Optimal regularity and exponential stability for the Blackstock–Crighton equation in Lp-spaces with Dirichlet and Neumann boundary conditions

The Blackstock–Crighton equation models nonlinear acoustic wave propagation in monatomic gases. In the present work, we investigate the associated inhomogeneous Dirichlet and Neumann boundary value problems in a bounded domain and prove long-time well-posedness and exponential stability for sufficiently small data. The solution depends analytically on the data. In the Dirichlet case, the solution decays to zero and the same holds for Neumann conditions if the data have zero mean. We choose an optimal Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L_p}$$\end{document}-setting, where the regularity of the initial and boundary data is necessary and sufficient for existence, uniqueness and regularity of the solution. The linearized model with homogeneous boundary conditions is represented as an abstract evolution equation for which we show maximal Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L_p}$$\end{document}-regularity. In order to eliminate inhomogeneous boundary conditions, we establish a general higher regularity result for the heat equation. We conclude that the linearized model induces a topological linear isomorphism and then solves the nonlinear problem by means of the implicit function theorem.


Introduction
An acoustic wave propagates through a medium as a local pressure change. Nonlinear effects typically occur in case of acoustic waves of high amplitude which are used for several medical and industrial purposes such as lithotripsy, thermotherapy, ultrasound cleaning or welding and sonochemistry. Research on mathematical aspects of nonlinear acoustic wave propagation is therefore not only interesting from a mathematicians point of view. In fact, enhancement of the mathematical understanding of the underlying models may help to solve problems arising in various applications.
The present work provides an analysis of the Blackstock-Crighton-Kuznetsov equation (1.1) and the Blackstock-Crighton-Westervelt equation for the acoustic velocity potential u, where c is the speed of sound, b is the diffusivity of sound and a is the heat conductivity of the fluid. Note that a = νPr −1 , where ν is the kinematic viscosity and Pr denotes the Prandtl number. The quantity B/A is known as the parameter of nonlinearity and is proportional to the ratio of the coefficients of the quadratic and linear terms in the Taylor series expansion of the variations of the pressure in a medium in terms of variations of the density, see [17,Section 4.2]. Equations (1.1) and (1.2) are derived from the full equations of motion for thermoviscous fluids by a small modification of Blackstock's approximation [4] and using Becker's assumption for monatomic gases which corresponds to the choice μ B = 0 and Pr = 3/4 for the bulk viscosity and the Prandtl number, respectively. The acoustic diffusivity then becomes b = 4γ ν/3, where γ is the specific heat ratio. Moreover, in case of monatomic gases we have B/A = γ − 1. Alternatively, (1.1) and (1.2) can be expressed in terms of the acoustic pressure p via the pressure density relation ρu t = p, where ρ denotes the mass density. Note that (1.2) is obtained from (1.1) by neglecting local nonlinear effects in the sense that the expression c 2 |∇u| 2 − (u t ) 2 is sufficiently small. With k = c −2 (1 + B/2 A), we represent (1.2) as which shows that potential degeneracy is an important feature of (1.1) and (1.2). We avoid this degeneracy by considering solutions for sufficiently small initial and boundary data.
For a rigorous derivation of the equations under consideration, we refer to [5, Chapter 1] and [7], whereas a detailed introduction to the theory and applications of nonlinear acoustics is provided by [17]. Summing up, we here just emphasize that (1.1) and (1.2) are approximate equations governing finite amplitude sound in monatomic gases (e.g., helium, xenon, argon).
While (1.1) and (1.2) are enhanced models in nonlinear acoustics, the Kuznetsov and the Westervelt equation conductivity, (1.3) and (1.4) can be regarded as simplifications of (1.1) and (1.2), respectively. The classical models (1.3) and (1.4) have recently been extensively investigated. In particular, results on well-posedness for the Kuznetsov and the Westervelt equation with homogeneous Dirichlet [19] and inhomogeneous Dirichlet [22], [21] and Neumann [20] boundary conditions have been shown in an L 2 ( )-setting on spatial domains ⊂ R n of dimension n ∈ {1, 2, 3}. Moreover, there are results on optimal regularity and long-time behavior of solutions for the Westervelt equation with homogeneous Dirichlet [26] and for the Kuznetsov equation with inhomogeneous Dirichlet [27] boundary conditions in L p ( )-spaces, where the spatial domain ⊂ R n is of arbitrary dimension.
On the contrary, mathematical research on higher-order partial differential equations arising in nonlinear acoustics is still in an early stage. Well-posedness and exponential decay results for the homogeneous Dirichlet boundary value problems associated with (1.1) and (1.2) in an L 2 ( )-setting where ⊂ R n , n ∈ {1, 2, 3}, have been shown in [6] and [8], respectively. In the present work, we consider (1.1) and (1.2) with inhomogeneous Dirichlet and Neumann boundary conditions in L p ( )-spaces, where the spatial domain is of dimension n ∈ N. We show global well-posedness and long-time behavior of solutions in an optimal functional analytic setting in the sense that the regularity of the initial and boundary data is both necessary and sufficient for the regularity of the solution. While in [6,8] the results were proved by means of energy estimates and Banach's fixed-point theorem, the techniques used in the present paper are based on maximal L p -regularity and the implicit function theorem.
We suppose that ⊂ R n , n ∈ N, is a bounded smooth domain, i.e., an open, connected and bounded subset of R n with smooth boundary . Let J = (0, T ) for some finite T > 0 or J = R + = (0, ∞). We consider the inhomogeneous Dirichlet boundary value problem 5) and the inhomogeneous Neumann boundary value problem where u 0 , u 1 , u 2 : → R and g, h : J × → R are given, u : J × → R is the unknown, u(t, x), and a, b, c > 0 and k ∈ R are positive constants. Moreover, ∂ ν u = ν · ∇u| denotes the normal derivative of u in terms of the outer unit normal vector ν. The parameter s ∈ {0, 1} allows us to switch between (1.1) and (1.2).

