On the Jordan-Moore-Gibson-Thompson equation: well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time

TL;DR

This paper analyzes the Jordan-Moore-Gibson-Thompson equation, establishing well-posedness with quadratic gradient nonlinearity and examining the singular limit as relaxation time vanishes, connecting it to classical nonlinear acoustics models.

Contribution

It proves well-posedness for the equation with gradient nonlinearity and studies the singular limit to classical models, using energy estimates and fixed-point methods.

Findings

Well-posedness in velocity potential formulation.

Convergence to Kuznetsov and Westervelt models as relaxation time vanishes.

Numerical experiments support theoretical results.

Abstract

In this paper, we consider the Jordan-Moore-Gibson-Thompson equation, a third order in time wave equation describing the nonlinear propagation of sound that avoids the infinite signal speed paradox of classical second order in time strongly damped models of nonlinear acoustics, such as the Westervelt and the Kuznetsov equation. We show well-posedness in an acoustic velocity potential formulation with and without gradient nonlinearity, corresponding to the Kuznetsov and the Westervelt nonlinearities, respectively. Moreover, we consider the limit as the parameter of the third order time derivative that plays the role of a relaxation time tends to zero, which again leads to the classical Kuznetsov and Westervelt models. To this end, we establish appropriate energy estimates for the linearized equations and employ fixed-point arguments for well-posedness of the nonlinear equations. The theoretical results are illustrated by numerical experiments.

Equations

$$\vec{v}={\vec{v}}_0+{\vec{v}}_{\sim },p=p_0+p_{\sim },\varrho ={\varrho }_0+{\varrho }_{\sim },$$

$${p_{\sim }}_{tt}-c^2\Delta p_{\sim }-\delta \Delta {p_{\sim }}_t={\left(\frac{1}{{\varrho }_0{c}^2}\frac{B}{2A}{p}_{\sim }^2+{\varrho }_0|\vec{v}{|}^2\right)}_{tt},$$

$$\delta=\frac{1}{\varrho_{0}}\left(\frac{4\mu_{V}}{3}+\zeta_{V}\Big{)}+\frac{\kappa}{\varrho_{0}}\Big{(}\frac{1}{c_{V}}-\frac{1}{c_{p}}\right),$$

$${\varrho }_0{\vec{v}}_t=-\nabla p.$$

$${p_{\sim }}_{tt}-c^2\Delta p_{\sim }-\delta \Delta {p_{\sim }}_t=\frac{{\beta }_a}{{\varrho }_0{c}^2}{p_{\sim }^2}_{tt},$$

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

$${\psi }_{tt}-c^2\Delta \psi -\delta \Delta {\psi }_t={\left(\frac{{\beta }_a}{c^2}({\psi }_t{)}^2\right)}_t,$$

$$\mathbf{q}=-K\nabla \vartheta$$

$$\tau {\mathbf{q}}_t+\mathbf{q}=-K\nabla \vartheta ,$$

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

$$b=\delta +\tau c^2.$$

$$\tau {\psi }_{ttt}+{\psi }_{tt}-c^2\Delta \psi -b\Delta {\psi }_t={\left(\frac{{\beta }_a}{c^2}({\psi }_t{)}^2\right)}_t,$$

$$\tau {\psi }_{ttt}+\alpha {\psi }_{tt}-c^2\Delta \psi -b\Delta {\psi }_t=f,$$

$$\gamma :=\alpha -\frac{\tau c^2}{b}.$$

$$E[\psi ](t)=\frac{1}{2}\left\{|{\psi }_t(t){|}^2+|\nabla \psi (t){|}^2+|{\psi }_{tt}(t){|}^2+|\nabla {\psi }_t(t){|}^2+|\Delta \psi (t){|}^2\right\}$$

$$\alpha {\psi }_{tt}-c^2\Delta \psi -\frac{b}{\alpha }\gamma \Delta {\psi }_t=f,$$

$$z:={\psi }_t+\frac{c^2}{b}\psi .$$

$$\tau z_{tt}+\gamma z_t-b\Delta z-\gamma \frac{c^2}{b}z+\gamma \frac{c^4}{b^2}\psi =f,$$

$$\tau {\psi }_{ttt}+(1-k{\psi }_t){\psi }_{tt}-c^2\Delta \psi -b\Delta {\psi }_t=2\nabla \psi \cdot \nabla {\psi }_t,$$

$$\psi =0\text{mbox}on\partial \Omega \times (0,T);$$

$$\psi (0)={\psi }_0,{\psi }_t(0)={\psi }_1,{\psi }_{tt}(0)={\psi }_2,$$

$$X^W:=$$

$$X^K:=$$

$${\bar{X}}^W:=$$

$${\bar{X}}^K:=$$

$$\Vert \psi {\Vert }_{X^W}^2:=$$

$$\Vert \psi {\Vert }_{X^K}^2:=$$

$$\Vert \psi {\Vert }_{{\bar{X}}^W}^2:=$$

$$\Vert \psi {\Vert }_{{\bar{X}}^K}^2:=$$

$$\Vert v{\Vert }_{L^6(\Omega )}\le C_{H^1,L^6}^{\Omega }\Vert v{\Vert }_{H^1(\Omega )},\Vert v{\Vert }_{L^{\infty }(\Omega )}\le C_{H^2,L^{\infty }}^{\Omega }\Vert v{\Vert }_{H^2(\Omega )}$$

$$\Vert v{\Vert }_{H^2(\Omega )}\le C_{(-\Delta {)}^{-1}}^{\Omega }\Vert -\Delta v{\Vert }_{L^2(\Omega )},\Vert v{\Vert }_{H^1(\Omega )}\le C_{PF}^{\Omega }\Vert \nabla v{\Vert }_{L^2(\Omega )},$$

$$\Vert \nu \cdot \nabla v{\Vert }_{H^{-1/2}(\partial \Omega )}\le C_{tr}^{\Omega }\Vert v{\Vert }_{H^1(\Omega )},\Vert v{\Vert }_{H^{1/2}(\partial \Omega )}\le C_{tr}^{\Omega }\Vert v{\Vert }_{H^1(\Omega )},$$

$$\{\begin{array}{l}\tau {\psi }_{ttt}+\alpha (x,t){\psi }_{tt}-c^2\Delta \psi -b\Delta {\psi }_t=f\text{mbox}in\Omega \times (0,T),\\ \psi =0\text{mbox}on\partial \Omega \times (0,T),\\ (\psi ,{\psi }_t,{\psi }_{tt})=({\psi }_0,{\psi }_1,{\psi }_2)\text{mbox}in\Omega \times \{0\},\end{array}$$

Full text

\ccode

AMS Subject Classification: 35L77, 35L72, 35L80,

1 Introduction

Nonlinear propagation of sound arises in numerous applications. We here especially mention high-intensity ultrasound used in medical imaging and therapy, but also for industrial purposes, such as ultrasound cleaning or welding; see, e.g., Refs. \refciteabramov,dreyer2000,uffc2002, and the given references therein. For the physical fundamentals of nonlinear acoustics, we refer to, e.g., Refs. \refciteCrighton79,EnfloHedberg,HamiltonBlackstock98,Kuznetsov71,MakarovOchmann96,MakarovOchmann97b,MakarovOchmann97a,Pierce89.

Its physical and mathematical description involves the acoustic particle velocity $\vec{v}$, the acoustic pressure $p$, as well as the mass density $\varrho$, which can be decomposed into constant and fluctuating components

[TABLE]

where in the applications mentioned above, the ambient flow vanishes; i.e, $\vec{v}_{0}=0$. Furthermore, we have the balances of mass, momentum and sometimes of energy, complemented with an equation of state that relates the mass density to the pressure. Combination of these balance and material laws yields wave-type partial differential equations that are often second order in space and time, but higher order in time equations play an important role as well. It is one of these third order in time equations that we focus on in this paper.

