# Spectral Analysis and Asymptotic Decay of the Solutions to Multilayered
Structure-Stokes Fluid Interaction PDE System

Pelin G. Geredeli
School of Mathematical and Statistical Sciences,
Clemson University, Clemson-SC 29634, USA
<EMAIL_ADDRESS>
###### Abstract

In this work, the dynamics of a multilayered structure-fluid interaction (FSI)
PDE system is considered. Here, the coupling of 3D Stokes and 3D elastic
dynamics is realized via an additional 2D elastic equation on the boundary
interface. Such modeling PDE systems appear in the mathematical modeling of
eukaryotic cells and vascular blood flow in mammalian arteries. We analyze the
long time behavior of solutions to such FSI coupled system in the sense of
strong stability.

Our proof is based on an analysis of the spectrum of the associated semigroup
generator $\mathbf{A}$ which in particular entails the elimination of all
three parts of the spectrum of $\mathbf{A}$ from the imaginary axis. In order
to avoid steady states in our stability analysis, we firstly show that zero is
an eigenvalue for the operator $\mathbf{A}$, and we provide a characterization
of the (one dimensional) zero eigenspace $Null(\mathbf{A}).$ In turn, we
address the issue of asymptotic decay of the solution to the zero state for
any initial data taken from the orthogonal complement of the zero eigenspace
$Null(\mathbf{A})^{\bot}.$

Key terms: Fluid-Structure Interaction, Stokes Flow, Stability

Mathematics Subject Classification (MSC): 35B35, 35B40, 35G35

##  1 Introduction

The multilayered (composite) structure-Stokes flow interaction PDE model which
we consider in this manuscript arises in the context of the blood
transportation process in mammalian arteries and shape deformation of cells in
cellular dynamics. The physiological interaction between arterial walls (or
the membranes of the eukaryotic cells) and blood flow (or the surrounding
fluids) plays a crucial role in the physiology and pathophysiology of the
human cardiovascular system and cells [10, 13, 23, 27, 28], and can be
mathematically realized by fluid structure interaction (FSI) PDE. In
particular, all these dynamics are governed via multi-physics PDE models.

In the existing literature of FSI dynamics, it is often the case that only a
single layer structural PDE component is present in the model; i.e., the
displacement along the interaction interface is not modeled via any elastic
equation. For example, coupled heat-wave PDE systems (and some variations)
have been considered in the past where the heat equation component is regarded
as a simplification of the fluid flow component of the fluid-structure
interaction (FSI) dynamics, and the wave equation is regarded as a
simplification of the structural (elastic) component; see e.g., [26, Section
9] and [30]. Also, the FSI dynamics in which the fluid PDE component of fluid-
structure interactions is governed by Stokes or Navier-Stokes flow were
studied in various settings [4, 5, 9, 16, 18, 20, 25, 26]. However, some
degree of physical realism is lost if an arterial wall is taken to have no
composite layers. Because the vascular wall structures are typically not
single-layered; they are in fact of multi-layered type ([14, 15, 28]).

The first contribution to the analysis of such multilayered FSI dynamics is
the pioneering paper [28], where multilayered FSI is composed of 2D (thick
layer) wave equation and 1D wave equation (thin layer) coupled to a 2D fluid
PDE across a boundary interface. In [28], the authors proved the wellposedness
of the said multilayered coupled system using a partitioned, loosely coupled
scheme. This work represented a crucial result for future studies, since it
showed that the presence of a thin structure with mass at the fluid–structure
interface indeed regularizes the FSI dynamics. Later on, the authors in [6],
considered a multilayered “canonical” 3D heat-2D wave-3D wave coupled system
with the objective of analyzing the existence-uniqueness and the asymptotic
behavior of the corresponding solutions. In [6], although the considered
multilayered heat-wave-wave system constitutes a simplification somewhat of
the FSI model in [28]– in particular, the 2D wave equation takes the place of
a fourth order plate or shell PDE – our results in [6] still remained valid if
one replaces the 2D wave equation with the corresponding linear fourth order
equation. With an objective of better understanding the regularizing effects
of the underlying fluid dissipation and the coupling effects at the boundary
interface in higher dimension, the authors of [6] analyzed the wellposedness
and stability properties of the said multilayered FSI PDE system. In [6], to
obtain their results, the authors appealed to the semigroup approach, and the
spectral theory of linear operators.

Subsequently, in [7], a multilayered Lam$\acute{e}$-heat PDE system was
considered, and the authors gave an alternate proof for the analogous strong
stability result which they obtained in [6]. The main reasoning to pursue a
different approach then that in [6] was to eschew the need for deriving
analogous energy identities for the thick Lam$\acute{e}$ solution component of
the multilayered structure-fluid interaction PDE system. To this end, the
strong stability proof in [7] was predicated upon ultimately invoking the
resolvent criterion introduced by Y. Tomilov [32].

The strong stability result obtained in [6] for the solution to the
multilayered “canonical” 3D heat - 2D wave - 3D wave coupled system has been
improved in [8], where the authors proved that this solution actually
satisfies a rational decay rate with respect to smooth initial data.

Very recently, in [19], a more realistic multilayered FSI PDE model was
analyzed for which the canonical heat component was replaced with the Stokes
flow and coupling of the 3D Stokes flow and 3D elastic (structural) PDE
components is realized, via an additional 2D elastic system on the boundary
interface. The main result of [19] was the semigroup wellposedness of the
coupled Stokes-wave-Lam$\acute{e}$ PDE model. Unlike the papers mentioned
above, dealing with the pressure term (due to the Stokes fluid) was one of the
main challenges–not seen in the aforecited works– in determining the existence
and uniqueness of solutions, and this term indeed required a nonstandard
elimination of the pressure term for semigroup generation of the corresponding
dynamical system. This difficulty was overcome by expressing the fluid
solution variable via a decoupling of the Stokes equation, and constructing
the elastic solution variables by solving a mixed variational formulation via
a Babuska-Brezzi approach.

In the current manuscript, our main goal is to analyze the long time behavior
of the solution to the 3D Stokes flow-2D elastic-3D elastic coupled PDE system
whose existence-uniqueness was established in [19]. In this regard, we show
that the associated semigroup is strongly stable. That is, there is asymptotic
decay to the zero state. To the best of the author’s knowledge there is no
result about the strong stability of such multilayered structure-Stokes fluid
interaction PDE systems with an elastic interface.

At this point, we should point out the main challenges encountered throughout
the manuscript and novelties:

(i) Spectral Analysis for the generator $\mathbf{A}:$ In order to derive our
stability result, we adopt a frequency domain approach. Since the domain
$D(\mathbf{A})$ of the associated $C_{0}-$semigroup generator $\mathbf{A}$
(see 10) is not compactly embedded into finite energy space $\mathbf{H}$ (see
6 below), classical weak stabilizability approaches are not availing; see
[12]. Instead, we appeal to well known strong stability criterion introduced
by Arend-Batty [1]. However, the spectral analysis required for the generator
$\mathbf{A}$ entails a great amount of technicalities. For example, one needs
to analyze all parts of the spectrum $\sigma(\mathbf{A}),$ and show that it
does not intersect the imaginary axis.

(ii) Zero eigenvalue for the generator $\mathbf{A}:$ Since there is a need to
avoid steady states in our stability analysis, as a prerequisite result, we
show that zero is an eigenvalue for the generator $\mathbf{A}$ of the 3D
Stokes flow-2D elastic-3D elastic coupled PDE system. This gives us an alert
that the presence of the zero eigenvalue disallows the consideration of any
sort of decay problem in the entire phase space $\mathbf{H}$. Accordingly, we
are led here to consider our strong stability analysis on the orthogonal
complement $\mathbf{N}^{\bot}$ of the zero eigenspace $\mathbf{N}$ that is
defined in (18) below. In this regard, we provide clean characterization for
the zero eigenspace $\mathbf{N},$ which is only one dimensional, and the
orthogonal complement $\mathbf{N}^{\bot}-$ultimately, the Hilbert space of
stabilizable initial data.

(iii) Overdetermined eigenvalue problem for the thick 3D-elastic component: As
we mentioned in (i), our strong stability proof entails the elimination of all
three parts of the spectrum of the generator $\mathbf{A}$ from the imaginary
axis. Accordingly, we analyze $\sigma_{p}(\mathbf{A})$ (point spectrum),
$\sigma_{r}(\mathbf{A})$ (residual spectrum), and $\sigma_{c}(\mathbf{A})$
(continuous spectrum). For this, we are essentially interested in static
equations with respect to the operators $\mathbf{A}$ and $(i\beta
I-\mathbf{A})$ for some $\beta\neq 0.$ In order to solve these resolvent
equations, we apply certain multipliers to obtain necessary estimates to solve
the solution variables. However, getting an estimate for the 3D-thick elastic
solution variable $\\{w_{0n}\\}$ complicates our analysis. In particular, we
see that the static equation obtained for $\\{w_{0n}\\}$ represents an
overdetermined eigenvalue problem. In order to solve such a problem, we are
forced to impose an assumption (see Assumption 4) to guarantee that the
spectrum of the associated generator $\mathbf{A}$ remains away from the
imaginary axis. Such (essentially geometric) assumptions have been seen in the
context of stabilization problems for other FSI. See e.g., [2, 3, 4].

Description of the Multilayered FSI PDE Model

We describe the mathematical modeling of the multilayered (composite)
structure-Stokes fluid interaction PDE system on the following geometry: the
fluid geometry $\Omega_{f}$ $\subseteq\mathbb{R}^{3}$ is a bounded domain of
class $C^{3}$ with exterior boundary $\Gamma_{f}$. The structure domain
$\Omega_{s}$ $\subseteq\mathbb{R}^{3}$ is “completely immersed” in
$\Omega_{f}$ (See figure below)

Figure: Geometry of the FSI Domain

The boundary $\Gamma_{s}=\partial\Omega_{s},$ between $\Omega_{f}$ and
$\Omega_{s}$ is of class $C^{2},$ and constitutes the interactive interface
between the fluid and structure dynamics. In addition, $\nu(.)$ is the unit
normal vector which is outward with respect to $\Omega_{f},$ and so inwards
with respect to $\Omega_{s}.$

With the geometry $\\{\Omega_{s},\Omega_{f}\\}$ above, the PDE system in
variables $[u,h,h_{t},w,w_{t}]$ is:

$\left\\{\begin{array}[]{l}u_{t}-\text{div}(\nabla u+\nabla^{T}u)+\nabla
p=0\text{ \ \ \ in \ }(0,T)\times\Omega_{f}\\\ \text{div}(u)=0\text{ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in \ }(0,T)\times\Omega_{f}\\\
u|_{\Gamma_{f}}=0\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ on \ }(0,T)\times\Gamma_{f};\end{array}\right.$ (1)
$h_{tt}-\Delta_{\Gamma_{s}}h=[\nu\cdot\sigma(w)]|_{\Gamma_{s}}-[\nu\cdot(\nabla
u+\nabla^{T}u)]|_{\Gamma_{s}}+p\nu\text{ \ \ \ on \ }(0,T)\times\Gamma_{s},$
(2) $\left\\{\begin{array}[]{l}w_{tt}-\text{div}\sigma(w)+w=0\text{ \ \ \ \ \
\ \ \ \ \ \ \ \ on \ }(0,T)\times\Omega_{s}\\\
w_{t}|_{\Gamma_{s}}=h_{t}=u|_{\Gamma_{s}}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ on \ }(0,T)\times\Gamma_{s}\end{array}\right.$ (3)
$[u(0),h(0),h_{t}(0),w(0),w_{t}(0)]=[\widetilde{u}_{0},\widetilde{h}_{0},\widetilde{h}_{1},\widetilde{w}_{0},\widetilde{w}_{1}]\in\mathbf{N}^{\bot}.$
(4)

