# Parabolic approximation of quasilinear wave equations with applications in
nonlinear acoustics

Barbara Kaltenbacher1 and Vanja Nikolić2 Department of Mathematics
Alpen-Adria-Universität Klagenfurt
9020 Klagenfurt, Austria<EMAIL_ADDRESS>Department of
Mathematics
Radboud University
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands<EMAIL_ADDRESS>
###### Abstract.

This work deals with the convergence analysis of parabolic perturbations to
quasilinear wave equations on smooth bounded domains. In particular, we
consider wave equations with nonlinearities of quadratic type, which cover the
two classical models of nonlinear acoustics, the Westervelt and Kuznetsov
equations. By employing a high-order energy analysis, we obtain convergence of
their solutions to the corresponding inviscid equations’ solutions in the
standard energy norm with a linear rate, assuming small data and a
sufficiently short time. The smallness of initial data can, however, be
imposed in a lower-order norm than the one needed in the energy analysis. It
arises only from ensuring the non-degeneracy of the studied wave equations. In
addition, we address the open questions of local well-posedness of the two
classical models on bounded domains with a non-negative, possibly vanishing
sound diffusivity.

###### Key words and phrases:

quasilinear wave equations, parabolic approximation, nonlinear acoustics,
Westervelt’s equation, Kuznetsov’s equation

###### 2010 Mathematics Subject Classification:

35L05, 35L72

1Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Austria
2Department of Mathematics, Radboud University, The Netherlands

##  1\. Introduction

This work is concerned with the convergence analysis of solutions to nonlinear
wave equations in the form of

(1.1) $\displaystyle\alpha(p)p_{tt}-c^{2}\Delta p-b\Delta p_{t}=f(p_{t})$

as well as

(1.2)
$\displaystyle\alpha(\psi_{t})\psi_{tt}-c^{2}\Delta\psi-b\Delta\psi_{t}=f(\nabla\psi,\nabla\psi_{t}),$

as the damping parameter $b$ tends to zero. These equations can be understood
as perturbations of quasi-linear hyperbolic models with $b=0$ into parabolic-
like equations with $b>0$. The error analysis of such perturbations is of
interest, for example, in numerical simulations, where artificial viscosity is
needed to resolve the nonlinear steepening of quasilinear waves by taming
spurious oscillations at the wave peaks; see, for example, [9, 26, 22] for
more details on this issue. The present work also addresses the well-posedness
of (1.1) and (1.2) on bounded domains which holds uniformly for $b\geq 0$,
under the assumption of having smooth data that are small in possibly low-
order topology.
We are particularly motivated in our research by applications of high-
intensity ultrasonic waves, whose behavior is inherently nonlinear and modeled
by nonlinear wave equations of this type. These applications range from non-
invasive treatments of different medical disorders to non-destructive material
testing and urge the need to accurately model and analyze ultrasonic waves in
different settings. The thermoviscous dissipation of fluids is known to play
an important role in the propagation of sound waves. In classical acoustic
models, such as the Westervelt and Kuznetsov equations, these effects are
encompassed within the medium parameter $b$, often referred to as the sound
diffusivity; see [16, 28, 15, 13]. Thus, the present work can also be
understood as an investigation into how the behavior of solutions to nonlinear
acoustic equations that model ultrasonic propagation changes with the sound
diffusivity. We refer to Section 2 below for further details on modeling in
nonlinear acoustics. Formally, setting $b=0$ in (1.1) and (1.2) corresponds to
lossless nonlinear sound propagation through inviscid fluids or gases; see,
e.g., [2] and the references therein. Corresponding to the applications in
nonlinear acoustics, we here focus on quadratic nonlinearities

$f(p_{t})=kp_{t}^{2},\quad\alpha(p)=1-kp$

in equation (1.1) and

$f(\nabla\psi,\nabla\psi_{t})=\nabla\psi\cdot\nabla\psi_{t},\quad\alpha(\psi_{t})=1-\kappa\psi_{t}$

in equation (1.2), but wish to emphasize that our analysis carries over to
more general nonlinearities under appropriate growth conditions on $f$.
We structure the paper as follows. In Section 2, we set up the problem,
provide an overview of related results, and discuss the main contributions of
the present work. Section 3 is dedicated to the well-posedness of a linear
version of equation (1.1). In particular, the main aim of that section is to
arrive at a higher-order energy bound that is uniform with respect to the
parameter $b$. In Section 4, we employ Banach’s fixed-point theorem on a
suitably defined mapping to analyze (1.1) when the data are sufficiently
smooth and small. Section 5 contains the convergence analysis for this
equation in the standard energy norm as well as a higher-order norm as
$b\rightarrow 0^{+}$ for small data. In Section 6, we extend our energy
analysis to (1.2). Section 7 is then devoted to the convergence rate of
solutions to this equation as $b\rightarrow 0^{+}$. We conclude the paper with
a discussion on the obtained results and an outlook on future work.

##  2\. Modeling and problem setting

Propagation of nonlinear sound waves through thermoviscous fluids is often
modeled by the Kuznetsov equation

(2.1)
$\psi_{tt}-c^{2}\Delta\psi-b\Delta\psi_{t}=\left(\frac{1}{c^{2}}\frac{B}{2A}\psi_{t}^{2}+|\nabla\psi|^{2}\right)_{t},$

given in terms of the acoustic velocity potential $\psi=\psi(x,t)$; see [15,
13] for its derivation. The coefficient $c>0$ stands for the speed of sound in
the fluid, whereas the ratio $B/A$ arises from the equation of state, which
relates variations of the acoustic pressure to the ones of medium density. The
dissipative mechanisms due to the viscosity and thermal conductivity of the
medium are accounted for via the coefficient $b$, known as the sound
diffusivity; cf. [16].
In many instances, cumulative nonlinear effects in sound propagation dominate
the local ones. This the case, for example, when the traveling distance
becomes much greater than a wavelength; see [8] for a detailed discussion on
this topic. The Westervelt equation is then a suitable second-order
approximation of the full set of governing equations. In terms of the acoustic
potential, it is given by

(2.2)
$\psi_{tt}-c^{2}\Delta\psi-b\Delta\psi_{t}=\frac{1}{c^{2}}\left(\frac{B}{2A}+1\right)(\psi_{t}^{2})_{t}.$

However, more commonly it is expressed via the acoustic pressure
$p=\varrho\psi_{t}$ as

(2.3) $p_{tt}-c^{2}\Delta p-b\Delta p=\frac{1}{\varrho
c^{2}}\left(\frac{B}{2A}+1\right)(p^{2})_{tt},$

where $\varrho$ denotes the medium density; see [28] for its original
derivation in the lossless form. The case $b=0$ corresponds to sound
propagation through lossless media.
Motivated by these classical models in nonlinear acoustics, we here study the
behavior of solutions to nonlinear wave equations in the form of

(2.4)
$\psi_{tt}-c^{2}\Delta\psi-b\Delta\psi_{t}=\frac{1}{2}(\kappa\psi_{t}^{2}+\sigma|\nabla\psi|^{2})_{t},$

and

(2.5) $p_{tt}-c^{2}\Delta p-b\Delta p=\frac{1}{2}(kp^{2})_{tt}$

as the positive parameter $b$ vanishes. Particular choices of $k$, $\kappa$,
$\sigma\in\mathbb{R}$ will lead to one of the above classical models of
nonlinear acoustics. Henceforth, we will refer to (2.4) and (2.5) as the
Kuznetsov and Westervelt equation, respectively.

###  2.1. Related literature

The global well-posedness of equations (2.4) and (2.5) for fixed positive
$b>0$ in the case of homogeneous Dirichlet boundary conditions and small,
smooth initial data has been studied in [18, 14] and [11, 17], respectively.
In the inviscid case with $b=0$, smooth solutions can be expected to exist
only locally in time. This observation has been made rigorous in [5] for the
Westervelt equation. The Kuznetsov and Westervelt equations with $b>0$ have
also been analyzed in [1] and [12], respectively, in the limit of third-order
acoustic models in time for a vanishing thermal relaxation parameter.
In $\mathbb{R}^{n}$, local well-posedness of the inviscid Westervelt and
Kuznetsov equations in potential form follows as a particular case of the
analysis performed in [10]. The Cauchy problem for the Kuznetsov equation and
its inviscid version has been further studied in [4] for small enough data. In
[23], convergence of solutions in $\mathbb{R}^{3}$ in the vanishing sound
diffusivity limit for an enhanced version of the Kuznetsov equation that
involves additionally a $\psi_{t}\Delta\psi$ term on the right-hand side has
been studied. Compared to our results in Theorem 7.1 below, convergence is
obtained in $d=3$ in a higher-order norm for small initial data in
$H^{2+l}(\mathbb{R}^{3})\times H^{1+l}(\mathbb{R}^{3})$, where $l>1$,
$l-1/2\notin 2\mathbb{N}$; see [23, Theorem 4].

###  2.2. Main contributions

The novel contributions of our work pertain to the convergence analysis of the
solutions to (2.4) and (2.5) with homogeneous Dirichlet conditions as
$b\rightarrow 0^{+}$ and where data are allowed to be large in higher-order
topologies. We prove convergence of solutions in the standard energy norm with
a linear rate to the solutions of the corresponding inviscid equations:

(2.6) $\displaystyle\|p^{(b)}-p^{(0)}\|_{\textup{E}}\lesssim b,$
$\displaystyle\|\psi^{(b)}-\psi^{(0)}\|_{\textup{E}}\lesssim b,$

as $b\rightarrow 0^{+}$. These results are contained in Theorems 5.1 and 7.1
below. Provided that the final time is short enough, the smallness conditions
on the initial data can be imposed in a lower-order norm than the one needed
in the energy analysis. We expect these estimates to be of practical relevance
in numerical simulations of quasilinear waves, where artificial dissipation is
commonly introduced to combat the spurious oscillations that arise as the
wavefront steepens, also known as Gibbs oscillations; cf. [27, 13]. As a by-
product of our analysis, we also address the questions of local well-posedness
of (2.4) and (2.5) on bounded domains for $b\geq 0$, which, to the best of our
knowledge, have not been answered before for $b=0$ in the Kuznetsov case
(2.4). Further, compared to the existing result on analysis of the inviscid
Westervelt equation [5], we allow for arbitrarily large data in
$H^{2}(\Omega)\times H^{1}(\Omega)$, as relevant in practical ultrasonic
applications. We point the readers to Theorems 4.1 and 6.1 below for the
respective results.
The convergence analysis of quasi-linear wave equations appears to be
fundamentally different from the linear analysis performed in [21] in that it
requires stronger regularity of initial data to obtain bounds that are
independent of $b$. Namely,

$(p(0),p_{t}(0))\in H^{3}(\Omega)\times H^{2}(\Omega)$

in the Westervelt, and

$(\psi(0),\psi_{t}(0))\in H^{4}(\Omega)\times H^{3}(\Omega)$

in the Kuznetsov case, as compared to

$(p(0),p_{t}(0))\in H^{1}(\Omega)\times L^{2}(\Omega)$

in the linear case. This higher regularity assumption allows us to obtain an
$O(b)$ convergence rate compared to [21, Proposition 7], where an
$O(\sqrt{b})$ convergence rate is established with $H^{1}(0,T;H^{1}(\Omega))$
regularity of the solution (that is, $H^{2}(\Omega)\times H^{1}(\Omega)$ –
actually a bit less – regularity of the initial data $(p(0),p_{t}(0))$). Note
that we also provide an $O(\sqrt{b})$ rate under the initial regularity
$(p(0),p_{t}(0))\in H^{3}(\Omega)\times H^{2}(\Omega)$ for the Westervelt
equation in a stronger norm than the standard energy one; cf. Remark 2.

###  2.3. Notation and useful inequalities

For simplicity of notation, we often omit the time interval and the spatial
domain when writing norms; i.e., $\|\cdot\|_{L^{p}(L^{q})}$ denotes the norm
on $L^{p}(0,T;L^{q}(\Omega))$. While $H^{s}(\Omega)$ denotes the usual Sobolev
spaces, we will denote by

(2.7) $\displaystyle{H_{\diamondsuit}^{2}(\Omega)}=$
$\displaystyle\,H_{0}^{1}(\Omega)\cap H^{2}(\Omega),$
$\displaystyle{H_{\diamondsuit}^{3}(\Omega)}=$ $\displaystyle\,\left\\{u\in
H^{3}(\Omega)\,:\,\mbox{tr}_{\partial\Omega}u=0,\
\mbox{tr}_{\partial\Omega}\Delta u=0\right\\},$
$\displaystyle{H_{\diamondsuit}^{4}(\Omega)}=$ $\displaystyle\,\left\\{u\in
H^{4}(\Omega)\,:\,\mbox{tr}_{\partial\Omega}u=0,\
\mbox{tr}_{\partial\Omega}\Delta u=0\right\\},$

spaces of functions
$H_{0}^{2}(\Omega)\subset{H_{\diamondsuit}^{2}(\Omega)}\subset H^{2}(\Omega)$,
$H_{0}^{3}(\Omega)\subset{H_{\diamondsuit}^{3}(\Omega)}\subset H^{3}(\Omega)$,
and $H_{0}^{4}(\Omega)\subset{H_{\diamondsuit}^{4}(\Omega)}\subset
H^{4}(\Omega)$ satisfying certain boundary conditions. By
$\hookrightarrow\hookrightarrow$ we denote compact embeddings, by
$\hookrightarrow$ continuous ones.
Throughout the paper, we often employ boundedness of the operator
$(-\Delta)^{-1}:L^{2}(\Omega)\to{H_{\diamondsuit}^{2}(\Omega)}$ and the
Poincaré–Friedrichs inequality:

(2.8) $\displaystyle\|v\|_{H^{2}(\Omega)}\leq C_{(-\Delta)^{-1}}\|-\Delta
v\|_{L^{2}(\Omega)},\quad$ $\displaystyle v\in{H_{\diamondsuit}^{2}(\Omega)};$
$\displaystyle\|v\|_{H^{1}(\Omega)}\leq C_{\textup{PF}}\|\nabla
v\|_{L^{2}(\Omega)},\quad$ $\displaystyle v\in H^{1}_{0}(\Omega).$

Furthermore, we rely on the continuous embeddings
$H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$ and $H^{2}(\Omega)\hookrightarrow
L^{\infty}(\Omega)$:

(2.9) $\displaystyle\|v\|_{L^{6}(\Omega)}\leq C_{H^{1},L^{6}}\|\nabla
v\|_{L^{2}(\Omega)},\quad$ $\displaystyle v\in H^{1}_{0}(\Omega)$
$\displaystyle\|v\|_{L^{\infty}(\Omega)}\leq C_{H^{2},L^{\infty}}\|\Delta
v\|_{L^{2}(\Omega)},\quad$ $\displaystyle v\in{H_{\diamondsuit}^{2}(\Omega)}.$

We often use $x\lesssim y$ to denote $x\leq Cy$, where the constant $C>0$ does
not depend on the sound diffusivity $b$, but may depend on other medium
parameters and the final time $T$.

##  3\. Energy analysis of the corresponding linear problem for the Westervelt
equation

In this section, we study the linear version of the initial-boundary value
problem for the Westervelt equation:

(3.1) $\left\\{\begin{aligned} \alpha p_{tt}-b\Delta p_{t}-c^{2}\Delta
p-kq_{t}p_{t}=&\,0&&\quad\text{ in }\Omega\times(0,T),\\\\[2.84526pt]
p=&\,0&&\quad\text{ on }\partial\Omega\times(0,T),\\\\[2.84526pt]
(p,p_{t})=&\,(p_{0},p_{1})&&\quad\mbox{ in
}\Omega\times\\{0\\},\end{aligned}\right.$

where $q$ is a given function, $\alpha=1-kq$, and $k\in\mathbb{R}$. Our aim is
to obtain a unique solution that is uniformly bounded with respect to $b$. We
analyze the problem under the non-degeneracy assumption on the coefficient
$\alpha=\alpha(x,t)$. In other words, we assume that there exist
$\overline{\alpha}$, $\underline{\alpha}>0$, such that

(3.2) $\underline{\alpha}\leq\alpha(x,t)\leq\overline{\alpha}\ \mbox{ on
}\Omega\ \ \text{ a.e. in }\Omega\times(0,T).$

The coefficient $\alpha$ is required to have the following regularity:

(3.3) $\displaystyle\alpha\in
X^{\textup{W}}_{\alpha}=L^{\infty}(0,T;H^{3}(\Omega))\cap
W^{1,\infty}(0,T;H^{2}(\Omega)).$

Note that then

(3.4) $q_{t}=-\frac{1}{k}\alpha_{t}\in L^{\infty}(0,T;H^{2}(\Omega).$

The space $X^{\textup{W}}_{\alpha}$ is equipped with the norm

(3.5)
$\displaystyle\|\alpha\|_{X^{\textup{W}}_{\alpha}}=\operatorname*{ess\,sup}_{t\in(0,T)}\|\alpha(t)\|_{H^{3}}+\operatorname*{ess\,sup}_{t\in(0,T)}\|\alpha_{t}(t)\|_{H^{2}}.$

Moreover, we impose the following regularity on the initial data:

(3.6) $(p_{0},p_{1})\in
X^{\textup{W}}_{0}={H_{\diamondsuit}^{3}(\Omega)}\times{H_{\diamondsuit}^{2}(\Omega)}.$

We claim that problem (3.1) has a unique solution which satisfies an energy
bound that does not degenerate as $b\rightarrow 0^{+}$. Note that since we are
interested in the limit as $b\rightarrow 0^{+}$, we may focus our attention on
an interval $[0,\bar{b})$ for some $\bar{b}>0$ without loss of generality.

######  Proposition 3.1.

Assume that the domain $\Omega\subset\mathbb{R}^{n}$, where $n\in\\{1,2,3\\}$,
is bounded and $C^{3}$ regular. Let $b\in[0,\bar{b})$ and let $T>0$ be a given
final time. Assume that the non-degeneracy condition (3.2) and the regularity
assumptions (3.3) and (3.6) hold. Then there exists a unique solution $p$ of
the problem (3.1), such that

(3.7) $\displaystyle p\in\,X^{\textup{W}}=$
$\displaystyle\,\begin{multlined}L^{\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)})\cap
W^{1,\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\\\ \cap
H^{2}(0,T;H^{1}(\Omega)).\end{multlined}L^{\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)})\cap
W^{1,\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\\\ \cap
H^{2}(0,T;H^{1}(\Omega)).$

If $b>0$, then additionally $p\in H^{1}(0,T;{H_{\diamondsuit}^{3}(\Omega)})$.
On the other hand, if $b=0$, then additionally $p\in
W^{2,\infty}(0,T;H^{1}(\Omega))$.
Furthermore, the solution fulfills the estimate

(3.8) $\displaystyle\begin{multlined}\|\nabla
p_{tt}\|_{L^{2}(L^{2})}^{2}+\operatorname*{ess\,sup}_{t\in(0,T)}\|\Delta
p_{t}(t)\|_{L^{2}}^{2}\,+\operatorname*{ess\,sup}_{t\in(0,T)}\|\nabla\Delta
p(t)\|_{L^{2}}^{2}+b\|\nabla\Delta
p_{t}\|_{L^{2}(L^{2})}^{2}\end{multlined}\|\nabla
p_{tt}\|_{L^{2}(L^{2})}^{2}+\operatorname*{ess\,sup}_{t\in(0,T)}\|\Delta
p_{t}(t)\|_{L^{2}}^{2}\,+\operatorname*{ess\,sup}_{t\in(0,T)}\|\nabla\Delta
p(t)\|_{L^{2}}^{2}+b\|\nabla\Delta p_{t}\|_{L^{2}(L^{2})}^{2}$
$\displaystyle\lesssim$ $\displaystyle\,\begin{multlined}\|\nabla\Delta
p_{0}\|_{L^{2}}^{2}+\|\Delta p_{1}\|_{L^{2}}^{2},\end{multlined}\|\nabla\Delta
p_{0}\|_{L^{2}}^{2}+\|\Delta p_{1}\|_{L^{2}}^{2},$

where the hidden constant tends to infinity as $T\rightarrow\infty$, but does
not depend on $b$.

If additionally $p_{1}\in{H_{\diamondsuit}^{3}(\Omega)}$, $q$, $\alpha\in
X^{\textup{W}}$ with

(3.9) $q(0)=p_{0}\mbox{ and }q_{t}(0)=p_{1},$

and the compatibility condition

(3.10) $p_{tt}(0)=\alpha(0)^{-1}(b\Delta p_{1}+c^{2}\Delta
p_{0}+kq_{t}(0)p_{1})$

holds, we obtain the enhanced estimate

(3.11)
$\displaystyle\begin{multlined}\|p_{ttt}\|_{L^{2}(L^{2})}^{2}+\operatorname*{ess\,sup}_{t\in(0,T)}\|\nabla
p_{tt}(t)\|_{L^{2}}^{2}+\operatorname*{ess\,sup}_{t\in(0,T)}\|\Delta
p_{t}(t)\|_{L^{2}}^{2}\\\ +\operatorname*{ess\,sup}_{t\in(0,T)}\|\nabla\Delta
p(t)\|_{L^{2}}^{2}+b\int_{0}^{T}\|\Delta
p_{tt}\|_{L^{2}}^{2}\,\textup{d}t+b\int_{0}^{T}\|\nabla\Delta
p_{t}\|_{L^{2}}^{2}\,\textup{d}t\end{multlined}\|p_{ttt}\|_{L^{2}(L^{2})}^{2}+\operatorname*{ess\,sup}_{t\in(0,T)}\|\nabla
p_{tt}(t)\|_{L^{2}}^{2}+\operatorname*{ess\,sup}_{t\in(0,T)}\|\Delta
p_{t}(t)\|_{L^{2}}^{2}\\\ +\operatorname*{ess\,sup}_{t\in(0,T)}\|\nabla\Delta
p(t)\|_{L^{2}}^{2}+b\int_{0}^{T}\|\Delta
p_{tt}\|_{L^{2}}^{2}\,\textup{d}t+b\int_{0}^{T}\|\nabla\Delta
p_{t}\|_{L^{2}}^{2}\,\textup{d}t$ $\displaystyle\lesssim$
$\displaystyle\,\begin{multlined}\|\nabla\Delta p_{0}\|_{L^{2}}^{2}+\|\Delta
p_{1}\|_{L^{2}}^{2}+b\|\nabla\Delta
p_{1}\|_{L^{2}}^{2}.\end{multlined}\|\nabla\Delta p_{0}\|_{L^{2}}^{2}+\|\Delta
p_{1}\|_{L^{2}}^{2}+b\|\nabla\Delta p_{1}\|_{L^{2}}^{2}.$

In case $b=0$, the estimate (3.11) holds without the additional assumption
$p_{1}\in{H_{\diamondsuit}^{3}(\Omega)}$. If $b>0$, then additionally $p\in
W^{1,\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)})$.