Function spaces, operators, embeddings and traces
The space BU C k ( ) contains all k-times Fréchet differentiable functions → R, whose derivatives up to order k are bounded and uniformly continuous. For p ∈ (1, ∞), let L p ( ) denote the space of (equivalence classes of) Lebesgue measurable p-integrable functions → R. We write W m p ( ) for the Sobolev-Slobodeckij space and H m p ( ) for the Bessel potential space of order m ∈ [0, ∞), where we have We always write X → Y if the Banach space X is continuously embedded into the Banach space Y . Moreover, let L(X, Y ) be the space of all bounded linear operators between X and Y . A linear operator A : X → Y is called an isomorphism if it is bounded and bijective. Then the closed graph theorem implies that A −1 : Y → X is also bounded, and therefore, A : X → Y is a homeomorphism. Now, let X and X be Banach spaces such that X → L 1,loc (J ; X ) where L 1,loc (J ; X ) is the space of locally integrable functions J → X . For any ω ∈ R, we define the exponentially weighted space e ω X = {u ∈ L 1,loc (J ; X ) : e −ωt u ∈ X}, equipped with the norm u e ω X = e −ωt u X , where e −ωt u denotes the mapping We recall that the spectrum σ (− D ) is a discrete subset of (0, ∞) consisting only of eigenvalues λ D n = λ n (− D ), n ∈ N 0 with finite multiplicity. We write λ D 0 > 0 for the smallest eigenvalue of − D . Moreover, the negative Neumann Laplacian − N : We shall use the embeddings W 1 p (J ) → BU C(J ) and W s p ( ) → W t p ( ) for s ≥ t. We always write γ D = ·| and γ N = ∂ ν · | = ν · (∇·)| for the Dirichlet and the Neumann trace, respectively. Moreover, γ t = · | t=0 denotes the temporal trace. Let B ∈ {D, N }, j D = 0, j N = 1. For p ∈ (1, ∞), k ∈ N and l ∈ N 0 , the spatial trace is bounded, see Lemma B.6. Furthermore, the trace where p ∈ (1, ∞), t, s ∈ [0, ∞), τ , σ ∈ (0, ∞), θ ∈ (0, 1). The embeddings (2.4) and (2.5) remain valid when is replaced by its boundary and they are frequently used for checking the continuity of differential operators in anisotropic spaces.

Maximal L p -regularity
Let J = (0, T ) or J = R + = (0, ∞) and assume p ∈ (1, ∞). We say that a closed linear operator A : for v 0 = 0. Furthermore, the inhomogeneous Cauchy problem (2.6) is said to admit maximal L p -regularity, if is a homeomorphism. Then its inverse is the solution map If A : D(A) → X has maximal L p -regularity on J , then the Cauchy problem (2.6) has maximal L p -regularity on J , cf. Section III.1.5 in [2]. The following result is very useful and will be used several times throughout this paper.
is a homeomorphism. Then is a homeomorphism.