Here, $\mathbf{N}$ represents the zero eigenspace of the semigroup generator
which is associated to the above PDE system (see the definition of
$\mathbf{A}:\mathbf{H}\rightarrow\mathbf{H}$ in (10) below.) Also, $\lambda=0$
is indeed an eigenvalue of $\mathbf{A}$, see Theorem 6 below. Moreover,
orthogonal complement $\mathbf{N}^{\bot}$ is characterized as

$\mathbf{N}^{\bot}=\\{[u_{0},h_{0},h_{1},w_{0},w_{1}]\in\mathbf{H}:\int_{\Gamma_{s}}(\nu\cdot
h_{0})d\Gamma_{s}=0\\},$ (5)

(see Theorem 6 below,) where

$\begin{array}[]{l}\mathbf{H}=\mathbf{N}\bigoplus\mathbf{N}^{\bot}=\\{[u_{0},h_{0},h_{1},w_{0},w_{1}]\in[L^{2}(\Omega_{f})]^{3}\times[H^{1}(\Gamma_{s})]^{2}\times[L^{2}(\Gamma_{s})]^{2}\times\\\
\text{ \ \ \ \ \ \ \ \ \ \ \
}\times[H^{1}(\Omega_{s})]^{3}\times[L^{2}(\Omega_{s})]^{3}:\text{
div}(u_{0})=0,\text{ \ \ }u_{0}\cdot\nu|_{\Gamma_{f}}=0\text{, ~{}and~{}~{}
}w_{0}|_{\Gamma_{s}}=h_{0}\text{ }\\}\end{array}$ (6)

is the associated finite energy Hilbert space with the inner product

$\left\langle\Phi_{0},\widetilde{\Phi}_{0}\right\rangle_{\mathbf{H}}=\left\langle
u_{0},\widetilde{u}_{0}\right\rangle_{\Omega_{f}}+\left\langle\nabla_{\Gamma_{s}}(h_{0}),\nabla_{\Gamma_{s}}(\widetilde{h}_{0})\right\rangle_{\Gamma_{s}}+\left\langle
h_{1},\widetilde{h}_{1}\right\rangle_{\Gamma_{s}}$ $\text{\ \ \ \ \ \ \ \ \
}+\left\langle\sigma(w_{0}),\epsilon(\widetilde{w}_{0})\right\rangle_{\Omega_{s}}+\left\langle
w_{0},\widetilde{w}_{0}\right\rangle_{\Omega_{s}}+\left\langle
w_{1},\widetilde{w}_{1}\right\rangle_{{}_{\Omega_{s}}},$ (7)

for any

$\Phi_{0}=\left[u_{0},h_{0},h_{1},w_{0},w_{1}\right]\in\mathbf{H}\text{; \
}\widetilde{\Phi}_{0}=\left[\widetilde{u}_{0},\widetilde{h}_{0},\widetilde{h}_{1},\widetilde{w}_{0},\widetilde{w}_{1}\right]\in\mathbf{H}.$
(8)

We also note here that $\Delta_{\Gamma_{s}}(.)$ is the Laplace Beltrami
operator, and the stress tensor $\sigma(.)$ constitutes the Lam$\acute{e}$
system of elasticity on the “thick” layer. Namely, for function $v$ in
$\Omega_{s},$

$\sigma(v)=2\mu\epsilon(v)+\lambda[I_{3}\cdot\epsilon(v)]I_{3},$

where strain tensor $\epsilon(\cdot)$ is given by

$\epsilon_{ij}(v)=\frac{1}{2}\left(\frac{\partial v_{j}}{\partial
x_{i}}+\frac{\partial v_{i}}{\partial x_{j}}\right),\text{ \ \ }1\leq i,j\leq
3.$

###### Remark 1

For the sake of numerical computation, the structure geometry
$\Omega_{s}\subset\mathbb{R}^{3},$ can also be taken to be a convex polyhedral
domain with polygonal boundary faces $\Gamma_{j},$ $1\leq j\leq K$, where
$\Gamma_{i}\cap\Gamma_{j}\neq\emptyset$ for $i\neq j,$ and,

$\Gamma_{s}=\cup_{j=1}^{K}\overline{\Gamma}_{j}.$

Figure: Alternative Geometry of the FSI Domain

In this case, the thin wave equation can be modeled for $j=1,...,K$ as

$\left\\{\begin{array}[]{l}\frac{\partial^{2}}{\partial t^{2}}h_{j}-\Delta
h_{j}=[\nu\cdot\sigma(w)]|_{\Gamma_{j}}-[\nu\cdot(\nabla
u+\nabla^{T}u)]|_{\Gamma_{j}}+p\nu\text{ \ \ \ on \ }(0,T)\times\Gamma_{j}\\\
h_{j}|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}=h_{l}|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}\text{
on \ }(0,T)\times(\partial\Gamma_{j}\cap\partial\Gamma_{l})\text{,\
}\forall~{}1\leq l\leq K\text{ s.t.
}\partial\Gamma_{j}\cap\partial\Gamma_{l}\neq\emptyset\\\
\left.\dfrac{\partial h_{j}}{\partial
n_{j}}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}=-\left.\dfrac{\partial
h_{{}_{l}}}{\partial
n_{l}}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}\text{on \
}(0,T)\times(\partial\Gamma_{j}\cap\partial\Gamma_{l})\text{,\
}\forall~{}1\leq l\leq K\text{ s.t
}\partial\Gamma_{j}\cap\partial\Gamma_{l}\neq\emptyset.\text{\
}\end{array}\right.$

where the Laplace Beltrami Operator $\Delta_{\Gamma_{s}}(.)$ in (2) is
replaced with the standard Laplace operator with the imposition of additional
continuity and boundary conditions in order to satisfy the surface
differentiation (See [6, 8].)

###  1.1 Notation

For the remainder of the text, norms $||\cdot||_{D}$ are taken to be
$L^{2}(D)$ for the domain $D$. Inner products in $L^{2}(D)$ are written
$<\cdot,\cdot>_{D}$, and the inner products $L_{2}(\partial D)$ are written
$\langle\cdot,\cdot\rangle_{\partial D}$. The space $H^{s}(D)$ will denote the
Sobolev space of order $s$, defined on a domain $D$, and $H_{0}^{s}(D)$
denotes the closure of $C_{0}^{\infty}(D)$ in the $H^{s}(D)$ norm which we
denote by $\|\cdot\|_{H^{s}(D)}$ or $\|\cdot\|_{s,D}$.

For the sake of simplicity, we denote

$\mathbf{L^{2}}(\Omega_{f})=[L^{2}(\Omega_{f})]^{3},~{}~{}\mathbf{L^{2}}(\Omega_{s})=[L^{2}(\Omega_{s})]^{3},~{}~{}\mathbf{H^{1}}(\Omega_{s})=[H^{1}(\Omega_{s})]^{3},~{}~{}\mathbf{H^{1}}(\Gamma_{s})=[H^{1}(\Gamma_{s})]^{2}.$

Also, we make use of the standard notation for the trace of functions defined
on a Lipschitz domain $D$; i.e. for a scalar function $\phi\in H^{1}(D)$, we
denote $\gamma(w)$ to be the trace mapping from $H^{1}(D)$ to
$H^{1/2}(\partial D)$. That is, $\gamma(w)=w|_{\partial D},$ for $w\in
C^{\infty}(D).$ We will also denote pertinent duality pairings as
$(\cdot,\cdot)_{X\times X^{\prime}}$.

##  2 Preliminaries

The PDE system given in (1)-(4) may be associated with an abstract ODE in
Hilbert space $\mathbf{H};$ Namely,

$\left\\{\begin{array}[]{l}\frac{d}{dt}\Phi(t)=\mathbf{A}\Phi(t)\\\
\Phi(0)=\Phi_{0}\end{array}\right.$ (9)

where $\Phi(t)=\left[u(t),h(t),h_{t}(t),w(t),w_{t}(t)\right],$ and
$\Phi_{0}=[u_{0},h_{0},h_{1},w_{0},w_{1}].$ Here, the operator
$\mathbf{A}:D(\mathbf{A})\subset\mathbf{H}\rightarrow\mathbf{H}$ is defined by

$\mathbf{A}=\left[\begin{array}[]{ccccc}\text{div}(\nabla(.)+\nabla^{T}(.))&0&0&0&0\\\
0&0&I&0&0\\\
-[\nu\cdot(\nabla(.)+\nabla^{T}(.))]|_{\Gamma_{s}}&\Delta_{\Gamma_{s}}(.)&0&\nu\cdot\sigma(\cdot)|_{\Gamma_{s}}&0\\\
0&0&0&0&I\\\ 0&0&0&\text{div}\sigma(\cdot)-I&0\end{array}\right]$
$+\left[\begin{array}[]{ccccc}\text{-}\nabla\mathcal{P}_{1}(.)&\text{-}\nabla\mathcal{P}_{2}(.)&0&\text{-}\nabla\mathcal{P}_{3}(.)&0\\\
0&0&0&0&0\\\
\mathcal{P}_{1}(.)\nu&\mathcal{P}_{2}(.)\nu&0&\mathcal{P}_{3}(.)\nu&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\end{array}\right].$ (10)

Here, the “pressure” operators $\mathcal{P}_{i}$ are as defined below in
(13)-(15). As established in [19], the domain $D(\mathbf{A})$ of the generator
$\mathbf{A}$ is characterized as follows:
$\left[u_{0},h_{0},h_{1},w_{0},w_{1}\right]\in
D(\mathbf{A})\Longleftrightarrow$

(A.i) $u_{0}\in\mathbf{H}^{1}(\Omega_{f}),$
$h_{1}\in\mathbf{H}^{1}(\Gamma_{s}),$ $w_{1}\in\mathbf{H}^{1}(\Omega_{s});$

(A.ii) There exists an associated $L^{2}(\Omega_{f})$-function
$p_{0}=p_{0}(u_{0},h_{0},w_{0})$ such that

$[\text{div}(\nabla u_{0}+\nabla^{T}u_{0})-\nabla p_{0}]\in
L^{2}(\Omega_{f}).$

Consequently, $p_{0}$ is harmonic and so one has the boundary traces

(a) $[p_{0}|_{\partial\Omega_{f}},\frac{\partial
p_{0}}{\partial\nu}|_{\partial\Omega_{f}}]\in
H^{-\frac{1}{2}}(\partial\Omega_{f})\times
H^{-\frac{3}{2}}(\partial\Omega_{f});$

(b) $(\nabla u_{0}+\nabla^{T}u_{0})\cdot\nu\in
H^{-\frac{3}{2}}(\partial\Omega_{f}),$

(A.iii) div$\sigma(w_{0})\in L^{2}(\Omega_{s});$ consequently,
$\nu\cdot\sigma(w_{0})\in H^{-\frac{1}{2}}(\Gamma_{s}),$

(A.iv)
$\Delta_{\Gamma_{s}}(h_{0})+[\nu\cdot\sigma(w_{0})]|_{\Gamma_{s}}-[(\nabla
u_{0}+\nabla^{T}u_{0})\cdot\nu]|_{\Gamma_{s}}+[p_{0}\nu]|_{\Gamma_{s}}\in
L^{2}(\Gamma_{s}),$