One of the most established models of nonlinear acoustics is Kuznetsov’s equation[24, 28]

[TABLE]

where $c$ is the speed of sound, $\delta$ is the diffusivity of sound

[TABLE]

and the velocity is related to the pressure via some balance of forces,

[TABLE]

By ignoring local nonlinear effects modeled by the quadratic velocity term, we arrive at the Westervelt equation[47]

[TABLE]

with $\beta_{a}=1+B/(2A)$. In terms of the acoustic velocity potential $\psi$ satisfying $\vec{v}=-\nabla\psi$ and $p=\varrho_{0}\psi_{t}$, these equations can be rewritten as

[TABLE]

and

[TABLE]

respectively.

As has been observed in, e.g., Ref. \refciteJordanMaxwellCattaneo14, the use of classical Fourier’s law

[TABLE]

where $\vartheta$, $\mathbf{q}$, and $K$ denote the absolute temperature, heat flux vector, and thermal conductivity, respectively, leads to an infinite signal speed paradox, which appears to be unnatural in wave propagation. Therefore in Ref. \refciteJordanMaxwellCattaneo14, several other constitutive relations for the heat flux are considered within the derivation of nonlinear acoustic wave equations. Among these is the Maxwell–Cattaneo law

[TABLE]

where $\tau$ is a positive constant accounting for relaxation (the relaxation time), whose combination with the above mentioned balance equations and the equation of state leads to the third order in time PDE model:

[TABLE]

where

[TABLE]

This model is known in the literature as the Jordan–Moore–Gibson–Thompson (JMGT) equation[20] and we refer to Ref. \refciteJordanMaxwellCattaneo14 for its derivation. If one neglects local nonlinear effects modeled by the quadratic velocity term in (6), one arrives at

[TABLE]

analogously to the reduction of the Kuznetsov to the Westervelt equation[20]. We will refer to equation (6) as the Kuznetsov-type and to (8) as the Westervelt-type JMGT equation. Obviously, equations (6) and (8) formally reduce to (4) and (5) upon setting $\tau=0$. The present work is, in part, devoted to the rigorous justification of passing to the limit $\tau\to 0$ in (6) and (8).

In Ref. \refciteKLM12_MooreGibson and, more comprehensively, in Ref. \refciteMarchandMcDevittTriggiani12, the linearized equation

[TABLE]

often called the Moore–Gibson–Thompson equation, is studied using semigroup techniques; see also Refs. \refciteDellOroPata and \refcitePellicerSolaMorales. As it turns out, the exponential stability of the trajectories depends on the critical parameter given by

[TABLE]

In the case of a constant coefficient $\alpha$, exponential decay of the energy function

[TABLE]

requires $\gamma$ to be strictly positive. The case $\gamma<0$ is unstable and the case $\gamma=0$ marginally stable. An intuitive explanation for this phenomenon is the following: According to the linear wave part of the equation, we can trade $\alpha\psi_{tt}$ for $c^{2}\Delta\psi$, thus also $\tau\psi_{ttt}$ for $\frac{\tau c^{2}}{\alpha}\Delta\psi_{t}$ in order to relate (9) back to the linearization of (5)

[TABLE]

which is a strongly damped wave equation.

We mention that the Moore–Gibson–Thompson equation (9) is also studied in Ref. \refciteLiuTriggiani13, where the problem of identifying $\gamma(t)$ from boundary measurements is considered, and in Ref. \refciteLasieckaWang15a and \refciteLasieckaWang15b, where the effect of additive convolution memory terms acting on $\Delta u$ and $\Delta u_{t}$, and their combination, respectively, is investigated.

For a reformulation of the nonlinear equation (8) in terms of the acoustic pressure $p=\varrho_{0}\psi_{t}$, global in time well-posedness and exponential decay of the energy $E[p](t)$ is proven in Ref. \refciteKLP12_JordanMooreGibson for small initial data $(p_{0},p_{1},p_{2})\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\times H_{0}^{1}(\Omega)\times L^{2}(\Omega)$, where $\Omega$ is a bounded $C^{2}$ smooth domain.

One of the key elements in the above cited works on the analysis of equations (8) and (9) is introduction of the auxiliary state

[TABLE]

Indeed, in this manner, the third order in time equation (9) is reduced to a linear (weakly) damped wave equation for $z$,

[TABLE]

where $-\gamma\tfrac{c^{2}}{b}z+\gamma\frac{c^{4}}{b^{2}}\psi$ is a lower order term.

This approach, first of all, illustrates the fact that $\gamma$ should be non-negative to guarantee a damping behaviour of the term $\gamma z_{t}$. Secondly, it displays the key difference to the strongly damped second order equations (4) and (5). As pointed out in Subsection 6.2.1 of Ref. \refciteMarchandMcDevittTriggiani12, equation (9) does not give rise to an analytic semigroup; see also Remark 1.3 in Ref. \refciteKLM12_MooreGibson. Consequently, the operator driving the evolution does not exhibit maximal parabolic regularity[27] and the Implicit function theorem argument from, e.g., Ref. \refciteMW13 cannot be transferred to the present setting.

2 Main results

This paper contributes to the analysis of the JMGT equation in two ways.

Firstly, we prove well-posedness with a quadratic gradient nonlinearity arising when taking into account local nonlinear effects; cf. the additional $(|\nabla\psi|^{2})_{t}$ term on the right hand side in (6) compared to (8). We base our approach on energy estimates for a reformulation of (6) in the form

[TABLE]

where we use the abbreviation $k=\frac{2}{c^{2}}\frac{B}{2A}$. The sign of $k$ will not matter in what follows, whereas we assume the coefficients $b$ and $c^{2}$ to be strictly positive. We rely on the formulation of the equations in terms of the acoustic velocity potential $\psi$ and not the acoustic pressure[20], since it allows to include more easily the quadratic velocity term $(|\nabla\psi|^{2})_{t}$ on the right-hand side. We note that the energy estimates required for this purpose differ from those provided in Ref. \refciteKLP12_JordanMooreGibson for the equation (8) without the quadratic gradient nonlinearity.

Secondly, we consider the limit $\tau\to 0$ and prove that solutions of the Kuznetsov-type equation (6) converge to a solution of equation (4) as $\tau\to 0$, and analogously for the Westervelt-type equation. For this purpose, the energy estimates we will derive are crucial. These estimates differ for the Kuznetsov-type and for the Westervelt-type version of the JMGT equation, which is why we treat these models in separate sections.

The rest of the paper is organized as follows. In Section 3, we consider the linearized equation (9) with a fixed positive coefficient $\alpha$ that is possibly space and time dependent, but bounded away from zero, and an inhomogeneity $f$, as well as a fixed positive $\tau$. We prove well-posedness of this linearized model together with an energy estimate.

Section 4 contains a well-posedness proof for the Westervelt version (8) of the equation by setting $\alpha=1-k\psi_{t}$ and $f=0$. The proof is based on the equation (14), but with zero right-hand side, as the gradient nonlinearity is not present in (8). This fact allows to prove local in time well-posedness for small inital data, even without any sign condition on the parameter $\gamma=\alpha-\frac{\tau c^{2}}{b}$. However, the energy estimates from Section 3 do not cover the gradient nonlinearity in the Kuznetsov-type version of the JMGT equation, so that higher order energy estimates are needed. We derive them in Section 5. These involve the auxiliary function $z$ and require positivity of both $\alpha$ and $\gamma$, where the latter follows from positivity and boundedness away from zero of $\alpha$ for $\tau$ sufficiently small.

Section 6 provides the corresponding well-posedness result for the equation (6) based on reformulation (14); i.e., setting $\alpha=1-k\psi_{t}$ and $f=2\nabla\psi\cdot\nabla\psi_{t}$. Starting from a sufficiently small positive value and letting $\tau$ tend to zero clearly preserves the sign structure of the coefficients, in particular of $\gamma$ and $b$, so that the energy bounds from Sections 4 and 6 can be used for justifying the limiting process $\tau\to 0$ in Section 7. There we also provide a brief comparison of the resulting regularity for the limiting equations (4), (5) to those already present in the literature. Finally, numerical experiments in Section 8 illustrate the theoretical findings.

