{2},w_{2})\Big{)}(t)+\int_{0}^{t}\int_{\Omega^{w_{1}}(t)}\mathbb{S}(\nabla\mathbf{u}_{1}-\nabla\mathbf{u}_{2}):(\nabla\mathbf{u}_{1}-\nabla\mathbf{u}_{2})+\alpha\int_{0}^{t}\int_{\Gamma}|\partial_{t}\nabla
w_{1}-\partial_{t}\nabla w_{2}|^{2}$
$\displaystyle\leq\mathcal{E}\Big{(}(\rho_{1},\mathbf{u}_{1},w_{1})\Big{|}(\rho_{2},\mathbf{u}_{2},w_{2})\Big{)}(0)+\int_{0}^{t}\mathcal{R}\Big{(}\rho_{1},\mathbf{u}_{1},w_{1},\rho_{2},\mathbf{u}_{2},w_{2}\Big{)}-\int_{0}^{t}\langle\mathcal{F}(w_{1}),\partial_{t}w_{1}-\partial_{t}w_{2}\rangle$

It is easy to prove that any finite energy weak solution satisfies this
inequality. Next, to obtain the regularity estimates in Theorem
$\ref{main2}(1)$, in $\eqref{aaaa}$ additional term
$-\int_{\Gamma_{T}}\langle\mathcal{F}(w),D_{h}^{s}w\rangle$ appears, which can
be controlled as

$\displaystyle-\int_{0}^{t}\langle\mathcal{F}(w),D_{-h}^{s}D_{h}^{s}w\rangle\leq\int_{0}^{t}\Big{[}||\mathcal{F}(w)-\mathcal{F}(0)||_{H^{-a}(\Gamma)}+||\mathcal{F}(0)||_{H^{-a}(\Gamma)}\Big{]}||D_{-h}^{s}D_{h}^{s}w||_{H^{a}(\Gamma)}$
$\displaystyle\leq\int_{0}^{t}C_{R}(||w||_{H^{2}(\Gamma)}+1)||w||_{H^{2}(\Gamma)}\leq
C$

by $(A2)$ and for any $s<\frac{1}{2}$, where we have used the fact that the
constant $C_{R}$ in $\eqref{assumption1}$ is uniform, since $w_{1},w_{2}$ are
uniformly bounded in $L^{\infty}(0,T;H_{0}^{2}(\Gamma))$.

Finally, to obtain the Theorem 2.3 in this case, in $\eqref{estimate}$ an
additional term appears
$-\int_{\Gamma_{T}}\langle\mathcal{F}(w_{1})-\mathcal{F}(w_{2}),\partial_{t}w_{1}-\partial_{t}w_{2}\rangle$
on the right-hand side, which can be controlled as

$\displaystyle-\int_{0}^{t}\langle\mathcal{F}(w_{1})-\mathcal{F}(w_{2}),\partial_{t}w_{1}-\partial_{t}w_{2}\rangle\leq\int_{0}^{t}||\mathcal{F}(w_{1})-\mathcal{F}(w_{2})||_{H^{-a}(\Gamma)}||\partial_{t}w_{1}-\partial_{t}w_{2}||_{H^{a}(\Gamma)}$
$\displaystyle\leq
C\int_{0}^{t}||w_{1}-w_{2}||_{H^{2}(\Gamma)}||\partial_{t}w_{1}-\partial_{t}w_{2}||_{H^{a}(\Gamma)}$
$\displaystyle\leq
C(\delta)\int_{0}^{t}\mathcal{E}\Big{(}\big{[}(\rho_{1},\mathbf{u}_{1},w_{1})|(\rho_{2},\mathbf{u}_{2},w_{2})\big{]}\Big{)}+\delta\int_{Q_{t}}J|\mathbb{S}(\nabla_{1}\mathbf{U}_{1}-\nabla_{1}\mathbf{U}_{2})|^{2},$