(A.v) $u_{0}|_{\Gamma_{f}}=0,\ \
u_{0}|_{\Gamma_{s}}=h_{1}=w_{1}|_{\Gamma_{s}}$

We note that the elimination of the pressure variable is very crucial in order
to formulate the PDE system (1)-(4) as an ODE problem given in (9). For this,
we basically apply the divergence operator to the Stokes equation in (1), and
use the fact that $u$ is solenoidal. This gives that the (pointwise) pressure
variable $p(t)$ is harmonic; i.e.,

$\Delta p(t)=0\text{ \ \ \ in }\Omega_{f}.$ (11)

Subsequently, we multiply (2) by $\nu|_{\Gamma_{s}}$ and use the matching
velocity condition in (3) along with system (1) to obtain the following
boundary condition for the pressure variable $p:$

$p+\frac{\partial
p}{\partial\nu}=\text{div}(\nabla(u)+\nabla^{T}(u))\cdot\nu|_{\Gamma_{s}}+[(\nabla
u+\nabla^{T}u)\cdot\nu-\Delta_{\Gamma_{s}}(h)-\nu\cdot\sigma(w)|_{\Gamma_{s}}]\cdot\nu|_{\Gamma_{s}}.$
(12)

Also, since $u$ is divergence free, if we take the inner product of both sides
of (1)${}_{1},$ with an extension of the normal vector, and subsequently take
the trace of this relation on $\Gamma_{f},$ we have

$\frac{\partial p}{\partial\nu}=[\text{div}(\nabla
u+\nabla^{T}u)]\cdot\nu~{}~{}on~{}~{}\Gamma_{f}.$

Accordingly, the pressure variable $p(t),$ as the solution of (11)-(12), can
formally be written pointwise in time as

$p(t)=\mathcal{P}_{1}(u(t))+\mathcal{P}_{2}(h(t))+\mathcal{P}_{3}(w(t))$

where the harmonic functions $\mathcal{P}_{1}(u(t)),$ $\mathcal{P}_{2}(h(t))$,
and $\mathcal{P}_{3}(w(t))$ solve the following elliptic BVPs:

$\left\\{\begin{array}[]{l}\Delta\mathcal{P}_{1}(u)=0\text{ \ \ \ \ \ in \ \
}\Omega_{f}\\\
\mathcal{P}_{1}(u)=\text{div}(\nabla(u)+\nabla^{T}(u))\cdot\nu|_{\Gamma_{s}}+([(\nabla
u+\nabla^{T}u)]\cdot\nu)\cdot\nu|_{\Gamma_{s}}\text{\ \ \ \ \ \ on \ \
}\Gamma_{s},\\\
\frac{\partial\mathcal{P}_{1}(u)}{\partial\nu}=\text{div}(\nabla(u)+\nabla^{T}(u))\cdot\nu|_{\Gamma_{f}}\text{\
\ \ \ \ on \ \ }\Gamma_{f},\end{array}\right.$ (13)

$\left\\{\begin{array}[]{l}\Delta\mathcal{P}_{2}(h)=0\text{ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ in \ \ }\Omega_{f}\\\
\mathcal{P}_{2}(h)=-\Delta_{\Gamma_{s}}(h)\cdot\nu|_{\Gamma_{s}}\text{\ \ \ \
\ on \ \ }\Gamma_{s},\\\
\frac{\partial\mathcal{P}_{2}(h)}{\partial\nu}=0\text{\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ on \ \ }\Gamma_{f},\end{array}\right.$ (14)

and

$\left\\{\begin{array}[]{l}\Delta\mathcal{P}_{3}(w)=0\text{ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in \ \ }\Omega_{f}\\\
\mathcal{P}_{3}(w)=-[\nu\cdot\sigma(w)|_{\Gamma_{s}}]\cdot\nu|_{\Gamma_{s}}\text{\
\ \ \ \ on \ \ }\Gamma_{s}\\\
\frac{\partial\mathcal{P}_{3}(w)}{\partial\nu}=0\text{\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ on \ \ }\Gamma_{f},\end{array}\right.$ (15)

The construction of these $\mathcal{P}_{i}$ functions, defined as the
solutions to above harmonic equations, allows for the elimination of the
pressure term in the system (1)-(4). As such, the pressure-free system can
indeed be associated with the abstract ODE (9) in Hilbert space $\mathbf{H},$
and the associated pressure function $p_{0}$ in (A.ii) can be identified
explicitly, via

$p_{0}=\mathcal{P}_{1}(u_{0})+\mathcal{P}_{2}(h_{0})+\mathcal{P}_{3}(w_{0}).$
(16)

(See [19]). The wellposedness of the PDE system (1)-(4), or equivalently the
abstract ODE system (9) was proved in [19] by means of a nonstandard mixed
variational formulation and the use of Lumer Philips Theorem. For reader’s
reference, we state here the recent wellposedness result:

###### Theorem 2

[19] With reference to problem (1)-(4), the operator
$\mathbf{A}:D(\mathbf{A})\subset\mathbf{H}\rightarrow\mathbf{H}$, defined in
(10), generates a $C_{0}$-semigroup of contractions on $\mathbf{H}$.
Consequently, the solution
$\Phi(t)=\left[u(t),h(t),h_{t}(t),w(t),w_{t}(t)\right]$ of (1)-(4), or
equivalently (9), is given by

$\Phi(t)=e^{\mathbf{A}t}\Phi_{0}\in C([0,T];\mathbf{H})\text{,}$

where $\Phi_{0}=\left[u_{0},h_{0},h_{1},w_{0},w_{1}\right]\in\mathbf{H}$.

###### Remark 3

Given the form of the adjoint
$A^{*}:D(A^{*})\subset\mathbf{H}\rightarrow\mathbf{H},$ in Proposition 9
below, it is readily seen that $\lambda>0$ is also an eigenvalue of $A^{*}$
with

$Null(\mathbf{A}^{*})=Null(\mathbf{A})=\mathbf{N}.$

Consequently, one has “invariance of the flow” on $\mathbf{N}^{\bot}.$ That
is, if $\Phi_{0}\in\mathbf{N}^{\bot},$ then

$e^{\mathbf{A}t}\Phi_{0}\in C([0,T];\mathbf{N}^{\bot}).$

##  3 Main Result: Strong Stability of the Multilayered structure-Stokes fluid
PDE System

This section is devoted to addressing the issue of asymptotic behavior of the
solution whose existence-uniqueness is guaranteed by Theorem 2. In this
regard, we show that the system given in (1)-(4) is strongly stable in the
space $\mathbf{N}^{\bot}$ (see (5) for definition). Before giving our main
result, we state the following Assumption which is crucial in our proof:

###### Assumption 4

Given a fixed $\beta\neq 0,$ suppose that the function
$w_{0}\in\mathbf{H}^{1}(\Omega_{s})$ satisfies the following static PDE
problem:

$\left\\{\begin{array}[]{l}-\beta^{2}w_{0}-\text{div}~{}\sigma(w_{0})+w_{0}=0\text{
\ \ \ in \ }\Omega_{s}\\\ w_{0}|_{\Gamma_{s}}=0\text{ \ \ \ on \
}\Gamma_{s}\\\ \nu\cdot\sigma(w_{0})=-c_{0}\nu\text{ \ \ \ on \
}\Gamma_{s}.\end{array}\right.$

Then, the solution of this overdetermined problem is $w_{0}=0,$ and so
necessarily $c_{0}=0.$

Such overdetermined eigenvalue boundary value problems also play a part in
considering stability properties of other fluid-structure PDE systems. See,
e.g. [2, 4, 5]. Now, we give our stability result:

###### Theorem 5

Let Assumption 4 hold. Then, with reference to the dynamical system (1)-(4),
for any
$[\widetilde{u}_{0},\widetilde{h}_{0},\widetilde{h}_{1},\widetilde{w}_{0},\widetilde{w}_{1}]\in\mathbf{N}^{\bot},$
the corresponding solution $\left[u(t),{h}(t),h_{t}(t),w(t),w_{t}(t)\right]\in
C([0,T];\mathbf{N}^{\bot})$ of (1)-(4) satisfies

$\lim_{t\rightarrow\infty}||\left[u(t),{h}(t),h_{t}(t),w(t),w_{t}(t)\right]||_{\mathbf{N}^{\bot}}=0.$

The domain $D(\mathbf{A})$ of the semigroup generator is not compactly
embedded in $\mathbf{H},$ and so classical weak stability approaches are
inapplicable here. See e.g., [12, page 378, Corollary 3.1]. The proof of
Theorem 5 is based on the well known spectral criterion given by W. Arendt and
C.J.K. Batty [1] (see Appendix, Theorem 12). The application of Theorem 12 on
the dynamical system (1)-(4) relies on analyzing the spectral properties of
the operator $\mathbf{A}$ defined in (10). We will give our analysis in a few
steps:

Zero eigenvalue for the generator $\mathbf{A}$ and explicit characterization
of $\mathbf{N}=Null(\mathbf{A})$

In order to analyze the long term dynamics of the system (1)-(4), we need to
avoid steady states so as to reasonably consider the possibility of finite
energy solutions tending to the zero state at infinity. For this reason, in
the process of applying Theorem 12, we firstly show that zero is an eigenvalue
for the operator $\mathbf{A}$. Moreover, we give an explicit characterization
for the corresponding zero eigenspace $\mathbf{N}=Null(\mathbf{A})$ and its
orthogonal complement $\mathbf{N}^{\bot}=Null(\mathbf{A})^{\bot}.$

###### Theorem 6

With reference to the PDE system (1)-(4) and the corresponding generator
$\mathbf{A}$ defined in (10), the point zero is an eigenvalue for
$\mathbf{A}.$ Moreover, the following explicit characterizations follow:

$\mathbf{N}=Null(\mathbf{A})=span\\{\phi\in\mathbf{H}:\phi=\left[\begin{array}[]{c}0\\\
h_{0}(1)\\\ 0\\\ w_{0}(1)\\\ 0\end{array}\right]\\}.$ (17)

Here, for any scalar $\alpha,$ the pair

$[h_{0}(\alpha),w_{0}(\alpha)]\in\mathbf{S}=\\{[f,g]\in\mathbf{H}^{1}(\Gamma_{s})\times\mathbf{H}^{1}(\Omega_{s}):f=g|_{\Gamma_{s}}\\}$

satisfies the variational relation

$\left\langle\nabla_{\Gamma_{s}}(h_{0}),\nabla_{\Gamma_{s}}(f)\right\rangle_{\Gamma_{s}}+\left\langle\sigma(w_{0}),\epsilon(g)\right\rangle_{\Omega_{s}}+\left\langle
w_{0},g\right\rangle_{\Omega_{s}}=\alpha\left\langle\nu,f\right\rangle_{\Gamma_{s}},~{}~{}\forall~{}[f,g]\in\mathbf{S}.$

Also,

$\mathbf{N}^{\bot}=Null(\mathbf{A})^{\bot}=\\{\widetilde{\phi}=\left[\begin{array}[]{c}\widetilde{u_{0}}\\\
\widetilde{h_{0}}\\\ \widetilde{h_{1}}\\\ \widetilde{w_{0}}\\\
\widetilde{w_{1}}\end{array}\right]\in\mathbf{H}:\int_{\Gamma_{s}}(\nu\cdot\widetilde{h_{0}})~{}d\Gamma_{s}=0\\}.$
(18)