Remark 2.1** (On medium parameters).**

We require strict positivity of constants $c$, $\delta$, and $\tau$ (hence, also of $b=\delta+\tau c^{2}$) appearing in the equations for proving well-posedness of initial-boundary value problems (6) and (8). These are, indeed, very natural assumptions from a physical point of view; typical values of these parameters in different media can be found, for example, in Refs. \refcitecrocker1998handbook and \refciteRossing. It is also known that, in order to establish global well-posedness for the limiting problems (4) and (5) in space dimensions higher than one, strict positivity of $\delta$ is needed[17, 18]. The constant $k$ does not need to have a particular sign in our analysis, but will typically be non-negative in applications.

2.1 Theoretical preliminaries and assumptions

We set here the notation and collect some useful theoretical results that we often use in the analysis. Throughout the paper, the spatial domain $\Omega\subset\mathbb{R}^{d}$, where $d\in\{1,2,3\}$, is assumed to be sufficiently smooth to admit integration by parts as well as second order elliptic regularity.

We consider the PDEs on a bounded space time cylinder $\Omega\times(0,T)$ and impose homogeneous Dirichlet boundary conditions on $\partial\Omega$ for simplicity

[TABLE]

i.e., $(-\Delta):H_{0}^{1}(\Omega)\to H^{-1}(\Omega)$ is the Laplace operator equipped with homogeneous Dirichlet boundary conditions. We expect that Neumann and impedance (absorbing) boundary conditions can be treated analogously, but lead to modifications in the energy estimates.

The third order in time equations (6) and (8) are complemented with initial conditions

[TABLE]

whereas for the limiting second order in time equations (4), (5) as $\tau\to 0$, the initial condition on $\psi_{tt}$ naturally disappears, which will be seen also in the energy estimates.

2.1.1 Notation

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 $L^{p}(0,T;L^{q}(\Omega))$. We also abbreviate the $L^{2}(\Omega)$ inner product by $(\cdot,\cdot)_{L^{2}}$ and the $L^{2}(\Omega)$ norm (as well as the absolute value) by $|\cdot|$. We employ the notation $L^{p}(0,T;Z)$, $W^{s,p}(0,T;Z)$ for the Bochner-Sobolev spaces of time dependent functions.

More specifically, we use dedicated function spaces for the solutions of the considered equations. We collect their notation here for future reference:

[TABLE]

with partly $\tau$-dependent norms induced by the energy estimates to be derived

[TABLE]

2.1.2 Helpful inequalities

Throughout the paper, we often employ the continuous embeddings $H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$, $H^{2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$

[TABLE]

as well as boundedness of the operator $(-\Delta)^{-1}:L^{2}(\Omega)\to H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$, the Poincaré-Friedrichs inequality,

[TABLE]

and the trace theorem

[TABLE]

see, e.g., Lemma 4.3 in Ref. \refciteMcLean00, where $\nu$ denotes the outward unit normal on the boundary of $\Omega$.

3 Analysis of the linear damped wave equation (13)

We now turn our attention to the linear equation (9) and study the following initial-boundary value problem:

[TABLE]

under the non-degeneracy assumption that for some $\underline{\alpha}>0$

[TABLE]

We assume that the coefficient $\alpha$ and the source term $f$ have the following regularity:

[TABLE]

Moreover, $\psi$ is assumed to satisfy the initial conditions (16) with

[TABLE]

For an analysis of equation (9) with constant coefficient $\alpha$, under the slightly weaker assumptions

[TABLE]

and $f\in L^{1}(0,T;L^{2}(\Omega))$, we refer to Corollary 1.2 in Ref. \refciteKLM12_MooreGibson. There it is shown that $\psi\in C(0,T;H_{0}^{1}(\Omega)\cap H^{2}(\Omega))\cap\,C^{1}(0,T;H_{0}^{1}(\Omega))\cap\,C^{2}(0,T;L^{2}(\Omega))$ by means of semigroup techniques. The assumptions (24) and (25) naturally arise from the energy estimates in the well-posedness proof below and lead to the stronger (as compared to Ref. \refciteKLM12_MooreGibson) regularity stated in (26) below.

Theorem 3.1**.**

Let $c^{2}$, $b$, $\tau>0$, and let $T>0$ be a fixed time horizon. Let the non-degeneracy assumption (23) and the regularity assumptions (24) and (25) hold. Then there exists a unique solution $\psi$ of the problem (22) that satisfies

[TABLE]

Furthermore, the solution fullfils the estimate

[TABLE]

The constant above is given by

[TABLE]

where $C_{1}$, $C_{2}>0$ do not depend on $\tau,T$, or $\alpha$.

If additionally we assume that $\|\nabla\alpha\|_{L^{\infty}L^{3}}$ is sufficiently small so that

[TABLE]

holds, then (27) is valid with an upper bound that is independent of $\tau$, i.e.,

[TABLE]

Proof 3.2**.**

We carry out the proof by via Galerkin approximations in space, relying on energy estimates; cf. Refs. \refciteEvansBook,Roubicek. Note that the initial data are meaningful since regularity (26) implies

[TABLE]

*where $C_{w}$ denotes the space of weakly continuous functions; see Lemma 3.3 in Ref. \refciteTemam.

Step 1: Discretization in space. Let $\{w_{i}\}_{i\in\mathbb{N}}$ denote the eigenfunctions of the Dirichlet-Laplacian operator $-\Delta$. Then $\{w_{i}\}_{i\in\mathbb{N}}$ can be normalized to form an orthogonal basis of $H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ and to be orthonormal with respect to the $L^{2}(\Omega)$ scalar product.

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*

[TABLE]

where $\xi_{i}:(0,T)\rightarrow\mathbb{R}$, $i\in[1,n]$. The initial data are chosen as

[TABLE]

where the coefficients $\xi_{i,0}$, $\xi_{i,1}$, $\xi_{i,2}\in\mathbb{R}$ are given by

[TABLE]

for $i\in[1,n]$. In this way, we have by construction that

[TABLE]

see Lemma 7.5 in Ref. \refciteRobinson. We then consider the following approximation of our problem

[TABLE]

We introduce matrices $I^{n}=[I_{ij}]$, $M^{n}=[M_{ij}]$, $K^{n}=[K_{ij}]$, $C^{n}=[C_{ij}]$, and vector $F^{n}=[F_{i}]$, where

[TABLE]

for $i,j\in[1,n]$, where $\delta_{ij}$ denotes the Kronecker delta. By setting $\xi^{n}=[\xi_{1}\ldots\xi_{n}]^{T}$, $\xi_{0}^{n}=[\xi_{1,0}\ldots\xi_{n,0}]^{T}$, $\xi_{1}^{n}=[\xi_{1,1}\ldots\xi_{n,n}]^{T}$, and $\xi_{2}^{n}=[\xi_{1,2}\ldots\xi_{n,2}]^{T}$, problem (32) can be rewritten as

[TABLE]

After additionally rewriting (34) as a first-order system, existence of an absolutely continuous solution $[\xi^{n},\xi^{n}_{t},\xi^{n}_{tt}]^{T}$ on $[0,T_{n}]$ for some $T_{n}\leq T$ follows from standard ODE theory; see, for example, Chapter 1 in Ref. \refciteRoubicek. To see that $\xi^{n}\in H^{3}(0,T_{n})$, we can employ a bootstrap argument,

[TABLE]

*We, therefore, conclude that (32) has a solution $\psi^{n}\in H^{3}(0,T_{n};V_{n})$. The upcoming energy estimate will allow us to extend the existence interval to $[0,T]$.

Step 2: Energy estimates. Our next goal is to obtain a bound for $\psi^{n}$ that is uniform with respect to $n$. To this end, we test our problem (32) with a suitable test function.