by $\eqref{tracedis}$. If $\alpha>0$, then we can choose $a\leq 1$ and bound

$\displaystyle-\int_{\Gamma_{T}}\langle\mathcal{F}(w_{1})-\mathcal{F}(w_{2}),\partial_{t}w_{1}-\partial_{t}w_{2}\rangle\leq\int_{0}^{t}||\mathcal{F}(w_{1})-\mathcal{F}(w_{2})||_{H^{-a}(\Gamma)}||\partial_{t}w_{1}-\partial_{t}w_{2}||_{H^{a}(\Gamma)}$
$\displaystyle\leq
C\int_{0}^{t}||w_{1}-w_{2}||_{H^{2}(\Gamma)}||\partial_{t}w_{1}-\partial_{t}w_{2}||_{H^{1}(\Gamma)}$
$\displaystyle\leq
C(\delta)\int_{0}^{t}\mathcal{E}\Big{(}\big{[}(\rho_{1},\mathbf{u}_{1},w_{1})|(\rho_{2},\mathbf{u}_{2},w_{2})\big{]}\Big{)}+\alpha\delta\int_{0}^{t}||\partial_{t}w_{1}-\partial_{t}w_{2}||_{H^{1}(\Gamma)}.$

###  5.2 A thermoelastic semilinear plate

Here, the equation $\eqref{structureeqs}$ is replaced by a system777Note that
in this case, there is an additional equation in the weak formulation.

$\displaystyle\partial_{t}^{2}w+\Delta^{2}w-\alpha\partial_{t}\Delta
w+\mathcal{F}(w)$ $\displaystyle=$ $\displaystyle
S^{w}\mathbf{f}_{fl}\cdot\mathbf{e_{3}},$
$\displaystyle\partial_{t}\theta-\Delta\theta-\partial_{t}\Delta w$
$\displaystyle=$ $\displaystyle 0,$

where $\theta:\Gamma\to\mathbb{R}$ is the temperature of the plate and
$\mathcal{F}$ satisfies the assumptions $(A1)-(A3)$. Here, the relative
entropy takes the form

$\displaystyle\mathcal{E}\Big{(}(\rho_{1},\mathbf{u}_{1},w_{1})\Big{|}(\rho_{2},\mathbf{u}_{2},w_{2})\Big{)}(t)$
$\displaystyle=\frac{1}{2}\int_{\Omega^{w_{1}}(t)}(\rho_{1}|\mathbf{u}_{1}-\mathbf{u}_{2}|^{2})(t)+\frac{1}{\gamma-1}\int_{\Omega^{w_{1}}(t)}\big{(}\rho_{1}^{\gamma}-\gamma\rho_{2}^{\gamma-1}(\rho_{1}-\rho_{2})-\rho_{2}^{\gamma}\big{)}(t)$
$\displaystyle\quad+\frac{1}{2}\int_{\Gamma}|\partial_{t}w_{1}-\partial_{t}w_{2}|^{2}(t)+\frac{1}{2}\int_{\Gamma}|\Delta
w_{1}-\Delta
w_{2}|^{2}(t)+\frac{1}{2}\int_{\Gamma}|\theta_{1}-\theta_{2}|^{2}(t)\quad\quad\quad\quad$

while the relative energy inequality is replaced by

$\displaystyle\mathcal{E}\Big{(}(\rho_{1},\mathbf{u}_{1},w_{1})\Big{|}(\rho_{2},\mathbf{u}_{2},w_{2})\Big{)}(t)+\int_{0}^{t}\int_{\Omega^{w_{1}}(t)}J\mathbb{S}(\nabla\mathbf{u}_{1}-\nabla\mathbf{u}_{2}):(\nabla\mathbf{u}_{1}-\nabla\mathbf{u}_{2})$
$\displaystyle\quad+\alpha\int_{0}^{t}\int_{\Gamma}|\partial_{t}\nabla
w_{1}-\partial_{t}\nabla
w_{2}|^{2}+\int_{0}^{t}\int_{\Gamma}|\nabla\theta_{1}-\nabla\theta_{2}|^{2}$
$\displaystyle\leq\mathcal{E}\Big{(}\big{[}(\rho_{1},\mathbf{u}_{1},w_{1})|(\rho_{2},\mathbf{u}_{2},w_{2})\big{]}\Big{)}(0)+\int_{0}^{t}\mathcal{R}\Big{(}\rho_{1},\mathbf{u}_{1},w_{1},\rho_{2},\mathbf{u}_{2},w_{2}\Big{)}-\int_{0}^{t}\langle\mathcal{F}(w_{1}),\partial_{t}w_{1}-\partial_{t}w_{2}\rangle$
$\displaystyle\quad-\int_{0}^{t}\int_{\Gamma}(\theta_{2}-\theta_{1})\partial_{t}\theta_{2}-\int_{0}^{t}\int_{\Gamma}(\theta_{2}-\theta_{1})\Delta\theta_{2}.$

The proofs of theorems 2.2 and 2.3 in this case can be carried out in the same
way as for the semilinear plate case so we omit them here.

