# Rational Decay of A Multilayered Structure-Fluid PDE System

George Avalos
Department of Mathematics
University of Nebraska-Lincoln  Pelin G. Geredeli
Department of Mathematics
Iowa State University  Boris Muha
Department of Mathematics
University of Zagreb

###### Abstract

In this work, we consider a certain multilayered (thick layer) wave–(thin
layer) wave–heat (fluid) interactive PDE system. Such coupled PDE systems have
been used in the literature to describe the blood transport process in
mammalian vascular systems. In particular, the deformations of the boundary
interface (thin layer) are described via the two dimensional elastic equation.
The present work constitutes an investigation of the extent of the stabilizing
effects of the underlying fluid dissipation – across the boundary interface –
upon both the thick and thin structural components. (All three PDE components
evolve on their respective geometries.) In this regard, our main result is the
derivation of uniform decay rates for classical solutions of this multilayered
PDE model. To obtain these estimates, necessary a priori inequalities for
certain static multilayered PDE models are generated here to ultimately allow
an application of a wellknown resolvent criterion for rational decay.

Key terms: Fluid-Structure Interaction, Multilayered System, Semigroup,
Rational Decay

##  1  Introduction

The fluid structure interaction (FSI) phenomena constitutes a broad area of
research with applications in variety of real world problems [16, 29, 33]. In
particular, mammalian blood vascular walls, being composed of viscoelastic
materials, undergo large deformations due to hemodynamic forces generated
during the blood transport process. As such, there is a coupling of respective
blood flow and wall deformation dynamics. This physiological interaction
between arterial walls and blood flow plays a crucial role in the physiology
and pathophysiology of the human cardiovascular system, and can be
mathematically realized by multilayered FSI PDE. In such FSI modeling, the
blood flow is governed by the fluid flow PDE component (incompressible Stokes
or Navier Stokes); the displacements along the elastic vascular wall are
described by the structural PDE component (e.g., systems of elasticity). In
this regard, the multilayered FSI modeling with a view to understanding the
incidence of aneurysm caused by arterial wall deformations during the blood
transportation process has recently been a topic of great interest, see e.g.
[18, 41] and reference within.

In this paper, we consider a simplified multilayered structure-fluid
interaction (FSI) system where the coupling of the 3D fluid (blood flow) and
3D elastic (structural vascular wall) PDE components is realized via an
additional 2D elastic system on the boundary interface.

The PDE Model

Let the fluid geometry $\Omega_{f}\subseteq\mathbb{R}^{3}$ be Lipschitz, and
the structure domain $\Omega_{s}\subseteq\mathbb{R}^{3}$ be a convex
polyhedron which is strictly contained in $\Omega_{f}$ (See Figure 1).
Moreover, fluid boundary is decomposed via
$\partial\Omega_{f}=\Gamma_{f}\cup\Gamma_{s},$ where
$\Gamma_{s}=\partial\Omega_{s}$, and so $\Gamma_{f}\cap\Gamma_{s}=\emptyset$.
Thus, $\Gamma_{s}$ is the boundary interface between fluid geometry
$\Omega_{f}$ and structure geometry $\Omega_{s}.$ The boundary interface is
further decomposed via
$\Gamma_{s}=\bigcup\limits_{j=1}^{K}\overline{\Gamma_{j}},$ where each
$\Gamma_{j}$ is an open polygonal domain, with
$\Gamma_{i}\cap\Gamma_{j}=\emptyset$ for $i\neq j$. In addition, for $1\leq
j\leq K,$ $n_{j}$ denotes the unit normal vector which is exterior to
$\partial\Gamma_{j}$. Also, as pictured in Figure 1, $\nu(x)$ denotes the unit
outward normal with respect to $\Omega_{f}$ (and so $\nu(x)$ is inward with
respect to $\Omega_{s}).$

Figure 1: Multilayered structure-fluid interaction domain

For said {$\Omega_{f},\Gamma_{s},\Omega_{s}$}, the multilayered structure-
fluid FSI system in solution variables $u(t,x)$ (corresponding to the fluid
velocity), $h_{j}(t,x)\text{ \ }(1\leq j\leq K)$ (thin layers displacements),
and $w(t,x)$ (thick layer displacement) is as follows:

$\displaystyle\left\\{\begin{array}[]{l}u_{t}-\Delta u=0\text{ \ in
}(0,T)\times\Omega_{f}\\\ \left.u\right|_{\Gamma_{f}}=0\text{ \ on \
}(0,T)\times\Gamma_{f}\end{array}\right.$ (3)
$\displaystyle\left\\{\begin{array}[]{l}\text{For }1\leq j\leq K\text{,}\\\
\frac{\partial^{2}}{\partial t^{2}}h_{j}-\Delta h_{j}+h_{j}=\frac{\partial
w}{\partial\nu}|_{\Gamma_{j}}-\frac{\partial
u}{\partial\nu}|_{\Gamma_{j}}\text{\ \ on }\left(0,T\right)\times\Gamma_{j}\\\
\\\
\left\\{\begin{array}[]{l}\left.h_{j}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}=\left.h_{l}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}\\\
\left.\frac{\partial h_{j}}{\partial
n_{j}}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}=-\left.\frac{\partial
h_{l}}{\partial
n_{l}}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}\end{array}\right.\text{on
}\left(0,T\right)\times(\partial\Gamma_{j}\cap\partial\Gamma_{l})\text{, \
}\forall~{}~{}1\leq l\leq
K;~{}~{}\partial\Gamma_{j}\cap\partial\Gamma_{l}\neq\emptyset\end{array}\right.$
(10) $\displaystyle\left\\{\begin{array}[]{l}w_{tt}-\Delta w=0\text{ \ in
}(0,T)\times\Omega_{s}\\\
\left.w_{t}\right|_{\Gamma_{j}}=\frac{\partial}{\partial
t}h_{j}=\left.u\right|_{\Gamma_{j}}\text{ \ on
}\left(0,T\right)\times\Gamma_{j}\text{, \ for }j=1,...,K\end{array}\right.$
(13) $\displaystyle\begin{array}[]{c}[u(0),h_{1}(0),\frac{\partial
h_{1}(0)}{\partial t},...,h_{K}(0),\frac{\partial h_{1}(0)}{\partial
t},w(0),w_{t}(0)]=[u_{0},h_{01},h_{02},...,h_{0K},h_{0K},w_{0},w_{1}]\in\mathbf{H}\end{array}$
(15)

where the finite energy space $\mathbf{H}$ is given by

$\begin{array}[]{l}\mathbf{H}\equiv\left\\{[u_{0},h_{01},h_{02},...,h_{0K},h_{0K},w_{0},w_{1}]\in\mathbf{L}^{2}(\Omega_{f})\times\prod\limits_{j=1}^{K}\left[H^{1}(\Gamma_{j})\times
L^{2}(\Gamma_{j})\right]\times\mathbf{H}^{1}(\Omega_{s})\times\mathbf{L}^{2}(\Omega_{s})\right.:\\\
\text{(i) }\left.w_{0}\right|_{\Gamma_{j}}=h_{0j};\\\ \text{(ii)
}\left.h_{j}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}=\left.h_{l}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}\text{
on
}\left(0,T\right)\times(\partial\Gamma_{j}\cap\partial\Gamma_{l})~{}~{}\forall~{}~{}1\leq
l\leq K\text{ such that
}\partial\Gamma_{j}\cap\partial\Gamma_{l}\neq\emptyset\Big{\\}}.\end{array}$
(16)

The $\mathbf{H}$-inner product is

$\displaystyle\left(\Phi_{0},\tilde{\Phi}_{0}\right)_{\mathbf{H}}$
$\displaystyle=$
$\displaystyle\left(u_{0,}\tilde{u}_{0}\right)_{\Omega_{f}}+\sum_{j=1}^{K}\left[\left(\nabla
h_{0j},\nabla\tilde{h}_{0j}\right)_{\Gamma_{j}}+\left(h_{0j},\tilde{h}_{0j}\right)_{\Gamma_{j}}\right]$
(17)
$\displaystyle+\sum_{j=1}^{K}\left(h_{1j},\tilde{h}_{1j}\right)_{\Gamma_{j}}+\left(\nabla
w_{0},\nabla\tilde{w}_{0}\right)_{\Omega_{s}}+\left(w_{1},\tilde{w}_{1}\right)_{\Omega_{s}}$

for
$\Phi_{0}=\left[u_{0},h_{01},h_{11}(t)....,h_{0K},h_{1K},w_{0},w_{1}\right]$
and
$\tilde{\Phi}_{0}=\left[\tilde{u}_{0},\tilde{h}_{01},\tilde{h}_{11}(t)....,\tilde{h}_{0K},\tilde{h}_{1K},\tilde{w}_{0},\tilde{w}_{1}\right]\in\mathbf{H}.$

We should note that the boundary interface does not evolve with time. However,
it is well-accepted that if the boundary interface displacements between
structure and fluid are small relative to the scale of the geometry, the
resulting FSI models are physically relevant and reliable; (see [27, 34]).

Notation

Throughout, for a given domain $D$, the norm of corresponding space $L^{2}(D)$
will be denoted as $||\cdot||_{D}$ (or simply $||\cdot||$ when the context is
clear). Inner products in $L^{2}(\mathcal{O})$ or
$\mathbf{L}^{2}(\mathcal{O})$ will be denoted by
$(\cdot,\cdot)_{\mathcal{O}}$, whereas inner products
$L^{2}(\partial\mathcal{O})$ will be written as
$\langle\cdot,\cdot\rangle_{\partial\mathcal{O}}$. We will also denote
pertinent duality pairings as $\left\langle\cdot,\cdot\right\rangle_{X\times
X^{\prime}}$ for a given Hilbert space $X$. The space $H^{s}(D)$ will denote
the Sobolev space of order $s$, defined on a domain $D$; $H_{0}^{s}(D)$ will
denote the closure of $C_{0}^{\infty}(D)$ in the $H^{s}(D)$-norm
$\|\cdot\|_{H^{s}(D)}$. We make use of the standard notation for the boundary
trace of functions defined on $\mathcal{O}$, which are sufficently smooth:
i.e., for a scalar function $\phi\in H^{s}(\mathcal{O})$,
$\frac{1}{2}<s<\frac{3}{2}$, $\gamma(\phi)=\phi\big{|}_{\partial\mathcal{O}},$
which is a well-defined and surjective mapping on this range of $s$, owing to
the Sobolev Trace Theorem on Lipschitz domains (see e.g., [40], or Theorem
3.38 of [36]). Also, $C>0$ will denote a generic constant.