First estimate. Testing the first equation in (32) with $\phi=-\Delta\psi^{n}_{tt}\in V_{n}$ and integrating over $(0,t)$, where $t\leq T_{n}$, yields the energy identity*

[TABLE]

where we have skipped the argument $(s)$ under the time integral for notational simplicity and used the abbreviation $L_{t}^{2}L^{2}$ for $L^{2}(0,t;L^{2}(\Omega))$. To derive (36), we have used the following three identities:

[TABLE]

and

[TABLE]

as well as

[TABLE]

We note that $f\in H^{1}(0,T;L^{2}(\Omega))\hookrightarrow C([0,T];L^{2}(\Omega))$. We next estimate $\textup{rhs}_{1}(t)$ from above. We introduce here a constant that depends on the initial data to simplify the notation:

[TABLE]

By applying Hölder’s inequality, we get

[TABLE]

We further estimate the right-hand side with the help of Young’s $\varepsilon$-inequality

[TABLE]

and choosing $\varepsilon=\frac{b}{4}$ or $\varepsilon=1$, and the embedding results to obtain for a.e. $t\in[0,T_{n}]$,

[TABLE]

We can estimate the term $\|-\Delta\psi^{n}(t)\|_{L^{2}}$ appearing on the right-hand side as follows

[TABLE]

Altogether, we get

[TABLE]

*If the smallness assumption (28) holds, then the term containing $\|\nabla\alpha\|_{L^{\infty}L^{3}}$ can be absorbed into the left-hand side.

A priori bound for $\psi^{n}$. Applying Gronwall’s inequality to (40), and taking the supremum over $t\in(0,T_{n})$ then yields*

[TABLE]

By employing the upper bounds for the approximate initial data stated in (31) and the inequality $T_{n}\leq T$, we further have

[TABLE]

The constant above is given by

[TABLE]

or, if assumption (28) holds, by

[TABLE]

*where $\tilde{C}_{1}$, $\tilde{C}_{2}>0$ do not depend on $n$ or $\tau$. Since the right-hand side of (42) does not depend on $T_{n}$, we can extend the existence interval to $[0,T]$, i.e. $T_{n}=T$.

Second estimate. By testing (32) with $\phi=\tau\psi^{n}_{ttt}\in V_{n}$ and integrating over $(0,T)$, we obtain*

[TABLE]

from which we have

[TABLE]

The terms $\|-\Delta\psi^{n}\|^{2}_{L^{2}L^{2}}$, $\|-\Delta\psi^{n}_{t}\|^{2}_{L^{2}L^{2}}$ can be further estimated similarly to (39),

[TABLE]

*The term $\|\psi^{n}_{tt}\|_{L^{2}L^{2}}$ by means of the Poincaré-Friedrichs inequality, so that by using (42) we obtain energy estimate (27) with $\psi^{n}$ in place of $\psi$.

Step 3: Passing to the limit. On account of estimate (42) and standard compactness results, together with the fact that the spatial and temporal domains $(0,T)$ and $\Omega$ are bounded, we know that there exist a subsequence, denoted again by $\{\psi^{n}\}_{n\in\mathbb{N}}$, and a function $\psi$ such that*

[TABLE]

Our next task is to prove that $\psi$ solves (22). We test (32) with $\eta\in C_{c}^{\infty}(0,T)$ and integrate over time to obtain

[TABLE]

for all $i\in[1,n]$. Thanks to (45), letting $n\rightarrow\infty$ in (46) leads to

[TABLE]

for all $i\in\mathbb{N}$ and $\eta\in C^{\infty}(0,T)$. By construction, $\cup_{n\in\mathbb{N}}V_{n}$ is dense in $L^{2}(\Omega)$, so $\psi$ solves the PDE in (22) in the $L^{2}(0,T;L^{2}(\Omega))$ sense. Due to the embeddings

[TABLE]

we know that

[TABLE]

*Thanks to (31), we can then infer that $\psi(0)=\psi_{0}$, $\psi_{t}(0)=\psi_{1}$, and $\psi_{tt}(0)=\psi_{2}$. Altogether, we conclude that $\psi$ is a solution of the initial-boundary value problem (22).

Step 4: Energy inequality for $\psi$. We can take the limit inferior as $n\rightarrow\infty$ of (42), (44), and via the weak and the weak-$\star$ lower semi-continuity of norms obtain the final estimate (27). Uniqueness of a solution follows by the linearity of the equation, together with the fact that the homogeneous equation only has the zero solution, by the above energy estimates.*

4 Well-posedness of the nonlinear Westervelt-type wave equation for $\tau>0$

After having studied the linearized equation, we now proceed to the nonlinear model (8). For proving well-posedness of (8), we introduce the fixed-point operator $\mathcal{T}$ that maps $\phi$ to a solution $\psi$ of

[TABLE]

on some ball

[TABLE]

in the space $X^{W}$, defined in (17). Note that the operator is well-defined on account of Theorem 3.1.

For establishing $\mathcal{T}$ as a self-mapping on $B_{\rho}^{X^{W}}$, it is crucial to prove that $\alpha=1-k\phi_{t}$ is in $L^{\infty}(0,T;L^{\infty}(\Omega))\cap L^{\infty}(0,T;W^{1,3}(\Omega))$ and that the smallness condition (28) of Theorem 3.1 holds, provided $\phi\in B_{\rho}^{X}$. Smallness of $\phi$ will also be required for verifying the non-degeneracy condition $\alpha(t)\geq\underline{\alpha}>0$.

Note that the radius of the neighborhood in which the self-mapping property holds will be independent of $\tau$. In particular, it holds for arbitrarily small $\tau$ and therefore allows for taking limits as $\tau\to 0$ later on.

Contractivity of $\mathcal{T}$, based on the fact that $\hat{\psi}=\psi_{1}-\psi_{2}=\mathcal{T}(\phi_{1})-\mathcal{T}(\phi_{2})$ solves

[TABLE]

with homogeneous initial and boundary conditions (where $\hat{\phi}=\phi_{1}-\phi_{1}$), would require to prove that $\alpha_{1}=1-k\phi_{1\,t}$ and $f_{2}=k\hat{\phi}_{t}\psi_{2\,tt}$ are in $L^{\infty}(0,T;L^{\infty}(\Omega))\cap L^{\infty}(0,T;W^{1,3}(\Omega))$ and $H^{1}(0,T;L^{2}(\Omega))$, respectively. This regularity, however, would only be possible in an $O(\sqrt{\tau})$ neighborhood because of the $\phi_{2ttt}$ term arising in $f_{2t}$. Therefore, we do not prove contractivity, but base our existence proof on Schauder’s theorem, similarly to the approach in Ref. \refciteKT18_ModelsNlAcoustics.

Theorem 4.1**.**

Let $c^{2}$, $b>0$, $k\in\mathbb{R}$ and let $T>0$. Then there exist $\rho>0$ and $\rho_{0}>0$ such that for all $(\psi_{0},\psi_{1},\psi_{2})\in X^{W}_{0}=H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\times H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\times H_{0}^{1}(\Omega)$ satisfying

[TABLE]

there exists a solution $\psi\in X^{W}$ of

[TABLE]

such that it holds

[TABLE]

Proof 4.2**.**

Our proof relies on Schauder’s fixed-point theorem applied to the operator

[TABLE]

where $\psi$ solves (47). To obtain the self-mapping property of $\mathcal{T}$ , we have to verify the condition (28) of Theorem 3.1 as well as $\alpha(t)\geq\underline{\alpha}>0$. We thus estimate the $L^{\infty}(0,T;L^{\infty}(\Omega))\cap L^{\infty}(0,T;W^{1,3}(\Omega))$ norm of $\alpha=1-k\phi_{t}$. In view of the bounds

[TABLE]

the smallness condition (28) and the non-degeneracy condition $\alpha(t)\geq\underline{\alpha}>0$ can be satisfied by choosing

[TABLE]

The self-mapping property follows from the estimate (27), with $f=0$ and

[TABLE]