Proof. Suppose $\Phi=\left[u_{0},h_{0},h_{1},w_{0},w_{1}\right]\in
D(\mathbf{A})$ is a solution of

$\mathbf{A}\Phi=0,$ (19)

where $\mathbf{A},$ is as given in (10). With pressure term $p_{0}$ as defined
in (16), this equation generates the following PDEs:

$\left\\{\begin{array}[]{l}\text{div}(\nabla u_{0}+\nabla^{T}u_{0})-\nabla
p_{0}=0\text{ \ \ \ in \ }\Omega_{f}\\\
\text{div}(u_{0})=0~{}~{}~{}in~{}~{}~{}\Omega_{f}\\\ h_{1}=0\text{ \ \ \ in \
}\Gamma_{s}\\\ -[\nu\cdot(\nabla
u_{0}+\nabla^{T}u_{0})]|_{\Gamma_{s}}+\Delta_{\Gamma_{s}}(h_{0})+[\nu\cdot\sigma(w_{0})]|_{\Gamma_{s}}+p_{0}\nu=0\text{
\ \ \ in \ }\Gamma_{s}\\\ w_{1}=0\text{ \ \ \ in \ }\Omega_{s}\\\
\text{div}\sigma(w_{0})-w_{0}=0\text{ \ \ \ in \ }\Omega_{s}\\\
u_{0}|_{\Gamma_{f}}=0,~{}~{}u_{0}|_{\Gamma_{s}}=h_{1}=0.\end{array}\right.$
(20)

Multiplying (19) by $\Phi,$ we take

$\left\langle\mathbf{A}\Phi,\Phi\right\rangle_{\mathbf{H}}=\left\langle\text{div}(\nabla(u_{0})+\nabla^{T}(u_{0})),u_{0}\right\rangle_{\Omega_{f}}+\left\langle\nabla_{\Gamma_{s}}(h_{1}),\nabla_{\Gamma_{s}}(h_{0})\right\rangle_{\Gamma_{s}}$
$+\left\langle-(\nabla
u_{0}+\nabla^{T}u_{0})\cdot\nu|_{{}_{\Gamma_{s}}},h_{1}\right\rangle_{\Gamma_{s}}+\left\langle\Delta_{\Gamma_{s}}(h_{0}),h_{1}\right\rangle_{\Gamma_{s}}+\left\langle\sigma(w_{0})\cdot\nu|_{\Gamma_{s}},h_{1}\right\rangle_{\Gamma_{s}}$
$+\left\langle\sigma(w_{1}),\epsilon(w_{0})\right\rangle_{\Omega_{s}}+\left\langle
w_{1},w_{0}\right\rangle_{\Omega_{s}}+\left\langle\text{div}\sigma(w_{0}),w_{1}\right\rangle_{\Omega_{s}}-\left\langle
w_{0},w_{1}\right\rangle_{\Omega_{s}}$
$-\left\langle[\nabla\mathcal{P}_{1}(u_{0})+\nabla\mathcal{P}_{2}(h_{0})+\nabla\mathcal{P}_{3}(w_{0})],u_{0}\right\rangle_{\Omega_{f}}$
$+\left\langle[\mathcal{P}_{1}(u_{0})\cdot\nu+\mathcal{P}_{2}(h_{0})\cdot\nu+\mathcal{P}_{3}(w_{0})\cdot\nu],h_{1}\right\rangle_{\Gamma_{s}}=0.$

Applying Green’s theorem, using the fact that $u_{0}$ is solenoidal, $u_{0}=0$
on $\partial\Omega_{f},$ and $h_{1}=w_{1}|_{\Gamma_{s}}=0$ on $\Gamma_{s}$ we
get

$\left\langle\mathbf{A}\Phi,\Phi\right\rangle_{\mathbf{H}}=-\frac{1}{2}\left\|\nabla(u_{0})+\nabla^{T}(u_{0})\right\|^{2}+\left\langle\sigma(w_{1}),\epsilon(w_{0})\right\rangle_{\Omega_{s}}+\left\langle
w_{1},w_{0}\right\rangle_{\Omega_{s}}-\left\langle\sigma(w_{0}),\epsilon(w_{1})\right\rangle_{\Omega_{s}}-\left\langle
w_{0},w_{1}\right\rangle_{\Omega_{s}}=0.$

or,

$\left\langle\mathbf{A}\Phi,\Phi\right\rangle_{\mathbf{H}}=-\frac{1}{2}\left\|\nabla(u_{0})+\nabla^{T}(u_{0})\right\|^{2}+2i\text{Im}\\{\left\langle\sigma(w_{1}),\epsilon(w_{0})\right\rangle_{\Omega_{s}}+\left\langle
w_{1},w_{0}\right\rangle_{\Omega_{s}}\\}=0.$

Hence

$\text{Re}\left\langle\mathbf{A}\Phi,\Phi\right\rangle_{\mathbf{H}}=-\frac{1}{2}\left\|\nabla(u_{0})+\nabla^{T}(u_{0})\right\|^{2},$
(21)

which, considering also the boundary condition $u_{0}|_{\Gamma_{f}}=0,$ gives
us

$u_{0}=0~{}~{}in~{}~{}\Omega_{f}.$ (22)

In turn, we have from (20)1 that

$p_{0}=c_{0}~{}(constant).$

Now, define the space

$\mathbf{S}=\\{[f,g]\in\mathbf{H}^{1}(\Gamma_{s})\times\mathbf{H}^{1}(\Omega_{s}):f=g|_{\Gamma_{s}}\\}.$

Multiplying (20)4 by $f$ and (20)6 by $g,$ we have

$-\left\langle\nabla_{\Gamma_{s}}(h_{0}),\nabla_{\Gamma_{s}}(f)\right\rangle_{\Gamma_{s}}+\left\langle\nu\cdot\sigma(w_{0})|_{\Gamma_{s}},f\right\rangle_{\Gamma_{s}}+\left\langle
c_{0}\nu,f\right\rangle_{\Gamma_{s}}$
$-\left\langle\sigma(w_{0})\cdot\nu|_{\Gamma_{s}},g\right\rangle_{\Gamma_{s}}-\left\langle\sigma(w_{0}),\epsilon(g)\right\rangle_{\Omega_{s}}-\left\langle
w_{0},g\right\rangle_{\Omega_{s}}=0.$ (23)

Using the fact that $f=g|_{\Gamma_{s}},$ we take the variational form in terms
of the solution variables $\\{h_{0},w_{0}\\}:$

$\mathbf{a}([h_{0},w_{0}],[f,g])=\mathbf{F}([f,g]),\text{ \ \ \ }\forall\text{
}[f,g]\in\textbf{S},$ (24)

where the bilinear form
$\mathbf{a}(.,.):\textbf{S}\times\textbf{S}\rightarrow\mathbb{R}$ is defined
as

$\mathbf{a}([h_{0},w_{0}],[f,g])=\left\langle\nabla_{\Gamma_{s}}(h_{0}),\nabla_{\Gamma_{s}}(f)\right\rangle_{\Gamma_{s}}+\left\langle\sigma(w_{0}),\epsilon(g)\right\rangle_{\Omega_{s}}+\left\langle
w_{0},g\right\rangle_{\Omega_{s}}$

and

$\mathbf{F}([f,g])=c_{0}\left\langle\nu,f\right\rangle_{\Gamma_{s}}.$

Since it can easily be seen that the bilinear form $\mathbf{a}(.,.)$ is
continuous and $\mathbf{S}$-elliptic, the application of Lax Milgram Theorem
gives the existence and uniqueness of a solution $[h_{0},w_{0}]\in\mathbf{S}$
to the variational equation (24). To conclude the proof of Theorem 6, we must
show that the derived solution $\left[h_{0},w_{0}\right]\in D(\mathbf{A})$ and
satisfies the equations in (20). To this end, if we take $g\in D(\Omega_{s}),$
and $f=0$ in (23) then we have

$\left\langle-\text{div}\sigma(w_{0})+w_{0},g\right\rangle=0,~{}~{}~{}\forall~{}g\in
D(\Omega_{s}),$

and hence

$-\text{div}\sigma(w_{0})+w_{0}=0,~{}~{}~{}\text{in}~{}~{}L^{2}(\Omega_{s}).$

In consequence, we have

$||\sigma(w_{0})\cdot\nu||_{H^{-1/2}(\Gamma_{s})}\leq
C||w_{0}||_{H^{1}(\Omega_{s})}\leq C|c_{0}|.$ (25)

In turn: let
$\gamma^{+}_{0}\in\mathcal{L}(H^{1/2}(\Gamma_{s}),H^{1}(\Omega_{s}))$ be the
right inverse of the Dirichlet trace map

$\gamma_{0}:H^{1}(\Omega_{s})\rightarrow H^{1/2}(\Gamma_{s}).$ Therewith,
setting

$[f,g]\equiv[f,\gamma^{+}_{0}(f)]$

in (23) where $f\in\mathbf{H}^{1}(\Gamma_{s}),$ we have

$\left\langle\nabla_{\Gamma_{s}}(h_{0}),\nabla_{\Gamma_{s}}(f)\right\rangle_{\Gamma_{s}}+\left\langle\sigma(w_{0}),\epsilon(\gamma^{+}_{0}(f))\right\rangle_{\Omega_{s}}+\left\langle
w_{0},\gamma^{+}_{0}(f)\right\rangle_{\Omega_{s}}=c_{0}\left\langle\nu,f\right\rangle_{\Gamma_{s}}$

or

$\left\langle\nabla_{\Gamma_{s}}(h_{0}),\nabla_{\Gamma_{s}}(f)\right\rangle_{\Gamma_{s}}+\left\langle\sigma(w_{0}),\epsilon(\gamma^{+}_{0}(f))\right\rangle_{\Omega_{s}}+\left\langle
w_{0},\gamma^{+}_{0}(f)\right\rangle_{\Omega_{s}}+\left\langle\text{div}\sigma(w_{0})-w_{0},\gamma^{+}_{0}(f)\right\rangle_{\Gamma_{s}}=c_{0}\left\langle\nu,f\right\rangle_{\Gamma_{s}}.$

Application of the Green’s Theorem gives then

$\left\langle-\Delta_{\Gamma_{s}}(h_{0})+\nu\cdot\sigma(w_{0})\right\rangle_{\Gamma_{s}}=c_{0}\left\langle\nu,f\right\rangle_{\Gamma_{s}}=0$

or

$-\Delta_{\Gamma_{s}}(h_{0})+\nu\cdot\sigma(w_{0})=c_{0}\nu~{}~{}~{}in~{}~{}~{}L^{2}(\Gamma_{s}).$
(26)

In sum, we have that the obtained vector $\phi=\left[\begin{array}[]{c}0\\\
h_{0}(1)\\\ 0\\\ w_{0}(1)\\\ 0\end{array}\right]\in D(\mathbf{A})$ solves

$\mathbf{A}\phi=0,$

and is indeed the zero eigenvector with the corresponding zero eigenspace
$\mathbf{N}=Null\mathbf{(}A)$ could be characterized as in (17). Subsequently,
recalling the inner product introduced in (7), and using the relations
(25)-(26), the orthogonal complement $\mathbf{N}^{\bot}$ follows as in (18).
This completes the proof of Theorem (6).