###### Proof.

We carry out the proof via energy analysis of suitable Galerkin approximations
in space. We refer to, for example, [6, 19] for general details on the method
and [12] for its use in the analysis of nonlinear acoustic models. Compared to
similar energy arguments in the literature, the challenge here is to arrive at
a bound that does not degenerate as sound diffusivity approaches zero.
We note that $p\in X^{\textup{W}}$ implies

(3.12) $\displaystyle p\in C_{w}([0,T];{H_{\diamondsuit}^{3}(\Omega)}),\qquad
p_{t}\in C_{w}([0,T];{H_{\diamondsuit}^{2}(\Omega)}),$

and so the initial conditions are meaningful.

#### Semi-discretization in space

Take $\\{w_{i}\\}_{i\in\mathbb{N}}$ to be the complete set of (smooth)
eigenfunctions of $-\Delta$ on $H_{0}^{1}(\Omega)$. Fix $n\in\mathbb{N}$ and
denote $V_{n}=\text{span}\\{w_{1},\ldots,w_{n}\\}$. We seek an approximate
solution in the form of

(3.13) $\displaystyle p^{n}=$
$\displaystyle\,\sum_{i=1}^{n}\xi_{i}(t)w_{i}(x),$

where the unknown coefficients are $\xi_{i}:(0,T)\rightarrow\mathbb{R}$,
$i\in[1,n]$. We choose the approximate initial data

(3.14) $\displaystyle(p^{n}(0),p^{n}_{t}(0))=(p^{n}_{0},p^{n}_{1})\in
V_{n}\times V_{n}$

such that

(3.15) $\displaystyle p^{n}_{0}\rightarrow p_{0}\quad\text{ in
}{H_{\diamondsuit}^{3}(\Omega)},$ $\displaystyle p^{n}_{1}\rightarrow
p_{1}\quad\text{ in }{H_{\diamondsuit}^{2}(\Omega)},$

which is possible since $V_{n}$ is dense in ${H_{\diamondsuit}^{3}(\Omega)}$.
We then study the following semi-discrete approximation of our problem:

(3.16) $\displaystyle\begin{cases}(p_{tt}^{n}-\tfrac{b}{\alpha}\Delta
p_{t}^{n}-\tfrac{c^{2}}{\alpha}\Delta
p^{n}-\tfrac{k}{\alpha}q_{t}p_{t}^{n},\phi)_{L^{2}}=0,\\\\[2.84526pt]
\text{for every $\phi\in V_{n}$ pointwise a.e. in $(0,T)$},\\\\[2.84526pt]
(p^{n}(0),p_{t}^{n}(0))=(p^{n}_{0},p^{n}_{1}).\end{cases}$

To arrive at a matrix equation, we also introduce matrices $M^{n}=[M_{ij}]$,
$K^{n}=[K_{ij}]$ and $C^{n}=[C_{ij}]$ as

(3.17) $\displaystyle M^{n}_{ij}=(w_{i},w_{j})_{L^{2}},$ $\displaystyle
K^{n}_{ij}(t)=-(\tfrac{c^{2}}{\alpha(t)}\Delta w_{i},w_{j})_{L^{2}},\quad
C^{n}_{ij}(t)=-(\tfrac{b}{\alpha(t)}\Delta
w_{i},w_{j})_{L^{2}}-(\tfrac{k}{\alpha(t)}q_{t}(t)w_{i},w_{j})_{L^{2}},$

for $i,j\in[1,n]$. By setting $\xi^{n}=[\xi_{1}\ldots\xi_{n}]^{T}$,
$\xi_{0}^{n}=[\xi_{1,0}\ldots\xi_{n,0}]^{T}$, and
$\xi_{1}^{n}=[\xi_{1,1}\ldots\xi_{n,n}]^{T}$, problem (3.16) can be further
rewritten as

(3.18)
$\displaystyle\begin{cases}M^{n}\xi^{n}_{tt}+C^{n}(t)\xi_{t}^{n}+K^{n}(t)\xi^{n}=0,\\\
(\xi^{n}(0),\xi^{n}_{t}(0))=(\xi^{n}_{0},\xi^{n}_{1}).\end{cases}$

After additionally rewriting (3.18) as a first-order system and owing to the
fact that the matrix $M^{n}$ is positive definite, existence of an absolutely
continuous solution $[\xi^{n},\xi^{n}_{t}]^{T}$ on $[0,T_{n}]$ for some
$T_{n}\leq T$ follows by ODE existence theory; see, for example, [19, §1]. To
see that $\xi^{n}\in H^{2}(0,T_{n})$, we can employ a standard bootstrap
argument; cf. [12, §3]. We thus conclude that problem (3.16) has a solution
$p^{n}\in H^{2}(0,T_{n};V_{n})\subset C^{1}([0,T_{n}];V_{n})$.

#### A priori estimates

Our next goal is to derive a uniform in $n$ estimate for the solution of
(3.16). Keeping in mind that we wish to arrive at a higher-order bound, we
test the semi-discrete problem with $\phi=\Delta^{2}p^{n}_{t}\in V_{n}$ to
arrive at

(3.19) $\displaystyle(p_{tt}^{n}-\tfrac{b}{\alpha}\Delta
p_{t}^{n}-\tfrac{c^{2}}{\alpha}\Delta
p^{n}-\tfrac{k}{\alpha}q_{t}p_{t}^{n},\Delta^{2}p^{n}_{t})_{L^{2}}=0,$

a.e. in time. Whenever the temporal argument is skipped here and in the
following, we suppose the stated (in)equality to hold for almost every time
instance in $(0,T_{n})$. We have chosen our discrete space so that
$p_{tt}^{n}=\Delta p^{n}_{t}=0$ on $\partial\Omega$ and, therefore, we have

(3.20) $\displaystyle-(\alpha^{-1}\Delta p^{n},\Delta^{2}p_{t}^{n})_{L^{2}}=$
$\displaystyle\,(\Delta p^{n}\nabla[\alpha^{-1}]+\alpha^{-1}\nabla\Delta
p^{n},\nabla\Delta p_{t}^{n})_{L^{2}}$ $\displaystyle=$
$\displaystyle\,\begin{multlined}(\Delta p^{n}\nabla[\alpha^{-1}],\nabla\Delta
p_{t}^{n})_{L^{2}}\\\
+\frac{1}{2}\frac{\textup{d}}{\textup{d}t}(\alpha^{-1}\nabla\Delta
p^{n},\nabla\Delta p^{n})_{L^{2}}-\frac{1}{2}([\alpha^{-1}]_{t}\nabla\Delta
p^{n},\nabla\Delta p^{n})_{L^{2}}\end{multlined}(\Delta
p^{n}\nabla[\alpha^{-1}],\nabla\Delta p_{t}^{n})_{L^{2}}\\\
+\frac{1}{2}\frac{\textup{d}}{\textup{d}t}(\alpha^{-1}\nabla\Delta
p^{n},\nabla\Delta p^{n})_{L^{2}}-\frac{1}{2}([\alpha^{-1}]_{t}\nabla\Delta
p^{n},\nabla\Delta p^{n})_{L^{2}}$ $\displaystyle=$
$\displaystyle\,\begin{multlined}-(\nabla\Delta
p^{n}\cdot\nabla[\alpha^{-1}]+\Delta p^{n}\Delta[\alpha^{-1}],\Delta
p_{t}^{n})_{L^{2}}\\\
+\frac{1}{2}\frac{\textup{d}}{\textup{d}t}(\alpha^{-1}\nabla\Delta
p^{n},\nabla\Delta p^{n})_{L^{2}}-\frac{1}{2}([\alpha^{-1}]_{t}\nabla\Delta
p^{n},\nabla\Delta p^{n})_{L^{2}}.\end{multlined}-(\nabla\Delta
p^{n}\cdot\nabla[\alpha^{-1}]+\Delta p^{n}\Delta[\alpha^{-1}],\Delta
p_{t}^{n})_{L^{2}}\\\
+\frac{1}{2}\frac{\textup{d}}{\textup{d}t}(\alpha^{-1}\nabla\Delta
p^{n},\nabla\Delta p^{n})_{L^{2}}-\frac{1}{2}([\alpha^{-1}]_{t}\nabla\Delta
p^{n},\nabla\Delta p^{n})_{L^{2}}.$

Furthermore, it holds

(3.21) $\displaystyle\Delta[\alpha^{-1}q_{t}\,p_{t}^{n}]$
$\displaystyle=2\alpha^{-1}\nabla q_{t}\cdot\nabla
p_{t}^{n}+\alpha^{-1}q_{t}\Delta p_{t}^{n}+\alpha^{-1}p_{t}^{n}\Delta q_{t}$
$\displaystyle\qquad+\Delta[\alpha^{-1}]\,q_{t}p_{t}^{n}+2\nabla[\alpha^{-1}]\cdot\nabla
q_{t}\,p_{t}^{n}+2\nabla[\alpha^{-1}]\cdot\nabla p_{t}^{n}\,q_{t}.$

Thus, we arrive at the following energy identity:

(3.22) $\displaystyle\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\Delta
p_{t}^{n}\|_{L^{2}}^{2}+b\|\sqrt{\alpha^{-1}}\nabla\Delta
p_{t}^{n}\|_{L^{2}}^{2}+c^{2}\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\sqrt{\alpha^{-1}}\nabla\Delta
p^{n}\|_{L^{2}}^{2}$ $\displaystyle=k(2\alpha^{-1}\nabla q_{t}\cdot\nabla
p_{t}^{n}+\alpha^{-1}q_{t}\Delta p_{t}^{n}+\alpha^{-1}p_{t}^{n}\Delta
q_{t}+2\nabla[\alpha^{-1}]\cdot\nabla
p_{t}^{n}\,q_{t}+2\nabla[\alpha^{-1}]\cdot\nabla q_{t}\,p_{t}^{n}$
$\displaystyle\qquad+\Delta[\alpha^{-1}]\,q_{t}p_{t}^{n},\Delta
p_{t}^{n})_{L^{2}}$ $\displaystyle\quad-b(\Delta
p_{t}^{n}\nabla[\alpha^{-1}],\nabla\Delta
p_{t}^{n})_{L^{2}}+c^{2}(\nabla\Delta p^{n}\cdot\nabla[\alpha^{-1}]+\Delta
p^{n}\Delta[\alpha^{-1}],\Delta p_{t}^{n})_{L^{2}}$
$\displaystyle\quad+c^{2}\frac{1}{2}([\alpha^{-1}]_{t}\nabla\Delta
p^{n},\nabla\Delta p^{n})_{L^{2}}.$

We can rely on the following estimates to handle the $\alpha$ terms on the
right:

$\displaystyle\|\alpha^{-1}\|_{L^{\infty}}\leq$
$\displaystyle\,\tfrac{1}{\underline{\alpha}},$
$\displaystyle\|\nabla[\alpha^{-1}]\|_{L^{\infty}}=$
$\displaystyle\,\|\alpha^{-2}\nabla\alpha\|_{L^{\infty}}\leq\tfrac{1}{\underline{\alpha}^{2}}|k|C_{H^{2},L^{\infty}}\|\nabla\Delta
q\|_{L^{2}}=:\tilde{C}_{1}\,|k|\|\nabla\Delta q\|_{L^{2}}$
$\displaystyle\|\Delta[\alpha^{-1}]\|_{L^{4}}=$
$\displaystyle\,\|-2\alpha^{-3}|\nabla\alpha|^{2}+\alpha^{-2}\Delta\alpha\|_{L^{4}},$
$\displaystyle\leq$
$\displaystyle\,\tfrac{2}{\underline{\alpha}^{3}}|k|^{2}C_{H^{2},L^{8}}\|\nabla\Delta
q\|_{L^{2}}^{2}+\tfrac{1}{\underline{\alpha}^{2}}|k|C_{H^{1},L^{4}}\|\nabla\Delta
q\|_{L^{2}}$ $\displaystyle=$ $\displaystyle\,:|k|\|\nabla\Delta
q\|_{L^{2}}(\tilde{C}_{2}+\tilde{C}_{3}\,|k|\|\nabla\Delta q\|_{L^{2}})$

as well as

$\displaystyle\|[\alpha^{-1}]_{t}\|_{L^{\infty}}$
$\displaystyle=\|\alpha^{-2}\alpha_{t}\|_{L^{\infty}}\leq\tfrac{1}{\underline{\alpha}^{2}}|k|C_{H^{2},L^{\infty}}\|\Delta
q_{t}\|_{L^{2}}=:\tilde{C}_{4}|k|\|\Delta q_{t}\|_{L^{2}}.$

Combined with Young’s inequality, this approach yields the energy estimate

(3.23) $\displaystyle\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\Delta
p_{t}^{n}\|_{L^{2}}^{2}+b\frac{1}{\overline{\alpha}}\,\|\nabla\Delta
p_{t}^{n}\|_{L^{2}}^{2}+c^{2}\frac{1}{\overline{\alpha}}\,\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\nabla\Delta
p^{n}\|_{L^{2}}^{2}$ $\displaystyle\leq$
$\displaystyle\,\begin{multlined}\left(2\tfrac{|k|}{\underline{\alpha}}((C_{H^{1},L^{4}})^{2}+C_{H^{2},L^{\infty}})\right.\\\
\left.+|k|\|\nabla\Delta
q\|_{L^{2}}\left(4\tilde{C}_{1}C_{\textup{PF}}C_{H^{2},L^{\infty}}+\bigl{(}\tilde{C}_{2}+\tilde{C}_{3}\,|k|\|\nabla\Delta
q\|_{L^{2}}\bigr{)}C_{H^{2},L^{8}}\right)\right)\\\ \times\|\Delta
q_{t}\|_{L^{2}}\,\|\Delta p_{t}^{n}\|_{L^{2}}^{2}\\\
+\frac{b\overline{\alpha}}{2}\tilde{C}_{1}^{2}\,|k|^{2}\|\nabla\Delta
q\|_{L^{2}}^{2}\|\Delta
p_{t}^{n}\|_{L^{2}}^{2}+\frac{b}{2\overline{\alpha}}\|\nabla\Delta
p_{t}^{n}\|_{L^{2}}^{2}\\\ +c^{2}|k|\|\nabla\Delta
q\|_{L^{2}}\bigl{(}\tilde{C}_{1}+(\tilde{C}_{2}+\tilde{C}_{3}\,|k|\|\nabla\Delta
q\|_{L^{2}})C_{H^{1},L^{4}}\bigr{)}\\\ \times\|\nabla\Delta
p^{n}\|_{L^{2}}\,\|\Delta p_{t}^{n}\|_{L^{2}}\\\
+c^{2}\tilde{C}_{4}|k|\|\Delta q_{t}\|_{L^{2}}\|\nabla\Delta
p^{n}\|_{L^{2}}^{2}.\end{multlined}\left(2\tfrac{|k|}{\underline{\alpha}}((C_{H^{1},L^{4}})^{2}+C_{H^{2},L^{\infty}})\right.\\\
\left.+|k|\|\nabla\Delta
q\|_{L^{2}}\left(4\tilde{C}_{1}C_{\textup{PF}}C_{H^{2},L^{\infty}}+\bigl{(}\tilde{C}_{2}+\tilde{C}_{3}\,|k|\|\nabla\Delta
q\|_{L^{2}}\bigr{)}C_{H^{2},L^{8}}\right)\right)\\\ \times\|\Delta
q_{t}\|_{L^{2}}\,\|\Delta p_{t}^{n}\|_{L^{2}}^{2}\\\
+\frac{b\overline{\alpha}}{2}\tilde{C}_{1}^{2}\,|k|^{2}\|\nabla\Delta
q\|_{L^{2}}^{2}\|\Delta
p_{t}^{n}\|_{L^{2}}^{2}+\frac{b}{2\overline{\alpha}}\|\nabla\Delta
p_{t}^{n}\|_{L^{2}}^{2}\\\ +c^{2}|k|\|\nabla\Delta
q\|_{L^{2}}\bigl{(}\tilde{C}_{1}+(\tilde{C}_{2}+\tilde{C}_{3}\,|k|\|\nabla\Delta
q\|_{L^{2}})C_{H^{1},L^{4}}\bigr{)}\\\ \times\|\nabla\Delta
p^{n}\|_{L^{2}}\,\|\Delta p_{t}^{n}\|_{L^{2}}\\\
+c^{2}\tilde{C}_{4}|k|\|\Delta q_{t}\|_{L^{2}}\|\nabla\Delta
p^{n}\|_{L^{2}}^{2}.$

We next estimate $\|\nabla p_{tt}^{n}\|_{L^{2}}$. Since $p^{n}$ satisfies the
semi-discrete equation, we have

(3.24) $\displaystyle\|\nabla p_{tt}^{n}\|_{L^{2}}^{2}=$
$\displaystyle\,(\nabla p_{tt}^{n},\nabla
p_{tt}^{n})_{L^{2}}=-(p_{tt}^{n},\Delta p_{tt}^{n})_{L^{2}}$ $\displaystyle=$
$\displaystyle\,-(\alpha^{-1}(b\Delta p_{t}^{n}+c^{2}\Delta
p^{n}+kq_{t}p_{t}^{n}),\Delta p_{tt}^{n})_{L^{2}}.$

Therefore,

(3.25) $\displaystyle\|\nabla p_{tt}^{n}\|_{L^{2}}^{2}=$
$\displaystyle\,\begin{multlined}(\alpha^{-1}(b\nabla\Delta
p_{t}^{n}+c^{2}\nabla\Delta p^{n}+k\nabla[q_{t}p_{t}^{n}]),\nabla
p_{tt}^{n})_{L^{2}}\\\\[2.84526pt] +(\nabla[\alpha^{-1}](b\Delta
p_{t}^{n}+c^{2}\Delta p^{n}+kq_{t}p_{t}^{n}),\nabla
p_{tt}^{n})_{L^{2}}.\end{multlined}(\alpha^{-1}(b\nabla\Delta
p_{t}^{n}+c^{2}\nabla\Delta p^{n}+k\nabla[q_{t}p_{t}^{n}]),\nabla
p_{tt}^{n})_{L^{2}}\\\\[2.84526pt] +(\nabla[\alpha^{-1}](b\Delta
p_{t}^{n}+c^{2}\Delta p^{n}+kq_{t}p_{t}^{n}),\nabla p_{tt}^{n})_{L^{2}}.$

The above identity together with the Cauchy–Schwarz inequality allow us to
conclude that

(3.26) $\displaystyle\|\nabla p_{tt}^{n}\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\,\begin{multlined}\underline{\alpha}^{-1}(b\|\nabla\Delta
p_{t}^{n}\|_{L^{2}}+c^{2}\|\nabla\Delta p^{n}\|_{L^{2}}+|k|\|\nabla
q_{t}\|_{L^{2}}C_{H^{2},L^{\infty}}\|\Delta p_{t}^{n}\|_{L^{2}}\\\
+|k|C_{H^{1},L^{4}}^{2}\|\nabla q_{t}\|_{L^{2}}\|\Delta
p_{t}^{n}\|_{L^{2}})\\\
+\underline{\alpha}^{-2}|k|C_{H^{2},L^{\infty}}\|\nabla\Delta
q\|_{L^{2}}(b\|\Delta p_{t}^{n}\|_{L^{2}}+c^{2}\|\Delta p^{n}\|_{L^{2}}\\\
+|k|\|q_{t}\|_{L^{2}}C_{H^{2},L^{\infty}}\|\Delta
p_{t}^{n}\|_{L^{2}}).\end{multlined}\underline{\alpha}^{-1}(b\|\nabla\Delta
p_{t}^{n}\|_{L^{2}}+c^{2}\|\nabla\Delta p^{n}\|_{L^{2}}+|k|\|\nabla
q_{t}\|_{L^{2}}C_{H^{2},L^{\infty}}\|\Delta p_{t}^{n}\|_{L^{2}}\\\
+|k|C_{H^{1},L^{4}}^{2}\|\nabla q_{t}\|_{L^{2}}\|\Delta
p_{t}^{n}\|_{L^{2}})\\\
+\underline{\alpha}^{-2}|k|C_{H^{2},L^{\infty}}\|\nabla\Delta
q\|_{L^{2}}(b\|\Delta p_{t}^{n}\|_{L^{2}}+c^{2}\|\Delta p^{n}\|_{L^{2}}\\\
+|k|\|q_{t}\|_{L^{2}}C_{H^{2},L^{\infty}}\|\Delta p_{t}^{n}\|_{L^{2}}).$

Applying Gronwall’s inequality to (3.23) and taking the supremum over
$t\in(0,T_{n})$ yield

(3.27) $\displaystyle\begin{multlined}\sup_{t\in(0,T_{n})}\|\Delta
p_{t}^{n}(t)\|_{L^{2}}^{2}\,+\sup_{t\in(0,T_{n})}\|\nabla\Delta
p^{n}(t)\|_{L^{2}}^{2}+b\int_{0}^{T_{n}}\|\nabla\Delta
p_{t}^{n}\|_{L^{2}}^{2}\,\textup{d}t\end{multlined}\sup_{t\in(0,T_{n})}\|\Delta
p_{t}^{n}(t)\|_{L^{2}}^{2}\,+\sup_{t\in(0,T_{n})}\|\nabla\Delta
p^{n}(t)\|_{L^{2}}^{2}+b\int_{0}^{T_{n}}\|\nabla\Delta
p_{t}^{n}\|_{L^{2}}^{2}\,\textup{d}t$ $\displaystyle\leq$
$\displaystyle\,\begin{multlined}C(T)(\|\nabla\Delta
p_{0}\|_{L^{2}}^{2}+\|\Delta
p_{1}\|_{L^{2}}^{2}),\end{multlined}C(T)(\|\nabla\Delta
p_{0}\|_{L^{2}}^{2}+\|\Delta p_{1}\|_{L^{2}}^{2}),$

where we have additionally bounded the approximate initial data by the exact
data in the same norm. The constant above is given by

(3.28) $\displaystyle
C(T)=C_{1}\exp(C_{2}\|k|(\|q\|_{X^{W}_{\alpha}}+(1+b^{2})\|q\|_{X^{W}_{\alpha}}^{2}+\|q\|_{X^{W}_{\alpha}}^{3})T),$

where $C_{1}$, $C_{2}>0$ do not depend neither on $b$ nor on $T_{n}$, $n$.
Thanks to (3.26), we then also have a uniform bound on $p_{tt}^{n}$:

(3.29) $\displaystyle\int_{0}^{T_{n}}\|\nabla
p_{tt}^{n}\|_{L^{2}}^{2}\,\textup{d}t\lesssim$
$\displaystyle\,\begin{multlined}b^{2}\|\nabla\Delta
p_{t}^{n}\|_{L^{2}(L^{2})}+\|\nabla\Delta p^{n}\|_{L^{2}(L^{2})}^{2}\\\
+(1+\|\nabla q_{t}\|_{L^{\infty}(L^{2})}^{2})\|\Delta
p_{t}^{n}\|_{L^{2}(L^{2})}^{2}\\\ +\|\nabla\Delta
q\|_{L^{\infty}(L^{2})}^{2}\left(b^{2}\|\Delta
p_{t}^{n}\|_{L^{2}(L^{2})}^{2}+\|\Delta p^{n}\|_{L^{2}(L^{2})}^{2}\right.\\\
\left.+\|q_{t}\|_{L^{\infty}(L^{2})}^{2}\|\Delta
p_{t}^{n}\|_{L^{2}(L^{2})}^{2}\right).\end{multlined}b^{2}\|\nabla\Delta
p_{t}^{n}\|_{L^{2}(L^{2})}+\|\nabla\Delta p^{n}\|_{L^{2}(L^{2})}^{2}\\\
+(1+\|\nabla q_{t}\|_{L^{\infty}(L^{2})}^{2})\|\Delta
p_{t}^{n}\|_{L^{2}(L^{2})}^{2}\\\ +\|\nabla\Delta
q\|_{L^{\infty}(L^{2})}^{2}\left(b^{2}\|\Delta
p_{t}^{n}\|_{L^{2}(L^{2})}^{2}+\|\Delta p^{n}\|_{L^{2}(L^{2})}^{2}\right.\\\
\left.+\|q_{t}\|_{L^{\infty}(L^{2})}^{2}\|\Delta
p_{t}^{n}\|_{L^{2}(L^{2})}^{2}\right).$

If $b=0$, we even get a bound on $\sup_{(0,T_{n})}\|\nabla
p_{tt}^{n}\|_{L^{2}}$ here.