##  2  Literature

Stability analysis of fluid structure interaction (FSI) PDE systems have been
an ongoing object of study [2, 3, 4, 5, 28, 42]. Because of their utility in
mathematically describing fluid or flow dynamics as they interact with elastic
materials, such FSI models arise in biomedicine, biomechanics and
aeroelasticity, see e.g. [16, 26]. The main motivation of the current problem
comes from the mathematical modeling of vascular blood flow: the corresponding
modeling PDE dynamics accounts for the fact that blood-transporting vessels
are generally composed of several layers. Such multilayered FSI PDE models
have a crucial role in understanding the physiology of the human
cardiovascular system [12, 31, 39].

Examples of single layered FSI – i.e., only one elastic PDE (describing three
dimensional bulk elasticity or some lower-dimensional model of plate/shell
type) models the structural dynamics; the displacement along the interaction
interface is not modeled via any elastic equation– appear extensively in the
literature,see e.g. [9, 11, 13, 21, 22, 32, 27, 38] and references within.
However, many biomedical devices (such as stents) are being developed with the
view that vascular wall structures are composed of composite materials and not
of single layer; see [20, 24, 25, 39, 19]. In short, some degree of physical
realism is lost if the FSI PDE does not adequately describe arterial wall
layers of composite type.

Compared to the extensive work undertaken for single layered FSI, there is a
relative paucity of results for multilayered FSI systems. A multilayered FSI
(2D heat-1D wave- 2D wave) system was initially studied in [39] with a focus
on showing wellposedness. Therein, the authors exploited an underlying
regularity which was available by the presence of the additional wave
equation. A simplified $1D$ model was studied in [37] where the optimal
regularity result was proved. In [6], wellposedness and strong stability of a
higher dimensional linear version of the system considered in [39] were
studied. In particular, to prove strong decay, the authors of [6] appealed to
the wellknown spectral criteria in [1]. Very recently, an alternative
resolvent criterion approach to strong decay, with respect to a multilayered
Lam$\acute{e}$-heat system was given in [7]. However, up to the present time,
there has not been, to the best of our knowledge, any investigations into
_uniform decay properties_ – with respect to either finite energy or higher
norm – of multilayered FSI PDE models. Accordingly, the question in the
present work is, is the dissipation emanating from the thermal component of
the FSI system (3)-(15) strong enough to elicit polynomial decay?

##  3  Novelty and Challenges

As noted above, despite extensive research activity on single-layer FSI models
in the last twenty years or so, a comprehensive long term analysis theory for
multilayered FSI – in which the boundary interface coupling between fluid and
structure components is realized via an additional elastic equation – is
largely absent. Having established in [6] the strong (asymptotic) stability
for the multilayered FSI model (3)-(15), the authors in the present work
address the issue of obtaining rational decay rates for solutions of the
coupled 3D heat-2D wave-3D wave dynamics under consideration. These rational
decay rates will pertain to solutions of (3)-(15) which correspond to smooth
initial data; i.e., initial data drawn from the domain of the associated heat-
wave-wave $C_{0}$-semigroup generator
$\mathbf{A}:D(\mathbf{A})\subset\mathbf{H}\rightarrow\mathbf{H}$ in (18)
below.

Our approach here for solving the rational decay problem will entail an
appropriate estimation of the resolvent of the corresponding semigroup
generator, with a view of invoking the wellknown resolvent criterion in [17]
for polynomial decay. Ultimately, we will obtain an explicit decay rate of
$\mathcal{O}(t^{-\frac{2}{11}})$. This is Theorem 3 below. As as far as we can
tell, this is the first such polynomial stability result obtained for
multilayered FSI. By way of obtaining the rational decay result, we will
operate in the frequency domain and deal with a static FSI system – which is
essentially the Laplace transformed version of the original system (3)-(15)–
and the resolvent of the generator of the dynamical system. In this regard,
challenging issues associated with the analysis are as follows:

(i) Majorizing the solution to the resolvent equation in terms of the heat
component. Having obtained in our previous work [6] an understanding of the
spectral properties of the corresponding semigroup generator
$\mathbf{A}:D(\mathbf{A})\subset\mathbf{H}\rightarrow\mathbf{H}$ (of (18)
below), we proceed with considering the resolvent system in (22) below and
discern an inherent (static) fluid dissipative relation. Analogous to what the
analysis undertaken in the time domain for control of PDE’s in general – see
e.g., [43] – we will strive here to exploit the static dissipation by
_majorizing_ the solution $\Phi$ of (21) in terms of the heat component
$\mathbf{u}$. However the _key_ issue here will be an appropriate estimate for
the 3D thick wave component $w$.

(ii) Sharpening Poincar$\acute{e}$’s Inequality for the thermal component. In
order to deal with critical boundary trace terms and ultimately majorize them
with respect to the static heat dissipation, we will need to refine some of
the estimates concerning the boundary term $\left.\frac{\partial
u}{\partial\nu}\right|_{\Gamma_{s}}$. This will require us to prove a
sharpening of Poincar$\acute{e}$’s Inequality; in particular, we will need to
properly majorize the $L^{2}$-norm of the thermal component in such a way so
as to ultimately secure the given decay rate.

(iii) Control of the critical three wave boundary terms. As we pointed out in
(i), the main challenge here will be the appropriate estimate for the 3D thick
wave component $w.$ The use of a certain “Dirichlet” map will ultimately
enable us to homogenize this thick wave component of (22), via a new variable
$z$, which has zero Dirichlet bounday trace. To this new variable $z$, we
subsequently apply frequency domain versions of known vector identities for
the control of (uncoupled) waves. However, the obtained preliminary estimate
on the thick wave component will still contain the problematic boundary term
$\left.\frac{\partial z}{\partial\nu}\right|_{\Gamma_{s}}$, a term which
should be controlled in $L^{2}$-sense. The desired estimate of this flux term
will require the invocation of the thin layer $h$-equations in (22) and some
sharp interpolation inequalities. It is the estimation of this boundary term
which ultimately dictates the obtained rational decay rate.

##  4 Preliminaries

It is shown in [6] that the multilayered PDE system (3)-(15) can be described
via the matrix operator
$\mathbf{A}:D(\mathbf{A})\subset\mathbf{H}\rightarrow\mathbf{H}$ defined by

$\mathbf{A}=\left[\begin{array}[]{cccccccc}\Delta&0&0&\ldots&0&0&0&0\\\
0&0&I&\ldots&0&0&0&0\\\
-\left.\frac{\partial}{\partial\nu}\right|_{\Gamma_{1}}&(\Delta-I)&0&\ldots&0&0&\left.\frac{\partial}{\partial\nu}\right|_{\Gamma_{1}}&0\\\
\vdots&\vdots&\vdots&\ldots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&\ldots&0&I&0&0\\\
-\left.\frac{\partial}{\partial\nu}\right|_{\Gamma_{K}}&0&0&\ldots&(\Delta-I)&0&\left.\frac{\partial}{\partial\nu}\right|_{\Gamma_{K}}&0\\\
0&0&0&\ldots&0&0&0&I\\\ 0&0&0&\ldots&0&0&\Delta&0\end{array}\right],$ (18)

with

$D(\mathbf{A})=\left\\{\left[u_{0},h_{01},h_{11}(t)....,h_{0K},h_{1K},w_{0},w_{1}\right]\in\mathbf{H:}\right.$

(A.i)

(a) $u_{0}\in H^{1}(\Omega_{f})$, (b) $h_{1j}\in H^{1}(\Gamma_{j})$ for
$j=1,...,K$, (c) $w_{1}\in H^{1}(\Omega_{f});$

(A.ii)

(a) $\Delta u_{0}\in L^{2}(\Omega_{f})$, (b) $\Delta w_{0}\in
L^{2}(\Omega_{s})$, (c) $\Delta h_{0j}-\left.\frac{\partial
u_{0}}{\partial\nu}\right|_{\Gamma_{j}}+\left.\frac{\partial
w_{0}}{\partial\nu}\right|_{\Gamma_{j}}\in L^{2}(\Gamma_{j})$ for $j=1,...,K,$

(d) $\left.\frac{\partial h_{0j}}{\partial
n_{j}}\right|_{\partial\Gamma_{j}}\in H^{-\frac{1}{2}}(\partial\Gamma_{j})$
for $j=1,...,K;$

(A.iii)

(a) $\left.u_{0}\right|_{\Gamma_{f}}=0$, (b) $\left.u_{0}\right|_{\Gamma
j}=h_{1j}=\left.w_{1}\right|_{\Gamma j}$ for $j=1,...,K;$

(A.iv)

For $j=1,...,K$ such that
$\partial\Gamma_{j}\cap\partial\Gamma_{l}\neq\emptyset$:

$\left.\text{(a)}\left.h_{1j}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}=\left.h_{1l}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}};~{}~{}~{}(b)\left.\frac{\partial
h_{0j}}{\partial
n_{j}}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}=-\left.\frac{\partial
h_{0l}}{\partial
n_{l}}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}\right\\}.$

This is to say, $\Phi(t)=\left[u(t),h_{1}(t),\frac{\partial}{\partial
t}h_{1}(t)....,h_{K}(t),\frac{\partial}{\partial t}h_{K},w(t),w_{t}(t)\right]$
satisfies the PDE model
(3)-(15) if and only if these variables solve the following ODE in Hilbert
space $\mathbf{H}:$

$\frac{d}{dt}\Phi(t)=\mathbf{A}\Phi(t)~{}~{}\text{on}~{}(0,T);~{}~{}~{}~{}~{}\Phi(0)=\Phi_{0}=[u_{0},h_{01},h_{02},...,h_{0K},h_{0K},w_{0},w_{1}]\in\mathbf{H}.$
(19)

We recall the wellposedness result given in [6]:

###### Theorem 1

The linear operator
$\mathbf{A}:D(\mathbf{A})\subset\mathbf{H}\rightarrow\mathbf{H},$ as defined
in (18), generates a $C_{0}$-semigroup
$\left\\{e^{\mathbf{A}t}\right\\}_{t\geq 0}$ of contractions on $\mathbf{H}.$
Thus, for
$\Phi_{0}=[u_{0},h_{01},h_{02},...,h_{0K},h_{0K},w_{0},w_{1}]\in\mathbf{H},$
the solution $\Phi(t)=\left[u(t),h_{1}(t),\frac{\partial}{\partial
t}h_{1}(t)....,h_{K}(t),\frac{\partial}{\partial t}h_{K},w(t),w_{t}(t)\right]$
of (3)-(15) is given (continuously) by