The proof of Theorem (5) will rely on the ultimate application of the spectral
criterion of W. Arendt and C. Batty for strong decay (see Appendix, Theorem
12). This will entail the elimination of all three parts of the spectrum of
the generator $\mathbf{A}$ from the imaginary axis: In this connection, we now
proceed with the analysis of the point spectrum $\sigma_{p}(\mathbf{A}).$

###### Lemma 7

Let Assumption 4 hold. With reference to the PDE system (1)-(4) and the
corresponding generator $\mathbf{A}$ defined in (10), given for $\beta\neq 0,$
$i\beta\notin\sigma_{p}(\mathbf{A}).$

Proof. Suppose $\Phi=\left[u_{0},h_{0},h_{1},w_{0},w_{1}\right]\in
D(\mathbf{A})$ satisfies the relation

$(i\beta I-\mathbf{A})\Phi=0.$ (27)

In PDE terms, we then have

$\left\\{\begin{array}[]{l}i\beta u_{0}-\text{div}(\nabla
u_{0}+\nabla^{T}u_{0})+\nabla p_{0}=0\text{ \ \ \ in \ }\Omega_{f}\\\
\text{div}(u_{0})=0\text{ \ \ \ \ \ \ in \ }\Omega_{f}\\\
u_{0}|_{\Gamma_{f}}=0\text{ \ \ \ on \ }\Gamma_{f};\end{array}\right.$ (28)

$\left\\{\begin{array}[]{c}\begin{array}[]{l}i\beta h_{0}-h_{1}=0\text{ \ \ in
}\Gamma_{s}\end{array}\\\ i\beta h_{1}+[\nu\cdot(\nabla
u_{0}+\nabla^{T}u_{0})]|_{\Gamma_{s}}-\Delta_{\Gamma_{s}}(h_{0})-[\nu\cdot\sigma(w_{0})]|_{\Gamma_{s}}-p_{0}\nu=0\text{
\ \ \ in\ \ }\Gamma_{s},\text{ \ \ }\end{array}\right.$ (29)
$\left\\{\begin{array}[]{l}i\beta w_{0}-w_{1}=0\text{ \ \ in \ \ \
}\Omega_{s}\\\ i\beta w_{1}-\text{div}\sigma(w_{0})+w_{0}=0\text{ \ \ \ in \
}\Omega_{s}\\\ w_{1}|_{\Gamma_{s}}=h_{1}=u_{0}|_{\Gamma_{s}}\text{ \ \ \ on \
}\Gamma_{s}\end{array}\right.$ (30)

with $p_{0}$ being the associated pressure of the PDE system. Therewith, the
usual energy method and the relation (21) gives

$\text{Re}\left\langle(i\beta
I-\mathbf{A})\Phi,\Phi\right\rangle_{\mathbf{H}}=-\text{Re}\left\langle\mathbf{A}\Phi,\Phi\right\rangle_{\mathbf{H}}=\frac{1}{2}\left\|\nabla(u_{0})+\nabla^{T}(u_{0})\right\|^{2}=0$
(31)

which, considering also the boundary condition $u_{0}|_{\Gamma_{f}}=0,$ gives

$u_{0}=0~{}~{}in~{}~{}\Omega_{f}.$ (32)

Also, taking into account (32) in (30)3, we have

$w_{1}|_{\Gamma_{s}}=h_{1}=0~{}~{}on~{}~{}\Gamma_{s}.$ (33)

Let $p_{0}=q_{0}+c_{0},$ where $q_{0}\in\hat{L}^{2}(\Omega_{f}),$ where

$\hat{L}^{2}(\Omega_{f})=\\{f\in
L^{2}(\Omega_{f}):\int_{\Omega_{f}}fd\Omega_{f}=0\\},$ (34)

and $c_{0}$ is a constant. Then by (32) and (28)${}_{1},$

$q_{0}=0~{}~{}in~{}~{}\Omega_{f}.$ (35)

Also, from (33) and (29)1, we have

$h_{0}=0~{}~{}in~{}~{}\Gamma_{s}.$ (36)

Now, if we consider (33) and (36) in the thin layer equation (29)${}_{2},$ as
well as (30)1 in (30)${}_{2},$ we get that if
$\Phi=\left[u_{0},h_{0},h_{1},w_{0},w_{1}\right]\in D(\mathbf{A})$ is an
eigenfunction corresponding to eigenvalue $i\beta~{}~{}(\beta\neq 0),$ then
$w_{0}$ solves the following overdetermined eigenvalue problem:

$\left\\{\begin{array}[]{l}-\beta^{2}w_{0}-\text{div}\sigma(w_{0})+w_{0}=0\text{
\ \ \ in \ }\Omega_{s}\\\ w_{0}|_{\Gamma_{s}}=0\text{ \ \ \ on \
}\Gamma_{s}\\\ \nu\cdot\sigma(w_{0})=-c_{0}\nu\text{ \ \ \ on \
}\Gamma_{s}.\end{array}\right.$ (37)

Exploiting Assumption 4 for the problem (37) gives that $w_{0}=0$ and
$c_{0}=0,$ which then yield that $\sigma_{p}(\mathbf{A})\bigcap
i\mathbb{R}=\emptyset.$ This completes the proof of Lemma 7.

In our second step, we continue with the analysis of the residual spectrum
$\sigma_{r}(\mathbf{A}):$

###### Lemma 8

Let Assumption 4 hold. With reference to the PDE system (1)-(4) and the
corresponding generator $\mathbf{A}$ defined in (10), given $\beta\neq 0,$
$i\beta\notin\sigma_{r}(\mathbf{A}).$

Proof. The proof of Lemma 8 relies on the wellknown fact that “For any closed,
densely defined operator $\mathbf{A}$, if $\lambda\in\sigma_{r}(\mathbf{A})$
then $\overline{\lambda}\in\sigma_{p}(\mathbf{A}^{*})."$ (See e.g., [21, page
127]). Accordingly, we firstly give a representation of the adjoint operator
$\mathbf{A}^{*}:D(\mathbf{A}^{*})\rightarrow\mathbf{H}.$ In fact, a standard
computation yields:

###### Proposition 9

For the generator operator $\mathbf{A}$ defined in (10), the Hilbert space
adjoint
$\mathbf{A}^{*}:D(\mathbf{A}^{*})\rightarrow\mathbf{H}$ is given by

$\mathbf{A}^{*}=\left[\begin{array}[]{ccccc}\text{div}(\nabla(.)+\nabla^{T}(.))&0&0&0&0\\\
0&0&-I&0&0\\\
-[\nu\cdot(\nabla(.)+\nabla^{T}(.))]|_{\Gamma_{s}}&-\Delta_{\Gamma_{s}}(.)&0&-\nu\cdot\sigma(\cdot)|_{\Gamma_{s}}&0\\\
0&0&0&0&-I\\\ 0&0&0&-\text{div}~{}\sigma(\cdot)+I&0\end{array}\right]$
$+\left[\begin{array}[]{ccccc}\text{-}\nabla\mathcal{P}_{1}(.)&\nabla\mathcal{P}_{2}(.)&0&\nabla\mathcal{P}_{3}(.)&0\\\
0&0&0&0&0\\\
\mathcal{P}_{1}(.)\nu&-\mathcal{P}_{2}(.)\nu&0&-\mathcal{P}_{3}(.)\nu&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\end{array}\right].$ (38)

The domain $D(\mathbf{A}^{*})$ of $\mathbf{A}^{*}$ is characterized as
follows:
$\left[\widetilde{u}_{0},\widetilde{h}_{0},\widetilde{h}_{1},\widetilde{w}_{0},\widetilde{w}_{1}\right]\in
D(\mathbf{A}^{*})\Longleftrightarrow$

(A.i) $\widetilde{u}_{0}\in\mathbf{H}^{1}(\Omega_{f}),$
$\widetilde{h}_{1}\in\mathbf{H}^{1}(\Gamma_{s}),$
$\widetilde{w}_{1}\in\mathbf{H}^{1}(\Omega_{s}),$

(A.ii) There exists an associated $L^{2}(\Omega_{f})$-function
$\widetilde{p}_{0}=\widetilde{p}_{0}(\widetilde{u}_{0},\widetilde{h}_{0},\widetilde{w}_{0})$
such that

$[\text{div}(\nabla\widetilde{u}_{0}+\nabla^{T}\widetilde{u}_{0})-\nabla\widetilde{p}_{0}]\in
L^{2}(\Omega_{f}).$

Consequently, $\widetilde{p}_{0}$ is harmonic and so

(a)
$[\widetilde{p}_{0}|_{\partial\Omega_{f}},\frac{\partial\widetilde{p}_{0}}{\partial\nu}|_{\partial\Omega_{f}}]\in
H^{-\frac{1}{2}}(\partial\Omega_{f})\times
H^{-\frac{3}{2}}(\partial\Omega_{f});$
(b) $(\nabla\widetilde{u}_{0}+\nabla^{T}\widetilde{u}_{0})\cdot\nu\in
H^{-\frac{3}{2}}(\partial\Omega_{f}),$

(A.iii) div$\sigma(\widetilde{w}_{0})\in L^{2}(\Omega_{s});$ consequently,
$\nu\cdot\sigma(\widetilde{w}_{0})\in H^{-\frac{1}{2}}(\Gamma_{s}),$

(A.iv)
$-\Delta_{\Gamma_{s}}(\widetilde{h}_{0})-[\nu\cdot\sigma(\widetilde{w}_{0})]|_{\Gamma_{s}}-[(\nabla\widetilde{u}_{0}+\nabla^{T}\widetilde{u}_{0})\cdot\nu]|_{\Gamma_{s}}+[\widetilde{p}_{0}\nu]|_{\Gamma_{s}}\in
L^{2}(\Gamma_{s}),$

(A.v) $\widetilde{u}_{0}|_{\Gamma_{f}}=0,\ \
\widetilde{u}_{0}|_{\Gamma_{s}}=\widetilde{h}_{1}=\widetilde{w}_{1}|_{\Gamma_{s}}$

It is readily discerned that $Null(\mathbf{A})=Null(\mathbf{A}^{*}).$ In turn,
$\lambda=0$ is an eigenvalue of $A^{*}.$ Moreover, under Assumption 4,
$i\beta~{}~{}(\beta\neq 0)$ is not an eigenvalue for the adjoint operator
$\mathbf{A}^{*}$ and hence it is not in the residual spectrum of $\mathbf{A}.$
This completes the proof of Lemma 8.