The uniform bounds with respect to $T_{n}$ allow us to extend the existence
interval to $[0,T]$.

#### Passing to the limit

Thanks to the uniform bounds on $p^{n}$ and the fact that our spatial and
temporal domains are bounded, we can employ standard compactness arguments and
conclude that there exist a subsequence, which we do not relabel, and a
function $p$ such that

(3.30) $\displaystyle p_{tt}^{n}$ $\displaystyle\relbar\joinrel\rightharpoonup
p_{tt}$ weakly in $\displaystyle L^{2}(0,T;H_{0}^{1}(\Omega)),$ $\displaystyle
p_{t}^{n}$ $\displaystyle\relbar\joinrel\rightharpoonup p_{t}$ weakly-$\star$
in $\displaystyle L^{\infty}(0,T;H_{0}^{1}(\Omega)\cap H^{2}(\Omega)),$
$\displaystyle p_{t}^{n}$ $\displaystyle\relbar\joinrel\rightharpoonup p_{t}$
weakly in $\displaystyle L^{2}(0,T;{H_{\diamondsuit}^{3}(\Omega)})\ \mbox{ if
}b>0,$ $\displaystyle p^{n}$ $\displaystyle\relbar\joinrel\rightharpoonup p$
weakly-$\star$ in $\displaystyle
L^{\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)}).$

We next wish to prove that this limit solves (3.1). We multiply the semi-
discrete equation by $\eta\in C_{0}^{\infty}(0,T)$ and integrate over time to
obtain

(3.31)
$\displaystyle\int_{0}^{T}(p^{n}_{tt},w_{i})_{L^{2}}\,\eta(t)\,\textup{d}t=\,\int_{0}^{T}(c^{2}\Delta
p^{n}+b\Delta
p_{t}^{n}+kq_{t}p_{t}^{n},\alpha^{-1}w_{i})_{L^{2}}\eta(t)\,\textup{d}t$

for $i\in[1,n]$. Letting $n\rightarrow\infty$ in the above equation and
employing (3.30) leads to

(3.32)
$\displaystyle\int_{0}^{T}(p_{tt},w_{i})_{L^{2}}\,\eta(t)\,\textup{d}t=\,\int_{0}^{T}(c^{2}\Delta
p+b\Delta p_{t}+kq_{t}p_{t},\alpha^{-1}w_{i})_{L^{2}}\eta(t)\,\textup{d}t,$

for all $i\in\mathbb{N}$ and $\eta\in C^{\infty}_{0}(0,T)$. Since
$\displaystyle\bigcup_{n\in\mathbb{N}}V_{n}$ is dense in $L^{2}(\Omega)$, and
by the Fundamental Lemma of Calculus of Variations, $p$ solves the original
problem. For $b\neq 0$, due to the embedding

(3.33) $\displaystyle p^{n}\in
H^{1}(0,T;{H_{\diamondsuit}^{3}(\Omega)})\hookrightarrow
C([0,T];{H_{\diamondsuit}^{3}(\Omega)}),$

we know that $p^{n}(0)\rightharpoonup p(0)\quad\text{in
}{H_{\diamondsuit}^{3}(\Omega)}$. On the other hand, $p^{n}(0)\rightarrow
p_{0}$ in ${H_{\diamondsuit}^{3}(\Omega)}$, and so $p(0)=p_{0}$. We also have

(3.34) $\displaystyle p^{n}_{t}\in$
$\displaystyle\,\begin{multlined}L^{\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\cap
H^{1}(0,T;H_{0}^{1}(\Omega))\\\ \hookrightarrow
C_{w}([0,T];{H_{\diamondsuit}^{2}(\Omega)})\cap
C([0,T];H_{0}^{1}(\Omega)),\end{multlined}L^{\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\cap
H^{1}(0,T;H_{0}^{1}(\Omega))\\\ \hookrightarrow
C_{w}([0,T];{H_{\diamondsuit}^{2}(\Omega)})\cap C([0,T];H_{0}^{1}(\Omega)),$

and, therefore, also $p_{t}(0)=p_{1}$.

#### Energy estimate for $p$

We can take the limit inferior as $n\rightarrow\infty$ of the discrete energy
bound, and by virtue of the weak and the weak-$\star$ lower semi-continuity of
norms obtain the final estimate. Uniqueness of a solution in $X^{\textup{W}}$
follows by testing the homogeneous problem (i.e., with $p_{0}=p_{1}=0)$ with
$p_{t}$ and relying on the bound

(3.35)
$\displaystyle\frac{1}{2}\|\sqrt{\alpha(t)}p_{t}(t)\|^{2}_{L^{2}}+c^{2}\|\nabla
p(t)\|^{2}_{L^{2}}+b\int_{0}^{t}\|\nabla p_{t}\|^{2}\,\textup{d}x\textup{d}s$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{t}(\alpha_{t}p_{t},p_{t})_{L^{2}}\,\textup{d}s+k\int_{0}^{t}(q_{t}p_{t},p_{t})\,\textup{d}s$
$\displaystyle\lesssim$
$\displaystyle\,(\|\alpha_{t}\|_{L^{\infty}(L^{\infty})}+\|q_{t}\|_{L^{\infty}(L^{\infty})})\|p_{t}\|^{2}_{L^{2}(L^{2})},$

for $t\in[0,T]$, together with Gronwall’s inequality.

#### Additional regularity

The already established existence of the solution $p\in X^{\textup{W}}$, via
the PDE implies

$p_{tt}=\alpha^{-1}(b\Delta p_{t}+c^{2}\Delta p+kq_{t}p_{t})\in
H^{1}(0,T;H^{-1}(\Omega))$

and, therefore, $p\in H^{3}(0,T;H^{-1}(\Omega))\cap X^{\textup{W}}$. Since by
interpolation (with interpolation parameter
$\theta=\frac{1-m}{2-m}\in[0,\frac{1}{2})$, where $m=\min\\{s,1\\}>0$ and
$r=\frac{m}{2(2-m)}>0$) we have

$\displaystyle H^{3}(0,T;H^{s}(\Omega))\cap
H^{2}(0,T;{H_{\diamondsuit}^{2}(\Omega)})$ $\displaystyle\hookrightarrow
H^{2+(1-\theta)}(0,T;H^{m+\theta(2-m)}(\Omega))$
$\displaystyle=H^{5/2+r}(0,T;H^{1}(\Omega))\hookrightarrow
C^{2}([0,T];H^{1}(\Omega)),$

the imposed initial conditions on $p$, $p_{t}$, and $p_{tt}$ make sense in the
spaces $H^{3}(\Omega)$, $H^{3}(\Omega)$, and $H^{1}(\Omega)$, respectively.

We now derive a uniform bound on $\|\nabla p_{tt}\|_{L^{\infty}(L^{2})}$ also
in the case $b>0$ for sufficiently regular initial data. Note that we have
already established such a bound when $b=0$, right after estimate (3.29).
To justify the derivation of this bound, we will employ a Galerkin
discretization of the time-differentiated version of (3.1):

(3.36)
$\displaystyle\begin{cases}(\alpha\tilde{p}^{n}_{ttt}+\alpha_{t}\tilde{p}^{n}_{tt}-b\Delta\tilde{p}^{n}_{tt}-c^{2}\Delta\tilde{p}^{n}_{t}-kq_{t}\tilde{p}^{n}_{tt}-{kq_{tt}\tilde{p}^{n}_{t}},\phi)_{L^{2}}=0,\\\\[2.84526pt]
\text{for every $\phi\in V_{n}$ pointwise a.e. in $(0,T)$},\\\\[2.84526pt]
(\tilde{p}^{n}(0),\tilde{p}^{n}_{t}(0),\tilde{p}^{n}_{tt}(0))=(\tilde{p}^{n}_{0},\tilde{p}^{n}_{1},\tilde{p}^{n}_{2}),\end{cases}$

where $\tilde{p}^{n}_{0}$ , $\tilde{p}^{n}_{1}$, $\tilde{p}^{n}_{2}$ are the
Galerkin projections of $p_{0}$, $p_{1}$, and $\alpha(0)^{-1}(b\Delta
p_{1}+c^{2}\Delta p_{0}+kq_{t}(0)p_{1})$ (for the latter see the compatibility
condition (3.10)). To relate the solution of the time-differentiated equation
to the original one, the compatibility condition (3.10) is needed.
Testing (3.36) with $\tilde{p}^{n}_{ttt}-\Delta\tilde{p}^{n}_{tt}$ results in

(3.37)
$\displaystyle(\alpha\tilde{p}^{n}_{ttt}+\alpha_{t}\tilde{p}^{n}_{tt}-b\Delta\tilde{p}^{n}_{tt}-c^{2}\Delta\tilde{p}^{n}_{t}-kq_{t}\tilde{p}^{n}_{tt}-kq_{tt}\tilde{p}^{n}_{t},\tilde{p}^{n}_{ttt}-\Delta\tilde{p}^{n}_{tt})_{L^{2}}=0.$

We can then rely on the identity

(3.38)
$\displaystyle(\alpha\tilde{p}^{n}_{ttt},\tilde{p}^{n}_{ttt}-\Delta\tilde{p}^{n}_{tt})_{L^{2}}=$
$\displaystyle\,(\alpha\tilde{p}^{n}_{ttt},\tilde{p}^{n}_{ttt})_{L^{2}}+(\nabla[\alpha\tilde{p}^{n}_{ttt}],\nabla\tilde{p}^{n}_{tt})_{L^{2}}$
$\displaystyle=$
$\displaystyle\,(\alpha\tilde{p}^{n}_{ttt},\tilde{p}^{n}_{ttt})_{L^{2}}+(\alpha\nabla\tilde{p}^{n}_{ttt}+\tilde{p}^{n}_{ttt}\nabla\alpha,\nabla\tilde{p}^{n}_{tt})_{L^{2}}$
$\displaystyle=$
$\displaystyle\,\begin{multlined}(\alpha\tilde{p}^{n}_{ttt},\tilde{p}^{n}_{ttt})_{L^{2}}+\frac{1}{2}\frac{\textup{d}}{\textup{d}t}(\alpha\nabla\tilde{p}^{n}_{tt},\nabla\tilde{p}^{n}_{tt})\\\
-\frac{1}{2}(\alpha_{t}\nabla\tilde{p}^{n}_{tt},\nabla\tilde{p}^{n}_{tt})+(\tilde{p}^{n}_{ttt}\nabla\alpha,\nabla\tilde{p}^{n}_{tt}),\end{multlined}(\alpha\tilde{p}^{n}_{ttt},\tilde{p}^{n}_{ttt})_{L^{2}}+\frac{1}{2}\frac{\textup{d}}{\textup{d}t}(\alpha\nabla\tilde{p}^{n}_{tt},\nabla\tilde{p}^{n}_{tt})\\\
-\frac{1}{2}(\alpha_{t}\nabla\tilde{p}^{n}_{tt},\nabla\tilde{p}^{n}_{tt})+(\tilde{p}^{n}_{ttt}\nabla\alpha,\nabla\tilde{p}^{n}_{tt}),$

and, similarly, the identity

(3.39)
$\displaystyle(\alpha_{t}\tilde{p}^{n}_{tt},\tilde{p}^{n}_{ttt}-\Delta\tilde{p}^{n}_{tt})_{L^{2}}=$
$\displaystyle\,(\alpha_{t}\tilde{p}^{n}_{tt},\tilde{p}^{n}_{ttt})_{L^{2}}+(\nabla[\alpha_{t}\tilde{p}^{n}_{tt}],\nabla\tilde{p}^{n}_{tt})_{L^{2}}$
$\displaystyle=$
$\displaystyle\,\begin{multlined}(\alpha_{t}\tilde{p}^{n}_{tt},\tilde{p}^{n}_{ttt})_{L^{2}}+(\alpha_{t}\nabla\tilde{p}^{n}_{tt},\nabla\tilde{p}^{n}_{tt})_{L^{2}}+(\tilde{p}^{n}_{tt}\nabla\alpha_{t},\nabla\tilde{p}^{n}_{tt})_{L^{2}}.\end{multlined}(\alpha_{t}\tilde{p}^{n}_{tt},\tilde{p}^{n}_{ttt})_{L^{2}}+(\alpha_{t}\nabla\tilde{p}^{n}_{tt},\nabla\tilde{p}^{n}_{tt})_{L^{2}}+(\tilde{p}^{n}_{tt}\nabla\alpha_{t},\nabla\tilde{p}^{n}_{tt})_{L^{2}}.$

In this manner, we obtain

(3.40)
$\displaystyle\begin{multlined}\|\sqrt{\alpha}\tilde{p}^{n}_{ttt}\|_{L^{2}}^{2}+\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\sqrt{\alpha}\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}\\\
+\frac{1}{2}c^{2}\frac{\textup{d}}{\textup{d}t}\|\Delta\tilde{p}^{n}_{t}\|_{L^{2}}^{2}+b\|\Delta\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}+\frac{1}{2}b\frac{\textup{d}}{\textup{d}t}\|\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}\end{multlined}\|\sqrt{\alpha}\tilde{p}^{n}_{ttt}\|_{L^{2}}^{2}+\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\sqrt{\alpha}\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}\\\
+\frac{1}{2}c^{2}\frac{\textup{d}}{\textup{d}t}\|\Delta\tilde{p}^{n}_{t}\|_{L^{2}}^{2}+b\|\Delta\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}+\frac{1}{2}b\frac{\textup{d}}{\textup{d}t}\|\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}$
$\displaystyle=$
$\displaystyle\,\begin{multlined}\frac{1}{2}(\alpha_{t}\nabla\tilde{p}^{n}_{tt},\nabla\tilde{p}^{n}_{tt})_{L^{2}}-(\tilde{p}^{n}_{ttt}\nabla\alpha,\nabla\tilde{p}^{n}_{tt})_{L^{2}}-(\alpha_{t}\tilde{p}^{n}_{tt},\tilde{p}^{n}_{ttt})_{L^{2}}\\\
-(\tilde{p}^{n}_{tt}\nabla\alpha_{t},\nabla\tilde{p}^{n}_{tt})_{L^{2}}+k(q_{t}\tilde{p}^{n}_{tt}+q_{tt}\tilde{p}^{n}_{t},\tilde{p}^{n}_{ttt}-\Delta\tilde{p}^{n}_{tt})_{L^{2}}+c^{2}(\Delta\tilde{p}^{n}_{t},\tilde{p}^{n}_{ttt})_{L^{2}}=:\textup{rhs}.\end{multlined}\frac{1}{2}(\alpha_{t}\nabla\tilde{p}^{n}_{tt},\nabla\tilde{p}^{n}_{tt})_{L^{2}}-(\tilde{p}^{n}_{ttt}\nabla\alpha,\nabla\tilde{p}^{n}_{tt})_{L^{2}}-(\alpha_{t}\tilde{p}^{n}_{tt},\tilde{p}^{n}_{ttt})_{L^{2}}\\\
-(\tilde{p}^{n}_{tt}\nabla\alpha_{t},\nabla\tilde{p}^{n}_{tt})_{L^{2}}+k(q_{t}\tilde{p}^{n}_{tt}+q_{tt}\tilde{p}^{n}_{t},\tilde{p}^{n}_{ttt}-\Delta\tilde{p}^{n}_{tt})_{L^{2}}+c^{2}(\Delta\tilde{p}^{n}_{t},\tilde{p}^{n}_{ttt})_{L^{2}}=:\textup{rhs}.$

Furthermore, we have

(3.41)
$\displaystyle(q_{t}\tilde{p}^{n}_{tt}+q_{tt}\tilde{p}^{n}_{t},\tilde{p}^{n}_{ttt}-\Delta\tilde{p}^{n}_{tt})_{L^{2}}=$
$\displaystyle\,(q_{t}\tilde{p}^{n}_{tt}+q_{tt}\tilde{p}^{n}_{t},\tilde{p}^{n}_{ttt})_{L^{2}}+(\nabla(q_{t}\tilde{p}^{n}_{tt}+q_{tt}\tilde{p}^{n}_{t}),\nabla\tilde{p}^{n}_{tt})_{L^{2}}.$

Thus, we can estimate the right-hand side of (3.40) as follows:

(3.42) $\displaystyle|\textup{rhs}|\lesssim$
$\displaystyle\,\begin{multlined}(1+\|\alpha_{t}\|_{L^{\infty}})\|\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}+\varepsilon\|\tilde{p}^{n}_{ttt}\|_{L^{2}}^{2}+(1+\|\nabla\alpha\|_{L^{\infty}}^{2}+\|q_{t}\|_{L^{\infty}})\|\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}\\\
+(\|\alpha_{t}\|_{L^{3}}^{2}+\|\nabla\alpha_{t}\|_{L^{3}}^{2}+\|q_{t}\|_{L^{3}}^{2}+\|\nabla
q_{t}\|_{L^{3}}^{2})\|\tilde{p}^{n}_{tt}\|_{L^{6}}^{2}\\\
+\|q_{tt}\|_{L^{3}}^{2}\|\nabla\tilde{p}^{n}_{t}\|_{L^{6}}^{2}+\|q_{tt}\|_{L^{3}}^{2}\|\tilde{p}^{n}_{t}\|_{L^{6}}^{2}+\|\nabla
q_{tt}\|_{L^{2}}^{2}\|\tilde{p}^{n}_{t}\|_{L^{\infty}}^{2}+\|\Delta\tilde{p}^{n}_{t}\|_{L^{2}}^{2}\end{multlined}(1+\|\alpha_{t}\|_{L^{\infty}})\|\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}+\varepsilon\|\tilde{p}^{n}_{ttt}\|_{L^{2}}^{2}+(1+\|\nabla\alpha\|_{L^{\infty}}^{2}+\|q_{t}\|_{L^{\infty}})\|\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}\\\
+(\|\alpha_{t}\|_{L^{3}}^{2}+\|\nabla\alpha_{t}\|_{L^{3}}^{2}+\|q_{t}\|_{L^{3}}^{2}+\|\nabla
q_{t}\|_{L^{3}}^{2})\|\tilde{p}^{n}_{tt}\|_{L^{6}}^{2}\\\
+\|q_{tt}\|_{L^{3}}^{2}\|\nabla\tilde{p}^{n}_{t}\|_{L^{6}}^{2}+\|q_{tt}\|_{L^{3}}^{2}\|\tilde{p}^{n}_{t}\|_{L^{6}}^{2}+\|\nabla
q_{tt}\|_{L^{2}}^{2}\|\tilde{p}^{n}_{t}\|_{L^{\infty}}^{2}+\|\Delta\tilde{p}^{n}_{t}\|_{L^{2}}^{2}$

for all $t\in[0,T]$, with $\varepsilon$ small enough, but independent of $b$.
From the compatibility condition (3.10) and boundedness of the projection
operator we can obtain a bound on $\nabla\tilde{p}^{n}_{tt}(0)$ as follows:

(3.43) $\displaystyle\|\nabla\tilde{p}^{n}_{tt}(0)\|_{L^{2}}\lesssim$
$\displaystyle\,\begin{multlined}b\|\nabla\Delta
p_{1}\|_{L^{2}}+\|\nabla\Delta p_{0}\|_{L^{2}}+\|\nabla
q_{t}(0)\|_{L^{2}}\|\Delta p_{1}\|_{L^{2}}+\|\nabla q_{t}(0)\|_{L^{2}}\|\Delta
p_{1}\|_{L^{2}}\\\ +\|\nabla\Delta q(0)\|_{L^{2}}(b\|\Delta
p_{1}\|_{L^{2}}+\|\Delta p_{0}\|_{L^{2}}+\|q_{t}(0)\|_{L^{2}}\|\Delta
p_{1}\|_{L^{2}}).\end{multlined}b\|\nabla\Delta p_{1}\|_{L^{2}}+\|\nabla\Delta
p_{0}\|_{L^{2}}+\|\nabla q_{t}(0)\|_{L^{2}}\|\Delta p_{1}\|_{L^{2}}+\|\nabla
q_{t}(0)\|_{L^{2}}\|\Delta p_{1}\|_{L^{2}}\\\ +\|\nabla\Delta
q(0)\|_{L^{2}}(b\|\Delta p_{1}\|_{L^{2}}+\|\Delta
p_{0}\|_{L^{2}}+\|q_{t}(0)\|_{L^{2}}\|\Delta p_{1}\|_{L^{2}}).$

Applying Gronwall’s inequality thus yields the energy estimate

(3.44)
$\displaystyle\begin{multlined}\|\sqrt{\alpha}\tilde{p}^{n}_{ttt}\|_{L^{2}}^{2}+\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\sqrt{\alpha}\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}\\\
+\frac{1}{2}c^{2}\frac{\textup{d}}{\textup{d}t}\|\Delta\tilde{p}^{n}_{t}\|_{L^{2}}^{2}+b\|\Delta\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}+\frac{1}{2}b\frac{\textup{d}}{\textup{d}t}\|\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}\end{multlined}\|\sqrt{\alpha}\tilde{p}^{n}_{ttt}\|_{L^{2}}^{2}+\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\sqrt{\alpha}\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}\\\
+\frac{1}{2}c^{2}\frac{\textup{d}}{\textup{d}t}\|\Delta\tilde{p}^{n}_{t}\|_{L^{2}}^{2}+b\|\Delta\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}+\frac{1}{2}b\frac{\textup{d}}{\textup{d}t}\|\nabla\tilde{p}^{n}_{tt}\|_{L^{2}}^{2}$
$\displaystyle\lesssim$ $\displaystyle\begin{multlined}\|\nabla\Delta
p_{0}\|_{L^{2}}^{2}+\|\Delta p_{1}\|_{L^{2}}^{2}+b\|\nabla\Delta
p_{1}\|_{L^{2}}^{2},\end{multlined}\|\nabla\Delta p_{0}\|_{L^{2}}^{2}+\|\Delta
p_{1}\|_{L^{2}}^{2}+b\|\nabla\Delta p_{1}\|_{L^{2}}^{2},$

where we have used that also (3.9) holds.
Due to the uniform bound (3.44), $\tilde{p}^{n}$ has a weakly convergent
subsequence whose limit $\tilde{p}$ inherits this bound. Moreover, $\tilde{p}$
solves the time-differentiated PDE and satisfies the compatibility condition
(3.10). By integration with respect to time, it therefore satisfies the
original initial-value problem (3.1) and thus by uniqueness has to coincide
with $p$.