The set $B_{\rho}^{X^{W}}$ is a weak compact and convex subset of the Banach space $X^{W}$, defined in (17). The weak* continuity of $\mathcal{T}$ can be established as follows: For any sequence $(\phi_{n})_{n\in\mathbb{N}}\subseteq B_{\rho}^{X^{W}}$ that weakly* converges to $\phi\in B_{\rho}^{X^{W}}$ in $X^{W}$, we also have*

[TABLE]

Thus, by compactness of the embedding $X^{W}\to W^{1,\infty}(0,T;L^{\infty}(\Omega))$, there exists a subsequence $(\phi_{n_{\ell}})_{\ell\in\mathbb{N}}$ such that $1-k\phi_{n_{\ell}\,t}$ converges to $1-k\phi_{t}$ strongly in $L^{\infty}(0,T;L^{\infty}(\Omega))$ and $\mathcal{T}(\phi_{n_{\ell}})$ converges weakly in $X^{W}$ to some $\psi\in B_{\rho}^{X^{W}}$, which by definition of $B_{\rho}^{X^{W}}$ satisfies the initial and homogeneous Dirichlet boundary conditions. It is readily checked that $\psi$ also solves the PDE (47), which, by uniqueness in Theorem 3.1, implies $\psi=\mathcal{T}(\phi)$. A subsequence-subsequence argument yields weak* convergence in $X^{W}$ of the whole sequence $(\mathcal{T}(\phi_{n}))_{n\in\mathbb{N}}$ to $\mathcal{T}(\phi)$.

We can therefore conclude existence of a fixed point of $\mathcal{T}$ in $B_{\rho}^{X^{W}}$ from the general version of Schauder’s fixed-point theorem in locally convex topological spaces; see Ref \refciteFan1952, which we here quote for the convenience of the reader*

Let $L$ be a locally convex topological linear space and $K$ a compact convex set in $L$. Let $M(K)$ be the family of all closed convex (non- empty) subsets of $K$. Then for any upper semicontinuous point-to-set transformation $f$ from $K$ into $M(K)$, there exists a point $x_{0}\in K$ such that $x_{0}\in f(x_{0})$.

We use this theorem with the single valued point-to-set relation (i.e., mapping) $f=\mathcal{T}$, the weaktopology on $X^{W}$, and $K=B_{\rho}^{X^{W}}$.*

5 Higher energy estimates

Due to the appearance of $\|f_{t}\|_{L^{2}L^{2}}$ on the right-hand side of estimate (27) in Theorem 3.1, we cannot rely only on this estimate for the Kuznetsov-type JMGT equation (6) since

[TABLE]

Existence of solutions can still be based on Theorem 3.1, (case $\|\nabla\alpha\|_{L^{\infty}L^{3}}<\underline{\alpha}/C^{\Omega}_{H^{1},L^{6}}$ with a $\tau$-independent bound on the energy) because $f$ is still in the right space. However,

[TABLE]

can only be shown to be bounded by $\frac{1}{\sqrt{\tau}}$, so it might be large as $\tau\to 0$. This does not matter for proving existence according to Theorem 3.1, but excludes a fixed-point argument for proving well-posedness of the nonlinear equation (6) in this setting.

To be able to take limits as $\tau\to 0$, we thus need higher order energy estimates. In particular, we need to derive $\tau$-independent bounds on $\|\psi\|_{L^{\infty}H^{3}}$ which will enable us to estimate $f=2\nabla\psi\cdot\nabla\psi_{t}$ in the required norms. We replace estimate (27) by an estimate on the auxiliary function $z$ and complement this with a higher order in space estimate on $\psi$.

In order to carry out these new error estimates, we now turn our attention to studying the equation (9), restated again here for convenience:

[TABLE]

together with its equivalent reformulation (13) using (12), (10), i.e.,

[TABLE]

where we recall that the auxiliary state is given by

[TABLE]

We assume that for some $\underline{\alpha}>0$, $\underline{\gamma}>0$

[TABLE]

Theorem 5.1**.**

Let $c^{2}$, $b$, and let $T>0$. Assume that

• •

$\alpha\in W^{1,1}(0,T;H^{1}(\Omega))\cap L^{\infty}(0,T;W^{1,3}(\Omega)\cap L^{1}(0,T;H^{2}(\Omega)$,

• •

$f\in H^{1}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{1}(\Omega))$,

• •

$(\psi_{0},\psi_{1},\psi_{2})\in X^{K}_{0}:=H_{0}^{1}(\Omega)\cap H^{3}(\Omega)\times H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\times H_{0}^{1}(\Omega)$,

and that the non-degeneracy condition (53) holds. Then there exists $\bar{\tau}>0$ such that for $\tau\in(0,\bar{\tau})$ and sufficiently small $\|\nabla\gamma\|_{L^{\infty}L^{3}}$, there exists a unique solution $(\psi,z)$ of the problem

[TABLE]

that satisfies $(\psi,z)\in L^{\infty}(0,T;H^{3}(\Omega))\times(L^{\infty}(0,T;H^{2}(\Omega))\cap W^{1,\infty}(0;T;H_{0}^{1}(\Omega)))$; in other words,

[TABLE]

Furthermore, there exists a $C(\gamma,T)>0$, which does not depend on $\tau$, such that

[TABLE]

Note that while $f\in H^{1}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{1}(\Omega))$ is required, in the right-hand side of the energy estimate only $\|f\|_{L^{2}H^{1}}$, but not $\|f\|_{H^{1}L^{2}}$ appears.

Proof 5.2**.**

Step 1: Existence of a solution.* Theorem 3.1 implies existence of a solution $(\psi,z)$ of (54) with regularity as stated in (26) and*

[TABLE]

*Note that $z_{t}$ inherits the homogeneous Dirichlet boundary conditions from $\psi$. Thus it only remains to establish the higher order energy estimates. For this purpose, we return to the Galerkin approximation (32) and define $z^{n}=\psi^{n}+\tfrac{c^{2}}{b}\psi^{n}$. *

Step 2: A priori estimates. As in Section 3, our goal is to obtain a bound for $\psi^{n}$ that is uniform with respect to $n$. To this end, we test the spatially discretized version of our problem (54) with two test functions. *

The first energy identity. Problem (32) can be equivalently rewritten as*

[TABLE]

where $V_{n}$ is defined as in the proof of Theorem 3.1, as the span of the first $n$ eigenfunctions of the Laplacian with homogeneous Dirichlet boundary conditions. Multiplying the first equation in (57) by $-\Delta z^{n}_{t}\in V_{n}$ and integrating over $\Omega$ and $(0,t)$ yields the energy identity

[TABLE]

where we have skipped the argument $(s)$ under the time integral for notational simplicity. To derive (58), we have used the identity

[TABLE]

Furthermore, we have made use of

[TABLE]

as well as the identity

[TABLE]

The second energy identity.* Our aim is to obtain a bound on $\psi$ in the $H^{3}(\Omega)$ norm. To this, end we test (32) with $\phi=(-\Delta)^{2}\psi^{n}\in V_{n}$ (due to the fact that $\psi^{n}$ is a linear combination of eigenfunctions of $-\Delta$) which yields the second energy identity*

[TABLE]

Above, we have made use of

[TABLE]

and the fact that $-\Delta\psi^{n}_{t}=0$ on $\partial\Omega$. Morever, we rewrote the $\alpha$ term as follows

[TABLE]

*where we used again that $-\Delta\psi_{t}^{n}=0$ on $\partial\Omega$. Note that under the assumptions made on $\alpha$, we have that $(\alpha\psi^{n}_{t})(t)\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ for almost every $t\in(0,T)$, since $\psi^{n}_{t}(t)\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$. *

The left-hand sides of our two energy identities (58) and (59) can be estimated from below by

[TABLE]

and by

[TABLE]

respectively.

We will consider the weighted sum (58) plus $\lambda>0$ times (59), which therefore can be bounded from below by

[TABLE]

*It then remains to estimate the right-hand sides, $\text{rhs}_{1}(t)$ and $\lambda\text{rhs}_{2}(t)$.