Now, we continue with analyzing the continuous spectrum of the generator
$\mathbf{A}.$ In this connection, we give the following Lemma:

###### Lemma 10

Let Assumption 4 hold. With reference to the PDE system (1)-(4) and the
corresponding generator $\mathbf{A}$ defined in (10), for a given $\beta\neq
0,$ $i\beta\notin\sigma_{c}(\mathbf{A}).$

Proof. Our proof is based on a contradiction argument. To start with, asssume
that $i\beta$ ($\beta\neq 0$) is in the continuous spectrum
$\sigma_{c}(\mathbf{A})$ of $\mathbf{A}.$ Since
$\sigma_{c}(\mathbf{A})\subset\sigma_{app}(\mathbf{A})$ (approximate spectrum)
(see e.g., [21, page 128]) then there exist sequences

$\\{\Phi_{n}\\}=\left\\{\left[\begin{array}[]{c}u_{0n}\\\ h_{0n}\\\ h_{1n}\\\
w_{0n}\\\ w_{1n}\end{array}\right]\right\\}\subseteq D(\mathbf{A});\text{ \ \
\ \ \ \ \ }\\{(i\beta
I-\mathbf{A)}\Phi_{n}\\}\equiv\\{\Phi^{*}_{n}\\}=\left\\{\left[\begin{array}[]{c}u_{0n}^{\ast}\\\
h_{0n}^{\ast}\\\ h_{1n}^{\ast}\\\ w_{0n}^{\ast}\\\
w_{1n}^{\ast}\end{array}\right]\right\\}\subseteq\mathbf{H}\text{\ ,}$ (39)

which satisfy for $n=1,2,...,$

$\left\|\Phi_{n}\right\|_{\mathbf{H}}=1,\text{ and \ \ \ }\left\|(i\beta
I-\mathbf{A)}\Phi_{n}\right\|_{\mathbf{H}}<\frac{1}{n}.$ (40)

In PDE terms, each $\Phi_{n}$ solves the following static system, where again
$p_{n}$ is given via (16):

$\left\\{\begin{array}[]{l}i\beta u_{0n}-\text{div}(\nabla
u_{0n}+\nabla^{T}u_{0n})+\nabla p_{n}=u_{0n}^{\ast}\text{ \ \ \ in \
}\Omega_{f}\\\ \text{div}(u_{0n})=0\text{ \ \ \ \ \ \ in \ }\Omega_{f}\\\
u_{0n}|_{\Gamma_{f}}=0\text{ \ \ \ on \ }\Gamma_{f};\end{array}\right.$ (41)

$\left\\{\begin{array}[]{c}\begin{array}[]{l}i\beta
h_{0n}-h_{1n}=h_{0n}^{\ast}\text{ \ \ in }\Gamma_{s}\end{array}\\\ i\beta
h_{1n}+[\nu\cdot(\nabla
u_{0n}+\nabla^{T}u_{0n})]|_{\Gamma_{s}}-\Delta_{\Gamma_{s}}(h_{0n})-[\nu\cdot\sigma(w_{0n})]|_{\Gamma_{s}}-p_{n}\nu=h_{1n}^{\ast}\text{
\ \ \ in\ \ }\Gamma_{s},\text{ \ \ }\end{array}\right.$ (42)
$\left\\{\begin{array}[]{l}i\beta w_{0n}-w_{1n}=w_{0n}^{\ast}\text{ \ \ in \ \
\ }\Omega_{s}\\\ i\beta
w_{1n}-\text{div}\sigma(w_{0n})+w_{0n}=w_{1n}^{\ast}\text{ \ \ \ in \
}\Omega_{s}\\\ w_{1n}|_{\Gamma_{s}}=h_{1n}=u_{0n}|_{\Gamma_{s}}\text{ \ \ \ on
\ }\Gamma_{s}\end{array}\right.$ (43)

We proceed here stepwise:

Step I: Estimates for the fluid variable $\\{u_{0n}\\}$ and thin-layer
solution variables $\\{h_{0n},h_{1n}\\}$

To start, via the integration by parts, dissipativity relation (21), and (40)
we have

$\text{Re}\left\langle(\Phi^{*}_{n},\Phi_{n}\right\rangle_{\mathbf{H}}=\text{Re}\left\langle(i\beta
I-\mathbf{A})\Phi_{n},\Phi_{n}\right\rangle_{\mathbf{H}}=\frac{1}{2}\left\|\nabla(u_{0n})+\nabla^{T}(u_{0n})\right\|^{2}=\mathcal{O}(\frac{1}{n}),$
(44)

and hence

$||u_{0n}||_{H^{1}(\Omega_{f})}\rightarrow 0.$ (45)

In turn, we have by (43), (44) and Sobolev Trace Theory

$u_{0n}|_{\Gamma_{s}}=w_{1n}|_{\Gamma_{s}}=h_{1n}=\mathcal{O}(\frac{1}{n}).$
(46)

Let $p_{n}=q_{n}+c_{n},$ where $q_{n}\in\hat{L}^{2}(\Omega_{f})$ (as defined
in (34)), and $c_{n}$ is a constant. Then, by the Stokes Theory [31],
$\\{u_{0n},q_{n}\\}\in\mathbf{H}^{1}(\Omega_{f})\times\hat{L}^{2}(\Omega_{f})$
uniquely solve

$\left\\{\begin{array}[]{l}-\text{div}(\nabla u_{0n}+\nabla^{T}u_{0n})+\nabla
q_{n}=-i\beta u_{0n}+u^{\ast}_{0n}\text{\ \ \ in \ }\Omega_{f}\\\
\text{div}(u_{0n})=0\text{\ \ \ \ in \ }\Omega_{f}\\\
u_{0n}|_{\Gamma_{s}}=u_{0n}|_{\Gamma_{s}}\text{ \ \ \ on \ }\Gamma_{s}\\\
u_{0n}|_{\Gamma_{f}}=0\text{ \ \ \ on \ }\Gamma_{f},\end{array}\right.$ (47)

and so the following estimate holds:

$||q_{n}||_{L^{2}(\Omega_{f})}\leq
C[||\Phi^{*}_{n}||_{\mathbf{H}}+||u_{0n}||_{H^{1}(\Omega_{f})}]=\mathcal{O}(\frac{1}{n}).$
(48)

Subsequently, an energy method yields that

$\\{\nu\cdot(\nabla u_{0n}+\nabla^{T}u_{0n})-q_{n}\cdot\nu\\}\in
H^{-1/2}(\partial\Omega_{f})$ (49)

with

$||\nu\cdot(\nabla
u_{0n}+\nabla^{T}u_{0n})-q_{n}\cdot\nu||_{H^{-1/2}(\partial\Omega_{f})}\leq
C[||u_{0n}^{\ast}-i\beta
u_{0n}||_{\Omega_{f}}+||u_{0n}||_{H^{1}(\Omega_{f})}]=\mathcal{O}(\frac{1}{n}),$
(50)

after using (40) and (44). Moreover, since by the thin-layer resolvent
relation in (42)1 and the boundary condition (43)3

$h_{0n}=-\frac{i}{\beta}h_{1n}-\frac{i}{\beta}h_{0n}^{\ast}=-\frac{i}{\beta}u_{0n}-\frac{i}{\beta}h_{0n}^{\ast}.$
(51)

Using again (40) and (44), we get

$||h_{0n}||_{H^{1/2}(\Gamma_{s})}=\mathcal{O}(\frac{1}{n}).$ (52)

Step II: Estimate for the term $\\{\nu\cdot\sigma(w_{0n})\\}$

We start by invoking the “Dirichlet” map $D_{s}:L^{2}(\Gamma_{s})\rightarrow
L^{2}(\Omega_{s})$ via

$D_{s}g=f\Longleftrightarrow\left\\{\begin{array}[]{c}\text{div}~{}\sigma(f)=0\text{
\ in \ }\Omega_{s}\\\ f|_{\Gamma_{s}}=g\text{ \ on \
}\Gamma_{s}.\end{array}\right.$

We know by the Lax-Milgram Theorem

$D_{s}\in\mathcal{L}(H^{\frac{1}{2}}(\Gamma_{s}),H^{1}(\Omega_{s})).$ (53)

Now, with the Dirichlet map $D_{s}$ in hand, we make the change of variable

$z_{n}\equiv
w_{0n}+\frac{i}{\beta}D_{s}[u_{n}|_{\Gamma_{s}}+w_{0n}^{\ast}|_{\Gamma_{s}}].$
(54)

Considering the resolvent relations in (43)1-(43)${}_{2},$ $z_{n}$ then solves
the following boundary value problem (BVP):

$-\beta^{2}z_{n}-\text{div}\sigma(z_{n})=F_{\beta}\text{ \ \ in \ \
}\Omega_{s}$ $\hskip 62.59596ptz_{n}|_{\Gamma_{s}}=0\text{ \ \ \ on \ \
}\Gamma_{s}$ (55)

where

$\hskip 28.45274ptF_{\beta}=w_{1n}^{\ast}+i\beta w_{0n}^{\ast}-w_{0n}-i\beta
D_{s}[u_{n}|_{\Gamma_{s}}+w_{0n}^{\ast}|_{\Gamma_{s}}]$ (56)

Since this BVP has homogeneous boundary data then we have the estimate– see
e.g., [17, page 296, Theorem 6.3-6]:

$||z_{n}||_{H^{2}{(\Omega_{s})}}\leq||w_{1n}^{\ast}+i\beta
w_{0n}^{\ast}-w_{0n}-i\beta
D_{s}[u_{n}|_{\Gamma_{s}}+w_{0n}^{\ast}|_{\Gamma_{s}}]||_{\Omega_{s}}\leq
C_{1,\beta}$ (57)

after using (40). Consequently, there is the trace mapping– see, e.g. [29],

$||\frac{\partial z_{n}}{\partial\nu}||_{\Gamma_{s}}\leq
C||z_{n}||_{H^{2}{(\Omega_{s})}}\leq C_{2,\beta}$ (58)

(again, after using (57).) With this trace estimate in hand, we invoke the
following known expression for $\\{\sigma(z_{n})\cdot\nu|_{\Gamma_{s}}\\}$ in
terms of the normal and tangential derivatives (see [3, page 18, Proposition
A.1]):

$\sigma(z_{n})\cdot\nu=\lambda\Big{[}\frac{\partial
z_{n}}{\partial\nu}\cdot\nu+\frac{\partial
z_{n}}{\partial\tau}\cdot\tau+\frac{\partial z_{n}}{\partial e}\cdot
e\Big{]}\nu+2\mu\frac{\partial z_{n}}{\partial\nu}$ $\hskip
56.9055pt+\mu\Big{[}\frac{\partial z_{n}}{\partial\tau}\cdot\nu-\frac{\partial
z_{n}}{\partial\nu}\cdot\tau\Big{]}\tau+\mu\Big{[}\frac{\partial
z_{n}}{\partial e}\cdot\nu-\frac{\partial z_{n}}{\partial\nu}\cdot e\Big{]}e$
(59)

Here, unit (tangent) vectors $\\{e,\tau\\}$ and $\nu$ constitute an
orthonormal system on $\mathbb{R}^{3}.$ Since $z_{n}=0$ on $\Gamma_{s}$ then
(59) is simplified to

$\sigma(z_{n})\cdot\nu=\lambda(\frac{\partial
z_{n}}{\partial\nu}\cdot\nu)+2\mu\frac{\partial
z_{n}}{\partial\nu}-\mu(\frac{\partial
z_{n}}{\partial\nu}\cdot\tau)\tau-\mu(\frac{\partial z_{n}}{\partial\nu}\cdot
e)e.$ (60)

Applying the estimate (58) to the RHS of (60) now yields

$||\sigma(z_{n})\cdot\nu||_{\Gamma_{s}}\leq C_{3,\beta}.$ (61)

Moreover, an integration by parts yields the inference that

$\nu\cdot\sigma(D_{s}(\cdot))\in\mathcal{L}(H^{1/2}(\Gamma_{s}),H^{-1/2}(\Gamma_{s})).$

Combining this boundedness with (40), and (44) we have

$||\nu\cdot\sigma(D_{s}[u_{n}|_{\Gamma_{s}}+w_{0n}^{\ast}|_{\Gamma_{s}}])||_{H^{-1/2}(\Gamma_{s})}=\mathcal{O}(\frac{1}{n}).$
(62)

Applying now (61)-(62) to the relation (54), we get

$||\nu\cdot\sigma(w_{0n})||_{H^{-1/2}(\Gamma_{s})}\leq C.$ (63)

Step III: Estimate for the term $\\{c_{n}\\}$

At this step, we recall the definition of the pressure term
$p_{n}=q_{n}+c_{n},$ and read off the equation (42)2 to have