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

##  5  Main Result

Our present work mainly focuses on analyzing the long time behavior of
solutions to the given multilayered system (3)-(15), with a view of obtaining
rational decay rate of these solutions. Our main proof of stability will be
based on an ultimate appeal to wellknown resolvent criterion of A. Borichev
and Y. Tomilov [17, Theorem 2.4]:

###### Theorem 2

Let $\left\\{T(t)\right\\}_{t\geq 0}$ be a bounded $C_{0}$-semigroup on a
Hilbert space $H$ with generator $A$ such that $i\mathbb{R}\subset\rho(A)$.
Then for fixed $\alpha>0$ the following are equivalent:

(i) $\left\|\mathcal{R}(is;A)\right\|=\mathcal{O}(\left|s\right|^{\alpha})$,
$\left|s\right|\rightarrow\infty;$

(ii) $\left\|T(t)A^{-1}x\right\|=o(t^{-\frac{1}{\alpha}})$,
$t\rightarrow\infty$, $x\in\mathbf{H}.$

Given this operator theoretic result, it will suffice to establish the
“frequency domain” PDE estimate in (20); the proof of this estimate will
constitute the bulk of the effort in the present paper. Now, we give our main
result of polynomial decay for solutions which correspond to smooth initial
data as follows:

###### Theorem 3

In regard to the multilayered PDE system in (3)-(15) (or equivalently (19)):
if $\Phi_{0}=[u_{0},h_{01},h_{02},...,h_{0K},h_{0K},w_{0},w_{1}]\in
D(\mathbf{A})$, then the corresponding solution of (3)-(15) (or equivalently
(19)) satisfies the estimate,

$\left\|\Phi(t)\right\|_{\mathbf{H}}\leq\frac{C}{t^{\frac{2}{11}}}\left\|\Phi_{0}\right\|_{D(\mathbf{A})}.$
(20)

That is, the solution to (3)-(15) (or equivalently (19)), which corresponds to
smooth initial data, decays at a rate of $\mathcal{O}(t^{-\frac{2}{11}})$.

Proof of Theorem 3

The proof relies on the resolvent criterion given in Theorem 2, and
presupposes that there is no intersection of $\sigma(\mathbf{A})$ with the
imaginary axis. In fact it was shown in [6, Proposition 7, Lemma 9, Corollary
10] – See Section 4 therein – that $i\mathbb{R}\subset\rho(\mathbf{A})$.

Subsequently, given parameter $\beta\in\mathbb{R}$ and data
$\Phi_{0}^{\ast}=[u^{\ast},h_{01}^{\ast},h_{02}^{\ast},...,h_{0K}^{\ast},h_{0K}^{\ast},w_{0}^{\ast},w_{1}^{\ast}]\in\mathbf{H}$,
we consider the resolvent equation

$\left[i\beta I-\mathbf{A}\right]\Phi=\Phi_{0}^{\ast},$ (21)

with solution $\Phi=[u,h_{01},h_{02},...,h_{0K},h_{0K},w_{0},w_{1}]\in
D(\mathbf{A})$. From the definition of $\mathbf{A,}$ this abstract equation
can be written explicitly as

$\begin{array}[]{l}\left\\{\begin{array}[]{l}i\beta u-\Delta u=u^{\ast}\text{
\ \ in }\Omega_{f}\\\ u|_{\Gamma_{f}}=0\text{ on
}\Gamma_{f}\end{array}\right.\\\ \\\ \left\\{\begin{array}[]{l}i\beta
h_{0j}-h_{1j}=h_{0j}^{\ast}\text{ \ \ in \ }\Gamma_{j}\\\
-\beta^{2}h_{0j}-\Delta h_{0j}+h_{0j}+\frac{\partial
u}{\partial\nu}-\frac{\partial w_{0}}{\partial\nu}=h_{1j}^{\ast}+i\beta
h_{0j}^{\ast}\text{ \ \ in \ }\Gamma_{j}\end{array}\right.\\\
\left\\{\begin{array}[]{l}\text{For all }1\leq l\leq K\text{\ such that
}\partial\Gamma_{j}\cap\partial\Gamma_{l}\neq\emptyset\\\
h_{0j}|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}=h_{0l}|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}\text{
}\\\ \left.\frac{\partial h_{0j}}{\partial
n_{j}}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}}=-\left.\frac{\partial
h_{0l}}{\partial
n_{l}}\right|_{\partial\Gamma_{j}\cap\partial\Gamma_{l}},\end{array}\right.\\\
\\\ \left\\{\begin{array}[]{l}w_{1}=i\beta w_{0}-w_{0}^{\ast}\text{ \ in
}\Omega_{s}\text{\ }\\\ -\beta^{2}w_{0}-\Delta w_{0}=i\beta
w_{0}^{\ast}+w_{1}^{\ast}\text{ \ \ in }\Omega_{s}\\\ [i\beta
w_{0}-w_{0}^{\ast}]_{\Gamma_{j}}=h_{1j}=u|_{\Gamma_{j}}\text{ \ on \
}\Gamma_{j}.\end{array}\right.\end{array}$ (22)

We will give our proof step-wise, estimating each solution component
separately:

Step 1: A static dissipation relation for heat component $u$:

We will start with an inherent (static) fluid dissipative relation which will
be the key ingredient for future steps. First, we take the $\mathbf{H}$-inner
product of both sides of (21) with respect to pre-image $\Phi$. This gives

$\begin{array}[]{l}i\beta\left\|\Phi\right\|_{\mathbf{H}}^{2}+\left\|\nabla
u\right\|_{\Omega_{f}}^{2}-2i\sum_{j=1}^{K}\left[\text{Im}\left(\nabla
h_{1j},\nabla
h_{0j}\right)_{\Gamma_{j}}+\text{Im}\left(h_{1j},h_{0j}\right)_{\Gamma_{j}}\right]\\\
\text{ \ \ }-2i\text{Im}\left(\nabla w_{1},\nabla
w_{0}\right)_{\Omega_{s}}=\left(\Phi_{0}^{\ast},\Phi_{0}\right)_{\mathbf{H}},\end{array}$
(23)

and then the following dissipation relation:

$\left\|\nabla
u\right\|_{\Omega_{f}}^{2}=\text{Re}\left(\Phi_{0}^{\ast},\Phi\right)_{\mathbf{H}}.$
(24)

In view of relation (24), _we should strive to_ _majorize_ _solution_ $\Phi$
_of_ (21) _in norm_ _in terms of the static heat dissipation_. With this theme
in mind, from the mechanical compatibility conditions in (16), and the
resolvent relations and matching velocity BC’s in (22), we have for
$j=1,...,K$,

$\left[i\beta
h_{0j}-h_{0j}^{\ast}\right]_{\Gamma_{j}}=\left.h_{1j}\right|_{\Gamma_{j}}=\left.u\right|_{\Gamma_{j}}.$
(25)

Combining this relation with the Sobolev Trace Theorem (and Poincaré’s
Inequality), we then have for $j=1,...,K$,

$\left\|\beta h_{0j}\right\|_{H^{\frac{1}{2}}(\Gamma_{j})}\leq
C\left(\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right).$
(26)

Moreover, via an integration by parts we get

$\displaystyle\left\|\frac{\partial
u}{\partial\nu}\right\|_{H^{-\frac{1}{2}}\left(\partial\Omega_{f}\right)}$
$\displaystyle\leq$ $\displaystyle C\left(\left\|\Delta
u\right\|_{\Omega_{f}}+\left\|\nabla u\right\|_{\Omega_{f}}\right)$ (27)
$\displaystyle=$ $\displaystyle C\left(\left\|i\beta
u-u^{\ast}\right\|_{\Omega_{f}}+\left\|\nabla u\right\|_{\Omega_{f}}\right)$

which, in turn, gives

$\left\|\frac{\partial
u}{\partial\nu}\right\|_{H^{-\frac{1}{2}}\left(\partial\Omega_{f}\right)}\leq
C\left(\left|\beta\right|\left\|u\right\|_{\Omega_{f}}+\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right).$
(28)

We should note that this estimate can be refined with respect to $\beta$. In
fact, for our particular multilayered PDE model, we have the following
“sharpening” of Poincaré’s Inequality:

###### Proposition 4

For $\left|\beta\right|>0$, the heat solution component of (22) obeys the
estimate

$\left|\beta\right|^{\frac{1}{2}}\left\|u\right\|_{\Omega_{f}}\leq
C\left(\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right).$
(29)

Proof: The details of the proof are taken in large part from Lemma 5.2 of [9]
and [8]. Given heat component $u$ of (22), let variable $u_{1}$ solve

$\left\\{\begin{array}[]{l}\Delta u_{1}=i\beta u\text{ \ in }\Omega_{f}\\\
\left.u_{1}\right|_{\partial\Omega_{f}}=0\text{ \ on
}\partial\Omega_{f}.\end{array}\right.$ (30)

Taking the $L^{2}$-inner product of both sides of (30)1 by $u_{1}$, we
subsequently have, from the heat equation in (22):

$\displaystyle\left\|\nabla u_{1}\right\|_{\Omega_{f}}^{2}$ $\displaystyle=$
$\displaystyle-\left(\Delta u_{1},u_{1}\right)_{\Omega_{f}}$ $\displaystyle=$
$\displaystyle-\left(i\beta u,u_{1}\right)_{\Omega_{f}}$ $\displaystyle=$
$\displaystyle-\left(\Delta u+u^{\ast},u_{1}\right)_{\Omega_{f}}$
$\displaystyle=$ $\displaystyle\left(\nabla u,\nabla
u_{1}\right)_{\Omega_{f}}-\left(u^{\ast},u_{1}\right)_{\Omega_{f}};$

whence we obtain, via Poincaré’s and Young’s Inequalities,

$\left\|\nabla u_{1}\right\|_{\Omega_{f}}\leq C\left(\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right).$
(31)

Since the $u_{1}$-equation in (30) gives

$\left\|u\right\|_{H^{-1}(\Omega_{f})}=\frac{1}{\left|\beta\right|}\left\|\Delta
u_{1}\right\|_{{}_{H^{-1}(\Omega_{f})}},$

using again the Poincaré’s Inequality yields

$\left\|u\right\|_{H^{-1}(\Omega_{f})}\leq\frac{C}{\left|\beta\right|}\left\|\nabla
u_{1}\right\|_{{}_{\Omega_{f}}}\leq\frac{C}{\left|\beta\right|}\left(\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right),$
(32)

after using (31). Moreover, it is known– see e.g., Theorem B.8, Theorem 3.30
and Theorem 3.33 of [36]– that

$L^{2}(\Omega_{f})=\left[H^{-1}(\Omega_{f}),H^{1}(\Omega_{f})\right]_{\frac{1}{2}}$