Optimal regularity results
In order to prove optimal regularity for the linearized versions of (1.5) and (1.6), we need results on optimal regularity for the heat equation and the linearized Westervelt equation. First we consider Dirichlet boundary conditions. Recall that λ D 0 > 0 always denotes the smallest eigenvalue of the negative Dirichlet Laplacian in L p ( ). In the following, we always assume a, b, c ∈ (0, ∞).

LEMMA 2.2. ([24, Proposition 8])
Let p ∈ (1, ∞) and ω ∈ (0, aλ D 0 ). Then the initial boundary value problem for the heat equation has a unique solution if and only if the given data f , g and u 0 satisfy the regularity conditions

10)
if and only if the data satisfy the following conditions: Now we prove optimal regularity for the heat equation and the linearized Westervelt equation with Neumann boundary conditions. Recall that λ N 1 > 0 denotes the smallest eigenvalue of the negative homogeneous Neumann Laplacian in L p,0 ( ). Moreover, letū = | | −1 u dx denote the mean of a function u : . Then the inhomogeneous Neumann boundary value problem for the heat equation if and only if the data satisfy the following conditions: If in addition f (t, ·), u 0 , g(t, ·) have mean value zero over resp. for all t, then w = 0.
Proof. We first let ω = 0. By [ for every given f 3 ∈ L p (R + ; L p,0 ( )). Furthermore, problem (2.11) admits at most one solution. Indeed, let us construct it as u = u 1 + u 2 + u 3 where we first solve for some sufficiently large μ > 0 with [11, Theorem 2.1]. Next, we let u 2 solve the ordinary differential equation Vol. 16 (2016) Optimal regularity and exponential stability 953 , we obtain u 3 as above. It is easy to check that v = u 1 + u 3 and w = u 2 satisfy the assertion. The case ω > 0 can be reduced to the previous one by multiplying the functions u, f , g with t → e ωt and using that the spectrum of −a N ,0 + ω is contained in (0, ∞).
Based on Lemma 2.4, we arrive at our next intermediate result on the way to optimal regularity for the linearized Westervelt equation with Neumann boundary conditions.
if and only if the data satisfy the following conditions: Proof. We start by proving sufficiency. From the boundedness of ∂ t : W ν → H ν , we infer that g t ∈ e −ω H ν . Therefore, from Lemma 2.4 we obtain that the heat problem In particular, since ϕ 1 has zero mean over , we have Clearly u t = ϕ; hence, u tt − b u t = f in . Integrating the latter with respect to space, multiplying with | | −1 and using the identity To verify the necessity of (i)-(v), we assume that The desired regularity of the initial values u 0 and u 1 is obtained by using the embedding W 1 p (J ) → BU C(J ) and the temporal trace (2.4), respectively. In order to check (iii), we apply the spatial trace (2.2) with k = l = 1 to e ωt v ∈ W u,0 . Using hold in the sense of traces and (iv) is satisfied.
f (s) ds which, for t = 0, implies (v). Finally, we can easily deduce from Lemma 2.4 that (2.12) has at most one solution.

15)
admits a unique solution of the form u(t, if and only if the data satisfy the following conditions: Proof. From Lemma 2.6, we obtain uniqueness. Necessity of (i)-(iv) is shown as in Lemma 2.5. It therefore remains to show sufficiency. Let δ > ω. From Lemma 2.5, we obtain that has a unique solution θ ∈ e −ω W u,0 . Furthermore, we define θ w as the solution of Then v = ϕ v + θ v and w = ϕ w + θ w satisfy the assertion and we are done.

The Dirichlet boundary value problem
In this section, we prove global well-posedness and exponential stability for (1.5). First of all, we consider the linearized version of the inhomogeneous Dirichlet boundary value problem and represent it as an abstract evolution equation. We show that this abstract equation admits maximal L p -regularity and derive an optimal regularity result for the linearized equation associated with (1.5). Then we use the implicit function theorem to construct a solution of the nonlinear problem (1.5). Exponential decay of this solution is an immediate consequence.