Combined with the previous uniform estimate for smooth approximations, this
allows us to obtain the higher-order in time bound (3.11). ∎

##  4\. Uniform bounds for the Westervelt equation

We next analyze the Westervelt equation by introducing the fixed-point mapping

(4.1) $\mathcal{T}:q\mapsto p,$

where $q$ will belong to a suitably chosen ball in the space

$X^{\textup{W}}\cap L^{\infty}(0,T;L^{\infty}(\Omega)),$

with $X^{\textup{W}}$ defined in (3.7). The function $p$ will solve the
linearized Westervelt equation:

(4.2) $\alpha p_{tt}-b\Delta p_{t}-c^{2}\Delta p-kq_{t}p_{t}=0,$

with $\alpha=1-kq$ and initial conditions $p(0)=q(0)=p_{0}$,
$p_{t}(0)=q_{t}(0)=p_{1}$.
We first determine the conditions under which $\mathcal{T}$ is a self-mapping
on an appropriately chosen set $M$. More precisely, we define $M$ as a subset
of the solution space $X^{\textup{W}}$,

(4.3) $\displaystyle M=\left\\{\vphantom{\frac{1}{2|k|}}q\in
X^{\textup{W}}:\right.$ $\displaystyle\,q(0)=p_{0},\ q_{t}(0)=p_{1},\
\|q\|_{L^{\infty}(L^{\infty})}\leq\frac{1}{2|k|}$ $\displaystyle\left.\|\nabla
q_{tt}\|_{L^{2}(L^{2})}^{2}+\|\Delta
q_{t}\|_{L^{\infty}(L^{2})}^{2}+\|\nabla\Delta
q\|_{L^{\infty}(L^{2})}^{2}+b\|\nabla\Delta q_{t}\|_{L^{2}(L^{2})}^{2}\leq
R^{2}\vphantom{\frac{1}{2|k|}}\right\\}.$

The imposed conditions allow us to uniformly bound the hidden constant in
estimate (3.8) by

(4.4)
$C_{\textup{lin}}(T,R)=C_{1}\exp{\left(C_{2}|k|(R+R^{2}+R^{3})T\right)},$

where the positive constants $C_{1}$ and $C_{2}$ do not depend on
$b\in[0,\bar{b})$ for fixed $\bar{b}$ nor on $R$, $T$; cf. (3.28).

######  Proposition 4.1.

Let $\Omega\subset\mathbb{R}^{n}$, where $n\in\\{1,2,3\\}$, be bounded and
$C^{3}$ regular. Furthermore, let $b\in[0,\bar{b})$ and let $T>0$ be a given
final time. For initial data satisfying the regularity condition (3.6), the
mapping $\mathcal{T}:q\mapsto p$ is a self-mapping on the set $M$ defined in
(4.3), provided the $H_{0}^{1}(\Omega)\times L^{2}(\Omega)$ norm of the
initial data is small enough so that

(4.5)
${C_{\textup{A}}\tilde{C}_{1}\exp(\tilde{C}_{2}RT)(\|p_{1}\|^{2}_{L^{2}}+\|\nabla
p_{0}\|^{2}_{L^{2}})^{1/4}R^{1/2}\leq\frac{1}{2|k|}}$

and the radius $R$ and final time $T$ are chosen such that

(4.6) $C_{1}\exp{\left(C_{2}|k|(R+R^{2}+R^{3})T\right)}(\|\nabla\Delta
p_{0}\|_{L^{2}}^{2}+\|\Delta p_{1}\|_{L^{2}}^{2})\leq R^{2}.$

Furthermore, $p=\mathcal{T}(q)$ satisfies the energy estimate

(4.7) $\displaystyle\|\nabla
p_{tt}\|_{L^{2}(L^{2})}^{2}+\operatorname*{ess\,sup}_{t\in(0,T)}\|\Delta
p_{t}(t)\|_{L^{2}}^{2}\,+\operatorname*{ess\,sup}_{t\in(0,T)}\|\nabla\Delta
p(t)\|_{L^{2}}^{2}+b\|\nabla\Delta p_{t}\|_{L^{2}(L^{2})}^{2}$
$\displaystyle\leq$ $\displaystyle\,C_{\textup{lin}}(T,R)(\|\nabla\Delta
p_{0}\|_{L^{2}}^{2}+\|\Delta p_{1}\|_{L^{2}}^{2}).$

Here the constants $C_{\textup{A}}$, $\tilde{C}_{1}$, $\tilde{C}_{2}$,
$C_{1}$, $C_{2}$, and $C_{\textup{lin}}(T,R)$ are as in (4.4) and (4.12)
below.

######  Remark 1.

To satisfy condition (4.5), only smallness of the initial data in a weaker
norm is needed; namely $H_{0}^{1}(\Omega)\times L^{2}(\Omega)$ rather than the
${H_{\diamondsuit}^{3}(\Omega)}\times{H_{\diamondsuit}^{2}(\Omega)}$ norm
required for the higher-order energy estimates. Condition (4.6) can be
satisfied even for initial data with large higher-order norm, by, for example,
choosing

$R^{2}=2C_{1}(\|\nabla\Delta p_{0}\|_{L^{2}}^{2}+\|\Delta
p_{1}\|_{L^{2}}^{2})$

and assuming the final time to be short enough, so that

$T\leq\frac{\ln(2)}{C_{2}|k|(R+R^{2}+R^{3})}.$

Note that both smallness conditions (4.5) and (4.6) are additionally mitigated
by the fact that the nonlinearity parameter $k$ is typically small in
magnitude in ultrasonic applications; cf. [13, §5].

###### Proof.

We rely in the proof on our previous linear analysis, which, in particular,
implies $M\not=\emptyset$ and that $\mathcal{T}$ is well-defined on $M$. Let
$q\in M$. The bound

$\|q\|_{L^{\infty}(L^{\infty})}\leq\frac{1}{2|k|}$

immediately implies that the non-degeneracy assumption (3.2) holds with
$\underline{\alpha}=\frac{1}{2}$ and $\overline{\alpha}=\frac{3}{2}$.
Moreover, the regularity assumptions of Proposition 3.1 are satisfied with the
uniform bounds

(4.8)
$\displaystyle\|\alpha\|_{L^{\infty}(H^{3})}\leq|\Omega|^{1/2}+|k|C_{(-\Delta)^{-1}}R,\qquad\|\alpha_{t}\|_{L^{\infty}(H^{2})}\leq|k|C_{(-\Delta)^{-1}}R,$
$\displaystyle\|q_{t}\|_{L^{\infty}(H^{2})}\leq C_{(-\Delta)^{-1}}R\,.$

Hence Proposition 3.1 applied to $p=\mathcal{T}(q)$, together with the imposed
smallness condition (4.6) imply that $\mathcal{T}$ is well-defined and yield
the $R^{2}$ bound in (4.3) on $\mathcal{T}q=p$.
To also achieve the $\frac{1}{2|k|}$ bound on the
$L^{\infty}(0,T;L^{\infty}(\Omega))$ norm of $p$, we derive a simple energy
estimate for the linearization. Similarly to (3.35), by multiplying equation
(4.2) with $p_{t}$ and integrating over space and time, we arrive at

(4.9)
$\displaystyle\frac{1}{2}\left\\{\|\sqrt{\alpha(t)}p_{t}(t)\|^{2}_{L^{2}}+c^{2}\|\nabla
p(t)\|^{2}_{L^{2}}\right\\}\Big{|}_{0}^{t}+b\int_{0}^{t}\|\nabla
p_{t}\|^{2}\,\textup{d}x\textup{d}s$ $\displaystyle\lesssim$
$\displaystyle\,(\|\alpha_{t}\|_{L^{\infty}(L^{\infty})}+\|q_{t}\|_{L^{\infty}(L^{\infty})})\|p_{t}\|^{2}_{L^{2}(L^{2})}.$

From here by Gronwall’s inequality, we have

(4.10) $\displaystyle\|p_{t}(t)\|^{2}_{L^{2}}+\|\nabla
p(t)\|^{2}_{L^{2}}\leq\,C_{1}\exp(C_{2}RT)(\|p_{1}\|^{2}_{L^{2}}+\|\nabla
p_{0}\|^{2}_{L^{2}})$

for all $t\in[0,T]$. Following, e.g., [1, Theorem 1.4], the desired
$L^{\infty}$ bound on $p$ can be obtained by employing the above estimate and
Agmon’s interpolation inequality [3, Lemma 4.10]:

(4.11) $\|p(t)\|_{L^{\infty}}\leq
C_{\textup{A}}\|p(t)\|_{H^{1}}^{1/2}\|p(t)\|_{H^{2}}^{1/2}.$

Indeed, by also using that $\|p\|_{L^{\infty}(H^{2})}\lesssim R$, we have

(4.12) $\displaystyle\|p\|_{L^{\infty}(L^{\infty})}\leq
C_{\textup{A}}\tilde{C}_{1}\exp(\tilde{C}_{2}RT)(\|p_{1}\|^{2}_{L^{2}}+\|\nabla
p_{0}\|^{2}_{L^{2}})^{1/4}R^{1/2}.$

for some positive constants $\tilde{C}_{1}$ and $\tilde{C}_{2}$, independent
of $b$. Thus for $\mathcal{T}(M)\subset M$ to hold, we need to impose the
following condition:

$C_{\textup{A}}\tilde{C}_{1}\exp(\tilde{C}_{2}RT)(\|p_{1}\|^{2}_{L^{2}}+\|\nabla
p_{0}\|^{2}_{L^{2}})^{1/4}R^{1/2}\leq\frac{1}{2|k|},$

as claimed. ∎

We next prove that $\mathcal{T}$ is a contraction in a suitably chosen
topology.

######  Proposition 4.2.

Let the assumptions on $\Omega$ from Proposition 4.1 be satisfied. There exist
constants $C_{5}$, $C_{6}>0$ depending only on the domain $\Omega$ and the
constant $c^{2}$, but neither on $T$ nor on $b$, such that if either

(4.13)
$C_{5}\exp{(C_{6}T)}k^{2}\,(T+1)\,(\|p_{0}\|^{2}_{H^{3}}+\|p_{1}\|^{2}_{H^{2}})<1$

or $p_{1}\in{H_{\diamondsuit}^{3}(\Omega)}$, (3.10), $q$, $\alpha\in
X^{\textup{W}}$ with (3.9), and

(4.14)
$C_{5}\exp{(C_{6}T)}k^{2}\,T\,(\|p_{0}\|^{2}_{H^{3}}+\|p_{1}\|^{2}_{H^{3}})<1,$

then the mapping $\mathcal{T}$ is a contraction on $M$ in the topology induced
by

(4.15) $\|p\|^{2}_{X}=\|p_{tt}\|_{L^{2}(L^{2})}^{2}+\|\nabla
p_{t}\|_{L^{\infty}(L^{2})}^{2}+\|\Delta p\|_{L^{\infty}(L^{2})}^{2}.$

Fulfillment of condition “(4.13) or (4.14)” can be achieved by either imposing
smallness on the initial data in
${H_{\diamondsuit}^{3}(\Omega)}\times{H_{\diamondsuit}^{2}(\Omega)}$ for a
fixed given final time $T$, or by imposing short enough final time $T$ for
fixed given initial data in
${H_{\diamondsuit}^{3}(\Omega)}\times{H_{\diamondsuit}^{3}(\Omega)}$. For the
latter, in case of $b=0$, it suffices to have
${H_{\diamondsuit}^{3}(\Omega)}\times{H_{\diamondsuit}^{2}(\Omega)}$ regular
data.

###### Proof.

In order to prove strict contractivity, we take $q^{(1)}$ and $q^{(2)}$ in
$M$. Let then $p^{(1)}=\mathcal{T}q^{(1)}$ and $p^{(2)}=\mathcal{T}q^{(2)}$.
Furthermore, we introduce the differences $\overline{p}=p^{(1)}-p^{(2)}$ and
$\overline{q}=q^{(1)}-q^{(2)}$. Then we know that $\overline{p}$ solves the
equation

(4.16)
$(1-kq^{(1)})\overline{p}_{tt}-c^{2}\Delta\overline{p}-b\Delta\overline{p}_{t}=kq_{t}^{(1)}\overline{p}_{t}+\underbrace{k\overline{q}_{t}p^{(2)}_{t}+k\overline{q}p^{(2)}_{tt}}_{:=f}$

and has zero initial conditions.
Testing equation (4.16) with $-\Delta\overline{p}_{t}$, integrating over space
and $(0,t)$, and noting that $\overline{p}(0)=\overline{p}_{t}(0)=0$ yields

(4.17)
$\displaystyle\frac{1}{2}\|\nabla\overline{p}_{t}(t)\|_{L^{2}}^{2}+\frac{c^{2}}{2}\|\Delta\overline{p}(t)\|_{L^{2}}^{2}+b\int_{0}^{t}\|\Delta\overline{p}_{t}\|_{L^{2}}^{2}\,\textup{d}s$
$\displaystyle=$
$\displaystyle\,\begin{multlined}k\int_{0}^{t}\int_{\Omega}\nabla(q^{(1)}\overline{p}_{tt})\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s+k\int_{0}^{t}\int_{\Omega}\nabla(q^{(1)}_{t}\overline{p}_{t})\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s\\\
+\int_{0}^{t}\int_{\Omega}\nabla
f\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s\end{multlined}k\int_{0}^{t}\int_{\Omega}\nabla(q^{(1)}\overline{p}_{tt})\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s+k\int_{0}^{t}\int_{\Omega}\nabla(q^{(1)}_{t}\overline{p}_{t})\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s\\\
+\int_{0}^{t}\int_{\Omega}\nabla
f\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s$ $\displaystyle=$
$\displaystyle\,\,\begin{multlined}\frac{1}{2}k\int_{\Omega}q^{(1)}(t)|\nabla\overline{p}_{t}(t)|^{2}\,\textup{d}x-\frac{1}{2}k\int_{0}^{t}\int_{\Omega}q^{(1)}_{t}|\nabla\overline{p}_{t}|^{2}\,\textup{d}x\textup{d}s\\\
+k\int_{0}^{t}\int_{\Omega}\overline{p}_{tt}\nabla
q^{(1)}\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s+k\int_{0}^{t}\int_{\Omega}\nabla(q^{(1)}_{t}\overline{p}_{t})\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s\\\
+\int_{0}^{t}\int_{\Omega}\nabla
f\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s.\end{multlined}\frac{1}{2}k\int_{\Omega}q^{(1)}(t)|\nabla\overline{p}_{t}(t)|^{2}\,\textup{d}x-\frac{1}{2}k\int_{0}^{t}\int_{\Omega}q^{(1)}_{t}|\nabla\overline{p}_{t}|^{2}\,\textup{d}x\textup{d}s\\\
+k\int_{0}^{t}\int_{\Omega}\overline{p}_{tt}\nabla
q^{(1)}\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s+k\int_{0}^{t}\int_{\Omega}\nabla(q^{(1)}_{t}\overline{p}_{t})\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s\\\
+\int_{0}^{t}\int_{\Omega}\nabla
f\cdot\nabla\overline{p}_{t}\,\textup{d}x\textup{d}s.$

From here, we further have by Young’s inequality

(4.18)
$\displaystyle\frac{1}{2}\underline{\alpha}\|\nabla\overline{p}_{t}(t)\|_{L^{2}}^{2}+\frac{c^{2}}{2}\|\Delta\overline{p}(t)\|_{L^{2}}^{2}+b\|\Delta\overline{p}_{t}\|_{L^{2}(L^{2})}^{2}$
$\displaystyle\leq$
$\displaystyle\,\,\begin{multlined}\frac{1}{2}|k|\|q^{(1)}_{t}\|_{L^{\infty}(L^{\infty})}\|\nabla\overline{p}_{t}\|_{L^{2}(L^{2})}^{2}+|k|\|\overline{p}_{tt}\|_{L^{2}(L^{2})}\|\nabla
q^{(1)}\|_{L^{\infty}(L^{\infty})}\|\nabla\overline{p}_{t}\|_{L^{2}(L^{2})}\\\
+|k|\|\nabla
q^{(1)}_{t}\|_{L^{\infty}(L^{3})}\|\overline{p}_{t}\|_{L^{2}(L^{6})}\|\nabla\overline{p}_{t}\|_{L^{2}(L^{2})}+\|\nabla
f\|_{L^{2}(L^{2})}\|\nabla\overline{p}_{t}\|_{L^{2}(L^{2})}.\end{multlined}\frac{1}{2}|k|\|q^{(1)}_{t}\|_{L^{\infty}(L^{\infty})}\|\nabla\overline{p}_{t}\|_{L^{2}(L^{2})}^{2}+|k|\|\overline{p}_{tt}\|_{L^{2}(L^{2})}\|\nabla
q^{(1)}\|_{L^{\infty}(L^{\infty})}\|\nabla\overline{p}_{t}\|_{L^{2}(L^{2})}\\\
+|k|\|\nabla
q^{(1)}_{t}\|_{L^{\infty}(L^{3})}\|\overline{p}_{t}\|_{L^{2}(L^{6})}\|\nabla\overline{p}_{t}\|_{L^{2}(L^{2})}+\|\nabla
f\|_{L^{2}(L^{2})}\|\nabla\overline{p}_{t}\|_{L^{2}(L^{2})}.$

Note that we can derive a bound on the $\nabla f$ term as follows:

(4.19) $\displaystyle\|\nabla f\|_{L^{2}(L^{2})}^{2}=$
$\displaystyle\,k^{2}\int_{0}^{t}\int_{\Omega}|\nabla(\overline{q}_{t}p^{(2)}_{t}+\overline{q}p^{(2)}_{tt})|^{2}\,\textup{d}x\textup{d}s$
$\displaystyle\leq$
$\displaystyle\,\begin{multlined}2k^{2}(\|\nabla\overline{q}_{t}\|_{L^{\infty}(L^{2})}^{2}\|p^{(2)}_{t}\|_{L^{2}(L^{\infty})}^{2}+\|\overline{q}_{t}\|_{L^{\infty}(L^{3})}^{2}\|\nabla
p^{(2)}_{t}\|_{L^{2}(L^{6})}^{2}\\\
+\|\nabla\overline{q}\|_{L^{\infty}(L^{3})}^{2}\|p^{(2)}_{tt}\|_{L^{2}(L^{6})}^{2}+\|\overline{q}\|_{L^{\infty}(L^{\infty})}^{2}\|\nabla
p^{(2)}_{tt}\|_{L^{2}(L^{2})}^{2}).\end{multlined}2k^{2}(\|\nabla\overline{q}_{t}\|_{L^{\infty}(L^{2})}^{2}\|p^{(2)}_{t}\|_{L^{2}(L^{\infty})}^{2}+\|\overline{q}_{t}\|_{L^{\infty}(L^{3})}^{2}\|\nabla
p^{(2)}_{t}\|_{L^{2}(L^{6})}^{2}\\\
+\|\nabla\overline{q}\|_{L^{\infty}(L^{3})}^{2}\|p^{(2)}_{tt}\|_{L^{2}(L^{6})}^{2}+\|\overline{q}\|_{L^{\infty}(L^{\infty})}^{2}\|\nabla
p^{(2)}_{tt}\|_{L^{2}(L^{2})}^{2}).$

Testing (4.16) additionally with ${\overline{p}_{tt}}$ produces

(4.20)
$\displaystyle\underline{\alpha}\|\overline{p}_{tt}\|^{2}_{L^{2}L^{2}}+\frac{b}{2}\|\nabla\overline{p}_{t}(t)\|_{L^{2}}^{2}$
$\displaystyle\leq$
$\displaystyle\,\left(c^{2}\|\Delta\overline{p}\|_{L^{2}(L^{2})}+|k|\|q^{(1)}_{t}\|_{L^{\infty}(L^{3})}\|\overline{p}_{t}\|_{L^{2}(L^{6})}+\|f\|_{L^{2}(L^{2})}\right)\|\overline{p}_{tt}\|_{L^{2}(L^{2})}.$

Combining the estimates derived above and then employing Gronwall’s inequality
yields

(4.21)
$\displaystyle\|\overline{p}_{tt}\|_{L^{2}(L^{2})}^{2}+\|\nabla\overline{p}_{t}(t)\|_{L^{2}}^{2}+\|\Delta\overline{p}(t)\|_{L^{2}}^{2}+b\|\Delta\overline{p}_{t}\|_{L^{2}(L^{2})}^{2}+b\|\nabla\overline{p}_{t}(t)\|_{L^{2}}^{2}$
$\displaystyle\leq$
$\displaystyle\,C_{1}\exp{(C_{2}T)}\|f\|^{2}_{L^{2}(H^{1})},$

where the positive constants $C_{1}$ and $C_{2}$ do not depend on $b$. From
here we have

(4.22) $\displaystyle\|\overline{p}\|^{2}_{X}\leq$
$\displaystyle\,\begin{multlined}C_{3}\exp{(C_{2}T)}k^{2}(T\|p^{(2)}_{t}\|^{2}_{L^{\infty}(L^{\infty})}+T\|\nabla
p^{(2)}_{t}\|^{2}_{L^{\infty}(L^{6})}\\\
+\|p^{(2)}_{tt}\|^{2}_{L^{2}(L^{6})}+\|\nabla
p^{(2)}_{tt}\|^{2}_{L^{2}(L^{2})})\|\overline{q}\|^{2}_{X}\end{multlined}C_{3}\exp{(C_{2}T)}k^{2}(T\|p^{(2)}_{t}\|^{2}_{L^{\infty}(L^{\infty})}+T\|\nabla
p^{(2)}_{t}\|^{2}_{L^{\infty}(L^{6})}\\\
+\|p^{(2)}_{tt}\|^{2}_{L^{2}(L^{6})}+\|\nabla
p^{(2)}_{tt}\|^{2}_{L^{2}(L^{2})})\|\overline{q}\|^{2}_{X}$
$\displaystyle\leq$
$\displaystyle\,C_{4}\exp{(C_{2}T)}k^{2}(T+1)C_{\textup{lin}}(T)(\|p_{0}\|^{2}_{H^{3}}+\|p_{1}\|^{2}_{H^{2}})\|\overline{q}\|^{2}_{X},$
$\displaystyle\leq$
$\displaystyle\,C_{5}\exp{(C_{6}T)}k^{2}(T+1)(\|p_{0}\|^{2}_{H^{3}}+\|p_{1}\|^{2}_{H^{2}})\|\overline{q}\|^{2}_{X}.$

Alternatively, assuming the initial data to be smoother so that the estimate
(3.11) applies, we can estimate $\|p^{(2)}_{tt}\|^{2}_{L^{2}(L^{6})}$ by
$T\|p^{(2)}_{tt}\|^{2}_{L^{\infty}(L^{6})}$ and thus get

(4.23)
$\|\overline{p}\|^{2}_{X}\leq\,C_{5}\exp{(C_{6}T)}k^{2}\,T\,(\|p_{0}\|^{2}_{H^{3}}+\|p_{1}\|^{2}_{H^{3}})\|\overline{q}\|^{2}_{X},$

in place of (4.22). Here $C_{3},\ldots,C_{6}>0$ do not depend on $b$. We,
therefore, conclude that $\mathcal{T}$ is strictly contractive provided the
data is small enough so that (4.13) or (4.14) holds. ∎

The two previous results allow us to now employ a contraction principle and
obtain a local well-posedness result for the Westervelt equation with a
uniform bound in $b$ for small and smooth data.