(which is the Lipschitz domain version of the interpolation result Lemma 12.1
of [35]). Combining this with the estimate (32) gives

$\left\|u\right\|_{L^{2}(\Omega_{f})}\leq
C\left\|u\right\|_{H^{-1}(\Omega_{f})}^{\frac{1}{2}}\left\|\nabla
u\right\|_{\Omega_{f}}^{\frac{1}{2}}\leq\frac{C}{\left|\beta\right|^{\frac{1}{2}}}\left(\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right)$

which yields (29) and finishes the proof of Proposition 4. $\square$

Now, applying the estimate (29) to the right hand side of (28), we then have
the following normal derivative trace estimate for the heat component of the
static problem (22):

###### Corollary 5

For $\left|\beta\right|>1$, the heat solution component of (22) obeys the
estimate

$\left\|\frac{\partial
u}{\partial\nu}\right\|_{H^{-\frac{1}{2}}\left(\partial\Omega_{f}\right)}\leq
C\left|\beta\right|^{\frac{1}{2}}\left(\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right).$
(33)

Step 2: The thick wave displacement $w_{0}$

In what follows, we require “Dirichlet” map $D$, defined by having for given
boundary function $g\in H^{\frac{1}{2}}(\Gamma_{s})$,

$\Delta Dg=0\text{ \ in }\Omega_{s};\ \ \ \ \
\left.Dg\right|_{\Gamma_{s}}=g\text{ \ on }\Gamma_{s}.$ (34)

By the Lax-Milgram Theorem and an argument similar to that which resulted in
(28), we have

$D\in\mathcal{L}\left(H^{\frac{1}{2}}(\Gamma_{s}),H^{1}(\Omega_{s})\right)\text{,
\ }\frac{\partial
D}{\partial\nu}\in\mathcal{L}\left(H^{\frac{1}{2}}(\Gamma_{s}),H^{-\frac{1}{2}}(\Gamma_{s})\right).$
(35)

(The latter is the “Dirichlet to Neumann” map.) Therewith, and with respect to
the thick wave displacement $w_{0}$ in (22), we set

$z=w_{0}+\frac{i}{\beta}D\left(\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right).$
(36)

Then, via (34), the variable $z$ satisfies the following boundary value
problem:

$\displaystyle-\beta^{2}z-\Delta z$ $\displaystyle=$ $\displaystyle-i\beta
D\left[\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right]+w_{1}^{\ast}+i\beta
w_{0}^{\ast}\text{ \ in }\Omega_{s};$ $\displaystyle z$ $\displaystyle=$
$\displaystyle 0\text{ \ on }\Gamma_{s}.$ (37)

Since $\Omega_{s}$ is convex then $z\in H^{2}(\Omega_{s})$ (see e.g., Theorem
3.2.1.2, p. 147 of [30]). Subsequently, we can appeal to the known Sobolev
boundary regularity results for polyhedral domains; [14, p. 43, Theorem 6.9].
In short, we have the estimate, for $\left|\beta\right|\geq 1$,

$\displaystyle\left\|z\right\|_{H^{2}(\Omega_{s})}+\sum\limits_{j=1}^{K}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{\frac{1}{2}}\left(\Gamma_{j}\right)}$
$\displaystyle\leq$ $\displaystyle
C_{0}\left(\beta^{2}\left\|z\right\|_{\Omega_{s}}+\left|\beta\right|\left\|D\left[\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right]\right\|_{\Omega_{s}}+\left\|w_{1}^{\ast}+i\beta
w_{0}^{\ast}\right\|_{\Omega_{s}}\right)$ (38) $\displaystyle\leq$
$\displaystyle C_{0}\left|\beta\right|\left(\left\|\beta
z\right\|_{\Omega_{s}}+\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right),$

after using (37), (35) and the Sobolev Imbedding Theorem.

With respect to the $z$-wave equation in (37), we appeal to the “frequency
domain” version of the well-known wave identity which is synonymous with
boundary control of wave equations; see Proposition 7 (ii) of [4], and also
[23],[43]. We adopt those wave identities to our solution component $z$ in the
following Proposition:

###### Proposition 6

Let $\mathbf{m}(x)=[m_{1}(x),m_{2}(x),m_{3}(x)]$ be an arbitrary real-valued
$[C^{2}(\bar{\Omega}_{s})]^{3}$-vector field, with associated Jacobian matrix
$M(x)$. Then the wave component of the solution to the resolvent equation (21)
obeys the following relation:

$\displaystyle\text{(i) \ }\int_{\Omega_{s}}\left|M^{\frac{1}{2}}(x)\nabla
z\right|^{2}d\Omega_{s}$ $\displaystyle=$
$\displaystyle-\text{Re}\int_{\Gamma_{s}}\frac{\partial
z}{\partial\nu}\left(\mathbf{m}\cdot\nabla\overline{z}\right)d\Gamma_{s}+\frac{1}{2}\int_{\Gamma_{s}}\left|\frac{\partial
z}{\partial\nu}\right|^{2}\mathbf{m}\cdot\mathbf{\nu}d\Gamma_{s}$ (39)
$\displaystyle+\frac{1}{2}\int_{\Omega_{s}}\left\\{\left|\nabla
z\right|^{2}-\beta^{2}\left|z\right|^{2}\right\\}\text{div}(\mathbf{m})d\Omega_{s}h$
$\displaystyle-\text{Re}\int_{\Omega_{s}}\left(i\beta
D\left[\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right]-w_{1}^{\ast}-i\beta
w_{0}^{\ast}\right)\left(\mathbf{m}\cdot\nabla\overline{z}\right)d\Omega_{s}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{\Gamma_{s}}\left|\frac{\partial
z}{\partial\nu}\right|^{2}\mathbf{m}\cdot\mathbf{\nu}d\Gamma_{s}+\frac{1}{2}\int_{\Omega_{s}}\left\\{\left|\nabla
z\right|^{2}-\beta^{2}\left|z\right|^{2}\right\\}\text{div}(\mathbf{m})d\Omega_{s}$
$\displaystyle+i\beta\text{Re}\int_{\Omega_{s}}\nabla\left(D\left[\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right]-w_{0}^{\ast}\right)\cdot\mathbf{m}\bar{z}d\Omega_{s}$
$\displaystyle+i\beta\text{Re}\int_{\Omega_{s}}\left(D\left[\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right]-w_{0}^{\ast}\right)\bar{z}\text{div}(\mathbf{m})d\Omega_{s}$
$\displaystyle+\text{Re}\int_{\Omega_{s}}w_{1}^{\ast}\left(\mathbf{m}\cdot\nabla\overline{z}\right)d\Omega_{s}.$

(ii) If $\mathbf{\tilde{m}}(x)$ is an arbitrary real-valued
$[C^{2}(\bar{\Omega}_{s})]^{3}$-vector field, then the wave component of the
solution to the resolvent equation (21) satisfies the relation,

$\displaystyle\int_{\Omega_{s}}\left\\{\left|\nabla
z\right|^{2}-\beta^{2}\left|z\right|^{2}\right\\}\text{div}(\mathbf{\tilde{m}})d\Omega_{s}$
$\displaystyle=$ $\displaystyle-\text{Re}\int_{\Omega_{s}}\left(i\beta
D\left[\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right]-w_{1}^{\ast}-i\beta
w_{0}^{\ast}\right)\overline{z}\text{div}\left(\mathbf{\tilde{m}}\right)d\Omega_{s}$
(40) $\displaystyle-\text{Re}\int_{\Omega_{s}}\left[\nabla
z\cdot\nabla\text{div}(\mathbf{\tilde{m}})\right]\bar{z}d\Omega_{s}.$

(Note that the expressions (39)-(40) each reflect the fact that $z=0$ on
$\Gamma_{s}$ and/or the unit normal vector $\nu(x)$ is pointing inward with
respect to solid geometry $\Omega_{s}$.)

Now, let the vector fields $\mathbf{m}$ and $\mathbf{\tilde{m}}$ in (39) and
(40), respectively, be taken as

$\mathbf{m}(x)=\mathbf{\tilde{m}}(x)=x.$ (41)

Then, via (35), Young’s Inequality and the Sobolev Trace Theorem, we have for
$\left|\beta\right|\geq 1$,

$\left|\int_{\Omega_{s}}\left\\{\left|\nabla
z\right|^{2}-\beta^{2}\left|z\right|^{2}\right\\}d\Omega_{s}\right|\leq\epsilon\beta^{2}\left\|z\right\|_{\Omega_{s}}^{2}+C_{\epsilon}\left(\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right).$
(42)

In turn, with vector field $\mathbf{m}$ as specified in (41), applying (42) to
the right hand side of (39), using (35), the Sobolev Trace Theorem and again
Young’s Inequality, we have the following Proposition:

###### Proposition 7

For $\left|\beta\right|\geq 1$ and $\epsilon>0$, the variable $z$ of (36) and
(37) satisfies

$\int_{\Omega_{s}}\left|\nabla z\right|^{2}d\Omega_{s}\leq
C^{\ast}\int_{\Gamma_{s}}\left|\frac{\partial
z}{\partial\nu}\right|^{2}d\Gamma_{s}+\epsilon\left(\left\|\nabla
z\right\|_{\Omega_{s}}^{2}+\beta^{2}\left\|z\right\|_{\Omega_{s}}^{2}\right)+C_{\epsilon}\left(\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right)$
(43)

(with respect to (42) there has also been a rescaling of $\epsilon>0$).