###  5.3 A thermoelastic quasilinear plate

Here, the equation $\eqref{structureeqs}$ is replaced by a system

$\displaystyle\partial_{t}^{2}w+\Delta^{2}w-\alpha\partial_{t}\Delta
w+\Delta(\Delta w)^{3}$ $\displaystyle=$ $\displaystyle
S^{w}\mathbf{f}_{fl}\cdot\mathbf{e_{3}},$
$\displaystyle\partial_{t}\theta-\Delta\theta-\partial_{t}\Delta w$
$\displaystyle=$ $\displaystyle 0.$

######  Remark 5.2.

Such a thermoelastic plate model was first studied in [21] and later in [23].
This high order nonlinearity comes from a thermoelastic plate model in which a
nonlinear coupling between the elastic, magnetic and thermoelastic fields is
considered.

In this case, the relative entropy takes the form

$\displaystyle\mathcal{E}\Big{(}(\rho_{1},\mathbf{u}_{1},w_{1})\Big{|}(\rho_{2},\mathbf{u}_{2},w_{2})\Big{)}(t)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega^{w_{1}}(t)}(\rho_{1}|\mathbf{u}_{1}-\mathbf{u}_{2}|^{2})(t)+\frac{1}{\gamma-1}\int_{\Omega^{w_{1}}(t)}\big{(}\rho_{1}^{\gamma}-\gamma\rho_{2}^{\gamma-1}(\rho_{1}-\rho_{2})-\rho_{2}^{\gamma}\big{)}(t)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\int_{\Gamma}|\partial_{t}w_{1}-\partial_{t}w_{2}|^{2}(t)+\frac{1}{2}\int_{\Gamma}|\Delta
w_{1}-\Delta
w_{2}|^{2}(t)+\frac{1}{2}\int_{\Gamma}|\theta_{1}-\theta_{2}|^{2}(t)$
$\displaystyle+$ $\displaystyle\int_{\Gamma}\Big{[}(\Delta w_{1})^{3}-(\Delta
w_{2})^{3}\Big{]}(\Delta w_{1}-\Delta w_{2}),$

while the relative energy inequality is replaced by

$\displaystyle\mathcal{E}\Big{(}(\rho_{1},\mathbf{u}_{1},w_{1})\Big{|}(\rho_{2},\mathbf{u}_{2},w_{2})\Big{)}(t)+\int_{0}^{t}\int_{\Omega^{w_{1}}(t)}J\mathbb{S}(\nabla\mathbf{u}_{1}-\nabla\mathbf{u}_{2}):(\nabla\mathbf{u}_{1}-\nabla\mathbf{u}_{2})$
$\displaystyle\quad+\alpha\int_{0}^{t}\int_{\Gamma}|\partial_{t}\nabla
w_{1}-\partial_{t}\nabla
w_{2}|^{2}+\int_{0}^{t}\int_{\Gamma}|\nabla\theta_{1}-\nabla\theta_{2}|^{2}$
$\displaystyle\leq\mathcal{E}\Big{(}\big{[}(\rho_{1},\mathbf{u}_{1},w_{1})|(\rho_{2},\mathbf{u}_{2},w_{2})\big{]}\Big{)}(0)+\int_{0}^{t}\mathcal{R}\Big{(}\rho_{1},\mathbf{u}_{1},w_{1},\rho_{2},\mathbf{u}_{2},w_{2}\Big{)}-\int_{0}^{t}\int_{\Gamma}\mathcal{(}\Delta
w_{2})^{3}(\Delta w_{1}-\Delta w_{2})$
$\displaystyle\quad-\int_{0}^{t}\int_{\Gamma}(\theta_{2}-\theta_{1})\partial_{t}\theta_{2}-\int_{0}^{t}\int_{\Gamma}(\theta_{2}-\theta_{1})\Delta\theta_{2}.$

In this case, in the proof of Theorem $\ref{main1}(1)$, when we choose
$(\mathbf{q},\psi)=(\mathbf{R}[D_{-h}^{s}D_{h}^{s}w],D_{-h}^{s}D_{h}^{s}w)$ in
the coupled momentum equation $\eqref{weaksolmom}$, the nonlinear term can be
expressed as