######  Theorem 4.1.

Let $\Omega\subset\mathbb{R}^{n}$, where $n\in\\{1,2,3\\}$, be bounded and
$C^{3}$ regular. Let $k\in\mathbb{R}$. Furthermore, let $b\in[0,\bar{b})$.
There exist constants $C_{1}$, $C_{2}$, $C_{5}$, $C_{6}>0$ that depend only on
the domain $\Omega$ and the constant $c^{2}$, but neither on final time $T$
nor on $b$, such that if, for some $\delta$, $R>0$,

(4.24)
${C_{\textup{A}}\tilde{C}_{1}\exp(\tilde{C}_{2}RT)\sqrt{\delta}R^{1/2}\leq\frac{1}{2|k|},}$

as well as (4.13) or (4.14) hold, then for any initial data satisfying (3.6),
(4.6), and

(4.25) ${\|p_{1}\|_{L^{2}}^{2}+\|\nabla p_{0}\|_{L^{2}}^{2}\leq\delta^{2},}$

there exists a unique solution $p\in X^{\textup{W}}$ of problem

(4.26) $\left\\{\begin{aligned} p_{tt}-b\Delta p_{t}-c^{2}\Delta
p=&\,\frac{k}{2}(p^{2})_{tt}&&\quad\text{ in
}\Omega\times(0,T),\\\\[2.84526pt] p=&\,0&&\quad\text{ on
}\partial\Omega\times(0,T),\\\\[2.84526pt]
(p,p_{t})=&\,(p_{0},p_{1})&&\quad\mbox{ in
}\Omega\times\\{0\\},\end{aligned}\right.$

where $X^{\textup{W}}$ is defined in (3.7). Furthermore, solution $p$
satisfies the estimate (4.7).

###### Proof.

The proof follows by employing the Banach fixed-point theorem to the mapping

$\mathcal{T}:M\ni q\mapsto p,$

with the ball $M$ defined in (4.3) and $p$ being the solution of

(4.27) $(1-kq)p_{tt}-b\Delta p_{t}-c^{2}\Delta p-kq_{t}p_{t}=0,$

with initial conditions $p(0)=q(0)=p_{0}$, $p_{t}(0)=q_{t}(0)=p_{1}$.
We first note that the space $(M,d)$ with the metric $d(x,y)=\|x-y\|_{X}$ is a
closed subset of a complete normed space, where we recall that

(4.28) $\|p\|^{2}_{X}=\|p_{tt}\|_{L^{2}(L^{2})}^{2}+\|\nabla
p_{t}\|_{L^{\infty}(L^{2})}^{2}+\|\Delta p\|_{L^{\infty}(L^{2})}^{2}.$

Indeed, $M$ is a ball in $X^{\textup{W}}\cap
L^{\infty}(0,T;L^{\infty}(\Omega))$, thus it is weakly-$\star$ closed in
$X^{\textup{W}}$ by virtue of the Banach–Alaoglu theorem. Therefore, any
Cauchy sequence with respect to the $X$ norm in $M$ converges to some

$y\in L^{\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\cap
W^{1,\infty}(0,T;H_{0}^{1}(\Omega))\cap H^{2}(0,T;L^{2}(\Omega))$

and has a weakly-$\star$ in $X^{\textup{W}}$ convergent subsequence with some
limit $x\in M$. Due to uniqueness of limits we have $y=x\in M$.
Proposition 4.1 guarantees the $\mathcal{T}$ is well-defined self-mapping,
whereas Proposition 4.2 yields strict contractivity. The assertion then
follows by Banach’s fixed-point theorem; cf., e.g., [24, Theorem 2.1]. Note
that this implies uniqueness, first of all, only on $M$. Global uniqueness in
$X^{\textup{W}}$ follows analogously to the standard uniqueness argument in
the Picard-Lindelöf Theorem under a local Lipschitz condition; see, e.g., [24,
Theorem 1.3]. ∎

We emphasize that smallness of the initial data is only required in a weaker
norm than the one of $X^{\textup{W}}_{0}$. The other smallness constraints in
Theorem 4.1 can be achieved by making either
$\|(p_{0},p_{1})\|_{X^{\textup{W}}_{0}}$ small or final time $T$ short enough.
This theoretical framework thus agrees well with ultrasonic applications,
where typically the final time will be short and the data smooth, but not
necessarily small; see, for example, [13, §5] and [27].
When $n=1$, already the space $H^{1}(\Omega)$ embeds into
$L^{\infty}(\Omega)$, which allows us to simplify the theoretical framework.
Thus we expect that the results of Theorem 4.1 can be sharpened in that case;
however, given the real-world ultrasonic applications that motivate our work,
we do not pursue that path here.

##  5\. The inviscid limit of the Westervelt equation

Equipped with a uniform bound in $b$, we are now ready to prove a limiting
result for the Westervelt equation as $b\rightarrow 0^{+}$. To formulate the
result, we introduce the space

(5.1) $\textup{E}=W^{1,\infty}(0,T;L^{2}(\Omega))\cap
L^{\infty}(0,T;H_{0}^{1}(\Omega))$

equipped with the standard energy norm for the wave equation

(5.2)
$\|p\|^{2}_{\textup{E}}=\sup_{t\in(0,T)}\|p_{t}(t)\|_{L^{2}}^{2}+\sup_{t\in(0,T)}\|\nabla
p(t)\|_{L^{2}}^{2}.$

######  Theorem 5.1.

Under the conditions of Theorem 4.1, the family of solutions
$\\{p^{(b)}\\}_{b>0}$ to the Westervelt equation converges in the topology
induced by the energy norm for the wave equation to a solution $p$ of the
inviscid Westervelt equation at a linear rate

(5.3) $\|p^{(b)}-p\|_{\textup{E}}\lesssim b\mbox{ as }b\to 0.$

###### Proof.

Following the general approach of [23], the statement follows by proving that
$\\{p^{(b)}\\}$ is a Cauchy sequence in E.
Let $b,b^{\prime}\in(0,\overline{b})$. Let $p^{(b)}$ and $p^{(b^{\prime})}$ be
the solutions of the Westervelt equation with the sound diffusivity $b$ and
$b^{\prime}$, respectively, and with homogeneous Dirichlet data. We assume the
initial conditions and final time are chosen so that the assumptions of
Theorem 4.1 are satisfied. Since these conditions are independent of $b$,
solutions exist on a maximal common interval $[0,T]$ with $T>0$ for all
$b,b^{\prime}\in[0,\overline{b})$.
Then $\overline{p}=p^{(b)}-p^{(b^{\prime})}$ solves the equation

(5.4)
$(1-kp^{(b)})\overline{p}_{tt}-c^{2}\Delta\overline{p}-b\Delta\overline{p}_{t}=k\overline{p}_{t}(p_{t}^{(b)}+p^{(b^{\prime})}_{t})+k\overline{p}p^{(b^{\prime})}_{tt}+(b-b^{\prime})\Delta
p_{t}^{(b^{\prime})}.$

Testing with $\overline{p}_{t}$ and integrating over space and time leads to

(5.5)
$\displaystyle\frac{1}{2}\|\sqrt{\alpha(t)}\overline{p}_{t}(t)\|_{L^{2}}^{2}+\frac{c^{2}}{2}\|\nabla\overline{p}(t)\|_{L^{2}}^{2}+b\|\nabla\overline{p}_{t}\|_{L^{2}(L^{2})}^{2}$
$\displaystyle=$
$\displaystyle\,\begin{multlined}-\frac{1}{2}k\int_{0}^{t}\int_{\Omega}p^{(b)}_{t}\overline{p}_{t}^{2}\,\textup{d}x\textup{d}s+k\int_{0}^{t}\int_{\Omega}(p_{t}^{(b)}+p_{t}^{(b^{\prime})})\overline{p}_{t}^{2}\,\textup{d}x\textup{d}s+k\int_{0}^{t}\int_{\Omega}p_{tt}^{(b^{\prime})}\overline{p}\overline{p}_{t}\,\textup{d}x\textup{d}s\\\
+(b-b^{\prime})\int_{0}^{t}\int_{\Omega}\Delta
p_{t}^{(b^{\prime})}\overline{p}_{t}\,\textup{d}x\textup{d}s.\end{multlined}-\frac{1}{2}k\int_{0}^{t}\int_{\Omega}p^{(b)}_{t}\overline{p}_{t}^{2}\,\textup{d}x\textup{d}s+k\int_{0}^{t}\int_{\Omega}(p_{t}^{(b)}+p_{t}^{(b^{\prime})})\overline{p}_{t}^{2}\,\textup{d}x\textup{d}s+k\int_{0}^{t}\int_{\Omega}p_{tt}^{(b^{\prime})}\overline{p}\overline{p}_{t}\,\textup{d}x\textup{d}s\\\
+(b-b^{\prime})\int_{0}^{t}\int_{\Omega}\Delta
p_{t}^{(b^{\prime})}\overline{p}_{t}\,\textup{d}x\textup{d}s.$

From here, by using Young’s inequality and then taking the supremum over
$(0,\tau)$, we have the estimate

(5.6)
$\displaystyle\sup_{t\in(0,\tau)}\|\overline{p}_{t}(t)\|_{L^{2}}^{2}+\sup_{t\in(0,\tau)}\|\nabla\overline{p}(t)\|_{L^{2}}^{2}+b\int_{0}^{\tau}\|\nabla\overline{p}_{t}\|_{L^{2}}^{2}\,\textup{d}s$
$\displaystyle\lesssim$
$\displaystyle\,\begin{multlined}(\|p^{(b)}_{t}\|_{L^{\infty}(L^{\infty})}+\|p_{t}^{(b^{\prime})}\|_{L^{\infty}(L^{\infty})})\int_{0}^{\tau}\|\overline{p}_{t}\|_{L^{2}}^{2}\,\textup{d}s\\\
+\varepsilon\sup_{s\in(0,\tau)}\|\overline{p}_{t}(s)\|_{L^{2}}^{2}+\|p_{tt}^{(b^{\prime})}\|_{L^{2}(L^{4})}^{2}\int_{0}^{\tau}\|\overline{p}\|_{L^{4}}^{2}\,\textup{d}s\\\
+(b-b^{\prime})^{2}\|\Delta
p_{t}^{(b^{\prime})}\|_{L^{2}(L^{2})}^{2}+\int_{0}^{\tau}\|\overline{p}_{t}\|_{L^{2}}^{2}\,\textup{d}s,\end{multlined}(\|p^{(b)}_{t}\|_{L^{\infty}(L^{\infty})}+\|p_{t}^{(b^{\prime})}\|_{L^{\infty}(L^{\infty})})\int_{0}^{\tau}\|\overline{p}_{t}\|_{L^{2}}^{2}\,\textup{d}s\\\
+\varepsilon\sup_{s\in(0,\tau)}\|\overline{p}_{t}(s)\|_{L^{2}}^{2}+\|p_{tt}^{(b^{\prime})}\|_{L^{2}(L^{4})}^{2}\int_{0}^{\tau}\|\overline{p}\|_{L^{4}}^{2}\,\textup{d}s\\\
+(b-b^{\prime})^{2}\|\Delta
p_{t}^{(b^{\prime})}\|_{L^{2}(L^{2})}^{2}+\int_{0}^{\tau}\|\overline{p}_{t}\|_{L^{2}}^{2}\,\textup{d}s,$

with $\varepsilon>0$ small enough, but independent of $b$. Dominating the
$\varepsilon$ term on the right-hand side by the first term on the left-hand
side and using continuous embeddings as well as the uniform bounds
$\|p^{(b)}\|_{X^{\textup{W}}},\,\|p^{(b^{\prime})}\|_{X^{\textup{W}}}\,\lesssim
R$ resulting from Theorem 4.1, we conclude that

(5.7)
$\displaystyle\sup_{t\in(0,\tau)}\|\overline{p}_{t}(t)\|_{L^{2}}^{2}+\sup_{t\in(0,\tau)}\|\nabla\overline{p}(t)\|_{L^{2}}^{2}+b\int_{0}^{\tau}\|\nabla\overline{p}_{t}\|_{L^{2}}^{2}\,\textup{d}s$
$\displaystyle\lesssim$
$\displaystyle\,\begin{multlined}R\int_{0}^{\tau}\|\overline{p}_{t}\|_{L^{2}}^{2}\,\textup{d}s+R\int_{0}^{\tau}\|\nabla\overline{p}\|_{L^{2}}^{2}\,\textup{d}s+R^{2}(b-b^{\prime})^{2}\end{multlined}R\int_{0}^{\tau}\|\overline{p}_{t}\|_{L^{2}}^{2}\,\textup{d}s+R\int_{0}^{\tau}\|\nabla\overline{p}\|_{L^{2}}^{2}\,\textup{d}s+R^{2}(b-b^{\prime})^{2}$

since $p^{(b)}$, $p^{(b^{\prime})}\in M$. Gronwall’s inequality therefore
yields

(5.8)
$\|p^{(b)}-p^{(b^{\prime})}\|_{\textup{E}}=\|\overline{p}\|_{\textup{E}}\lesssim|b-b^{\prime}|.$

This estimate, which remains valid for $b^{\prime}=0$ also yields the
convergence rate

(5.9) $\|p^{(b)}-p\|_{\textup{E}}=\|\overline{p}\|_{\textup{E}}\lesssim b,$

where $p$ is a solution to the inviscid version of the Westervelt equation. ∎

######  Remark 2 (Convergence in a higher-order norm).

Similarly to the proof of contractivity in Proposition 4.2, we can also
establish convergence with respect to the topology used there

(5.10) $X=H^{2}(0,T;L^{2}(\Omega))\cap W^{1,\infty}(0,T;H_{0}^{1}(\Omega))\cap
L^{\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)});$

that is, a stronger topology than E, with a slower convergence rate

(5.11) $\|p^{(b)}-p\|_{X}\lesssim\sqrt{b}\mbox{ as }b\to 0.$

Indeed, estimating the right-hand side terms up to the last one in (5.4) goes
through analogously to the estimates for the difference equation (4.16); cf.
(4.19). For the last term we obtain

(5.12) $\|(b-b^{\prime})\Delta
p_{t}^{(b^{\prime})}\|_{L^{2}(L^{2})}\leq|b-b^{\prime}|\,b^{-1/2}R,$

with $R$ as in (4.3); see also (4.5) and (4.7). As in the proof of Proposition
4.2, this yields

$\|p^{(b)}-p\|_{X}\lesssim|b-b^{\prime}|b^{-1/2}.$

Thus, with $b^{\prime}=0$ above, we obtain the convergence rate (5.11).

##  6\. Analysis of the Kuznetsov equation

We wish to extend our considerations of the vanishing sound diffusivity
dynamics to the Kuznetsov equation next. As a by-product, we will also obtain
results for the Westervelt equation in the acoustic potential formulation
(2.2) by setting $\sigma=0$ and $\kappa=\frac{2}{c^{2}}(\frac{B}{2A}+1)$ in
(2.4).
Compared to the analysis of the Westervelt equation, now the factor next to
the second time derivative involves $\psi_{t}$. This implies that a uniform
bound for $\|\psi_{t}\|_{L^{\infty}(L^{\infty})}$ is needed. Together with the
presence of quadratic gradient nonlinearity, higher-order energy analysis is
required in this case to arrive in the end at a uniform bound with respect to
the parameter $b$.

###  6.1. Energy analysis of the linearized Kuznetsov equation

Similarly to before, we begin by studying a linear non-degenerate version of
the nonlinear equation in question. We consider the following linearization:

(6.1) $\left\\{\begin{aligned}
\alpha\psi_{tt}-c^{2}\Delta\psi-b\Delta\psi_{t}=&\,\sigma\nabla\phi\cdot\nabla\psi_{t}&\quad\text{
in }\Omega\times(0,T),\\\\[2.84526pt] \psi=&\,0&\quad\text{ on
}\partial\Omega\times(0,T),\\\\[2.84526pt]
(\psi,\psi_{t})=&\,(\psi_{0},\psi_{1})&\quad\mbox{ in
}\Omega\times\\{0\\},\end{aligned}\right.$

with $\alpha=1-\kappa\phi_{t}$ and $\kappa\in\mathbb{R}$. By abbreviating
$p=\psi_{t}$, $q=\phi_{t}$, the PDE above can be rewritten as

(6.2) $\alpha p_{t}-b\Delta p-c^{2}\Delta\psi=\sigma\nabla\phi\cdot\nabla p$

with $\alpha=1-\kappa q$.
The functions $\phi$ and $q=\phi_{t}$ are required to have the following
regularity:

(6.3) $\phi\in L^{\infty}(0,T;{H_{\diamondsuit}^{4}(\Omega)}),\ q\in
L^{\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)}),\ q_{t}\in
L^{\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)}),$

which implies that the coefficient $\alpha$ has the regularity

(6.4) $\displaystyle\alpha\in
X^{\textup{K}}_{\alpha}=L^{\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)})\cap
W^{1,\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)});$

cf. (2.7). Again, a compatibility condition on the initial data will be
needed, which here reads as

(6.5)
$\psi_{tt}(0)=\psi_{2}=\alpha(0)^{-1}(b\Delta\psi_{1}+c^{2}\Delta\psi_{0}+\sigma\nabla\phi(0)\cdot\nabla\psi_{1})\,.$

We are now ready to state a well-posedness result for this linear problem.

######  Proposition 6.1.

Assume that the domain $\Omega\subset\mathbb{R}^{n}$, where $n\in\\{1,2,3\\}$,
is bounded and $C^{4}$ regular. Let $b\in[0,\bar{b})$ and $T>0$. Let $\phi\in
L^{\infty}(0,T;{H_{\diamondsuit}^{4}(\Omega)})$. Assume that $\alpha\in
X^{\textup{K}}_{\alpha}$ and that there exist $\underline{\alpha}$,
$\overline{\alpha}>0$ such that

(6.6) $\underline{\alpha}\leq\alpha(x,t)\leq\overline{\alpha}\ \mbox{ on
}\Omega\ \ \text{ a.e. in }\Omega\times(0,T).$

Furthermore, assume that

(6.7) $(\psi_{0},\psi_{1})\in
X^{\textup{K}}_{0}={H_{\diamondsuit}^{4}(\Omega)}\times{H_{\diamondsuit}^{3}(\Omega)}$

and (6.5) holds. Then there exists a unique solution $\psi$ of the problem
(3.1) such that

(6.8) $\displaystyle\psi\in\,X^{\textup{K}}=$
$\displaystyle\,\begin{multlined}H^{3}(0,T;H_{0}^{1}(\Omega))\cap
W^{2,\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\\\ \cap
W^{1,\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)})\cap
L^{\infty}(0,T;{H_{\diamondsuit}^{4}(\Omega)}).\end{multlined}H^{3}(0,T;H_{0}^{1}(\Omega))\cap
W^{2,\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\\\ \cap
W^{1,\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)})\cap
L^{\infty}(0,T;{H_{\diamondsuit}^{4}(\Omega)}).$

Furthermore, this solution satisfies the estimate

(6.9)
$\displaystyle\begin{multlined}\|\nabla\psi_{ttt}\|_{L^{2}(L^{2})}^{2}\,+\operatorname*{ess\,sup}_{t\in(0,T)}\|\Delta\psi_{tt}(t)\|_{L^{2}}^{2}\,+\operatorname*{ess\,sup}_{t\in(0,T)}\|\nabla\Delta\psi_{t}(t)\|_{L^{2}}^{2}\\\
+\operatorname*{ess\,sup}_{t\in(0,T)}\|\Delta^{2}\psi(t)\|_{L^{2}}^{2}+b\|\nabla\Delta\psi_{tt}\|_{L^{2}(L^{2})}^{2}+b\|\Delta^{2}\psi_{t}\|_{L^{2}(L^{2})}^{2}\end{multlined}\|\nabla\psi_{ttt}\|_{L^{2}(L^{2})}^{2}\,+\operatorname*{ess\,sup}_{t\in(0,T)}\|\Delta\psi_{tt}(t)\|_{L^{2}}^{2}\,+\operatorname*{ess\,sup}_{t\in(0,T)}\|\nabla\Delta\psi_{t}(t)\|_{L^{2}}^{2}\\\
+\operatorname*{ess\,sup}_{t\in(0,T)}\|\Delta^{2}\psi(t)\|_{L^{2}}^{2}+b\|\nabla\Delta\psi_{tt}\|_{L^{2}(L^{2})}^{2}+b\|\Delta^{2}\psi_{t}\|_{L^{2}(L^{2})}^{2}$
$\displaystyle\lesssim$
$\displaystyle\,\begin{multlined}\|\Delta^{2}\psi_{0}\|_{L^{2}}^{2}+\|\nabla\Delta\psi_{1}\|_{L^{2}}^{2}+\|\Delta[\nabla\phi(0)\cdot\nabla\psi_{1}]\|_{L^{2}}^{2},\end{multlined}\|\Delta^{2}\psi_{0}\|_{L^{2}}^{2}+\|\nabla\Delta\psi_{1}\|_{L^{2}}^{2}+\|\Delta[\nabla\phi(0)\cdot\nabla\psi_{1}]\|_{L^{2}}^{2},$

where

$\|\Delta[\nabla\phi(0)\cdot\nabla\psi_{1}]\|_{L^{2}}\leq
2(C_{H^{2},L^{\infty}}+C_{H^{1},L^{4}}^{2})\|\nabla\Delta\phi(0)\|_{L^{2}}\|\nabla\Delta\psi_{1}\|_{L^{2}}.$

The hidden constant in (6.9) tends to infinity as $T\rightarrow\infty$, but
does not depend on the parameter $b$.