At this point, we also note in the $z$-wave relation (39) that $\mathbf{m}(x)$
could be specified to be the smooth vector field of Lemma 1.5.1.9, pg. 40 of
[30]: That is, for some $\delta>0$,
$\mathbf{m}(x)\in[C^{\infty}(\overline{\Omega}_{s})]$ satisfies

$-\mathbf{m}(x)\cdot\nu\geq\delta\text{ \ a.e. \ on }\Gamma_{s}.$ (44)

This gives in (39):

$\displaystyle\frac{\delta}{2}\int_{\Gamma_{s}}\left|\frac{\partial
z}{\partial\nu}\right|^{2}d\Gamma_{s}$ $\displaystyle\leq$
$\displaystyle\left|\int_{\Omega_{s}}\left|M^{\frac{1}{2}}(x)\nabla
z\right|^{2}d\Omega_{s}-\frac{1}{2}\int_{\Omega_{s}}\left\\{\left|\nabla
z\right|^{2}-\beta^{2}\left|z\right|^{2}\right\\}\text{div}(\mathbf{m})d\Omega_{s}\right.$
$\displaystyle-i\beta\text{Re}\int_{\Omega_{s}}\nabla\left(D\left[\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right]-w_{0}^{\ast}\right)\cdot\mathbf{m}\bar{z}d\Omega_{s}$
$\displaystyle\left.-i\beta\text{Re}\int_{\Omega_{s}}\left(D\left[\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right]-w_{0}^{\ast}\right)\bar{z}\text{div}(\mathbf{m})d\Omega_{s}-\text{Re}\int_{\Omega_{s}}w_{1}^{\ast}\left(\mathbf{m}\cdot\nabla\overline{z}\right)d\Omega_{s}\right|.$

Estimating right hand side of this relation by means of (35), Young’s
Inequality and the Sobolev Trace Theorem, we have the control of the trace
term $\frac{\partial z}{\partial\nu}|_{\Gamma_{s}}$ in the following
proposition:

###### Proposition 8

The variable $z$ of (36) and (37) satisfies

$\int_{\Gamma_{s}}\left|\frac{\partial
z}{\partial\nu}\right|^{2}d\Gamma_{s}\leq C\left(\left\|\nabla
z\right\|_{L^{2}(\Omega_{s})}^{2}+\beta^{2}\left\|z\right\|_{\Omega_{s}}^{2}+\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right).$
(45)

We should emphasize that given the right hand side of estimate (43), it is
apparent that a useful estimate of the thick wave energy component of (22)
will necessitate “decent” control of $\left.\frac{\partial
z}{\partial\nu}\right|_{\Gamma_{s}}$ in $L^{2}$-sense: that is, the positive
constant $C$ in (45) need not be “small”. However, while the estimate (45)
does not constitute such needed control, it will serve as an ingredient for
attaining that end.

Step 3: The thin wave displacement $h_{0}$

With respect to the $h$-wave equations in (22), we take the $L^{2}$-inner
product with respect to $h_{0j}$, for $j=1,...,K$. This gives

$\displaystyle-\left(\Delta
h_{0j},h_{0j}\right)_{\Gamma_{j}}+\left\|h_{0j}\right\|_{\Gamma_{j}}^{2}$
$\displaystyle=$
$\displaystyle\left(\beta^{2}h_{0j},h_{0j}\right)_{\Gamma_{j}}+\left\langle\frac{\partial
w_{0}}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}-\left\langle\frac{\partial
u}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}$ (46)
$\displaystyle+\left(h_{1j}^{\ast},h_{0j}\right)_{\Gamma_{j}}+i\beta\left(h_{0j}^{\ast},h_{0j}\right)_{\Gamma_{j}}.$

Subsequently, we invoke the Green’s Theorem (for $j=1,...,K$) to get

$\displaystyle\left\|\nabla
h_{0j}\right\|_{\Gamma_{j}}^{2}+\left\|h_{0j}\right\|_{\Gamma_{j}}^{2}-\left\langle\frac{\partial
h_{0j}}{\partial\nu},h_{0j}\right\rangle_{\partial\Gamma_{j}}$ (47)
$\displaystyle=$
$\displaystyle\beta^{2}\left\|h_{0j}\right\|_{\Gamma_{j}}^{2}+\left\langle\frac{\partial
w_{0}}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}-\left\langle\frac{\partial
u}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}$
$\displaystyle+\left(h_{1j}^{\ast},h_{0j}\right)_{\Gamma_{j}}+i\beta\left(h_{0j}^{\ast},h_{0j}\right)_{\Gamma_{j}}.$

Using the thin layer boundary conditions in (22), we then have upon summation

$\displaystyle\sum\limits_{j=1}^{K}\left[\left\|\nabla
h_{0j}\right\|_{\Gamma_{j}}^{2}+\left\|h_{0j}\right\|_{\Gamma_{j}}^{2}\right]$
$\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{K}\left[\beta^{2}\left\|h_{0j}\right\|_{\Gamma_{j}}^{2}+\left\langle\frac{\partial
w_{0}}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}-\left\langle\frac{\partial
u}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}\right.$
$\displaystyle\left.+\left(h_{1j}^{\ast},h_{0j}\right)_{\Gamma_{j}}+i\beta\left(h_{0j}^{\ast},h_{0j}\right)_{\Gamma_{j}}\right].$

Estimating right hand side by means of (33) and (26), and via the inequality

$\left\langle\frac{\partial
u}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}\leq\left(\frac{1}{\left|\beta\right|^{\frac{1}{2}}}\left\|\frac{\partial
u}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{j})}\right)\left(\left|\beta\right|^{\frac{1}{2}}\left\|h_{0j}\right\|_{H^{\frac{1}{2}}(\Gamma_{j})}\right),$

we then have for $\left|\beta\right|\geq 1$ that the thin wave solution
components of (22) obeys the following (intermediate) estimate:

$\sum\limits_{j=1}^{K}\left[\left\|\nabla
h_{0j}\right\|_{\Gamma_{j}}^{2}+\left\|h_{0j}\right\|_{\Gamma_{j}}^{2}\right]\leq\left|\sum\limits_{j=1}^{K}\left\langle\frac{\partial
w_{0}}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}\right|+C\left(\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right).$
(48)

###  5.1 An Appropriate Estimate for $\left.\frac{\partial
z}{\partial\nu}\right|_{\Gamma_{s}}$

Using the decomposition (36) and the thin wave equations in (22), we have for
$j=1,...,K,$

$\frac{\partial z}{\partial\nu}=-\beta^{2}h_{0j}-\Delta
h_{0j}+h_{0j}+\frac{\partial u}{\partial\nu}+\frac{i}{\beta}\frac{\partial
D}{\partial\nu}\left[\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right]-h_{1j}^{\ast}-i\beta
h_{0j}^{\ast}\text{ \ \ in \ }\Gamma_{j}.$ (49)

For the first term on right hand side, we use the matching velocities BC and
$h$-resolvent relation in (22) and a wellknown trace moment inequality – see
e.g., Theorem 1.6.6, p. 37 of [15] – so as to have

$\left\|\beta^{2}h_{0j}\right\|_{\Gamma_{j}}=\left\|\beta\left.u\right|_{\Gamma_{j}}+\beta\left.h_{0j}^{\ast}\right|_{\Gamma_{j}}\right\|_{\Gamma_{j}}\leq
C\left|\beta\right|\left(\left\|u\right\|_{\Omega_{f}}^{\frac{1}{2}}\left\|\nabla
u\right\|_{\Omega_{f}}^{\frac{1}{2}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right).$
(50)

Subsequently invoking the improvement over Poincaré’s Inequality in
Proposition 4, we have now for $\left|\beta\right|\geq 1$,

$\left\|\beta^{2}h_{0j}\right\|_{\Gamma_{j}}\leq
C\left(\left|\beta\right|^{\frac{3}{4}}\left\|\nabla
u\right\|_{\Omega_{f}}+\left|\beta\right|\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right).$

Applying this estimate to the right hand side of (49), along with (48), (33),
(35), and the Sobolev Trace Theorem, we then have for $\left|\beta\right|\geq
1$ (upon summing over $j=1,...,K$),

$\sum\limits_{j=1}^{K}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-1}(\Gamma_{j})}\leq
C_{1}\left(\sqrt{\sum\limits_{j=1}^{K}\left|\left\langle\frac{\partial
w_{0}}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}\right|}+C\left|\beta\right|^{\frac{3}{4}}\left\|\nabla
u\right\|_{\Omega_{f}}+\left|\beta\right|\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right).$
(51)

To refine the right hand side: Using again the decomposition (36), we have for
$\left|\beta\right|\geq 1$,

$\displaystyle\sqrt{\sum\limits_{j=1}^{K}\left|\left\langle\frac{\partial
w_{0}}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}\right|}$ $\displaystyle=$
$\displaystyle\sqrt{\sum\limits_{j=1}^{K}\left|\left\langle\frac{\partial}{\partial\nu}\left[z-\frac{i}{\beta}D([u+w_{0}^{\ast}]|_{\Gamma_{s}})\right],h_{0j}\right\rangle\right|_{\Gamma_{j}}}$
$\displaystyle=$
$\displaystyle\sqrt{\sum\limits_{j=1}^{K}\left|\left\langle\frac{1}{\beta}\frac{\partial
z}{\partial\nu},\beta
h_{0j}\right\rangle-\frac{i}{\beta}\left\langle\frac{\partial}{\partial\nu}D([u+w_{0}^{\ast}]|_{\Gamma_{s}}),h_{0j}\right\rangle\right|_{\Gamma_{j}}}.$

Applying (26), (35), Sobolev Trace Theorem and the Young’s Inequality, we then
obtain for $\left|\beta\right|\geq 1$,

$\sqrt{\sum\limits_{j=1}^{K}\left|\left\langle\frac{\partial
w_{0}}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}\right|}\leq\frac{\delta^{\ast}}{\left|\beta\right|C_{1}}\sum\limits_{j=1}^{K}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{j})}+C_{\delta^{\ast}}\left(\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right),$

where $C_{1}>0$ is the constant in (51). Applying this inequality to (51), we
have for $\left|\beta\right|\geq 1$,

$\sum\limits_{j=1}^{K}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-1}(\Gamma_{j})}\leq\frac{\delta^{\ast}}{\left|\beta\right|}\sum\limits_{j=1}^{K}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{j})}+C_{2,\delta^{\ast}}\left[\left|\beta\right|^{\frac{3}{4}}\left\|\nabla
u\right\|_{\Omega_{f}}+\left|\beta\right|\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right].$
(52)

Interpolating now between (45) and (52) we have for $j=1,...,K,$

$\begin{array}[]{l}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{j})}\leq
C\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-1}(\Gamma_{j})}^{\frac{1}{2}}\left\|\frac{\partial
z}{\partial\nu}\right\|_{\Gamma_{j}}^{\frac{1}{2}}\\\ \text{ \ }\leq
C_{3}\left\\{\left(\frac{\delta^{\ast}}{\left|\beta\right|}\sum\limits_{j=1}^{K}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{j})}+C_{2,\delta^{\ast}}\left[\left|\beta\right|^{\frac{3}{4}}\left\|\nabla
u\right\|_{\Omega_{f}}+\left|\beta\right|\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right]\right)^{\frac{1}{2}}\right.\\\
\text{ \ \ \ \ \ }\left.\times\left(\left\|\nabla
z\right\|_{\Omega_{s}}+\left\|\beta z\right\|_{\Omega_{s}}+\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right)^{\frac{1}{2}}\right\\}.\end{array}$
(53)

Subsequently, for $\left|\beta\right|\geq 1$, with constant $C_{3}$ as in (53)
and using the relation
$\left|ab\right|\leq\frac{\delta^{\ast}KC_{3}}{2\left|\beta\right|}a^{2}+\frac{\left|\beta\right|}{2\delta^{\ast}KC_{3}}b^{2}$,
we then have