$\displaystyle\int_{\Gamma_{T}}(\Delta w)^{3}D_{-h}^{s}D_{h}^{s}[\Delta
w]=\int_{\Gamma_{T}}D_{h}^{s}(\Delta w)^{3}D_{h}^{s}\Delta
w\geq\frac{1}{4}\int_{\Gamma_{T}}|D_{h}^{\frac{s}{2}}\Delta w|^{4}.$

This then gives us $\Delta w\in L^{4}(0,T;W^{\frac{s}{2},4}(\Gamma))\cap
L^{2}(0,T;H^{s}(\Gamma))$, where $s$ satisfies the same conditions as in
Theorem $\ref{main2}(1)$.

Acknowledgments: This research was partially carried out during the author’s
PhD studies at Shanghai Jiao Tong University, where it was supported by the
National Natural Science Foundation of China (NNSFC) under Grant No. 11631008.
The author is immensely grateful to the referees for their very detailed
reading of the manuscript and for many useful comments and suggestions.

## Appendix - the nonlinear plate models

The assumptions $(A1)-(A3)$ given in section 5.1 are satisfied by the
following models:

The Kirchhoff model: Here the nonlinear elastic force takes the form of the
Nemytskii operator:

$\displaystyle\mathcal{F}(w)=-\nu\cdot\text{div}\big{[}|\nabla w|^{q}\nabla
w-\mu|\nabla w|^{r}\nabla w\big{]}+f(w)-h(x),$

where $\nu\geq 0$, $q>r\geq 0$, and $\mu\in\mathbb{R}$ are parameters, $h\in
L^{2}(\Gamma)$ and

$\displaystyle
f\in\text{Lip}_{loc}(\mathbb{R})\quad\text{satisfies}\quad\lim\limits_{|s|\to\infty}\inf
f(s)s^{-1}>-\lambda_{1}^{2},$

where $\lambda_{1}$ is the first eigenvalue for the Laplacian with the
Dirichlet boundary condition. The corresponding potential is

$\displaystyle\Pi(w)=\int_{\Gamma}\Phi(w)+\frac{\nu}{q+2}\int_{\Gamma}|\nabla
w|^{q+2}-\frac{\nu\mu}{r+2}\int_{\Gamma}|\nabla w|^{r+2}-\int_{\Gamma}wh,$

where $\Phi(s)=\int_{0}^{s}f(x)dx$. This model is used for the plates with
small displacements and it is important for pre-buckling analysis. See [9] for
more details about this model.

The von Kármán model: Here, the nonlinear elastic force takes the form

$\displaystyle\mathcal{F}(w)=-[w,v(w)+F_{0}]-h$

where $F_{0}\in H^{4}(\Gamma)$ and $h\in L^{2}(\Gamma)$ are given functions,
and the von Karman bracket is defined as

$\displaystyle[w,u]=\partial_{x_{1}}^{2}w\cdot\partial_{x_{2}}^{2}u+\partial_{x_{2}}^{2}w\cdot\partial_{x_{1}}^{2}u-2\cdot\partial_{x_{1}x_{2}}^{2}w\cdot\partial_{x_{1}x_{2}}^{2}u,$

where $v=v(w)$ is determined by the following problem:

$\displaystyle\Delta v+[w,w]=0\text{ in }\Gamma,\quad\partial_{\nu}v=v=0\text{
on }\partial\Gamma.$

This model is used for the plates with large displacements and it is important
in post-buckling analysis. More about this model can be found in [8].
The Berger model: In this case, the nonlinear elastic force takes the form

$\displaystyle\mathcal{F}(w)=-\Big{[}\nu\int_{\Gamma}|\nabla
w|^{2}dx-G\Big{]}\Delta w-h,$

where $\nu>0$ and $G\in\mathbb{R}$ are parameters, $h\in L^{2}(\Gamma)$. The
corresponding potential is

$\displaystyle\Pi(w)=\frac{\nu}{4}\Big{[}\int_{\Gamma}|\nabla
w|^{2}\Big{]}^{2}-\frac{G}{2}\int_{\Gamma}|\nabla w|^{2}-\int_{\Gamma}wh.$

For more details about this model, see [7, Chapter 7].

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