###### Proof.

We start by proving existence of solutions to the time-differentiated version
of our original problem (6.1). We thus consider

(6.10) $\left\\{\begin{aligned} \alpha p_{tt}-b\Delta p_{t}-c^{2}\Delta
p=&-\alpha_{t}p_{t}+\sigma\nabla q\cdot\nabla p+\sigma\nabla\phi\cdot\nabla
p_{t}&&\ \text{ in }\Omega\times(0,T),\\\\[2.84526pt] p=&\,0&&\ \text{ on
}\partial\Omega\times(0,T),\\\\[2.84526pt] (p,p_{t})=&\,(\psi_{1},\psi_{2})&&\
\mbox{ in }\Omega\times\\{0\\},\end{aligned}\right.$

where the function $\psi_{2}$ is defined by (6.5).

The proof can be carried out as before via smooth Faedo–Galerkin
approximations in space, by projecting the problem (6.10) onto the span
$V_{n}$ of the first $n$ eigenfunctions of the Laplacian pointwise in time.
We focus here on deriving the crucial uniform bound for the approximate
solution since the remaining steps are analogous to the proof of Proposition
3.1. For simplicity of notation, we drop the superscript $n$ in the
approximate solution.
In the first step, we multiply (6.10) by $\Delta^{2}p_{t}$ and integrate over
$\Omega$:

(6.11) $(\alpha p_{tt}-b\Delta p_{t}-c^{2}\Delta
p,\Delta^{2}p_{t})_{L^{2}}=(-\alpha_{t}p_{t}+\sigma\nabla q\cdot\nabla
p+\sigma\nabla\phi\cdot\nabla p_{t},\Delta^{2}p_{t})_{L^{2}}.$

When deriving the energy bound, we can rely on the identity

$\displaystyle(\alpha p_{tt},\Delta^{2}p_{t})_{L^{2}}=$
$\displaystyle\,\,(\Delta[\alpha p_{tt}],\Delta p_{t})_{L^{2}}$
$\displaystyle=$
$\displaystyle\,\frac{\textup{d}}{\textup{d}t}\frac{1}{2}(\alpha\Delta
p_{t},\Delta p_{t})_{L^{2}}+\frac{1}{2}\kappa(q_{t}\Delta p_{t},\Delta
p_{t})_{L^{2}}-(\kappa\Delta q\,p_{tt}+\kappa\nabla q\cdot\nabla p_{tt},\Delta
p_{t})_{L^{2}}$ $\displaystyle=$
$\displaystyle\,\frac{\textup{d}}{\textup{d}t}\frac{1}{2}(\alpha\Delta
p_{t},\Delta p_{t})_{L^{2}}+(f_{1}(\phi,q,p),\Delta p_{t})_{L^{2}},$

where we have used the fact that $\Delta p$ vanishes on the boundary for
smooth approximations of the problem and introduced the short-hand notation

(6.12) $\displaystyle f_{1}(\phi,q,p)=\frac{1}{2}\kappa q_{t}\Delta
p_{t}-\kappa\Delta qp_{tt}-\kappa\nabla q\cdot\nabla p_{tt}.$

Recalling that $p_{tt}=\Delta p_{t}=\Delta p=0$ on $\partial\Omega$, which via
the PDE implies that also $\nabla q\cdot\nabla p+\nabla\phi\cdot\nabla p_{t}$
vanishes on the boundary, we further find that

(6.13) $\displaystyle(-\alpha_{t}p_{t}+\sigma\nabla q\cdot\nabla
p+\sigma\nabla\phi\cdot\nabla p_{t},\Delta^{2}p_{t})_{L^{2}}$ $\displaystyle=$
$\displaystyle\,\begin{multlined}(\Delta[-\alpha_{t}p_{t}],\Delta
p_{t})_{L^{2}}+\sigma(\Delta[\nabla q\cdot\nabla p+\nabla\phi\cdot\nabla
p_{t}],\Delta p_{t})_{L^{2}}.\end{multlined}(\Delta[-\alpha_{t}p_{t}],\Delta
p_{t})_{L^{2}}+\sigma(\Delta[\nabla q\cdot\nabla p+\nabla\phi\cdot\nabla
p_{t}],\Delta p_{t})_{L^{2}}.$

Furthermore, we have

$\displaystyle\Delta[-\alpha_{t}p_{t}+\sigma\nabla q\cdot\nabla
p+\sigma\nabla\phi\cdot\nabla p_{t}]$ $\displaystyle=$
$\displaystyle\,\begin{multlined}\kappa\Delta q_{t}\,p_{t}+2\kappa\nabla
q_{t}\cdot\nabla p_{t}+\kappa q_{t}\,\Delta p_{t}+\sigma\nabla\Delta
q\cdot\nabla p+2\sigma D^{2}q:D^{2}p\\\ +\sigma\nabla q\cdot\nabla\Delta
p+\sigma\nabla\Delta\phi\cdot\nabla p_{t}+2\sigma
D^{2}\phi:D^{2}p_{t}+\sigma\nabla\phi\cdot\nabla\Delta
p_{t}\end{multlined}\kappa\Delta q_{t}\,p_{t}+2\kappa\nabla q_{t}\cdot\nabla
p_{t}+\kappa q_{t}\,\Delta p_{t}+\sigma\nabla\Delta q\cdot\nabla p+2\sigma
D^{2}q:D^{2}p\\\ +\sigma\nabla q\cdot\nabla\Delta
p+\sigma\nabla\Delta\phi\cdot\nabla p_{t}+2\sigma
D^{2}\phi:D^{2}p_{t}+\sigma\nabla\phi\cdot\nabla\Delta p_{t}$ $\displaystyle=$
$\displaystyle\,f_{2}(\phi,q,p)+\sigma\nabla\phi\cdot\nabla\Delta p_{t},$

where we have introduced another short-hand notation

(6.14) $\displaystyle f_{2}(\phi,q,p)=\kappa\Delta q_{t}\,p_{t}+2\kappa\nabla
q_{t}\cdot\nabla p_{t}+\kappa q_{t}\,\Delta p_{t}+\sigma\nabla\Delta
q\cdot\nabla p+2\sigma D^{2}q:D^{2}p$ $\displaystyle+\sigma\nabla
q\cdot\nabla\Delta p+\sigma\nabla\Delta\phi\cdot\nabla p_{t}+2\sigma
D^{2}\phi:D^{2}p_{t}\,.$

By elliptic regularity the Hessian
$D^{2}v=(\partial_{x_{i}}\partial_{x_{j}}v)_{i,j}$ satisfies

$\|D^{2}v\|_{L^{p}}\leq C_{\textup{H}}\|\Delta v\|_{L^{p}}\quad\mbox{ for all
}v\in H_{0}^{1}(\Omega)\cap W^{2,p}(\Omega)\,\quad p\in(1,6],$

see, e.g., [7, Theorem 2.4.2.5]. Due to the PDE, we know that

(6.15) $\displaystyle p_{tt}=$ $\displaystyle\,\alpha^{-1}\left(b\Delta
p_{t}+c^{2}\Delta p+\kappa q_{t}p_{t}+\sigma\nabla q\cdot\nabla
p+\sigma\nabla\phi\cdot\nabla p_{t}\right),$ $\displaystyle\nabla p_{tt}=$
$\displaystyle\,\begin{multlined}\kappa\alpha^{-2}\nabla q\,\left(b\Delta
p_{t}+c^{2}\Delta p+\kappa q_{t}p_{t}+\sigma\nabla q\cdot\nabla
p+\sigma\nabla\phi\cdot\nabla p_{t}\right)\\\ +\alpha^{-1}\left(b\nabla\Delta
p_{t}+c^{2}\nabla\Delta p+\kappa\nabla q_{t}p_{t}+\kappa q_{t}\nabla
p_{t}+\sigma D^{2}q\,\nabla p\right.\\\ \left.+\sigma D^{2}p\,\nabla q+\sigma
D^{2}\phi\,\nabla p_{t}+\sigma
D^{2}p_{t}\,\nabla\phi\right).\end{multlined}\kappa\alpha^{-2}\nabla
q\,\left(b\Delta p_{t}+c^{2}\Delta p+\kappa q_{t}p_{t}+\sigma\nabla
q\cdot\nabla p+\sigma\nabla\phi\cdot\nabla p_{t}\right)\\\
+\alpha^{-1}\left(b\nabla\Delta p_{t}+c^{2}\nabla\Delta p+\kappa\nabla
q_{t}p_{t}+\kappa q_{t}\nabla p_{t}+\sigma D^{2}q\,\nabla p\right.\\\
\left.+\sigma D^{2}p\,\nabla q+\sigma D^{2}\phi\,\nabla p_{t}+\sigma
D^{2}p_{t}\,\nabla\phi\right).$

Altogether, we have the energy identity

$\displaystyle\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\sqrt{\alpha}\Delta
p_{t}\|_{L^{2}}^{2}+b\|\nabla\Delta
p_{t}\|_{L^{2}}^{2}+c^{2}\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\nabla\Delta
p\|_{L^{2}}^{2}$
$\displaystyle=\begin{multlined}(-f_{1}(\phi,q,p)+f_{2}(\phi,q,p)+\sigma\nabla\phi\cdot\nabla\Delta
p_{t},\Delta
p_{t})_{L^{2}}.\end{multlined}(-f_{1}(\phi,q,p)+f_{2}(\phi,q,p)+\sigma\nabla\phi\cdot\nabla\Delta
p_{t},\Delta p_{t})_{L^{2}}.$

We can estimate the right-hand side by employing the following bound:

$\displaystyle\|f_{1}(\phi,q,p)\|_{L^{2}}\leq$
$\displaystyle\,\begin{multlined}\frac{1}{2}|\kappa|C_{H^{2},L^{\infty}}\|\Delta
q_{t}\|_{L^{2}}\|\Delta p_{t}\|_{L^{2}}\\\
+(|{\kappa}|C_{H^{2},L^{\infty}}+|\kappa|C_{H^{1},L^{4}}^{2})\|\nabla\Delta
q\|_{L^{2}}\|\nabla
p_{tt}\|_{L^{2}}.\end{multlined}\frac{1}{2}|\kappa|C_{H^{2},L^{\infty}}\|\Delta
q_{t}\|_{L^{2}}\|\Delta p_{t}\|_{L^{2}}\\\
+(|{\kappa}|C_{H^{2},L^{\infty}}+|\kappa|C_{H^{1},L^{4}}^{2})\|\nabla\Delta
q\|_{L^{2}}\|\nabla p_{tt}\|_{L^{2}}.$

Furthermore, by virtue of identities (6.15), we have

$\displaystyle\|\nabla p_{tt}\|_{L^{2}}\leq$
$\displaystyle\,\begin{multlined}\frac{|\kappa|}{\underline{\alpha}^{2}}C_{H^{2},L^{\infty}}\|\nabla\Delta
q\|_{L^{2}}\left(b\|\Delta p_{t}\|_{L^{2}}+c^{2}\|\Delta
p\|_{L^{2}}+C_{H^{1},L^{4}}^{2}\left(|\kappa|\|\nabla q_{t}\|_{L^{2}}\|\nabla
p_{t}\|_{L^{2}}\right.\right.\\\ \left.\left.+|\sigma|\|\Delta
q\|_{L^{2}}\|\Delta p\|_{L^{2}}+|\sigma|\|\Delta\phi\|_{L^{2}}\|\Delta
p_{t}\|_{L^{2}}\right)\vphantom{C_{H^{2},L^{\infty}}}\right)\\\
+\frac{1}{\underline{\alpha}}\left(b\|\nabla\Delta
p_{t}\|_{L^{2}}+c^{2}\|\nabla\Delta
p\|_{L^{2}}+2|\kappa|C_{H^{1},L^{4}}C_{\textup{PF}}^{1/2}\|\Delta
q_{t}\|_{L^{2}}\|\Delta p_{t}\|_{L^{2}}\right.\\\
\left.+|\sigma|C_{\textup{H}}(C_{H^{2},L^{\infty}}+C_{H^{1},L^{4}}^{2})(\|\nabla\Delta
q\|_{L^{2}}\|\Delta p\|_{L^{2}}+\|\nabla\Delta\phi\|_{L^{2}}\|\Delta
p_{t}\|_{L^{2}})\right).\end{multlined}\frac{|\kappa|}{\underline{\alpha}^{2}}C_{H^{2},L^{\infty}}\|\nabla\Delta
q\|_{L^{2}}\left(b\|\Delta p_{t}\|_{L^{2}}+c^{2}\|\Delta
p\|_{L^{2}}+C_{H^{1},L^{4}}^{2}\left(|\kappa|\|\nabla q_{t}\|_{L^{2}}\|\nabla
p_{t}\|_{L^{2}}\right.\right.\\\ \left.\left.+|\sigma|\|\Delta
q\|_{L^{2}}\|\Delta p\|_{L^{2}}+|\sigma|\|\Delta\phi\|_{L^{2}}\|\Delta
p_{t}\|_{L^{2}}\right)\vphantom{C_{H^{2},L^{\infty}}}\right)\\\
+\frac{1}{\underline{\alpha}}\left(b\|\nabla\Delta
p_{t}\|_{L^{2}}+c^{2}\|\nabla\Delta
p\|_{L^{2}}+2|\kappa|C_{H^{1},L^{4}}C_{\textup{PF}}^{1/2}\|\Delta
q_{t}\|_{L^{2}}\|\Delta p_{t}\|_{L^{2}}\right.\\\
\left.+|\sigma|C_{\textup{H}}(C_{H^{2},L^{\infty}}+C_{H^{1},L^{4}}^{2})(\|\nabla\Delta
q\|_{L^{2}}\|\Delta p\|_{L^{2}}+\|\nabla\Delta\phi\|_{L^{2}}\|\Delta
p_{t}\|_{L^{2}})\right).$

We note that also

$\displaystyle\|f_{2}(\phi,q,p)\|_{L^{2}}\leq$
$\displaystyle\,\begin{multlined}2|\kappa|(C_{H^{2},L^{\infty}}+C_{H^{1},L^{4}}^{2})\|\Delta
q_{t}\|_{L^{2}}\|\Delta p_{t}\|_{L^{2}}\\\
+2|\sigma|(C_{H^{2},L^{\infty}}+C_{\textup{H}}^{2}C_{H^{1},L^{4}}^{2})\|\nabla\Delta
q\|_{L^{2}}\|\nabla\Delta p\|_{L^{2}}\\\
+|\sigma|(C_{\textup{H}}C_{H^{2},L^{\infty}}+2C_{H^{1},L^{4}}^{2})\|\Delta^{2}\phi\|_{L^{2}}\|\Delta
p_{t}\|_{L^{2}}\end{multlined}2|\kappa|(C_{H^{2},L^{\infty}}+C_{H^{1},L^{4}}^{2})\|\Delta
q_{t}\|_{L^{2}}\|\Delta p_{t}\|_{L^{2}}\\\
+2|\sigma|(C_{H^{2},L^{\infty}}+C_{\textup{H}}^{2}C_{H^{1},L^{4}}^{2})\|\nabla\Delta
q\|_{L^{2}}\|\nabla\Delta p\|_{L^{2}}\\\
+|\sigma|(C_{\textup{H}}C_{H^{2},L^{\infty}}+2C_{H^{1},L^{4}}^{2})\|\Delta^{2}\phi\|_{L^{2}}\|\Delta
p_{t}\|_{L^{2}}$

and

$\displaystyle 2(\nabla\phi\cdot\nabla\Delta p_{t},\Delta
p_{t})_{L^{2}}=(\nabla\phi,\nabla(\Delta p_{t})^{2})_{L^{2}}=$
$\displaystyle\,(\Delta\phi,(\Delta p_{t})^{2})_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\,C_{H^{2},L^{\infty}}\|\Delta^{2}\phi\|_{L^{2}}\|\Delta
p_{t}\|_{L^{2}}^{2}.$

Under the regularity assumptions (6.3), similarly to the proof of Proposition
3.1, by using Young’s and Gronwall’s inequalities, we arrive at an energy
estimate of the form

(6.16) $\displaystyle\begin{multlined}\|\nabla
p_{tt}\|_{L^{2}(L^{2})}^{2}\,+\sup_{t\in(0,T)}\|\Delta
p_{t}(t)\|_{L^{2}}^{2}\,+\sup_{t\in(0,T)}\|\nabla\Delta
p(t)\|_{L^{2}}^{2}+b\|\nabla\Delta
p_{t}\|_{L^{2}(L^{2})}^{2}\end{multlined}\|\nabla
p_{tt}\|_{L^{2}(L^{2})}^{2}\,+\sup_{t\in(0,T)}\|\Delta
p_{t}(t)\|_{L^{2}}^{2}\,+\sup_{t\in(0,T)}\|\nabla\Delta
p(t)\|_{L^{2}}^{2}+b\|\nabla\Delta p_{t}\|_{L^{2}(L^{2})}^{2}$
$\displaystyle\lesssim$
$\displaystyle\,\begin{multlined}\|\nabla\Delta\psi_{1}\|_{L^{2}}^{2}+\|\Delta\psi_{2}\|_{L^{2}}^{2},\end{multlined}\|\nabla\Delta\psi_{1}\|_{L^{2}}^{2}+\|\Delta\psi_{2}\|_{L^{2}}^{2},$

first of all, for the Galerkin approximations. Via weak limits we obtain
existence, uniqueness, and the energy estimate (6.16) also for a solution to
(6.10). Integrating with respect to time and using compatibility of the
initial data (6.5) yields a unique solution to (6.1).

#### Additional regularity

The terms containing $\|\Delta^{2}\phi\|_{L^{2}}\|\Delta p_{t}\|_{L^{2}}$
above and in $f_{2}$ show that another energy estimate is needed to obtain an
overall bound that is independent of $b$ for the nonlinear problem. To this
end, we exploit elliptic regularity and the fact that by the PDE, $\Delta\psi$
satisfies

(6.17) $(b\partial_{t}+c^{2})\Delta\psi=r\mbox{ with }r=\alpha
p_{t}-\sigma\nabla\phi\cdot\nabla p.$

Thus in case $b>0$, we have

(6.18)
$\Delta\psi(t)=g_{0}(t)\Delta\psi(0)+\int_{0}^{t}g^{\prime}(t-s)r(s)\,\textup{d}s$

with

$g(t)=-c^{-2}\exp{\left(-\frac{c^{2}}{b}t\right)},\qquad
g_{0}(t)=\exp{\left(-\frac{c^{2}}{b}t\right)}\in[0,1].$

By positivity of $g^{\prime}$, we have

$\|g^{\prime}\|_{L^{1}(0,t)}=\int_{0}^{t}g^{\prime}(s)\,\textup{d}s=g(t)-g(0)=c^{-2}\left(1-\exp{\left(-\frac{c^{2}}{b}t\right)}\right)\leq
c^{2}.$

Therefore,

(6.19)
$\|\Delta^{2}\psi(t)\|_{L^{2}}\leq\|\Delta^{2}\psi(0)\|_{L^{2}}+c^{-2}\|\Delta
r\|_{L^{\infty}(0,t;L^{2}(\Omega)}.$

In case $b=0$, estimate (6.19) (without the $\|\Delta^{2}\psi(0)\|_{L^{2}}$
term) immediately follows from (6.17). The right hand-side norm can be
estimated as follows:

(6.20) $\displaystyle\|\Delta r\|_{L^{\infty}(L^{2})}$ $\displaystyle=$
$\displaystyle\,\|\kappa\Delta q\,p_{t}+2\kappa\nabla q\cdot\nabla
p_{t}+\alpha\,\Delta p_{t}+\sigma\nabla\Delta\phi\cdot\nabla p+2\sigma
D^{2}\phi:D^{2}p+\sigma\nabla\phi\cdot\nabla\Delta p\|_{L^{\infty}(L^{2})}$
$\displaystyle\leq$
$\displaystyle\,\begin{multlined}{(1+|\kappa|(2C_{H^{2},L^{\infty}}+2C_{\textup{H}}^{2}C_{H^{1},L^{4}}^{2})}\|\Delta
q\|_{L^{\infty}(L^{2})})\|\Delta p_{t}\|_{L^{\infty}(L^{2})}\\\
+2|\sigma|({C_{\textup{H}}}C_{H^{2},L^{\infty}}+C_{\textup{H}}^{2}C_{H^{1},L^{4}}^{2})\|\nabla\Delta\phi\|_{L^{\infty}(L^{2})}\|\nabla\Delta
p\|_{L^{\infty}(L^{2})}\,,\end{multlined}{(1+|\kappa|(2C_{H^{2},L^{\infty}}+2C_{\textup{H}}^{2}C_{H^{1},L^{4}}^{2})}\|\Delta
q\|_{L^{\infty}(L^{2})})\|\Delta p_{t}\|_{L^{\infty}(L^{2})}\\\
+2|\sigma|({C_{\textup{H}}}C_{H^{2},L^{\infty}}+C_{\textup{H}}^{2}C_{H^{1},L^{4}}^{2})\|\nabla\Delta\phi\|_{L^{\infty}(L^{2})}\|\nabla\Delta
p\|_{L^{\infty}(L^{2})}\,,$

and therefore bounded by means of the already established estimate (6.16).
Here we have used the elliptic regularity result (see, e.g., [25, Lemma 1,
page 34])

$\|v\|_{H^{3}}\leq C_{\textup{H}}\|(-\Delta)^{3/2}v\|_{L^{2}}\quad\mbox{ for
all }v\in H_{0}^{1}(\Omega)\cap H^{3}(\Omega),$

and estimated

$\|\nabla p\|_{L^{\infty}}\leq C_{H^{2},L^{\infty}}\|p\|_{H^{3}}\leq
C_{H^{2},L^{\infty}}C_{\textup{H}}\|(-\Delta)^{3/2}p\|_{L^{2}}=C_{H^{2},L^{\infty}}C_{\textup{H}}\|\nabla\Delta
p\|_{L^{2}}\,,$

and in the same manner bounded $\|\nabla\phi\|_{L^{\infty}}$, due to the fact
that $\Delta p$ and $\Delta\phi$ vanish on $\partial\Omega$. Furthermore, by
(6.18), the function $\psi$ satisfies the boundary conditions

(6.21) $\displaystyle\psi|_{\partial\Omega}=0\,,$
$\displaystyle\Delta\psi|_{\partial\Omega}={\mathrm{bdy}}:=\begin{cases}g_{0}(t)\Delta\psi|_{\partial\Omega}(0)+\int_{0}^{t}g^{\prime}(t-s)(\sigma\nabla\phi\cdot\nabla
p)|_{\partial\Omega}(s)\,\textup{d}s&\mbox{ if }b>0\\\
(\sigma\nabla\phi\cdot\nabla p)|_{\partial\Omega}&\mbox{ if
}b=0\,,\end{cases}$

where

(6.22) $\displaystyle\|(\sigma\nabla\phi\cdot\nabla
p)|_{\partial\Omega}\|_{L^{\infty}(0,T;H^{3/2}(\partial\Omega))}\leq
C_{\textup{tr}}\|\sigma\nabla\phi\cdot\nabla
p\|_{L^{\infty}(0,T;{H^{2}(\Omega)})}$

can again be estimated by means of (6.16).
Thus we can invoke higher elliptic regularity [20, Theorem 8.14] to first
conclude

$\|\psi\|_{L^{\infty}(0,T;H^{4}(\Omega))}\leq
C_{\textup{H}}\Bigl{(}\|\Delta\psi\|_{L^{\infty}(0,T;H^{2}(\Omega)}+\|\psi\|_{L^{\infty}(L^{2})}\Bigr{)}$

and then further estimate

$\|\Delta\psi\|_{L^{\infty}(0,T;H^{2}(\Omega))}\leq
C_{\textup{H}}\Bigl{(}\|\Delta
r\|_{L^{\infty}(L^{2})}+{\|\mathrm{bdy}\|_{L^{\infty}(H^{3/2}(\partial\Omega))}}+\|\psi\|_{L^{\infty}(L^{2})}\Bigr{)}.$

Altogether, by taking into account the derived bounds, we obtain the energy
estimate (6.9); cf. (6.16), where we estimate $\|\Delta\psi_{2}\|_{L^{2}}^{2}$
by means of (6.5). ∎

###  6.2. Uniform bounds for the Kuznetsov equation

We next employ a fixed-point argument to analyze the Kuznetsov equation for
small data as well as obtain a uniform in $b$ energy bound for its solution.
To this end, we introduce the mapping

(6.23) $\mathcal{T}_{\textup{K}}:\phi\mapsto\psi,$

where, similarly to before, $\phi$ will belong to a suitably chosen ball in
the space $X^{\textup{K}}$, and $\psi$ will solve the linearized Kuznetsov
equation

(6.24)
$\alpha\psi_{tt}-c^{2}\Delta\psi-b\Delta\psi_{t}=\,\sigma\nabla\phi\cdot\nabla\psi_{t}$

with $\alpha=1-\kappa\phi_{t}$ and initial data
$(\psi(0),\psi_{t}(0))=(\phi(0),\phi_{t}(0))=(\psi_{0},\psi_{1})$. We note
that the mapping $\mathcal{T}_{\textup{K}}$ is well-defined on account of
Proposition 6.1, whose assumptions we verify below.