$\displaystyle\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{j})}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2K}\sum\limits_{j=1}^{K}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{j})}+\frac{\delta^{\ast}KC_{3}^{2}}{2\left|\beta\right|}\left(\left\|\nabla
z\right\|_{L^{2}(\Omega_{s})}+\left\|\beta z\right\|_{\Omega_{s}}\right)$
$\displaystyle+$ $\displaystyle
C_{\delta^{\ast}}\left[\left|\beta\right|^{\frac{7}{4}}\left\|\nabla
u\right\|_{\Omega_{f}}+\left|\beta\right|^{2}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right]\text{,
\ for }j=1,...,K.$

Summing this estimate over $j$ now gives

$\displaystyle\sum\limits_{j=1}^{K}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{j})}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\sum\limits_{j=1}^{K}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{j})}+\frac{\delta^{\ast}}{\left|\beta\right|}C_{4}\left(\left\|\nabla
z\right\|_{L^{2}(\Omega_{s})}+\left\|\beta z\right\|_{\Omega_{s}}\right)$
$\displaystyle+C_{\delta^{\ast}}\left(\left|\beta\right|^{\frac{7}{4}}\left\|\nabla
u\right\|_{\Omega_{f}}+\left|\beta\right|^{2}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right)$

(where constant $C_{4}$ is independent of $\delta^{\ast}>0$). We have then for
$j=1,...,K$,

$\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{j})}\leq\frac{\delta^{\ast}}{\left|\beta\right|}C_{5}\left(\left\|\nabla
z\right\|_{\Omega_{s}}+\left\|\beta
z\right\|_{\Omega_{s}}\right)+C_{\delta^{\ast}}\left(\left|\beta\right|^{\frac{7}{4}}\left\|\nabla
u\right\|_{\Omega_{f}}+\left|\beta\right|^{2}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right).$
(54)

Subsequently, we interpolate between (54) and (38) to have,

$\begin{array}[]{l}\left\|\frac{\partial
z}{\partial\nu}\right\|_{\Gamma_{j}}\leq C_{6}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{j})}^{\frac{1}{2}}\left\|\frac{\partial
z}{\partial\nu}\right\|_{H^{\frac{1}{2}}(\Gamma_{j})}^{\frac{1}{2}}\\\ \\\
\text{ \ }\leq
C_{6}\left\\{\left[\frac{\delta^{\ast}}{\left|\beta\right|}C_{5}\left(\left\|\nabla
z\right\|_{\Omega_{s}}+\left\|\beta
z\right\|_{\Omega_{s}}\right)+C_{\delta^{\ast}}\left(\left|\beta\right|^{\frac{7}{4}}\left\|\nabla
u\right\|_{\Omega_{f}}+\left|\beta\right|^{2}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right)\right]\right.\\\
\text{ \ \ \ \ \ \ \ }\left.\times C_{0}\left|\beta\right|\left(\left\|\beta
z\right\|_{\Omega_{s}}+\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right)\right\\}^{\frac{1}{2}}\\\
\\\ \leq C_{6}\left\\{\delta^{\ast}C_{0}C_{5}\left(\left[\left\|\nabla
z\right\|_{\Omega_{s}}+\left\|\beta z\right\|_{\Omega_{s}}+\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right]^{2}\right)\right.\\\
\left.+C_{\delta^{\ast}}C_{0}\left|\beta\right|\left(\left|\beta\right|^{\frac{7}{4}}\left\|\nabla
u\right\|_{\Omega_{f}}+\left|\beta\right|^{2}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right)\left(\left\|\nabla
z\right\|_{\Omega_{s}}+\left\|\beta z\right\|_{\Omega_{s}}+\left\|\nabla
u\right\|_{\Omega_{f}}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right)\right\\}^{\frac{1}{2}}\end{array}$

Invoking once more the relation
$\left|ab\right|\leq\delta^{\ast}a^{2}+C_{\delta^{\ast}}b^{2}$, for
$\delta^{\ast}>0$, we have now

$\left\|\frac{\partial
z}{\partial\nu}\right\|_{\Gamma_{j}}\leq\delta^{\ast}C_{7}\left(\left\|\nabla
z\right\|_{\Omega_{s}}+\left\|\beta
z\right\|_{\Omega_{s}}\right)+C_{\delta^{\ast}}\left(\left|\beta\right|^{\frac{11}{4}}\left\|\nabla
u\right\|_{\Omega_{f}}+\left|\beta\right|^{3}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right).$

Summing over $j=1,...,K$, and rescaling $\delta^{\ast}>0$, we have now the
following desired trace estimate for $\left.\frac{\partial
z}{\partial\nu}\right|_{\Gamma_{s}}$:

###### Lemma 9

For $\left|\beta\right|\geq 1$ and arbitrary $\delta^{\ast}>0$, the the
variable $z$ of (36) and (37) obeys the estimate

$\left\|\frac{\partial
z}{\partial\nu}\right\|_{\Gamma_{s}}\leq\delta^{\ast}\left(\left\|\nabla
z\right\|_{\Omega_{s}}+\left\|\beta
z\right\|_{\Omega_{s}}\right)+C_{\delta^{\ast}}\left(\left|\beta\right|^{\frac{11}{4}}\left\|\nabla
u\right\|_{\Omega_{f}}+\left|\beta\right|^{3}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}\right).$
(55)

Completion of the Proof of Theorem 3

Applying (55) to right hand side of (43) of Proposition 7 (and subsequently
rescaling), we have for $\left|\beta\right|\geq 1$,

$\int_{\Omega_{s}}\left|\nabla
z\right|^{2}d\Omega_{s}\leq\epsilon\left(\left\|\nabla
z\right\|_{\Omega_{s}}^{2}+\beta^{2}\left\|z\right\|_{\Omega_{s}}^{2}\right)+C_{\epsilon}\left(\left|\beta\right|^{\frac{11}{2}}\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left|\beta\right|^{6}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right).$
(56)

In turn, we take vector field $\mathbf{\tilde{m}}(x)$ in (40) of Proposition 6
to satisfy $\text{div}(\mathbf{\tilde{m}})=1$. Afterwards, we estimate this
relation by means of (56), (35), the Sobolev Trace Theorem and the Young’s
Inequality, $\left|ab\right|\leq\rho a^{2}+C_{\rho}b^{2}$ ($1>\rho>0$) to have

$\int_{\Omega_{s}}\left|\beta
z\right|^{2}d\Omega_{s}\leq\frac{\epsilon}{\left(1-\rho\right)}\left(\left\|\nabla
z\right\|_{L^{2}(\Omega_{s})}^{2}+\beta^{2}\left\|z\right\|_{L^{2}(\Omega_{s})}^{2}\right)+C_{\epsilon,\rho}\left(\left|\beta\right|^{\frac{11}{2}}\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left|\beta\right|^{6}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right).$
(57)

In sum, for $\left|\beta\right|\geq 1$, (56) and (57) give now,

$\left\|z\right\|_{H^{1}(\Omega_{s})}^{2}+\left\|\beta
z\right\|_{\Omega_{s}}^{2}+\left\|\frac{\partial
z}{\partial\nu}\right\|_{\Gamma_{s}}^{2}\leq
C\left(\left|\beta\right|^{\frac{11}{2}}\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left|\beta\right|^{6}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right).$
(58)

Subsequently, via the change of variable in (36), (58) and the regularity for
$D$ in (35), with respect to the thick layer wave solution components in (22),
we have that for $\left|\beta\right|\geq 1$,

$\displaystyle\text{(i) }\left\|w_{0}\right\|_{H^{1}(\Omega_{s})}^{2}$
$\displaystyle=$
$\displaystyle\left\|z-\frac{i}{\beta}D\left(\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right)\right\|_{H^{1}(\Omega_{s})}^{2}$
(59) $\displaystyle\leq$ $\displaystyle
C\left(\left|\beta\right|^{\frac{11}{2}}\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left|\beta\right|^{6}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right);$
$\displaystyle\text{(ii) }\left\|w_{1}\right\|_{L^{2}(\Omega_{s})}^{2}$
$\displaystyle=$
$\displaystyle\left\|i\beta\left[z-\frac{i}{\beta}D\left(\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right)\right]-w_{0}^{\ast}\right\|_{L^{2}(\Omega_{s})}^{2}$
(60) $\displaystyle\leq$ $\displaystyle
C\left(\left|\beta\right|^{\frac{11}{2}}\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left|\beta\right|^{6}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right).$

In turn, via (48) and the decomposition (36) we get

$\displaystyle\sum\limits_{j=1}^{K}\left[\left\|\nabla
h_{0j}\right\|_{\Gamma_{j}}^{2}+\left\|h_{0j}\right\|_{\Gamma_{j}}^{2}\right]$
$\displaystyle\leq$
$\displaystyle\left|\sum\limits_{j=1}^{K}\left\langle\frac{\partial(z-\frac{i}{\beta}D\left(\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right)}{\partial\nu},h_{0j}\right\rangle_{\Gamma_{j}}\right|+C\left(\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right)$
$\displaystyle\leq$ $\displaystyle C\left(\left\|\frac{\partial
z}{\partial\nu}\right\|_{\Gamma_{s}}+\frac{1}{\left|\beta\right|}\left\|\frac{\partial\frac{i}{\beta}D\left(\left.u\right|_{\Gamma_{s}}+\left.w_{0}^{\ast}\right|_{\Gamma_{s}}\right)}{\partial\nu}\right\|_{H^{-\frac{1}{2}}(\Gamma_{s})}\right)\sum\limits_{j=1}^{K}\left\|h_{0j}\right\|_{H^{\frac{1}{2}}(\Gamma_{j})}.$

Invoking now, (58), (35), the Sobolev Trace Theorem, and (26), we have for
$\left|\beta\right|\geq 1$,

$\sum\limits_{j=1}^{K}\left[\left\|\nabla
h_{0j}\right\|_{\Gamma_{j}}^{2}+\left\|h_{0j}\right\|_{\Gamma_{j}}^{2}\right]\leq
C\left(\left|\beta\right|^{\frac{11}{2}}\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left|\beta\right|^{6}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right).$
(61)

In addition, via the resolvent relation in (22), we have for $j=1,...,K$,

$\left\|h_{1j}\right\|_{H^{\frac{1}{2}}(\Gamma_{j})}^{2}=\left\|i\beta
h_{0j}-h_{0j}^{\ast}\right\|_{H^{\frac{1}{2}}(\Gamma_{j})}^{2},$

whence by (26),

$\left\|h_{1j}\right\|_{H^{\frac{1}{2}}(\Gamma_{j})}^{2}\leq
C\left(\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right).$
(62)

Finally, collecting (59), (60), (61) and (62) we have with the solution
variable
$\Phi=[u,h_{01},h_{02},...,h_{0K},h_{0K},w_{0},w_{1}]$ that

$\left\|\Phi\right\|_{\mathbf{H}}^{2}\leq
C\left(\left|\beta\right|^{\frac{11}{2}}\left\|\nabla
u\right\|_{\Omega_{f}}^{2}+\left|\beta\right|^{6}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right).$

Invoking now the static dissipation relation (24) and Young’s Inequality one
last time, we obtain

$\displaystyle\left\|\Phi\right\|_{\mathbf{H}}^{2}$ $\displaystyle\leq$
$\displaystyle
C\left(\left|\beta\right|^{\frac{11}{2}}\left|\left(\Phi_{0}^{\ast},\Phi\right)_{\mathbf{H}}\right|+\left|\beta\right|^{6}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}\right)$
$\displaystyle\leq$
$\displaystyle\epsilon\left\|\Phi\right\|_{\mathbf{H}}^{2}+\left|\beta\right|^{11}\left\|\Phi_{0}^{\ast}\right\|_{\mathbf{H}}^{2}.$

Since data $\Phi_{0}^{\ast}\in\mathbf{H}$ in (21) was arbitrary, this gives
the desired resolvent bound

$\left\|\mathcal{R}(i\beta;\mathbf{A})\right\|_{\mathcal{L(}\mathbf{H})}\leq
C\left|\beta\right|^{\frac{11}{2}}.$

An appeal to Theorem 2 now concludes the proof of Theorem 3.