Estimates of the right-hand sides. For estimating the right-hand sides in (58) and (59), we can then use the norms of $z$, $z_{t}$, and $\psi$ appearing in the lower bounds (60) and (61). Furthermore, we can employ the continuous embeddings (19) as well as boundedness of $(-\Delta)^{-1}:L^{2}(\Omega)\to H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ and the Poincaré-Friedrichs inequality (20). Additionally, we employ the identities*

[TABLE]

To simplify the notation, we introduce two constants depending on the initial data,

[TABLE]

as well as

[TABLE]

By applying Hölder’s inequality and the trace theorem, we get for the right-hand side in (58),

[TABLE]

a.e. in time. We further obtain, by making use of Young’s inequality (37) and the embedding results,

[TABLE]

since for the Galerkin discretization by eigenfunctions of the Laplacian, we have $(-\Delta)\psi^{n}\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ for smooth $\Omega$. All terms on the right-hand side except for

[TABLE]

can be absorbed into the left-hand side (62) by making $\|\nabla\gamma\|_{L^{\infty}L^{3}}$ small. The right-hand side in (59) can be estimated as follows

[TABLE]

where we have estimated the boundary term by means of the trace theorem

[TABLE]

We further have for $\lambda>0$ by Young’s inequality that

[TABLE]

where by making $\tau$ and $\lambda$ small, all terms except for those containing the initial data and the inhomogeneity,

[TABLE]

can be absorbed into the left-hand side given in (62).

We now combine the energy estimates obtained from (58), and $\lambda$ times (59) with a small constant $\lambda>0$ and absorb the indicated terms from the right-hand side estimates (63), (65) into the left-hand side so that only $\widetilde{\textbf{rhs}}_{1}$ and $\lambda\widetilde{\textbf{rhs}}_{2}$ remain on the right-hand side, cf. (62), (64), (66). Therewith, we end up with an inequality of the form

[TABLE]

for

[TABLE]

to which we employ Gronwall’s lemma.

To obtain a uniform bound on the full $H^{3}(\Omega)$ norm of $\psi^{n}$, we combine the $|\nabla(-\Delta)\psi^{n}|_{L^{2}}$ term with $|-\Delta z^{n}|_{L^{2}}$ and the fact that

[TABLE]

for $t\in(0,T)$. In this way, we have

[TABLE]

for a.e. $t\in(0,T)$. Altogether, we get the estimate

[TABLE]

*with a constant $C(\gamma,T)>0$ independent of $\tau$, provided $\|\nabla\gamma\|_{L^{\infty}L^{3}}$ is sufficiently small.

Step 3: Passing to the limit. On account of estimate (68) and the Banach-Alaoglu theorem, we know that there exists a subsequence, denoted again by $\{\psi^{n}\}_{n\in\mathbb{N}}$, and a function $\tilde{\psi}$ such that*

[TABLE]

By uniqueness of limits $\tilde{\psi}$ has to coincide with the solution $\psi$ according to Theorem 3.1, which thus satisfies (56).

6 Well-posedness for the nonlinear Kuznetsov-type wave equation for $\tau>0$ sufficiently small

We next intend to employ the Banach fixed-point theorem to prove well-posedness for equation (6). To this end, we introduce the operator $\mathcal{T}$ that maps $\phi$ to a solution $\psi$ of

[TABLE]

on some ball

[TABLE]

in the space $X^{K}$, defined in (17). Thus, for establishing $\mathcal{T}$ as a self-mapping on $B_{\rho}^{X^{K}}$, it is crucial to prove that $\alpha=1-k\phi_{t}$ and $f=2\nabla\phi\cdot\nabla\phi_{t}$ are in $W^{1,1}(0,T;H^{1}(\Omega))\cap L^{\infty}(0,T;W^{1,3}(\Omega)\cap L^{1}(0,T;H^{2}(\Omega)$ and $L^{2}(0,T;H^{1}(\Omega))$, respectively, and that the derivatives of $\alpha$ are small when $\phi\in B_{\rho}^{X^{K}}$.

Concerning non-degeneracy, we assume that $\tau\in(0,\bar{\tau}]$ with $\bar{\tau}<\tfrac{b}{c^{2}}$ so that

[TABLE]

Therefore, keeping

[TABLE]

with $m<\gamma^{*}$ allows to choose

[TABLE]

in (53) independently of $\tau$, which will also be important for the considerations in Section 7. Thus we also need to verify that (71) follows from $\phi\in B_{\rho}^{X^{K}}$.

To additionally obtain contractivity, based on the fact that the difference $\hat{\psi}=\psi_{1}-\psi_{2}=\mathcal{T}(\phi_{1})-\mathcal{T}(\phi_{2})$ solves

[TABLE]

with homogeneous initial and boundary conditions (where $\hat{\phi}=\phi_{1}-\phi_{1}$), we need to prove that $\alpha_{1}=1-k\phi_{1\,t}$ and $f_{2}=k\hat{\phi}_{t}\psi_{2\,tt}+2\nabla\hat{\phi}\cdot\nabla\phi_{1\,t}+2\nabla\phi_{2}\cdot\nabla\hat{\phi}_{t}$ are in $W^{1,1}(0,T;H^{1}(\Omega))\cap L^{\infty}(0,T;W^{1,3}(\Omega)\cap L^{1}(0,T;H^{2}(\Omega)$ and $L^{2}(0,T;H^{1}(\Omega))$, and that the derivatives of $\alpha_{1}$ are small, provided $\phi_{1},\phi_{2}\in B_{\rho}^{X^{K}}$. Moreover, $\|f_{2}\|_{L^{2}H^{1}}$ needs to be estimated by a multiple of $\|\hat{\phi}\|_{X^{K}}$ with a small factor.

Theorem 6.1**.**

Let $c^{2}$, $b$, $T>0$, $k\in\mathbb{R}$. Then there exist $\bar{\tau}$, $\rho>0$ , $\rho_{0}>0$ such that for all $(\psi_{0},\psi_{1},\psi_{2})\in X^{K}_{0}=H_{0}^{1}(\Omega)\cap H^{3}(\Omega)\times H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\times H_{0}^{1}(\Omega)$ satisfying

[TABLE]

and all $\tau\in(0,\bar{\tau})$, there exists a unique solution $\psi\in X^{K}$ of

[TABLE]

which satisfies the estimate

[TABLE]

Proof 6.2**.**

We first prove that $\mathcal{T}$ is a self-mapping on $B_{\rho}^{X^{K}}$. For this purpose, we estimate $\alpha$ and $f$, assuming that $\phi\in B_{\rho}^{X^{K}}$ and abbreviating $w=\phi_{t}+\frac{c^{2}}{b}\phi$:

[TABLE]

Moreover, we find that

[TABLE]

Therefore, energy estimate (56) yields

[TABLE]

provided that the initial data are small in the sense of (74) with

[TABLE]

which implies that $\mathcal{T}$ maps $B_{\rho}^{X^{K}}$ into itself.

For proving contractivity of $\mathcal{T}$, we estimate $\alpha_{1}$ analogously to (76), and with abbreviations $\hat{w}:=\hat{\phi}_{t}+\frac{c^{2}}{b}\hat{\phi}$, $w_{i}:=\phi_{i\,t}+\frac{c^{2}}{b}\phi_{i}$, $z_{i}:=\psi_{i\,t}+\frac{c^{2}}{b}\psi_{i}$, $i\in\{1,2\}$. We have

[TABLE]

From here it folllows that

[TABLE]

By applying estimate (56) to equation (73) with homogeneous initial conditions, we obtain

[TABLE]

which after possibly decreasing $\rho$ yields contractivity.

Since $B_{\rho}^{X^{K}}$ is closed, we can make use of Banach’s contraction principle to conclude existence and uniqueness of a solution.

Remark 6.3** (On global well-posedness).**

Note that $C(\gamma,T)$ in (56) depends on the final time due to the use of Gronwall’s inequality. We do not expect that global in time well-posedness can be proven in the nonlinear case due to the fact that we must deal with a quadratic nonlinearity and only have weak damping in the equation for $z$.