######  Theorem 6.1.

Let $\Omega\subset\mathbb{R}^{n}$, where $n\in\\{1,2,3\\}$, be bounded and
$C^{4}$ regular. Let $\kappa$, $\sigma\in\mathbb{R}$. Furthermore, let
$b\in[0,\bar{b})$ and let the initial data $(\psi_{0},\psi_{1})$ satisfy the
regularity assumption (6.7) with

(6.25) $\|\psi_{0}\|^{2}_{H^{4}}+\|\psi_{1}\|^{2}_{H^{3}}\leq\delta^{2}.$

There exist $\tilde{\delta}>0$ and a final time $T=T(\delta)>0$, independent
of $b$, such that if

(6.26)
${\|\psi_{0}\|^{2}_{H^{2}}+\|\psi_{1}\|^{2}_{H^{2}}\leq\tilde{\delta}^{2},}$

then there exists a unique solution $\psi\in X^{\textup{K}}$ of the problem

(6.27) $\left\\{\begin{aligned}
\psi_{tt}-b\Delta\psi_{t}-c^{2}\Delta\psi=&\,\frac{1}{2}(\kappa\psi_{t}^{2}+\sigma|\nabla\psi|^{2})_{t}&&\quad\text{
in }\Omega\times(0,T),\\\\[2.84526pt] \psi=&\,0&&\quad\text{ on
}\partial\Omega\times(0,T),\\\\[2.84526pt]
(\psi,\psi_{t})=&\,(\psi_{0},\psi_{1})&&\quad\mbox{ in
}\Omega\times\\{0\\},\end{aligned}\right.$

where $X^{\textup{K}}$ is defined in (6.8). Furthermore, the solution $\psi$
satisfies the estimate (6.9).

###### Proof.

We begin by proving that the mapping
$\mathcal{T}_{\textup{K}}:\phi\mapsto\psi$ is well-defined and a self-mapping
on

(6.28) $\displaystyle
M_{\textup{K}}=\left\\{\vphantom{\sup_{t\in(0,T)}\|\Delta^{2}\psi(t)\|_{L^{2}}^{2}\leq
R_{\textup{K}}^{2}}\right.\phi\in X^{\textup{K}}\,:$
$\displaystyle\phi(0)=\psi_{0},\ \phi_{t}(0)=\psi_{1},$ $\displaystyle\
\left.\vphantom{X^{\textup{K}}}\|\phi_{t}\|_{L^{\infty}(L^{\infty})}\leq\frac{1}{2|\kappa|},\right.\
\left.\|\phi\|_{X^{\textup{K}}}\leq R_{\textup{K}}\
\vphantom{\sup_{t\in(0,T)}\|\Delta^{2}\psi(t)\|_{L^{2}}^{2}\leq
R_{\textup{K}}^{2}}\right\\},$

where the radius $R_{\textup{K}}$ will be specified below. We first note that
the conditions of Proposition 6.1 are satisfied. Indeed, regularity
assumptions (6.3) and (6.7) hold with the uniform bounds

(6.29)
$\displaystyle\|\phi\|_{L^{\infty}(H_{\diamondsuit}^{4})}\leq\,R_{\textup{K}},\qquad\|\alpha\|_{X^{\textup{K}}_{\alpha}}\lesssim\,1+R_{\textup{K}}.$

Furthermore, we have

$\displaystyle\|1-\alpha\|_{L^{\infty}(L^{\infty})}\leq|\kappa|\|\phi_{t}\|_{L^{\infty}(L^{\infty})}\leq\frac{1}{2}.$

We can thus take $\underline{\alpha}=\frac{1}{2}$ and
$\overline{\alpha}=\frac{3}{2}$ in (6.6), and so the non-degeneracy condition
is fulfilled as well. Then Proposition 6.1 applied to
$\psi=\mathcal{T}_{\textup{K}}(\phi)$ implies that the mapping is well-defined
and, moreover,

(6.30)
$\displaystyle\|\psi\|^{2}_{X^{\textup{K}}}\leq\,C_{\textup{lin},\textup{K}}(T,R_{\textup{K}})(\|\Delta^{2}\psi_{0}\|_{L^{2}}^{2}+\|\nabla\Delta\psi_{1}\|_{L^{2}}^{2}+\|\Delta\nabla\psi_{0}\|_{L^{2}}^{2}),$

where $C_{\textup{lin},\textup{K}}(T,R_{\textup{K}})$ is determined by the
hidden constant in the linear bound (6.9) and by inspection of (6.16) can be
seen to have a form similar to the Westervelt case (4.4)

(6.31)
$C_{\textup{lin}}(T,R)=C_{1}\exp{\left(C_{2}(|\kappa|+|\sigma|)(R_{\textup{K}}+R_{\textup{K}}^{2}+R_{\textup{K}}^{3})T\right)},$

with constants $C_{1}$ and $C_{2}$ independent of $R_{\textup{K}}$, $T$, and
$b\in[0,\bar{b})$ for fixed $\bar{b}$. Therefore, provided that
$R_{\textup{K}}$ and final time $T$ are chosen so that

$C_{\textup{lin},\textup{K}}(T,R_{\textup{K}})(\|\Delta^{2}\psi_{0}\|_{L^{2}}^{2}+\|\nabla\Delta\psi_{1}\|_{L^{2}}^{2}+\|\Delta\nabla\psi_{0}\|_{L^{2}}^{2})\leq
R_{\textup{K}}^{2},$

the solution $\psi$ satisfies the $R_{\textup{K}}^{2}$ bound in (6.28).
To obtain the $1/{(2|\kappa|)}$ bound, we will, similarly to the Westervelt
case, derive a uniform in $b$ estimate of $\|\psi_{t}\|_{L^{\infty}(H^{1})}$
and then rely on Agmon’s interpolation inequality. By multiplying the time-
differentiated equation (6.24) with $p_{t}$ and integrating over space and
time, we arive at

(6.32)
$\displaystyle\frac{1}{2}\left\\{\|\sqrt{\alpha(t)}p_{t}(t)\|^{2}_{L^{2}}+c^{2}\|\nabla
p(t)\|^{2}_{L^{2}}\right\\}\Big{|}_{0}^{t}+b\int_{0}^{t}\|\nabla
p_{t}\|^{2}\,\textup{d}x\textup{d}s$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{0}^{t}(\alpha_{t}p_{t},p_{t})_{L^{2}}\,\textup{d}s+\sigma\int_{0}^{t}(\nabla
q\cdot\nabla p+\nabla\phi\cdot\nabla p_{t},p_{t})\,\textup{d}s$
$\displaystyle\lesssim$
$\displaystyle\,\begin{multlined}\|\alpha_{t}\|_{L^{\infty}(L^{\infty})}\|p_{t}\|^{2}_{L^{2}(0,t;L^{2})}+\|\nabla
q\|_{L^{\infty}(L^{\infty})}\|\nabla
p\|_{L^{2}(0,t;L^{2})}\|p_{t}\|_{L^{2}(0,t;L^{2})}\\\
+\sigma\int_{0}^{t}(\nabla\phi\cdot\nabla
p_{t},p_{t})\,\textup{d}s.\end{multlined}\|\alpha_{t}\|_{L^{\infty}(L^{\infty})}\|p_{t}\|^{2}_{L^{2}(0,t;L^{2})}+\|\nabla
q\|_{L^{\infty}(L^{\infty})}\|\nabla
p\|_{L^{2}(0,t;L^{2})}\|p_{t}\|_{L^{2}(0,t;L^{2})}\\\
+\sigma\int_{0}^{t}(\nabla\phi\cdot\nabla p_{t},p_{t})\,\textup{d}s.$

To estimate the last term on the right, we can first rely on the identity

(6.33) $\displaystyle\int_{0}^{t}\int_{\Omega}(\nabla\phi\cdot\nabla
p_{t})p_{t}\,\textup{d}x\textup{d}s$ $\displaystyle=$
$\displaystyle\,\int_{0}^{t}\int_{\Omega}(-\Delta\phi)p_{t}p_{t}\,\textup{d}x\textup{d}s-\int_{0}^{t}\int_{\Omega}p_{t}(\nabla\phi\cdot\nabla
p_{t})\,\textup{d}x\textup{d}s$

since $p_{t}=0$ on $\partial\Omega$, which then leads to the bound

(6.34)
$\displaystyle\left|2\int_{0}^{t}\int_{\Omega}(\nabla\phi\cdot\nabla\overline{\psi}_{t})p_{t}\,\textup{d}x\textup{d}s\right|=$
$\displaystyle\,\left|\int_{0}^{t}\int_{\Omega}(-\Delta\phi)\overline{\psi}_{t}p_{t}\,\textup{d}x\textup{d}s\right|$
$\displaystyle\leq$
$\displaystyle\,\|\Delta\phi\|_{L^{\infty}(0,t;L^{\infty})}\|p_{t}\|_{L^{2}(0,t;L^{2})}^{2}.$

Thus, utilizing the above estimate in (6.32) and employing Gronwall’s
inequality leads to

(6.35) $\displaystyle\|p_{t}(t)\|^{2}_{L^{2}}+\|\nabla p(t)\|^{2}_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\,\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{1}\exp(\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{2}R_{\textup{K}}T)(\|\psi_{2}\|^{2}_{L^{2}}+\|\nabla\psi_{1}\|^{2}_{L^{2}})$
$\displaystyle\leq$
$\displaystyle\,\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{1}\exp(\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{2}R_{\textup{K}}T)(\|\alpha(0)^{-1}(b\Delta\psi_{1}+c^{2}\Delta\psi_{0}+\sigma\nabla\psi_{0}\cdot\nabla\psi_{1})\|^{2}_{L^{2}}+\|\nabla\psi_{1}\|^{2}_{L^{2}}),$

where we have also relied on the compatibility of initial data (6.5) in the
last line. From here, noting that

$\|\nabla\psi_{0}\cdot\nabla\psi_{1}\|_{L^{2}}\leq\|\nabla\psi_{0}\|_{L^{4}}\|\nabla\psi_{1}\|_{L^{4}}\lesssim\|\nabla\psi_{0}\|_{H^{1}}\|\nabla\psi_{1}\|_{H^{1}},$

we conclude that there exist positive constants
$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{2}$ and
$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{3}$, independent of $b$, such
that

(6.36) $\displaystyle\|p_{t}(t)\|^{2}_{L^{2}}+\|\nabla
p(t)\|^{2}_{L^{2}}\leq\,\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{3}\exp(\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{2}R_{\textup{K}}T)(\|\Delta\psi_{1}\|^{2}_{L^{2}}+\|\Delta\psi_{0}\|^{2}_{L^{2}}+\|\nabla\psi_{1}\|^{2}_{L^{2}})$

for all $t\in[0,T]$. The non-degeneracy bound follows from the above estimate
and Agmon’s inequality:

$\|p\|_{L^{\infty}(L^{\infty})}\leq
C_{\textup{A}}\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{4}\exp(\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{5}RT)(\|\Delta\psi_{1}\|^{2}_{L^{2}}+\|\Delta\psi_{0}\|^{2}_{L^{2}}+\|\nabla\psi_{1}\|^{2}_{L^{2}})^{1/4}R_{\textup{K}}^{1/2},$

for some positive constants
$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{4}$,
$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{5}$ independent of $b$, if we
choose $\tilde{\delta}>0$ small enough, so that

$C_{\textup{A}}\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{4}\exp(\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{C}$}}_{5}RT)\sqrt{\tilde{\delta}}\,R_{\textup{K}}^{1/2}\leq\frac{1}{2|\kappa|}.$

Therefore,

$\mathcal{T}_{\textup{K}}(M_{\textup{K}})\subset M_{\textup{K}}.$

We next prove strict contractivity of the mapping $\mathcal{T}_{\textup{K}}$
in the energy norm for sufficiently small data. To this end, we take
$\phi^{(1)}$ and $\phi^{(2)}$ in $M_{\textup{K}}$, and denote their difference
by $\overline{\phi}=\phi^{(1)}-\phi^{(2)}$. Let then
$\psi^{(1)}=\mathcal{T}_{\textup{K}}(\phi^{(1)})$ and
$\psi^{(2)}=\mathcal{T}_{\textup{K}}(\phi^{(2)})$. The difference
$\overline{\psi}=\psi^{(1)}-\psi^{(2)}\in X^{\textup{K}}$ solves the equation

(6.37)
$\displaystyle(1-\kappa\phi_{t}^{(1)})\overline{\psi}_{tt}-c^{2}\Delta\overline{\psi}-b\Delta\overline{\psi}_{t}=\kappa\overline{\phi}_{t}\psi^{(2)}_{tt}+\sigma\nabla\overline{\phi}\cdot\nabla\psi^{(1)}_{t}+\sigma\nabla\phi^{(2)}\cdot\nabla\overline{\psi}_{t}$

with zero initial data. We test this equation with $\overline{\psi}_{t}$ and
integrate over space and $(0,t)$ to arrive at the identity

(6.38)
$\displaystyle\frac{1}{2}\|\sqrt{\alpha(t)}\,\overline{\psi}_{t}(t)\|_{L^{2}}^{2}+\frac{c^{2}}{2}\|\nabla\overline{\psi}(t)\|_{L^{2}}^{2}+b\|\nabla\overline{\psi}_{t}\|_{L^{2}(0,t;L^{2})}^{2}$
$\displaystyle=$
$\displaystyle\,\begin{multlined}-\frac{\kappa}{2}\int_{0}^{t}\int_{\Omega}\phi_{tt}^{(1)}\overline{\psi}^{2}_{t}\,\textup{d}x\textup{d}s+\kappa\int_{0}^{t}\int_{\Omega}\overline{\phi}_{t}\psi_{tt}^{(2)}\overline{\psi}_{t}\,\textup{d}x\textup{d}s\\\
+\sigma\int_{0}^{t}\int_{\Omega}(\nabla\overline{\phi}\cdot\nabla\psi^{(1)}_{t}+\nabla\phi^{(2)}\cdot\nabla\overline{\psi}_{t})\overline{\psi}_{t}\,\textup{d}x\textup{d}s,\end{multlined}-\frac{\kappa}{2}\int_{0}^{t}\int_{\Omega}\phi_{tt}^{(1)}\overline{\psi}^{2}_{t}\,\textup{d}x\textup{d}s+\kappa\int_{0}^{t}\int_{\Omega}\overline{\phi}_{t}\psi_{tt}^{(2)}\overline{\psi}_{t}\,\textup{d}x\textup{d}s\\\
+\sigma\int_{0}^{t}\int_{\Omega}(\nabla\overline{\phi}\cdot\nabla\psi^{(1)}_{t}+\nabla\phi^{(2)}\cdot\nabla\overline{\psi}_{t})\overline{\psi}_{t}\,\textup{d}x\textup{d}s,$

where $\alpha(t)=1-\kappa\phi^{(1)}_{t}(t)$. To treat the
$\nabla\overline{\phi}\cdot\nabla\psi^{(1)}_{t}$ term on the right, we employ
the following estimate:

(6.39)
$\displaystyle\left|2\int_{0}^{t}\int_{\Omega}\nabla\overline{\phi}\cdot\nabla\psi^{(1)}_{t}\overline{\psi}_{t}\,\textup{d}x\textup{d}s\right|\leq$
$\displaystyle\,2\|\nabla\psi^{(1)}_{t}\|_{L^{2}(0,t;L^{\infty})}\|\nabla\overline{\phi}\|_{L^{\infty}(0,t;L^{2})}\|\overline{\psi}_{t}\|_{L^{2}(0,t;L^{2})}.$

Similarly to (6.33)–(6.34), to estimate the
$\nabla\phi^{(2)}\cdot\nabla\overline{\psi}_{t}$ term, we can rely on

(6.40)
$\displaystyle\left|2\int_{0}^{t}\int_{\Omega}(\nabla\phi^{(2)}\cdot\nabla\overline{\psi}_{t})\overline{\psi}_{t}\,\textup{d}x\textup{d}s\right|=$
$\displaystyle\,\left|\int_{0}^{t}\int_{\Omega}(-\Delta\phi^{(2)})\overline{\psi}_{t}\overline{\psi}_{t}\,\textup{d}x\textup{d}s\right|$
$\displaystyle\leq$
$\displaystyle\,\|\Delta\phi^{(2)}\|_{L^{\infty}(0,t;L^{\infty})}\|\overline{\psi}_{t}\|_{L^{2}(0,t;L^{2})}^{2}.$

Additionally, we can bound the two $\kappa$ terms on the right-hand side of
(6.38) as follows:

(6.41)
$\displaystyle\left|-\frac{1}{2}\kappa\int_{0}^{t}\int_{\Omega}\phi_{tt}^{(1)}\overline{\psi}^{2}_{t}\,\textup{d}x\textup{d}s+\kappa\int_{0}^{t}\int_{\Omega}\psi_{tt}^{(2)}\overline{\phi}_{t}\overline{\psi}_{t}\,\textup{d}x\textup{d}s\right|$
$\displaystyle\leq$
$\displaystyle\,\begin{multlined}\frac{1}{2}|\kappa|\|\phi_{tt}^{(1)}\|_{L^{\infty}(0,t;L^{\infty})}\|\overline{\psi}_{t}\|_{L^{2}(0,t;L^{2})}^{2}+|\kappa|\|\psi_{tt}^{(2)}\|_{L^{2}(0,t;L^{\infty})}\|\overline{\phi}_{t}\|_{L^{\infty}(0,t;L^{2})}\|\overline{\psi}_{t}\|_{L^{2}(0,t;L^{2})}.\end{multlined}\frac{1}{2}|\kappa|\|\phi_{tt}^{(1)}\|_{L^{\infty}(0,t;L^{\infty})}\|\overline{\psi}_{t}\|_{L^{2}(0,t;L^{2})}^{2}+|\kappa|\|\psi_{tt}^{(2)}\|_{L^{2}(0,t;L^{\infty})}\|\overline{\phi}_{t}\|_{L^{\infty}(0,t;L^{2})}\|\overline{\psi}_{t}\|_{L^{2}(0,t;L^{2})}.$

Employing the above estimates in (6.38), using Young’s inequality, and then
taking the supremum over $t\in(0,\tau)$ leads to

(6.42)
$\displaystyle\frac{1}{2}\underline{\alpha}\sup_{t\in(0,\tau)}\|\overline{\psi}_{t}(t)\|_{L^{2}}^{2}+\frac{c^{2}}{2}\sup_{t\in(0,\tau)}\|\nabla\overline{\psi}(t)\|_{L^{2}}^{2}+b\|\nabla\overline{\psi}_{t}\|^{2}_{L^{2}(0,\tau;L^{2})}$
$\displaystyle\leq$
$\displaystyle\,\begin{multlined}\frac{1}{2}|\kappa|\|\phi_{tt}^{(1)}\|_{L^{\infty}(L^{\infty})}\|\overline{\psi}_{t}\|_{L^{2}(0,\tau;L^{2})}^{2}+\frac{1}{2}|\kappa\|\psi_{tt}^{(2)}\|_{L^{2}(L^{\infty})}|\|\overline{\psi}_{t}\|_{L^{2}(0,\tau;L^{2})}^{2}\\\
+\frac{1}{2}|\kappa|\|\psi_{tt}^{(2)}\|_{L^{2}(L^{\infty})}\|\overline{\phi}_{t}\|_{L^{\infty}(0,\tau;L^{2})}^{2}+\sigma\|\nabla\psi^{(1)}_{t}\|_{L^{2}(L^{\infty})}\|\nabla\overline{\phi}\|_{L^{\infty}(0,\tau;L^{2})}^{2}\\\
+\sigma\|\nabla\psi^{(1)}_{t}\|_{L^{2}(L^{\infty})}\|\overline{\psi}_{t}\|_{L^{2}(0,\tau;L^{2})}^{2}+\sigma\|\Delta\phi^{(2)}\|_{L^{\infty}(L^{\infty})}\|\overline{\psi}_{t}\|^{2}_{L^{2}(0,\tau;L^{2})}.\end{multlined}\frac{1}{2}|\kappa|\|\phi_{tt}^{(1)}\|_{L^{\infty}(L^{\infty})}\|\overline{\psi}_{t}\|_{L^{2}(0,\tau;L^{2})}^{2}+\frac{1}{2}|\kappa\|\psi_{tt}^{(2)}\|_{L^{2}(L^{\infty})}|\|\overline{\psi}_{t}\|_{L^{2}(0,\tau;L^{2})}^{2}\\\
+\frac{1}{2}|\kappa|\|\psi_{tt}^{(2)}\|_{L^{2}(L^{\infty})}\|\overline{\phi}_{t}\|_{L^{\infty}(0,\tau;L^{2})}^{2}+\sigma\|\nabla\psi^{(1)}_{t}\|_{L^{2}(L^{\infty})}\|\nabla\overline{\phi}\|_{L^{\infty}(0,\tau;L^{2})}^{2}\\\
+\sigma\|\nabla\psi^{(1)}_{t}\|_{L^{2}(L^{\infty})}\|\overline{\psi}_{t}\|_{L^{2}(0,\tau;L^{2})}^{2}+\sigma\|\Delta\phi^{(2)}\|_{L^{\infty}(L^{\infty})}\|\overline{\psi}_{t}\|^{2}_{L^{2}(0,\tau;L^{2})}.$