##  6 Acknowledgment

The authors G. Avalos and Pelin G. Geredeli would like to thank the National
Science Foundation, and acknowledge their partial funding from NSF Grant
DMS-1907823.

The author Boris Muha would like to thank the Croatian Science Foundation
(Hrvatska Zaklada za Znanost), and acknowledge their partial funding from
grant number IP-2018-01-3706.

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AAaffiliationtext: Dutch Tax and Customs Administration, The
NetherlandsBBaffiliationtext: Utrecht University, The Netherlands

# Human-centred explanation of rule-based decision-making systems in the legal
domain

Suzan Zuurmond  AnneMarie Borg  Matthijs van Kempen  Remi Wieten

###### Abstract

We propose a human-centred explanation method for rule-based automated
decision-making systems in the legal domain. Firstly, we establish a
conceptual framework for developing explanation methods, representing its key
internal components (content, communication and adaptation) and external
dependencies (decision-making system, human recipient and domain). Secondly,
we propose an explanation method that uses a graph database to enable
question-driven explanations and multimedia display. This way, we can tailor
the explanation to the user. Finally, we show how our conceptual framework is
applicable to a real-world scenario at the Dutch Tax and Customs
Administration and implement our explanation method for this scenario.

## Introduction

In the mid-eighties, governments worldwide started using automated decision-
making systems to improve the efficiency and effectiveness of public
administration [1]. Often, these systems are so-called rule-based systems,
with rules based on legislation. These systems have advantages over human
workers since they are consistent, cost-efficient and often expedite the
decision-making. However, they can also impact individuals’ rights or legal
quality [2], especially when there is no careful and transparent design
process [3]. 00footnotetext: This is the full version of a demo at the 36th
International Conference on Legal Knowledge and Information Systems
(JURIX’23).

As the ongoing digitisation of governance raises concerns globally, there is a
pressing need for innovation and digital technologies that improve
transparency [4]. A transparent decision-making process provides all the
relevant information on the input data, system design and processes and allows
individuals to understand this information as well as the system’s decisions
[5].

An example of an innovation that improves the transparency of legislative
automation is the Agile Law Execution Factory (ALEF) [6], in use at the Dutch
Tax and Customs Administration (DTCA). ALEF is a tool for developing and
generating service applications that can perform specific legal tasks like tax
calculations. It uses a domain-specific language called Regelspraak [7] to
allow for the specification of rules, data descriptions and test cases, which
are interpretable for legal experts and developers while being executable for
the software system itself [8]. Using Regelspraak, ALEF combines legislative
and policy analysis descriptions in interpretable knowledge representations.
While tools like ALEF could contribute to explaining automated decisions [9],
they often lack adequate explanation mechanisms.

In this paper, we introduce a human-centred explanation method for rule-based
decision-making systems within the legal domain. As a result, (i) we improve
the transparency of the design process by bridging the gap between the
transparency required by regulations and the complexity of practical
implementation, and (ii) we introduce an explanation mechanism for systems,
like the ones created using ALEF, offering a way for generating and conveying
explanations that are tailored to the needs of individuals involved with these
systems generated by such tools.

First, in Section 1, we propose a conceptual framework for explanation methods
for automated decision-making systems, based on internal components [10] and
external dependencies [11]. Then, in Section 2, we show how this framework can
be applied to the context of ALEF and the DTCA.

Subsequently, in Section 3, we develop an explanation method that integrates
graph databases with question-based explanations. This approach ensures
flexibility within the internal components and adaptability concerning the
external dependencies. Furthermore, we offer a variety of explanation formats,
including textual, tabular, and visual representations.

To illustrate the practical application of our explanation method, we
conducted a case study within the DTCA in Section 4, focusing on the tax
interest calculation system developed using the ALEF tool. We show how our
explanation method can be used to generate explanations tailored to different
target audiences. As a result, we improve the explainability of automated
decisions made at the DTCA and ensure that both the decision-making and the
applied legislation become transparent to the individuals involved with the
system

##  1 Conceptual Framework for Explanation Methods

In this section, we introduce a conceptual framework illustrating an
explanation methods’ internal components and external dependencies (see Figure
1). This framework will serve as a guide for this research.

In the literature, various analyses reveal three common aspects of
explanation: identifying relevant information and constructing justifications;
conveying explanations effectively; and tailoring explanations to the target
audience’s needs [12, 13, 10]. These components collectively form the core of
explanation methods and focus on the following questions, named after [10]:

Content

what should be included in the explanation?

Communication

how should the explanation be conveyed?

Adaptation

to whom and how should the explanation be tailored?

The effectiveness of an explanation method, as well as the explanations it
produces, are highly contextual [12]. More specifically, the objectives an
explanation needs to fulfil can differ greatly based on the domain, the
characteristics of the AI system, and the users involved [11]. So, when
developing an explanation method, one should consider the following external
dependencies:

Domain

the operational context in which the system operates;

System

properties as complexity, input or output type, and autonomy level;

Recipient

the goals and knowledge of the individuals receiving explanations.

This distinction of dependencies is represented in our primary objective: to
craft a human-centred (recipient) explanation method for a rule-based
decision-making system (system) in the legal domain (domain).

Figure 1: Overview of the components and dependencies of an explanation
method.

##  2 Automated decision-making systems at the DTCA

In this section, we leverage the conceptual framework to explore the external
dependencies within the context of the Dutch Tax and Customs Administration,
particularly focusing on decision-making systems developed using the ALEF
tool.

#### Domain.

The legal domain is complex, with continuously evolving laws and intricate
regulations. Therefore, the DTCA employs Law Analysis, outlined in [14],
ensuring clarity in translating legislation into practice, enabling
justifiable decisions, and facilitating adaptability to changing legislation
in ICT systems.

A key element of this approach is the legal analysis schema, drawing from [15,
16], offering a visual overview of legislative components and relationships
(see Figure 2). This schema identifies legal concepts like legal subject,
object, relationship, and fact, which are vital for understanding and applying
legislation. It also includes conditions and derivation rules. As these
concepts are the building blocks for constructing models used in decision-
making systems, this schema will be used to structure the explanations in
Section 3.2.

Figure 2: Diagram representing a framework for legal analysis, translated from
[14].

#### System.

The decision-making systems generated by the ALEF tool [6] function through a
rule-based reasoning process, utilising pre-defined rules that are explicitly
programmed into the systems. These rules typically take the form of
(propositional) if-then statements with distinct condition and action
components. For example, if the tax payment is not made by the specified due
date, then tax interest is applied to the unpaid amount over time. When
employing this rule within a system, once the condition of missing the
specified due date for payment is satisfied, the corresponding action entails
the imposition of tax interest.

We focus on creating automated explanations to be applicable and practical in
real-world settings like the DTCA. This means the method should generate
explanations with minimal human intervention after the initial setup.
Additional factors, such as efficiency and scalability, are beyond the scope
of this paper.

While the case study (Section 4) is aimed at explanations for decisions
derived by the ALEF tool, the explanation method is based on more generic
characteristics. This ensures that our method can also be applied to other
rule-based decision-making systems.

#### Recipient.

Individuals ask questions because they have a particular goal, such as
understanding a system’s behaviour or accomplishing specific tasks. A
question-driven framework, aided by visualisation, assists in connecting these
inquiries with their fundamental goals and provides an adaptable and effective
explanation method [17, 18].

We adapt the taxonomy of user needs from [18] to rule-based systems and the
legal domain. Table 2 presents the relevant explanation types for automated
legal decisions, divided into two categories. The _system_ category includes
global explanations on the general decision model, while the _decision_
category includes local explanations regarding a specific decision-making
instance. Moreover, we differentiate the purpose of explanation types based on
whether they describe something or clarify something [10].

Compared to [18], we add the _whether_ type to provide information on whether
certain requirements are met. The need for this question type builds on the
necessity of verification (see Section 4.1). Furthermore, we distinguish
between _how_ and _how to_ explanations, where the first clarifies how the
system generally makes certain (intermediate) decisions, while the latter
clarifies how to achieve a desired decision, given some (but not all
necessary) input values. This improves the adaptability of our explanations.

o — X[.1,c] — X[.4,l] — X[2,l] — | Type | Purpose
---|---|---
Decision | What | Describes what decision(s) it made and what input it used.
What if | Describes the system’s decision when an input value changes.
Why | Clarifies the reasoning behind the system’s decision, given the input values.
Why not | Clarifies why an alternative decision was not generated, given the input values.
How to | Clarifies how to achieve a desired decision, given some (but not all necessary) input values.
System | Input | Describes which input the system could use.
Output | Describes which decisions the system could make.
How | Clarifies how the system generally makes a certain decision.
Visualisation | Describes a simplified view of the system’s conceptual model.
Whether | Clarifies whether the system meets a certain requirement.

Table 1: Different explanation type questions for automated legal decisions,
building on the explanation types in [18].

This set of explanation types ensures that our explanation method effectively
addresses the unique goals and information needs of all individuals involved
in rule-based decision-making systems in the legal domain.