Maximal L p -regularity for the linearized equation
where u 0 , u 1 , u 2 : → R and g, h : J × → R are the given initial and boundary data, respectively. We say that the data (g, h, u 0 , u 1 , u 2 ) are compatible if u 0 | = g| t=0 , u 1 | = g t | t=0 , u 0 | = h| t=0 and, if p > 3/2, also u 1 | = h t | t=0 , u 2 | = g tt | t=0 . In order to address the problem of maximal L p -regularity for the linearized equation, we represent (3.1) with g = h = 0 as an abstract Cauchy problem ⎛ Vol. 16 (2016) Optimal regularity and exponential stability 957 This motivates us to consider the Banach space

.2) and the densely defined linear operator A D : D(A D ) → X D given by
Therewith, we may write (3.1) as an abstract evolution equation First of all, we will treat the issue of maximal L p -regularity of A D : D(A D ) → X D on R + . PROPOSITION 3.1. Let p ∈ (1, ∞). There is a constant μ > 0 such that μ + A D has maximal L p -regularity on R + .
First we show that A D 1 : D(A D ) → X D has maximal L p -regularity. To this end, we consider the Cauchy Since we know from Lemma 2.2 that the homogeneous heat equation admits maximal L p -regularity, we obtain that for all f 3 ∈ L p (R + × ) there exists a unique solution 958 R. Brunnhuber and S. Meyer J. Evol. Equ. v 3 ∈ W 1 p (R + ; L p ( )) ∩ L p (R + ; D( D )). Moreover, as f 2 + v 3 ∈ L p (R + ; D( D )), Lemma B.5 implies that there is a unique solution v 2 ∈ W 1 p (R + ; D( D )) ∩ L p (R + ; D(( D ) 2 )). Now, note that for α > 0 the operator ). Altogether, we conclude that A 1 : D(A D ) → X D admits maximal L p -regularity.
Moreover, since A D 2 : X D → X D is bounded, Proposition 4.3 and Theorem 4.4 in [10] imply that there exists some μ > 0 such that μ + A D 1 + A D 2 has maximal L p -regularity.
To prove maximal regularity for the operator A D , we need information about its spectrum.
Proof. We follow the proof of Theorem 2.5 in [26]. From Proposition 3.1, we know μ+ A D admits maximal regularity on R + for some μ > 0. Multiplying v D t + A D v D = F by e −μt shows that A D has maximal L p -regularity on bounded intervals J = (0, T ). Lemma 3.2 tells us that spectral bound s(−A D ) = −ω 0 of −A D is strictly negative. Hence, s(−A D + ω) = ω − ω D 0 < 0 as long as ω ∈ [0, ω D 0 ). From [14, Theorem 2.4], we deduce that A D − ω admits maximal L p -regularity on R + for every ω ∈ [0, ω D 0 ), that is, is an isomorphism. Now Lemma 2.1 implies the result.

if and only if
Vol. 16 (2016) Optimal regularity and exponential stability 959

Proof. Based on the choices of X D and D(A D ) in (3.2) and (3.3), it is straightforward to check that the condition
). Since the mixed derivative embedding gives us E u → W 2 p (R + ; W 2 p ( )), we arrive at u ∈ e −ω E u . Furthermore, F ∈ L p (R + ; X D ) leads to (i). Next, we determine (X,

Moreover, interpolation with boundary conditions as in [3, Section 4.9] yields u
We now arrive at the final result for this section and prove optimal regularity for the linear initial boundary value problem (3.1).
if and only if the data satisfy the conditions Moreover, the solution fulfills the estimate Proof. It is not difficult to check that (i)-(iv) are necessary for the regularity of the solution, by using Sobolev's embedding W 1 p (J ) → BU C(J ), the spatial trace theorem (2.3), the temporal trace theorem (2.4) and the mixed derivative embeddings (2.5). In particular, the compatibility conditions are understood in the following sense: It remains to show that conditions (i)-(iv) imply the existence of a unique solution u ∈ e −ω E u . As we are dealing with a linear partial differential equation with constant coefficients, we may interchange the order of differentiation and consider the subproblems From condition (ii), we obtain a u 0 − u 1 ∈ W 2 p ( ) and a u 1 − u 2 ∈ W 2−2/ p p ( ). Furthermore, (iii) implies ah − g t ∈ e −ω W . By Lemma 2.3, problem (3.5) admits a unique solution w ∈ e −ω W u . Now Corollary B.4 with l = 1 and k = 2 shows that (3.6) has a solution u ∈ e −ω E u . This yields sufficiency and uniqueness follows from Corollary 3.4.