Since the functions $\phi^{(1)}$ and $\phi^{(2)}$ belong to $M_{\textup{K}}$
and functions $\psi^{(1)}$ and $\psi^{(2)}$ solve the linear problem, we have
the uniform bounds

(6.43) $\displaystyle\|\phi_{tt}^{(1)}\|_{L^{\infty}(L^{\infty})}\leq$
$\displaystyle\,C_{H^{2},L^{\infty}}R_{\textup{K}},$
$\displaystyle\|\Delta\phi^{(2)}\|_{L^{\infty}(L^{\infty})}\leq$
$\displaystyle\,C_{H^{2},L^{\infty}}R_{\textup{K}},$
$\displaystyle\|\psi_{tt}^{(2)}\|_{L^{2}(L^{\infty})}\leq$
$\displaystyle\,C_{H^{2},L^{\infty}}\sqrt{T}C_{\textup{lin},\textup{K}}(T,R_{\textup{K}})(\|\psi_{0}\|_{H^{4}}+\|\psi_{1}\|_{H^{3}}),$
$\displaystyle\|\nabla\psi^{(1)}_{t}\|_{L^{2}(L^{\infty})}\leq$
$\displaystyle\,C_{H^{2},L^{\infty}}\sqrt{T}\|\nabla\psi^{(1)}_{t}\|_{L^{\infty}(H^{2})}$
$\displaystyle\leq$
$\displaystyle\,\begin{multlined}C_{H^{2},L^{\infty}}\sqrt{T}\left((1+T+T^{3/2}C_{\textup{lin},\textup{K}}(T,R_{\textup{K}}))^{2}+C_{\textup{lin},\textup{K}}(T,R_{\textup{K}})^{2}\right)^{1/2}\\\
\times(\|\psi_{0}\|_{H^{4}}+\|\psi_{1}\|_{H^{3}}),\end{multlined}C_{H^{2},L^{\infty}}\sqrt{T}\left((1+T+T^{3/2}C_{\textup{lin},\textup{K}}(T,R_{\textup{K}}))^{2}+C_{\textup{lin},\textup{K}}(T,R_{\textup{K}})^{2}\right)^{1/2}\\\
\times(\|\psi_{0}\|_{H^{4}}+\|\psi_{1}\|_{H^{3}}),$

where we have estimated

$\displaystyle\|\nabla\psi^{(1)}_{t}\|_{L^{\infty}(H^{2})}^{2}\leq$
$\displaystyle\,\|\nabla\psi^{(1)}_{t}\|_{L^{\infty}(L^{2})}^{2}+\|\nabla\Delta\psi^{(1)}_{t}\|_{L^{\infty}(L^{2})}^{2}$
$\displaystyle\leq$
$\displaystyle\,(\|\nabla\psi_{1}\|_{L^{2}}+T\|\nabla\psi_{2}\|_{L^{2}}+T^{3/2}\|\nabla\psi^{(1)}_{ttt}\|_{L^{2}(L^{2})})^{2}+\|\nabla\Delta\psi^{(1)}_{t}\|_{L^{\infty}(L^{2})}^{2}.$

Therefore, applying Gronwall’s inequality to (6.42) results in the bound

(6.44) $\displaystyle\|\overline{\psi}\|^{2}_{\textup{E}}\leq$
$\displaystyle\,\begin{multlined}C_{1}(1+T^{3/2})\,\sqrt{T}\,(\|\psi_{0}\|_{H^{4}}+\|\psi_{1}\|_{H^{3}})\\\
\times\exp(C_{2}(R_{\textup{K}}+(1+T^{3/2})\sqrt{T}(\|\psi_{0}\|_{H^{4}}+\|\psi_{1}\|_{H^{3}}))\|\overline{\phi}\|^{2}_{\textup{E}},\end{multlined}C_{1}(1+T^{3/2})\,\sqrt{T}\,(\|\psi_{0}\|_{H^{4}}+\|\psi_{1}\|_{H^{3}})\\\
\times\exp(C_{2}(R_{\textup{K}}+(1+T^{3/2})\sqrt{T}(\|\psi_{0}\|_{H^{4}}+\|\psi_{1}\|_{H^{3}}))\|\overline{\phi}\|^{2}_{\textup{E}},$

where the constants $C_{1}$, $C_{2}>0$ do not depend on $b$ nor $T$. By
sufficiently reducing the final time so that

$\displaystyle\begin{multlined}C_{1}(1+T^{3/2})\,\sqrt{T}\,(\|\psi_{0}\|_{H^{4}}+\|\psi_{1}\|_{H^{3}})\\\
\times\exp(C_{2}(R_{\textup{K}}+(1+T^{3/2})\sqrt{T}(\|\psi_{0}\|_{H^{4}}+\|\psi_{1}\|_{H^{3}}))<1,\end{multlined}C_{1}(1+T^{3/2})\,\sqrt{T}\,(\|\psi_{0}\|_{H^{4}}+\|\psi_{1}\|_{H^{3}})\\\
\times\exp(C_{2}(R_{\textup{K}}+(1+T^{3/2})\sqrt{T}(\|\psi_{0}\|_{H^{4}}+\|\psi_{1}\|_{H^{3}}))<1,$

we can guarantee that

$\|\psi^{(1)}-\psi^{(2)}\|_{\textup{E}}\leq
r\|\phi^{(1)}-\phi^{(2)}\|_{\textup{E}}\ \text{ for some }r\in(0,1).$

We note that it would be possible to impose smallness of data in the
$H^{4}\times H^{3}$ norm instead of reducing the final time, analogously to
the proof of well-posedness for the Westervelt equation.
We can argue similarly to the proof of Theorem 4.1 that
$(M_{\textup{K}},d_{\textup{E}})$ is a closed subset of a complete space with
the metric induced by the energy norm
$d_{\textup{E}}(x,y)=\|x-y\|_{\textup{E}}$. An application of Banach’s Fixed-
point theorem to $\mathcal{T}$ thus concludes the proof. ∎

##  7\. The inviscid limit of the Kuznetsov equation

Having obtained the uniform bounds with respect to $b$ in the $X^{\textup{K}}$
norm, we are now ready to derive the convergence rate for solutions of the
Kuznetsov equation as $b\rightarrow 0^{+}$.

######  Theorem 7.1.

Under the assumptions of Theorem 6.1, the family of solutions
$\\{\psi^{(b)}\\}_{b>0}$ of the Kuznetsov equation converges in the energy
norm to a solution $\psi$ of the inviscid Kuznetsov equation at a linear rate.
In other words,

(7.1) $\|\psi^{(b)}-\psi\|_{\textup{E}}\lesssim b\ \mbox{ as }b\to 0.$

###### Proof.

The proof follows along the lines of the proof of contractivity in the
previous theorem. Let $b,b^{\prime}\in(0,\overline{b})$. Let $\psi^{(b)}$ and
$\psi^{(b^{\prime})}$ be the solutions of the Kuznetsov equation with the
sound diffusivity $b$ and $b^{\prime}$, respectively, and with homogeneous
Dirichlet data, where we assume that initial conditions and final time are
chosen according to the well-posedness result of Theorem 6.1.
The difference $\overline{\psi}=\psi^{(b)}-\psi^{(b^{\prime})}$ then solves
the equation

(7.2)
$\displaystyle(1-\kappa\psi_{t}^{(b)})\overline{\psi}_{tt}-c^{2}\Delta\overline{\psi}-b\Delta\overline{\psi}_{t}$
$\displaystyle=$
$\displaystyle\,\begin{multlined}\kappa\overline{\psi}_{t}\psi^{(b^{\prime})}_{tt}+\sigma\nabla\overline{\psi}\cdot\nabla\psi^{(b)}_{t}+\sigma\nabla\psi^{(b^{\prime})}\cdot\nabla\overline{\psi}_{t}+(b-b^{\prime})\Delta\psi_{t}^{(b^{\prime})},\end{multlined}\kappa\overline{\psi}_{t}\psi^{(b^{\prime})}_{tt}+\sigma\nabla\overline{\psi}\cdot\nabla\psi^{(b)}_{t}+\sigma\nabla\psi^{(b^{\prime})}\cdot\nabla\overline{\psi}_{t}+(b-b^{\prime})\Delta\psi_{t}^{(b^{\prime})},$

supplemented by zero initial conditions. Testing with $\overline{\psi}_{t}$
and integrating with respect to space and time yields

(7.3)
$\displaystyle\frac{1}{2}\|\sqrt{\alpha(t)}\overline{\psi}_{t}(t)\|_{L^{2}}^{2}+\frac{c^{2}}{2}\|\nabla\overline{\psi}(t)\|_{L^{2}}^{2}+b\|\nabla\overline{\psi}_{t}\|_{L^{2}(0,t;L^{2})}^{2}$
$\displaystyle=$
$\displaystyle\,\begin{multlined}-\frac{\kappa}{2}\int_{0}^{t}\int_{\Omega}\psi_{tt}^{(b)}\overline{\psi}^{2}_{t}\,\textup{d}x\textup{d}s+\kappa\int_{0}^{t}\int_{\Omega}\psi_{tt}^{(b^{\prime})}\overline{\psi}^{2}_{t}\,\textup{d}x\textup{d}s\\\
+\sigma\int_{0}^{t}\int_{\Omega}(\nabla\overline{\psi}\cdot\nabla\psi^{(b)}_{t}+\nabla\psi^{(b^{\prime})}\cdot\nabla\overline{\psi}_{t})\overline{\psi}_{t}\,\textup{d}x\textup{d}s\\\
+(b-b^{\prime})\int_{0}^{t}\int_{\Omega}\Delta\psi_{t}^{(b^{\prime})}\overline{\psi}_{t}\,\textup{d}x\textup{d}s,\end{multlined}-\frac{\kappa}{2}\int_{0}^{t}\int_{\Omega}\psi_{tt}^{(b)}\overline{\psi}^{2}_{t}\,\textup{d}x\textup{d}s+\kappa\int_{0}^{t}\int_{\Omega}\psi_{tt}^{(b^{\prime})}\overline{\psi}^{2}_{t}\,\textup{d}x\textup{d}s\\\
+\sigma\int_{0}^{t}\int_{\Omega}(\nabla\overline{\psi}\cdot\nabla\psi^{(b)}_{t}+\nabla\psi^{(b^{\prime})}\cdot\nabla\overline{\psi}_{t})\overline{\psi}_{t}\,\textup{d}x\textup{d}s\\\
+(b-b^{\prime})\int_{0}^{t}\int_{\Omega}\Delta\psi_{t}^{(b^{\prime})}\overline{\psi}_{t}\,\textup{d}x\textup{d}s,$

where $\alpha(t)=1-\kappa\psi^{(b)}_{t}(t)$. We can rely on the following
estimate:

(7.4)
$\displaystyle\left|\int_{0}^{t}\int_{\Omega}\nabla\overline{\psi}\cdot\nabla\psi^{(b)}_{t}\overline{\psi}_{t}\,\textup{d}x\textup{d}s\right|\leq\|\nabla\psi^{(b)}_{t}\|_{L^{\infty}(0,t;L^{\infty})}\|\nabla\overline{\psi}\|_{L^{2}(0,t;L^{2})}\|\overline{\psi}_{t}\|_{L^{2}(0,t;L^{2})}.$

Similarly to (6.40), we also have the bound

(7.5)
$\displaystyle\left|\int_{0}^{t}\int_{\Omega}(\nabla\psi^{(b^{\prime})}\cdot\nabla\overline{\psi}_{t})\overline{\psi}_{t}\,\textup{d}x\textup{d}s\right|=$
$\displaystyle\,\left|\frac{1}{2}\int_{0}^{t}\int_{\Omega}(-\Delta\psi^{(b^{\prime})})\overline{\psi}_{t}\overline{\psi}_{t}\,\textup{d}x\textup{d}s\right|$
$\displaystyle\leq$
$\displaystyle\,\frac{1}{2}\|\Delta\psi^{(b^{\prime})}\|_{L^{\infty}(0,t;L^{\infty})}\|\overline{\psi}_{t}\|_{L^{2}(0,t;L^{2})}^{2}.$

The two $\kappa$ terms on the right-hand side of (7.3) can be estimated as
follows:

(7.6)
$\displaystyle\left|-\frac{1}{2}\kappa\int_{0}^{t}\int_{\Omega}\psi_{tt}^{(b)}\overline{\psi}^{2}_{t}\,\textup{d}x\textup{d}s+\kappa\int_{0}^{t}\int_{\Omega}\psi_{tt}^{(b^{\prime})}\overline{\psi}^{2}_{t}\,\textup{d}x\textup{d}s\right|$
$\displaystyle\leq$
$\displaystyle\,|\kappa|\left(\frac{1}{2}\|\psi_{tt}^{(b)}\|_{L^{\infty}(0,t;L^{\infty})}+\|\psi_{tt}^{(b^{\prime})}\|_{L^{\infty}(0,t;L^{\infty})}\right)\|\overline{\psi}_{t}\|_{L^{2}(0,t;L^{2})}^{2}.$

Thus, employing the above estimates in (7.3), Young’s inequality, and taking
the supremum over $t\in(0,\tau)$ leads to

(7.7)
$\displaystyle\frac{1}{2}\underline{\alpha}\sup_{t\in(0,\tau)}\|\overline{\psi}_{t}(t)\|_{L^{2}}^{2}+\frac{c^{2}}{2}\sup_{t\in(0,\tau)}\|\nabla\overline{\psi}(t)\|_{L^{2}}^{2}+b\|\nabla\overline{\psi}_{t}\|^{2}_{L^{2}(0,\tau;L^{2})}$
$\displaystyle\leq$
$\displaystyle\,\begin{multlined}|\kappa|\left(\frac{1}{2}\|\psi_{tt}^{(b)}\|_{L^{\infty}(L^{\infty})}+\|\psi_{tt}^{(b^{\prime})}\|_{L^{\infty}(L^{\infty})}\right)\|\overline{\psi}_{t}\|_{L^{2}(L^{2})}^{2}\\\
+\frac{1}{2}|\sigma|\|\nabla\psi^{(b)}_{t}\|^{2}_{L^{\infty}(L^{\infty})}\|\nabla\overline{\psi}\|_{L^{2}(L^{2})}^{2}+\frac{1}{2}|\sigma|\|\overline{\psi}_{t}\|^{2}_{L^{2}(L^{2})}\\\
+\frac{1}{2}|\sigma|\|\Delta\psi^{(b^{\prime})}\|_{L^{\infty}(L^{\infty})}\|\overline{\psi}_{t}\|_{L^{2}(L^{2})}^{2}+\frac{1}{2}(b-b^{\prime})^{2}\|\Delta\psi_{t}^{(b^{\prime})}\|_{L^{2}(L^{2})}^{2}+\frac{1}{2}\|\overline{\psi}_{t}\|_{L^{2}(L^{2})}^{2}.\end{multlined}|\kappa|\left(\frac{1}{2}\|\psi_{tt}^{(b)}\|_{L^{\infty}(L^{\infty})}+\|\psi_{tt}^{(b^{\prime})}\|_{L^{\infty}(L^{\infty})}\right)\|\overline{\psi}_{t}\|_{L^{2}(L^{2})}^{2}\\\
+\frac{1}{2}|\sigma|\|\nabla\psi^{(b)}_{t}\|^{2}_{L^{\infty}(L^{\infty})}\|\nabla\overline{\psi}\|_{L^{2}(L^{2})}^{2}+\frac{1}{2}|\sigma|\|\overline{\psi}_{t}\|^{2}_{L^{2}(L^{2})}\\\
+\frac{1}{2}|\sigma|\|\Delta\psi^{(b^{\prime})}\|_{L^{\infty}(L^{\infty})}\|\overline{\psi}_{t}\|_{L^{2}(L^{2})}^{2}+\frac{1}{2}(b-b^{\prime})^{2}\|\Delta\psi_{t}^{(b^{\prime})}\|_{L^{2}(L^{2})}^{2}+\frac{1}{2}\|\overline{\psi}_{t}\|_{L^{2}(L^{2})}^{2}.$

By virtue of Theorem 6.1, we have the uniform bound

(7.8)
$\displaystyle\begin{multlined}\|\psi_{tt}^{(b)}\|_{L^{\infty}(L^{\infty})}+\|\psi_{tt}^{(b^{\prime})}\|_{L^{\infty}(L^{\infty})}+\|\nabla\psi^{(b)}_{t}\|^{2}_{L^{\infty}(L^{\infty})}\\\
+\|\Delta\psi^{(b^{\prime})}\|_{L^{\infty}(L^{\infty})}\lesssim
R_{\textup{K}}+R_{\textup{K}}^{2},\end{multlined}\|\psi_{tt}^{(b)}\|_{L^{\infty}(L^{\infty})}+\|\psi_{tt}^{(b^{\prime})}\|_{L^{\infty}(L^{\infty})}+\|\nabla\psi^{(b)}_{t}\|^{2}_{L^{\infty}(L^{\infty})}\\\
+\|\Delta\psi^{(b^{\prime})}\|_{L^{\infty}(L^{\infty})}\lesssim
R_{\textup{K}}+R_{\textup{K}}^{2},$

and $\|\Delta\psi_{t}^{(b^{\prime})}\|_{L^{2}(L^{2})}^{2}\lesssim
TR_{\textup{K}}^{2}$. Therefore applying Gronwall’s inequality results in

(7.9)
$\|\overline{\psi}\|_{\textup{E}}=\|\psi^{(b)}-\psi^{(b^{\prime})}\|_{\textup{E}}\lesssim|b-b^{\prime}|.$

Finally, we note that the estimate remains valid for $b^{\prime}=0$ and thus
(7.1) holds, as claimed. ∎

## Conclusion

In this work, we have performed a convergence study of parabolic
approximations to quasilinear wave equations of second order as the damping
parameter $b$ tends to zero. Our research has been motivated by two classical
nonlinear acoustics models, the Westervelt and Kuznetsov equations. As the
analysis showed, the solutions of these equations converge in the energy norm
with a linear rate to the solutions of the respective inviscid equations (with
$b=0$), provided that the initial data are smooth and small and the final time
short enough. For a short enough final time, the smallness condition on the
initial data can be imposed in a lower-order space than the one needed to
derive higher-energy bounds. In particular, the smallness condition on the
data only arises from ensuring that the non-degeneracy assumption on $p$ and
$\psi_{t}$ holds in the Westervelt and Kuznetsov equations, respectively.

## Acknowledgment

The work of the first author was supported by the Austrian Science Fund fwf
under the grants P30054 and DOC 78.

## References

  * [1] M. Bongarti, S. Charoenphon, and I. Lasiecka, Vanishing relaxation time dynamics of the Jordan–Moore–Gibson–Thompson equation arising in nonlinear acoustics, Journal of Evolution Equations, (2021), pp. 1–32.
  * [2] I. Christov, C. I. Christov, and P. M. Jordan, Modeling weakly nonlinear acoustic wave propagation, The Quarterly Journal of Mechanics and Applied Mathematics, 60 (2007), pp. 473–495.
  * [3] P. Constantin and C. Foias, Navier-Stokes equations, University of Chicago Press, 1988.
  * [4] A. Dekkers and A. Rozanova-Pierrat, Cauchy problem for the Kuznetsov equation, Discrete & Continuous Dynamical Systems - A, 39 (2019), pp. 277–307.
  * [5] W. Dörfler, H. Gerner, and R. Schnaubelt, Local well-posedness of a quasilinear wave equation, Applicable Analysis, 95 (2016), pp. 2110–2123.
  * [6] L. C. Evans, Partial Differential Equations, vol. 2, Graduate Studies in Mathematics, AMS, 2010.
  * [7] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman Advanced Pub. Program Boston, 1985.
  * [8] M. F. Hamilton and D. T. Blackstock, Nonlinear acoustics, vol. 237, Academic press San Diego, 1998.
  * [9] J. Hoffelner, H. Landes, M. Kaltenbacher, and R. Lerch, Finite element simulation of nonlinear wave propagation in thermoviscous fluids including dissipation, IEEE transactions on ultrasonics, ferroelectrics, and frequency control, 48 (2001), pp. 779–786.
  * [10] T. J. Hughes, T. Kato, and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Archive for Rational Mechanics and Analysis, 63 (1977), pp. 273–294.
  * [11] B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discrete & Continuous Dynamical Systems-S, 2 (2009), p. 503.
  * [12] B. Kaltenbacher and V. Nikolić, The Jordan–Moore–Gibson–Thompson equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time, Mathematical Models and Methods in Applied Sciences, 29 (2019), pp. 2523–2556.
  * [13] M. Kaltenbacher, Numerical simulation of mechatronic sensors and actuators, vol. 3, Springer, 2014.
  * [14] S. Kawashima and Y. Shibata, Global existence and exponential stability of small solutions to nonlinear viscoelasticity, Communications in mathematical physics, 148 (1992), pp. 189–208.
  * [15] V. P. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics: Acoustics, 16 (1970), pp. 467–470.
  * [16] M. J. Lighthill and J. Lighthill, Waves in fluids, Cambridge university press, 2001.
  * [17] S. Meyer and M. Wilke, Global well-posedness and exponential stability for Kuznetsov’s equation in ${L}_{p}$-spaces, Evolution Equations & Control Theory, 2 (2013), pp. 365–378.
  * [18] K. Mizohata and S. Ukai, The global existence of small amplitude solutions to the nonlinear acoustic wave equation, Journal of Mathematics of Kyoto University, 33 (1993), pp. 505–522.
  * [19] T. Roubíček, Nonlinear partial differential equations with applications, vol. 153, Springer Science & Business Media, 2013.
  * [20] S. Salsa, Partial Differential Equations in Action: From Modelling to Theory - UNITEX), Springer Publishing Company, Incorporated, 2nd ed., 2015\.
  * [21] R. E. Showalter, Regularization and approximation of second order evolution equations, SIAM Journal on Mathematical Analysis, 7 (1976), pp. 461–472.
  * [22] J. Soneson, The parabolic approximation in therapeutic ultrasound, The Journal of the Acoustical Society of America, 146 (2019), pp. 3036–3036.
  * [23] A. Tani, Mathematical analysis in nonlinear acoustics, in AIP Conference Proceedings, vol. 1907, AIP Publishing LLC, 2017, p. 020003.
  * [24] G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate studies in mathematics, American Mathematical Soc., 2012.
  * [25] V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics, Springer Berlin Heidelberg, 2006.
  * [26] T. Tsuchiya and Y. Kagawa, A simulation study on nonlinear sound propagation by finite element approach, Journal of the Acoustical Society of Japan (E), 13 (1992), pp. 223–230.
  * [27] T. Walsh and M. Torres, Finite element methods for nonlinear acoustics in fluids, Journal of Computational Acoustics, 15 (2007), pp. 353–375.
  * [28] P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustical Society of America, 35 (1963), pp. 535–537.