7 Singular limit for vanishing relaxation time

We next focus on proving a limiting result for equations (6) and (8) as $\tau\to 0$. Recall that $b=\delta+\tau c^{2}$ and that the norms on the spaces $X^{W}$ and $X^{K}$, defined in (17), depend on $\tau$, whereas the radius $\rho$ of the balls (48) and (69) is independent of $\tau$.

As already stated in the notational preliminaries (18), we denote by $\|\cdot\|_{\bar{X}^{W}}$ and $\|\cdot\|_{\bar{X}^{K}}$ the respective $\tau$-independent part of the norms defined in (48) and (69),

[TABLE]

Moreover, we recall the spaces for the initial data

[TABLE]

the only difference being in the regularity of $\psi^{0}$. Therewith, we can formulate a limiting result for (6) and (8).

Theorem 7.1**.**

Let $c^{2}$, $b$, $T>0$, and $k\in\mathbb{R}$. Then there exist $\bar{\tau}$, $\rho_{0}>0$ such that for all $(\psi_{0},\psi_{1},\psi_{2})\in X_{0}^{W}$, the family $(\psi^{\tau})_{\tau\in(0,\bar{\tau})}$ of solutions to (51) according to Theorem 4.1 converges weakly in $\bar{X}^{W}$ to a solution $\bar{\psi}\in\bar{X}^{W}$ of (5) with homogeneous Dirichlet boundary conditions (15) and initial conditions $\bar{\psi}(0)=\psi_{0}$, $\bar{\psi}_{t}(0)=\psi_{1}$.*

The statement remains valid with the equations (51), (5), the spaces $X_{0}^{W}$, $\bar{X}^{W}$ and Theorem 4.1 replaced by the equations (75), (4), the spaces $X_{0}^{K}$, $\bar{X}^{K}$ and Theorem 6.1, respectively.

Proof 7.2**.**

From the energy estimates in Theorems 4.1 (or 6.1), we have uniform boundedness of $(\psi^{\tau})_{\tau\in(0,\bar{\tau})}$ in $\bar{X}^{W}$ (or in $\bar{X}^{K}$) and therefore existence of a weakly $\bar{X}^{W}$ (or $\bar{X}^{K}$) convergent sequence $(\psi^{\ell})_{\ell\in\mathbb{N}}$ with $\tau_{\ell}\searrow 0$. By compactness of embeddings, this sequence also converges strongly in $C^{1}(0,T;L^{4}(\Omega))\cap C(0,T;W^{1,4}(\Omega))$. Its limit $\bar{\psi}$ therefore lies in $\bar{X}^{W}$ (or in $\bar{X}^{K}$) and satisfies the initial conditions $\bar{\psi}(0)=\psi_{0}$, $\bar{\psi}_{t}(0)=\psi_{1}$.*

To prove that $\bar{\psi}$ also satisfies the respective PDEs, we test with arbitrary functions $v\in C_{0}^{\infty}(0,T;C_{0}^{\infty}(\Omega))$ and invoke the Fundamental Lemma of Calculus of Variations. To this end, we introduce an abbreviation for the nonlinear term in the respective equations, namely

[TABLE]

Now the $\tau$-dependent and the limiting equation can be rewritten in both Westervelt and Kuznetsov cases as

[TABLE]

and

[TABLE]

*respectively.

Note that by the regularity inherent in the spaces $X^{W}$, $X^{K}$ and $\bar{X}^{W}$, $\bar{X}^{K}$, cf. (17), $\psi^{\tau}$ satisfies equation (78) in $L^{2}(0;T;L^{2}(\Omega))$. Inserting $\bar{\psi}$ into the left-hand side of (79) yields an $L^{2}(0;T;L^{2}(\Omega))$ function. Therewith, we get, for $\hat{\psi}_{\ell}:=\bar{\psi}-\psi^{\ell}$ and any $v\in C_{0}^{\infty}(0,T;C_{0}^{\infty}(\Omega))$ that*

[TABLE]

Above, we have that

[TABLE]

due to the weak convergence to zero of $\hat{\psi}_{\ell}$ in $\bar{X}^{W}$ (or $\bar{X}^{K}$). Moreover,*

[TABLE]

due to the boundedness of $(\psi_{\ell})_{\ell\in\mathbb{N}}$ in $\bar{X}^{W}$ (or $\bar{X}^{K}$) by $\rho$, and the strong convergence to zero of $\hat{\psi}_{\ell}$ in $C^{1}(0,T;L^{4}(\Omega))\cap C(0,T;W^{1,4}(\Omega))$. Finally,

[TABLE]

due to the boundedness of $(\psi_{\ell})_{\ell\in\mathbb{N}}$ in $\bar{X}^{W}$ (or $\bar{X}^{K}$), and $\tau_{\ell}\to 0$.

A subsequence-subsequence argument, together with uniqueness of the solution to (79) according to results in, e.g., Refs. \refciteKL09Westervelt,KL12_Kuznetsov,MW11,MW13 yields convergence of the whole family $(\psi^{\tau})_{\tau\in(0,\bar{\tau})}$.

Remark 7.3** (On compatibility conditions).**

Note that, in contrast to Ref. \refciteKT18_ModelsNlAcoustics, no compatibility condition on $\psi_{2}$ is needed, since no continuity of the limit $\bar{\psi}_{tt}$ with respect to time arises and in the used energy estimates the $\psi_{2}$ term vanishes as $\tau\to 0$.

Remark 7.4** (On strong convergence).**

We could look directly at the equation solved by the difference $\hat{\psi}=\psi^{\tau}-\bar{\psi}$ of solutions to the JMGT and the Westervelt equation:

[TABLE]

However, showing that $\hat{\psi}$ tends to zero as $\tau\rightarrow 0$ appears to be beyond our theoretical reach, although suggested by the numerical results in Section 8. A particular challenge is to estimate the first term $\tau\psi^{\tau}_{ttt}$: we only know this it is bounded by $\rho$ in $L^{2}(0,T;L^{2}(\Omega))$ according to estimate (52), but not that it tends to zero. An analogous argument can be made for the Kuznetsov-type JMGT equation.

7.1 Comparison to the regularity results in the literature

We note that Theorem 7.1 also contains a regularity result on the solutions $\bar{\psi}\in\bar{X}^{W}$ and $\bar{\psi}\in\bar{X}^{K}$ of the Westervelt (5) and the Kuznetsov (4) equations with homogeneous Dirichlet boundary conditions (15) and initial conditions $\bar{\psi}(0)=\psi_{0}\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ or $H^{3}(\Omega)$, $\bar{\psi}_{t}(0)=\psi_{1}\in H_{0}^{1}(\Omega)$.

By comparing this regularity with the regularity results on the Westervelt equation from Refs. \refciteKL09Westervelt,MW11 and with those for the Kuznetsov equation from Ref. \refciteKL12_Kuznetsov,MW13,MizohataUkai, and noting that in Refs. \refciteKL09Westervelt,MW11,KL12_Kuznetsov,MW13, $u$ is the acoustic pressure, i.e., related to $\psi$ by $u=\varrho_{0}\psi_{t}$, we get

• •

Westervelt equation:

• –

Ref. \refciteKL09Westervelt: $(u_{0},u_{1})\in(H_{0}^{1}(\Omega)\cap H^{2}(\Omega))\times H_{0}^{1}(\Omega)$

and additionally $(1-ku_{0})^{-1}[c^{2}\Delta u_{0}+b\Delta u_{1}+ku_{1}^{2}]\in L^{2}(\Omega)$

$\Rightarrow$ $u\in C^{2}(0,T;L^{2}(\Omega))\cap H^{2}(0,T;H_{0}^{1}(\Omega))\cap C(0,T;H^{2}(\Omega))$;

• –

Ref. \refciteMW11: $(u_{0},u_{1})\in(H_{0}^{1}(\Omega)\cap H^{2}(\Omega))\times H_{0}^{1}(\Omega)$

$\Rightarrow$ $u\in H^{2}(0,T;L^{2}(\Omega))\cap H^{1}(0,T;H_{0}^{1}(\Omega)\cap H^{2}(\Omega))$;

• –

here: $(\psi_{0},\psi_{1})\in\left(H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\right)^{2}$

and additionally $(1-k\psi_{0})^{-1}[c^{2}\Delta\psi_{0}+b\Delta\psi_{1}]\in H_{0}^{1}(\Omega)$

$\Rightarrow$ $u\in H^{2}(0,T;H_{0}^{1}(\Omega))\cap W^{1,\infty}(0,T;H^{2}(\Omega))$.