$||-c_{n}\nu||_{H^{-1}(\Gamma_{s})}=||-i\beta
h_{1n}+h_{1n}^{\ast}-[\nu\cdot(\nabla
u_{0n}+\nabla^{T}u_{0n})]|_{\Gamma_{s}}+\Delta_{\Gamma_{s}}(h_{0n})+[\nu\cdot\sigma(w_{0}n)]|_{\Gamma_{s}}+q_{n}\nu||_{H^{-1}(\Gamma_{s})},$

whence we get via (40), (48), (50), and (63)

$|c_{n}|\leq C,$ (64)

where again $C>0$ depends on $\lambda,\beta,\mu.$

Step IV: Estimate for $\\{\nabla_{\Gamma_{s}}h_{0n}\\}$

We multiply both sides of the equation (42)2 by $h_{0n},$ integrate in space,
and then use the integration by parts to take

$||\nabla_{\Gamma_{s}}h_{0n}||^{2}_{\Gamma_{s}}=-i\beta\left\langle
h_{1n},h_{0n}\right\rangle_{\Gamma_{s}}+\left\langle
h_{1n}^{\ast},h_{0n}\right\rangle_{\Gamma_{s}}-\left\langle[\nu\cdot(\nabla
u_{0n}+\nabla^{T}u_{0n})]|_{\Gamma_{s}},h_{0n}\right\rangle_{\Gamma_{s}}$
$+\left\langle[\nu\cdot\sigma(w_{0n})]|_{\Gamma_{s}},h_{0n}\right\rangle_{\Gamma_{s}}+\left\langle
p_{n}\nu,h_{0n}\right\rangle_{\Gamma_{s}}.$

Invoking again (40), (44), (46), (48), (63)-(64), and moreover using the
resolvent relation (42)1 for the second term of the RHS of the last equality,
we have

$||\nabla_{\Gamma_{s}}h_{0n}||^{2}_{\Gamma_{s}}=\mathcal{O}(\frac{1}{n}).$
(65)

Now, collecting all the estimates obtained in (45), (46), (48), and (65) we
get the following convergences:

$\left\\{\begin{array}[]{l}u_{0n}\rightarrow
0~{}~{}in~{}~{}\mathbf{H}^{1}(\Omega_{f})\\\ q_{n}\rightarrow
0~{}~{}in~{}~{}L^{2}(\Omega_{f})\\\ q_{n}|_{\Gamma_{s}}\rightarrow
0~{}~{}in~{}~{}H^{-1/2}(\Gamma_{s})\\\ w_{0n}|_{\Gamma_{s}}\rightarrow
0~{}~{}in~{}~{}\mathbf{H}^{1/2}(\Gamma_{s})\\\ h_{0n}\rightarrow
0~{}~{}in~{}~{}\mathbf{H}^{1}(\Gamma_{s})\\\ h_{1n}\rightarrow
0~{}~{}in~{}~{}\mathbf{H}^{1/2}(\Gamma_{s}).\end{array}\right.$ (66)

Moreover, from (61), the sequence $\\{\nu\cdot\sigma(z_{n})\\}$ has a weakly
convergent subsequence (still denoted as itself) such that
$\\{\nu\cdot\sigma(z_{n})\\}$ converges (weakly) in $L^{2}(\Gamma_{s})$ and
since $L^{2}(\Gamma_{s})\subset_{>}H^{-1}(\Gamma_{s})$ is compact we have then
$\\{\nu\cdot\sigma(z_{n})\\}$ converges strongly in $H^{-1}(\Gamma_{s}).$

Lastly, from (64), we have that $\\{c_{n}\\}$ converges strongly to $c^{*},$
say. If we recall the definition of $z_{n}$ given in (54) and invoke the
resolvent relation (42)2 together with the limits in (66), we have

$lim_{n\rightarrow\infty}\nu\cdot\sigma(z_{n})=-c^{*}\nu.$ (67)

To finish our proof, we consider again the BVP given in (55): Initially, the
relations (40) and (54) provide for the weak convergence

$z_{n}\xrightarrow{w}z,~{}~{}(say)~{}~{}in~{}~{}\mathbf{H}^{1}(\Omega_{s}).$

Hence, if we pass to the limit in (55) when $n\rightarrow\infty,$ then we get
that $z\in\mathbf{H}^{1}(\Omega_{s})$ satisfies the following problem:

$-\beta^{2}\left\langle
z,\Psi\right\rangle_{\Omega_{s}}+\left\langle\sigma(z),\epsilon(\Psi)\right\rangle_{\Omega_{s}}+\left\langle
c^{*}\nu,\Psi\right\rangle_{\Omega_{s}}+\left\langle
z,\Psi\right\rangle_{\Omega_{s}}=0,~{}~{}\forall\Psi\in\mathbf{H}^{1}(\Omega_{s}).$

That is, $\\{-\beta^{2},z\\}$ solves the following overdetermined eigenvalue
problem:

$-\beta^{2}z-\text{div}\sigma(z)+z=0\text{ \ \ in \ \ }\Omega_{s}$ $\hskip
85.35826ptz=0\text{ \ \ on \ \ }\Gamma_{s}$ $\hskip 99.58464pt\frac{\partial
z}{\partial\nu}=-c^{*}\nu\text{ \ \ on \ \ }\Gamma_{s}.$ (68)

Now, under Assumption 4, the only solution to problem (68) is $z=0$ for
$c^{*}=0.$ Then from (54), (45), and (40), we have $w_{0n}\rightarrow 0.$ This
convergence, and those in (66), contradicts the assumption that

$||\Phi_{n}||_{\mathbf{H}}=1.$ (69)

As a result, for any $\beta\neq 0,$ $i\beta\notin\sigma_{c}(\mathbf{A}),$ and
this completes the proof of Lemma 10.

To proceed with the proof of Theorem 5, we recall by Theorem 6 that zero is an
eigenvalue for the generator $\mathbf{A}.$ We show in fact that $\lambda=0$ is
in the resolvent set of $\mathbf{A}|_{N^{\bot}}:$

###### Lemma 11

$\lambda=0$ is in the resolvent set $\rho(\mathbf{A}|_{N^{\bot}})$ of
$\mathbf{A}|_{N^{\bot}}:D(\mathbf{A}|_{N^{\bot}})\rightarrow N^{\bot}.$ That
is,

$\Big{(}\mathbf{A}|_{N^{\bot}}\Big{)}^{-1}\in\mathcal{L}(N^{\bot}).$

Proof. As before, we use the denotations

$\Phi=\left[u_{0},h_{0},h_{1},w_{0},w_{1}\right]\in D(\mathbf{A})\cap
N^{\bot},~{}~{}~{}\Phi^{*}=\left[u^{*}_{0},h^{*}_{0},h^{*}_{1},w^{*}_{0},w^{*}_{1}\right]\in
N^{\bot},$

and consider solving the following relation

$\mathbf{A}\Phi=\Phi^{*}.$ (70)

Then, in PDE terms, this equation generates the following static system

$\left\\{\begin{array}[]{l}\text{div}(\nabla u_{0}+\nabla^{T}u_{0})-\nabla
q=u^{*}_{0}\text{ \ \ \ in \ }\Omega_{f}\\\ \text{div}(u_{0})=0\\\
h_{1}=h^{*}_{0}\text{ \ \ \ in \ }\Gamma_{s}\\\ -[\nu\cdot(\nabla
u_{0}+\nabla^{T}u_{0})]|_{\Gamma_{s}}+\Delta_{\Gamma_{s}}(h_{0})+[\nu\cdot\sigma(w_{0})]|_{\Gamma_{s}}+(q+c_{0}^{*})\nu=h^{*}_{1}\text{
\ \ \ in \ }\Gamma_{s}\\\ w_{1}=w^{*}_{0}\text{ \ \ \ in \ }\Omega_{s}\\\
\text{div}\sigma(w_{0})-w_{0}=w^{*}_{1}\text{ \ \ \ in \ }\Omega_{s}\\\
u_{0}|_{\Gamma_{f}}=0,~{}~{}u_{0}|_{\Gamma_{s}}=h_{1}=w_{1}|_{\Gamma_{s}}\end{array}\right.$
(71)

where $p=q+c^{*}_{0}$ is the associated pressure as described in
$D(\mathbf{A}).$ Firstly, the fluid component $u_{0}$ of (71) and the pressure
term $q$ can be recovered via the Stokes Theory (See [31, pg 22, Theorem 2.4])
and the pair $\\{u_{0},q\\}\in[\mathbf{H}^{1}(\Omega_{f})\cap
Null(\text{div})]\times\hat{L}^{2}(\Omega_{f})$ solves the following static
problem

$\left\\{\begin{array}[]{l}-\text{div}(\nabla u_{0}+\nabla^{T}u_{0})+\nabla
q=-u_{0}^{\ast}\text{\ \ \ in \ }\Omega_{f}\\\ \text{div}(u_{0})=0\text{\ \ \
\ in \ }\Omega_{f}\\\ u_{0}|_{\Gamma_{s}}=h^{*}_{0},\end{array}\right.$ (72)

with the estimate

$||\nu\cdot(\nabla
u_{0}+\nabla^{T}u_{0})-q\nu||_{H^{-1/2}(\Gamma_{s})}+||\nabla
u_{0}+\nabla^{T}u_{0}||_{\Omega_{f}}+||q||_{\Omega_{f}}\leq
C||\Phi^{*}||_{\mathbf{H}}.$ (73)

Now, with the associated pressure $p=q+c^{*}_{0},$ the constant component
$c^{*}_{0}$ is to be determined. We turn our attention to the thick and thin
elastic PDE component in (71). Define the space

$\textbf{S}=\left\\{(\varphi,\psi)\in\mathbf{H}^{1}(\Gamma_{s})\times\mathbf{H}^{1}(\Omega_{s}):\varphi=\psi|_{\Gamma_{s}}\right\\}.$

In order to generate a mixed variational formulation for the static “thin” and
“thick” solution variables in (71), we respectively multiply (71)4 and (71)6
by functions $\varphi\in[\mathbf{H}^{1}(\Gamma_{s})],$ and
$\psi\in[\mathbf{H}^{1}(\Omega_{s})]$ in the space S, use Green’s Theorem, and
add the subsequent relations to get:

$-\left\langle\nabla_{\Gamma_{s}}h_{0},\nabla_{\Gamma_{s}}\varphi\right\rangle_{\Gamma_{s}}+\left\langle\nu\sigma(w_{0})|_{\Gamma_{s}},\varphi\right\rangle_{\Gamma_{s}}+\left\langle
c_{0}^{*}\nu,\varphi\right\rangle_{\Gamma_{s}}$
$-\left\langle\nu\sigma(w_{0})|_{\Gamma_{s}},\psi\right\rangle_{\Gamma_{s}}-\left\langle\sigma(w_{0}),\epsilon(\psi)\right\rangle_{\Omega_{s}}-\left\langle
w_{0},\psi\right\rangle_{\Omega_{s}}$ $=\left\langle
h_{1}^{\ast},\varphi\right\rangle_{\Gamma_{s}}+\left\langle
w_{1}^{\ast},\psi\right\rangle_{\Omega_{s}}+\left\langle\nu\cdot(\nabla
u_{0}+\nabla^{T}u_{0})|_{\Gamma_{s}},\varphi\right\rangle_{\Gamma_{s}}-\left\langle
q\nu,\varphi\right\rangle_{\Gamma_{s}}.$ (74)

The last relation now gives us the following mixed variational formulation in
terms of the variables $h_{0}$ and $w_{0}:$ Namely,