##  3 Explanation Method

We can now present a human-centred explanation method for rule-based decision-
making systems in the legal domain. We first discuss using Graph Database
Management Systems and then provide a short implementation overview.

###  3.1 Explanation through Graph Database Management Systems

We propose an explanation method using Graph Database Management Systems
(GDBMS), specifically Neo4j [19]. These make it possible to store data in
graph structures and allow for intuitive questioning of relational data and
visualising relationships between data points.

In our explanation method, we utilise GDBMS to structure our explanations
according to the legal analysis schema (Figure 2) and we combine it with
question-driven user-system interaction and multimedia display to account for
the three internal components. More specific, in GDBMS, nodes can, for
example, represent legal subjects, legal objects, requirements, rules and
variables, while the connections represent their legal relationships.
Recipients can question both the decision model and a specific decision
(focus), filter information by, for instance, focusing on a specific part,
node or relationship (level of detail) and extract causal relations between
conditions, rules and derivations of these rules (causality), thereby varying
the content of the explanation. Explanations are answers to predefined
questions that can be communicated through visual graphs, verbal text answers
and customised filtering options. Finally, GDBMS enables the creation of
different perspectives for different recipient categories, thereby adapting
the explanation to the recipient, by assuming a certain level of knowledge
(both domain and reasoning knowledge) and linking a specified goal to a set of
specific questions.

###  3.2 Implementation of the Explanation Method

Due to space constraints, we only provide a high-level sketch of the
implementation.111For the full implementation, see
https://github.com/sjzuurmond/mps_explaineo. We start by importing all the
model elements in a structured form, creating an abstract syntax graph
containing the knowledge from all the decision models. From this graph, we
create a simplified graph representing the decision models’ key structural and
semantic information. This simplification is based on the goals and knowledge
of the recipient of the explanation (e.g., in our case study in Section 4 we
used the legal analysis schema from Figure 2). To represent a specific
decision, we can instantiate the simplified graph with the values of a model
instance.

#### Object, Rule and Service Model.

In practice, a decision system uses multiple models representing its
reasoning. ALEF uses three models: the Object Model, representing the entities
and their attributes involved in decision-making; the Rule Model, representing
rules and their dependencies for making decisions; and the Service Model,
representing services used by legal professionals for decision-making by
providing input and output variables.

Variables play a central role in the decision-making process and are the
different models’ overlapping factors. Hence, a clear representation of
variables aids in understanding the decision-making process. For example,
variables represent the attributes of objects in the Object Model, the input
and output fields of services in the Service Model are mapped onto variables,
and the rules in the Rule Model use variables in their conditions or
calculations or to derive new variable values.

The Object Model consists of variables and object types with “relates_to”
relationships between object types. In the decision-making system, a
distinction is made between Boolean features and attributes of objects.
However, in line with the diagram, we will refer to them as variables. The
Service Model consists of input and output messages with input and output
relationships to the connected variables. Finally, the Rule Model consists of
rules with calculation, derivation, and condition relationships from and
toward the corresponding variables.

#### Relation Between Legal Concepts and Models.

The legal concepts in the legal analysis diagram (Figure 2) can be related to
the elements in the models used by the decision-making system. For example,
the Object Model represents legal subjects, objects, and their relationships.
Legal subjects and objects correspond to the object nodes in the Object Model,
while the connections between these nodes can represent legal relationships.
The Rule Model can represent conditions, derivation rules, and requirements
that must be fulfilled. In addition, operators can also be represented as
rules in the Rule Model. Rule nodes correspond to specific rules or
conditions, while variable-type nodes in the Rule Model represent the
variables these rules use. At last, the Service Model can represent services
and their input and output variables. Service nodes correspond to specific
services used during the decision-making process, while variable type nodes in
the Service Model represent input and output variables for these services.

Note that not all the legal concepts are represented and captured in the
models, e.g. delegation power and delegation elaboration. However, the
modeller can add other relevant information. For example, they can set the
sources of rules. This added information is extracted from the model elements
and represented as properties in the summarising graph.

##  4 Case Study

We concentrate on an application within the domain of tax law – the tax
interest calculation system. This application plays a crucial role in
determining interest rates for tax assessments in the Dutch tax landscape,
where tax professionals rely on it to calculate rates with significant
financial implications for taxpayers. Therefore, explanations are essential
for all users involved; we consider two.

First, model experts implement tax regulations into decision models. These
experts are responsible for ensuring that the application accurately reflects
the complex web of tax laws and regulations. Clear explanations not only aid
in the model’s development but also enable experts to verify that the system
aligns with legal requirements, promoting regulatory compliance. Second, legal
support professionals bridge the gap between complex tax decisions and
taxpayers seeking clarity. When communicating with taxpayers, they require a
comprehensive understanding of the underlying decision-making process to
provide concise and accurate explanations. Thus, explanations serve as a vital
contribution, fostering effective help for model experts and legal support
professionals by promoting regulatory adherence and ensuring taxpayers receive
the clarity they deserve.

We show how our method creates explanations for these two target groups.

###  4.1 Model Experts

The model expert is involved in the creation and verification of the model.
When creating the model, our explanation method offers guidance in this
modelling process by providing a visual representation of each model to aid
the textual editor of ALEF. Such a visualisation contains the variables,
objects, rules and input/output messages as nodes as well as the relationships
between these nodes. What type of nodes and relations are included depends on
the type of question. For example, when asking _What elements has the Object
Model?_ the nodes are the variables and objects, while when asking _What
elements has the Rule Model?_ there are nodes for rules as well.

The verification process for a rule-based decision-making system is essential
to ensure its proper functioning. We distinguish three types of checks to
guide the verification process:

Path checks

verify if the system utilises an element and help identify redundant elements.

Assignment checks

ensure an element is assigned a value, confirming that these elements can
provide results.

Logical checks

confirm a rule’s absence of logical contradictions in its conditions and help
identify any inconsistencies.

This distinction generalises the verification documentation as drafted by
model experts at the DTCA for the ALEF tool used in the case study, but are
generally applicable in decision-making systems. The same documentation states
several questions a modeller should answer themselves to verify the models
they made:

  * •

_Is each in and output message used by the Service?_ (path check)

  * •

_Is all input used to create the output?_ and _Can all output be created,
given the input?_ (path check)

  * •

_Is each variable used?_ (path check)

  * •

_Are all variables assigned?_ (assignment check)

The answers to these questions contain (i) a text message stating whether the
answer to the question is positive or negative; (ii) a table with for each
message or variable whether it is used or a path has been found; and (iii) a
visualisation of the textual answers, similar to those for model creation.

For example, in order to understand the created system better, the model
expert can ask _Can all output be created?_. If this is not the case, the
system shows which variables cannot be derived (e.g., the start and end date
of the interest period), in text, in a table it shows all the variables and
whether there is a path and the visualisation allows the expert to understand
the relations and see how the variables could be assigned.222The GitHub page
https://github.com/sjzuurmond/mps_explaineo contains images of the three types
of answers.

###  4.2 Legal Support Professionals

Legal support professionals work within customer service or other relevant
departments to provide legal support and explanation to data subjects
regarding the decisions made. Hence, they need to be able to answer relevant
questions for a data subject. For these questions, a distinction is made
between two types of explanation (matching two of the explanation types made
by ICO [20]):

Data explanation

explaining the data used in the decision-making system.

Rationale explanation

explaining the reasoning behind a decision made by a decision-making system.

A data explanation answers: _What decisions did the system make?_ by providing
a table with the output message, variable and assigned values; _What
(personal) information was used for these decisions?_ by providing a table
with the input message, variable and given values; and _Which rules were used
for these decisions?_ by giving all rules used in the decisions. A rationale
explanation answers: _Why is this decision made?_ by providing a textual
description of the rule and conditions that are met, aided by a visualisation
which shows the variable representing the decision, the rule that determined
this variable and the derivation relationships from this rule to the variable
as well as the conditional variables and relationships; and _Why is this
decision made? (trace)_ by providing a textual description of the rules and
conditions that are met like a trace and again aided by a visualisation.

For example, when a taxpayer asks a legal support professional why they have
to pay tax interest, the system shows the professional, in text, the
applicable rules, including a link to the specific law (e.g., taxpayer paid
taxes too late, link), and the conditions that are satisfied (e.g., the
required payment specification was made at a specific date and at least one
payment was overdue) as well as a visualisation of the conclusion, the rules
and conditions and their relations, see Figure 3.333See the GitHub page
https://github.com/sjzuurmond/mps_explaineo for the visualisation as well as
the trace variant of the explanation. The professional can then communicate
this with the taxpayer.

Figure 3: Visualisation of the explanation for _why do they have to pay tax
interest?_ , with a trace from input messages to decision.

##  5 Discussion and Conclusion

In this paper, we have proposed a framework for developing explanation methods
that offer a human-centred approach to explaining rule-based decision-making
systems in the legal domain. By focusing on content, communication, and
adaptation, the method ensures that users can obtain the information they need
to understand the decision-making process and trust its decisions. Using graph
databases further enhances the explanatory value of the method, facilitating
more effective communication and adaptation to certain individuals’ needs.
Since our explanation method is connected to the legal analysis diagram,
explanations provide insight into the decision-making system and the
legislation. We also implemented the proposed method in a real-world scenario:
an automated decision-making system used by the DTCA.

While our primary focus has been on DTCA and ALEF, our explanation method
holds significant value for a broader spectrum of automated decision-making
systems. For instance, the conceptual framework introduced in Section 1 offers
a versatile foundation applicable to a wide range of explanation methods.
Moreover, our approach successfully bridges the gap between well-established
explanation methodologies [10, 11, 12, 13] and cutting-edge technologies [6,
21]. The integration of graph databases, combined with question-driven user-
system interaction and multimedia display, exemplifies how our method
harnesses both traditional principles of human explanation and emerging
technologies to craft more effective and human-friendly explanations.

Our explanation method lacks the ability to reason independently. That is,
other than revealing relations, the graph database lacks the capability to re-
calculate or reason about decisions independently. Consequently, for the model
expert, it can answer questions related to path and assignment checks, but it
cannot perform logical checks since it cannot compare two conditions.
Similarly, for the legal professional, our explanation method can answer
‘what’ and ‘why’ questions, but it cannot provide answers to ‘what if’ and
‘why not’ questions. Extending the explanation method to include reasoning
capacity or a live integration with the decision system is part of future
work.

There are many possible directions for future work, from extending our
explanation method to allow for, e.g., dynamics in the target group and other
explanations forms, to improving the visualisation. Additionally, implementing
the explanation method for the DTCA provides a good starting point for
applying the method to other domains. We can then further explore the
flexibility of the method and its ability to cater to the unique needs of
different user groups.

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