• •

Kuznetsov equation:

• –

Ref. \refciteMizohataUkai: $(\psi_{0},\psi_{1})\in H_{0}^{1}(\Omega)\cap H^{3}(\Omega)\times H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$

$\Rightarrow$ $\psi\in C^{1}(0,T;H_{0}^{1}(\Omega)\cap H^{2}(\Omega))\cap H^{1}(0,T;H^{3}(\Omega))$;

• –

Ref. \refciteKL12_Kuznetsov: $(u_{0},u_{1})\in(H_{0}^{1}(\Omega)\cap H^{2}(\Omega))\times H_{0}^{1}(\Omega)$

and additionally $(1-ku_{0})^{-1}[c^{2}\Delta u_{0}+b\Delta u_{1}+ku_{1}^{2}+2|\nabla u_{0}|^{2}]\in L^{2}(\Omega)$

$\Rightarrow$ $u\in C^{2}(0,T;L^{2}(\Omega))\cap H^{2}(0,T;H_{0}^{1}(\Omega))\cap C(0,T;H^{2}(\Omega))$;

• –

Ref. \refciteMW13: $(u_{0},u_{1})\in(H_{0}^{1}(\Omega)\cap H^{2}(\Omega))\times H_{0}^{1}(\Omega)$

$\Rightarrow$ $u\in H^{5/2}(0,T;H_{0}^{1}(\Omega))\cap H^{2}(0,T;H^{2}(\Omega))$;

• –

here: $(\psi_{0},\psi_{1})\in H_{0}^{1}(\Omega)\cap H^{3}(\Omega)\times H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$

and additionally $(1-k\psi_{1})^{-1}[c^{2}\Delta\psi_{0}+b\Delta\psi_{1}+\nabla\psi_{0}\cdot\nabla\psi_{1}]\in H_{0}^{1}(\Omega)$

$\Rightarrow$ $u\in H^{2}(0,T;H_{0}^{1}(\Omega))\cap W^{1,\infty}(0,T;H^{2}(\Omega))\cap L^{\infty}(0,T;H^{3}(\Omega))$.

We point out that these works also contain results on global in time existence and exponential decay of solutions, as well as, in case of Refs. \refciteDekkersRozanova,MizohataUkai, on the Cauchy problem, and, in case of Refs. \refciteMW11,MW13, in general, non-Hilbert $L^{p}(\Omega)$ and $W^{s,p}(\Omega)$ spaces. Moreover, we wish to point to Ref. \refciteDoerflerGernerSchnaubelt16, where local in time well-posedness of a class of quasilinear wave equations without strong damping was shown, which also comprises the Westervelt equation with $\delta=0$. It yields the regularity $u\in C^{2}(0,T;H^{1}(\Omega))\cap C^{1}(0,T;H^{2}(\Omega))\cap C(0,T;H^{3}(\Omega))$ for initial data $(u_{0},u_{1})\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)\times H_{0}^{1}(\Omega)\cap H^{3}(\Omega)$ with $\Delta u_{1}|_{\partial\Omega}=0$.

8 Numerical results

As an illustration of our theoretical findings, we solve and compare numerically equations (5) and (8) in a one-dimensional channel geometry. For the medium, we choose water with parameters

[TABLE]

cf. Chapter 5 in Ref. \refcitekaltenbacher2007numerical. Recall that $b=\delta+\tau c^{2}$; the choice of the relaxation parameter $\tau$ is given below. Discretization in space is performed by employing B-splines as basis functions within the framework of Isogeometric Analysis (IGA); see Refs. \refciteCottrell,HughesBook. For a detailed insight into the application of Isogeometric Analysis in nonlinear acoustics, we refer to Refs. \refciteFritz,2017isogeometricPaper. We use quadratic basis functions with the maximum $C^{1}$ global regularity and have $251$ degrees of freedom for the channel length $l=0.2\,$m. The nonlinearities are resolved by a fixed-point iteration with the tolerance set to $\textup{TOL}=10^{-8}$.

After discretizing in space, we end up with a semi-discrete matrix equation and proceed with a time-stepping scheme. For the Westervelt equation (8), we employ the standard Newmark relations[39] for second-order equations:

[TABLE]

realized through a predictor-corrector scheme analogously to Algorithm 1 in Ref.  \refcite2017isogeometricPaper. In (81), $\Delta t$ denotes the time step size. The vectors $\underline{\psi}^{n}$, $\underline{\dot{\psi}}^{n}$, and $\underline{\ddot{\psi}}^{n}$ denote the discrete acoustic potential, its first time derivative, and its second time derivative, respectively, at the time step $n$.

For the Jordan–Moore–Gibson–Thompson equation with Westervelt-type nonlinearity (8), we use an extension of the Newmark relations to third-order models similar to the one employed in Appendix B.2 of Ref. \refcitefathi2015time:

[TABLE]

The average acceleration scheme corresponds to taking the Newmark parameters $(\beta,\gamma)=(1/4,1/2)$ in (81) for the Westervelt equation and $(\beta,\gamma,\eta)=(1/12,1/4,1/2)$ in (82) for the Jordan–Moore–Gibson–Thompson equation (8), which is what we use in all the experiments.

We set the initial conditions to

[TABLE]

with $\mathcal{A}=8\cdot 10^{4}\,\textup{m}^{2}/\textup{s}^{2}$ and $\sigma=0.01$, meaning that we normalize potential (which is determined by $\vec{v}=-\nabla\psi$ only up to a constant) such that it vanishes at $t=0$, drive the system by an inital pressure (based on the idetity $\rho\psi_{t}=p$) concentrated at $x=0.1$, and assume vanishing initial acceleration. Discretization in time is performed with $800$ time steps for the final time $\textup{T}=45\,\mu$s. The spatial and temporal refinement always remain the same for both equations and different values of the relaxation time $\tau$. All the numerical results are obtained with the help of the GeoPDEs package in MATLAB[46].

Figure 1 displays on the left side snapshots of the pressure wave $u=\varrho\psi_{t}$ obtained by employing equation (8) with the relaxation time set to $\tau=0.1\,\mu$s. We observe the nonlinear steepening of the wave as it propagates. On the right, we see how the pressure profile changes with decreasing relaxation time. The pressure wave for $\tau=0\,\mu$s is computed by solving the Westervelt equation.

To further illustrate the results from Section 7, we solve equation (8) with the relaxation time varying over $\tau\in[10^{-4},1]\,\mu s$ and compute the difference to the solution of the Westervelt equation (5). We plot the relative errors in the $\bar{X}^{W}$ norm, defined in (18), and in the $C([0,T];H^{1}(\Omega))$ norm:

[TABLE]

see Figure 2. The numerical errors decrease with the parameter $\tau$, in agreement with the theoretical results of Theorem 7.1. Figure 2 even indicates a stronger result, i.e., strong convergence in the $\bar{X}^{\textup{W}}$ norm. For $\tau=10^{-10}\,$s, the errors amount to $\textup{error}_{CH^{1}}(\tau)\approx 6.43\cdot 10^{-5}$ and $\textup{error}_{\bar{X}^{W}}(\tau)\approx 8.5\cdot 10^{-4}$. The error plots also suggest a lower rate of convergence with respect to $\tau$ in the $\bar{X}^{W}$ norm.

Acknowledgments

The second author acknowledges the funding provided by the Deutsche Forschungsgemeinschaft under the grant number WO 671/11-1.