$\left\\{\begin{array}[]{l}\mathbf{a}([h_{0},w_{0}],[\varphi,\psi])+\mathbf{b}([\varphi,\psi],c^{*}_{0})=\mathbf{F}([\varphi,\psi]),\text{
\ \ \ }\forall\text{ }[\varphi,\psi]\in\textbf{S}\\\
\mathbf{b}([h_{0},w_{0}],r)=0,\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \
}\forall\text{ }r\in\mathbb{R}.\end{array}\right.$ (75)

Here, the bilinear forms
$\mathbf{a}(.,.):\textbf{S}\times\textbf{S}\rightarrow\mathbb{R}$ and
$\mathbf{b}(.,.):\textbf{S}\times\mathbb{R}\rightarrow\mathbb{R}$ are
respectively given as

$\displaystyle\mathbf{a}([\phi,\xi],[\widetilde{\phi},\widetilde{\xi}])$
$\displaystyle=$
$\displaystyle\left\langle\nabla_{\Gamma_{s}}h_{0},\nabla_{\Gamma_{s}}\varphi\right\rangle_{\Gamma_{s}}-\left\langle\nu\sigma(w_{0})|_{\Gamma_{s}},\varphi\right\rangle_{\Gamma_{s}}$
$\displaystyle+\left\langle\nu\sigma(w_{0})|_{\Gamma_{s}},\psi\right\rangle_{\Gamma_{s}}+\left\langle\sigma(w_{0}),\epsilon(\psi)\right\rangle_{\Omega_{s}}+\left\langle
w_{0},\psi\right\rangle_{\Omega_{s}},$
$\mathbf{b}([\widetilde{\phi},\widetilde{\xi}],r)=-r\int_{\Gamma_{s}}\nu\cdot\widetilde{\phi}d\Gamma_{s},$

and the functional $\mathbf{F}(.)$ is defined as

$\mathbf{F}([\widetilde{\phi},\widetilde{\xi}])=-\left\langle
h_{1}^{\ast},\varphi\right\rangle_{\Gamma_{s}}-\left\langle
w_{1}^{\ast},\psi\right\rangle_{\Omega_{s}}-\left\langle\nu\cdot(\nabla
u_{0}+\nabla^{T}u_{0})|_{\Gamma_{s}},\varphi\right\rangle_{\Gamma_{s}}+\left\langle
q\nu,\varphi\right\rangle_{\Gamma_{s}}.$

In order to solve this variational formulation we appeal to the Babuska-Brezzi
Theorem (see Appendix, Theorem 13). It is clear that the bilinear forms
$\mathbf{a}(.,.)$ and $\mathbf{b}(.,.)$ are continuous, and moreover
$\mathbf{a}(.,.)$ is S-elliptic. In order to conclude that the variational
problem (75) has a unique solution, we need to show that the bilinear form
$\mathbf{b}(.,.)$ satisfies the “inf-sup” condition given in Theorem 13. For
this, we consider the following problem:

Given $r\in\mathbb{R}$, let $\eta\in\mathbf{H}^{1}(\Gamma_{s})$ satisfy

$\Delta_{\Gamma_{s}}\eta=sgn(r)\nu\text{ \ \ \ in \ }\Gamma_{s}$

It is easily seen that
$\left\|\nabla_{\Gamma_{s}}\eta\right\|_{\Gamma_{s}}\leq
C\left\|\nu\right\|_{\Gamma_{s}}.$ Now, taking into account that
$\gamma:\mathbf{H}^{1}(\Omega_{s})\rightarrow\mathbf{H}^{1/2}(\Gamma_{s})$ is
a surjective map, and so it has a continuous right inverse $\gamma^{+}(\eta),$
we have

$\displaystyle\sup_{[\theta,\varsigma]\in\textbf{S}}\frac{b([\theta,\varsigma],r)}{\left\|[\theta,\varsigma]\right\|_{\textbf{S}}}$
$\displaystyle\geq$
$\displaystyle\frac{b([\eta,\gamma^{+}(\eta)],r)}{\left\|[\eta,\gamma^{+}(\eta)]\right\|_{\textbf{S}}}$
$\displaystyle=$ $\displaystyle\frac{-r\int\limits_{\Gamma_{s}}\nu\cdot\eta
d\Gamma_{s}}{\left\|[\eta,\gamma^{+}(\eta)]\right\|_{\textbf{S}}}$
$\displaystyle=$ $\displaystyle-
rsgn(r)\frac{\int\limits_{\Gamma_{s}}\Delta_{\Gamma_{s}}\eta\cdot\eta
d\Gamma_{s}}{\left\|[\eta,\gamma^{+}(\eta)]\right\|_{\textbf{S}}}$
$\displaystyle=$
$\displaystyle\left|r\right|\frac{\int\limits_{\Gamma_{s}}|\nabla_{\Gamma_{s}}\eta|^{2}d\Gamma_{s}}{\left\|[\eta,\gamma^{+}(\eta)]\right\|_{\textbf{S}}}$
$\displaystyle\geq$ $\displaystyle
C\left|r\right|\frac{\int\limits_{\Gamma_{s}}|\nabla_{\Gamma_{s}}\eta|^{2}d\Gamma_{s}}{\left\|\eta\right\|_{\mathbf{H}^{1}(\Gamma_{s})}}$
$\displaystyle=$ $\displaystyle
C\left|r\right|\left\|\eta\right\|_{\mathbf{H}^{1}(\Gamma_{s})},$

which yields that the inf-sup condition holds with the constant
$\beta=C\left\|\eta\right\|_{\mathbf{H}^{1}(\Gamma_{s})}.$ Consequently, the
existence and uniqueness of the solution $[h_{0},w_{0}]\in\textbf{S}$ and
$c^{*}_{0}\in\mathbb{R}$ to the mixed variational problem (75) follows from
Theorem 13, and satisfy

$||[h_{0},w_{0}]||_{\textbf{S}}+|c^{*}_{0}|\leq C||\Phi^{*}||_{\mathbf{H}}.$
(76)

Subsequently, if we take $[\varphi,\psi]=[0,\psi]$ in (75) where
$\psi\in[D(\Omega_{s})]^{3},$ we infer that the obtained $w_{0}$ solves:

$-\text{div}\sigma(w_{0})+w_{0}=-w^{*}_{1}\text{ \ \ \ in \ }\Omega_{s}.$ (77)

In turn, via an energy method, we have the estimate

$||\nu\cdot\sigma(w_{0}))||_{H^{-1/2}(\Gamma_{s})}\leq
C||\Phi^{*}_{0}||_{\mathbf{H}}.$ (78)

With this estimate in hand, $\\{[h_{0},w_{0}],c^{*}_{0}\\}$ solves

$\left\langle\nabla_{\Gamma_{s}}h_{0},\nabla_{\Gamma_{s}}\varphi\right\rangle_{\Gamma_{s}}+\left\langle\sigma(w_{0}),\epsilon(\psi)\right\rangle_{\Omega_{s}}+\left\langle
w_{0},\psi\right\rangle_{\Omega_{s}}-\left\langle
c^{*}_{0}\nu,\varphi\right\rangle_{\Gamma_{s}}$ $=-\left\langle
h_{1}^{\ast},\varphi\right\rangle_{\Gamma_{s}}-\left\langle
w_{1}^{\ast},\psi\right\rangle_{\Omega_{s}}-\left\langle\nu\cdot(\nabla
u_{0}+\nabla^{T}u_{0})|_{\Gamma_{s}},\varphi\right\rangle_{\Gamma_{s}}+\left\langle
q\nu,\varphi\right\rangle_{\Gamma_{s}}.$

An integration by parts and consideration of (77) then yields

$-\Delta_{\Gamma_{s}}(h_{0})+[\nu\cdot(\nabla
u_{0}+\nabla^{T}u_{0})]|_{\Gamma_{s}}-[\nu\cdot\sigma(w_{0})]|_{\Gamma_{s}}-p\nu=-h^{*}_{1}\text{
\ \ \ in \ }\Gamma_{s}.$

If we read off the last equation then we get

$||-\Delta_{\Gamma_{s}}(h_{0})+[\nu\cdot(\nabla
u_{0}+\nabla^{T}u_{0})]|_{\Gamma_{s}}-[\nu\cdot\sigma(w_{0})]|_{\Gamma_{s}}-p\nu||_{\Gamma_{s}}\leq
C||\Phi^{*}||_{\mathbf{H}}.$

Finally, the pressure term $p$ can be reconstructed via the maps
$(\mathcal{P}_{i})s,$ as defined in (16), which then yields that
$\Phi=\left[u_{0},h_{0},h_{1},w_{0},w_{1}\right]\in D(\mathbf{A})\cap
N^{\bot}$ indeed solves (70). Hence $0\in\rho(\mathbf{A}|_{N^{\bot}}).$

As a result, combining Theorem 6, Lemma 7, Lemma 8, Lemma 10 and Lemma 11
finishes the proof of our main result Theorem 5.

##  4 Appendix

For the reader’s convenience, we provide the statements of the following
theorems which are critically used in this manuscript.

###### Theorem 12

[1] Let ${T(t)}_{t\geq 0}$ be a bounded $C_{0}$-semigroup on a reflexive
Banach space X, with generator $\mathbf{A}$. Assume that
$\sigma_{p}(\mathbf{A})\cap i\mathbb{R}=\emptyset$, where
$\sigma_{p}(\mathbf{A})$ is the point spectrum of $\mathbf{A}$. If
$\sigma(\mathbf{A})\cap i\mathbb{R}$ is countable then ${T(t)}_{t\geq 0}$ is
strongly stable.

In order to establish our strong stability result, one of the key tools that
we use to show that zero is in the resolvent set of the operator
$\\{\mathbf{A}|_{N^{\bot}}\\}$ is the Babuska-Brezzi Theorem whose statement
here is recalled for the reader’s convenience:

###### Theorem 13

[24] (Babuska-Brezzi) Let $\Sigma,$ $V$ be Hilbert spaces and
$a:\Sigma\times\Sigma\rightarrow\mathbb{R},$ $b:\Sigma\times
V\rightarrow\mathbb{R}$ bilinear forms which are continuous. Let

$Z=\left\\{\sigma\in\Sigma:b(\sigma,v)=0,\text{ \ for every }v\in V\right\\}.$

Assume that $a(\cdot,\cdot)$ is $Z$-elliptic, i.e., there exists a constant
$\alpha>0$ such that

$a(\sigma,\sigma)\geq\alpha\left\|\sigma\right\|_{\Sigma}^{2},\text{ \ \ for
every }\alpha\in Z.\text{ }$

Assume further that there exists a constant $\beta>0$ such that

$\sup_{\tau\in\Sigma}\frac{b(\tau,v)}{\left\|\tau\right\|_{\Sigma}}\geq\beta\left\|v\right\|_{V},\text{
\ \ for every }v\in V.$

Then if $\kappa\in\Sigma$ and $l\in V,$ there exists a unique pair
$(\sigma,u)\in\Sigma\times V$ such that

$\left\\{\begin{array}[]{l}a(\sigma,\tau)+b(\tau,u)=(\kappa,\tau)\text{ \ \ \
}\forall\text{ }\tau\in\Sigma\\\ b(\sigma,v)=(l,v)\text{ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ }\forall\text{ }v\in V.\end{array}\right.$

##  5 Acknowledgement

The author Pelin G. Geredeli would like to thank the National Science
Foundation, and acknowledge her partial funding from NSF Grant DMS-2348312.

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