Global well-posedness and exponential stability
Based on Proposition 3.5, we now show that there exists a unique global solution of the nonlinear initial boundary value problem (1.5) which depends continuously (in fact, even analytically) on the (sufficiently small) initial and boundary data. Moreover, we prove that the equilibrium u = 0 is exponentially stable. For a detailed treatment of analytic mappings in Banach spaces and the analytic version of the implicit function theorem, we refer to [9, Section 15.1]. THEOREM 3.6. (Global well-posedness: the Dirichlet case) Let p > max{n/4 + 1/2, n/3}, p = 3/2 and define ω D and assume that the data (g, h, u 0 , u 1 , u 2 ) are compatible. Then for every ω ∈ (0, ω 0 ) there exists some ρ > 0 such that if Vol. 16 (2016) Optimal regularity and exponential stability 961 the nonlinear initial boundary value problem (1.5) admits a unique solution which depends analytically on the data (3.7) with respect to the corresponding topologies. Moreover, conditions (3.7) are necessary for the regularity of the solution (3.9).
Proof. Employing the results on the linearized problem (3.1) from Sect. 3.1, we will now construct a solution of the nonlinear initial boundary value problem (1.5) which we linearize at u = 0. Hence, the solution will be of the form u = u + u • , where u solves the linearized problem (3.1) for the data ( f = 0, g, h, u 0 , u 1 , u 2 ) and u • satisfies homogeneous boundary and initial conditions. We will find the (small) deviation u • from u by application of the implicit function theorem to the map Step 1: The implicit function theorem applies. First of all, we will now verify the assumptions of the analytic version of the implicit function theorem.
For the remaining bilinear operators, we employ the pointwise multiplication estimates f g p ≤ f p g ∞ and f g p ≤ f 2 p g 2 p and suitable embeddings as in the proof of Lemma 6 in [27]. In order to arrive at the · ∞ -norm, we infer from the mixed derivative embeddings and Sobolev's embedding that the embedding is valid if ε ∈ (0, 1 − 1/ p) and 4 − 2/ p − 2ε − n/ p > 0. Such a number ε exists if p > n/4 + 1/2. Sobolev's embedding also yields the embedding if p > n/3. In order to arrive at the · 2 p -norm, we conclude that for ε > 0, provided that ε > 0 is sufficiently small and p > n/4 + 1/2. Similarly, we obtain for some ε > 0, provided that p > n/6 + 1/3. Furthermore, since e ωt ≤ e 2ωt for ω ≥ 0, we observe that e −2ω L p (R + × ) → e −ω L p (R + × ).
By setting w = u • + u , f = e ωt u • and g = e ωt u we are done. Step Step 2: Construction of the solution. On the strength of the implicit function theorem, there exists a ball B ρ (0) ⊂ e −ω E u with sufficiently small radius ρ > 0 and an analytic map ϕ : . Hence, whenever u satisfies the boundary conditions u | = g, u | = h and initial conditions u | t=0 = u 0 , u ,t | t=0 = u 1 , u ,tt | t=0 = u 2 which is the case if we define u ∈ e −ω E u to be the unique solution of (3.1) with ( f = 0, u 0 , u 1 , u 2 , g, h), then u • + u = ϕ(u ) + u solves (1.5).
Step 3: Dependence of the solution on the data. It remains to show that the solution u ∈ e −ω E u depends analytically on (g, h, u 0 , u 1 , u 2 ). To this end, we define the spaces From Proposition 3.5 with f = 0, we obtain that u depends linearly and continuously and thus analytically on (g, h, u 0 , u 1 , u 2 ) ∈ D. Moreover, u → u • = ϕ(u ) is analytic on B ρ (0), and therefore, u • ∈ e −ω E u,h depends analytically on the data (g, h, u 0 , u 1 , u 2 ) ∈ D. Altogether, u = u • + u enjoys the same property which concludes the proof.
An immediate consequence of Theorem 3.6 is that the solution decays to zero exponentially. COROLLARY 3.7. (Exponential stability: the Dirichlet case) Under the same assumptions as in Theorem 3.6, the solution u decays exponentially fast to zero as t → ∞, in the sense that for some C ≥ 0 depending on the boundary and initial data g, h, u 0 , u 1 and u 2 .

The Neumann boundary value problem
In this section, we treat the Neumann boundary value problem (1.6). We proceed analogously to the Dirichlet case; that is, we first consider the linearized equation and then construct a solution of the nonlinear problem (1.6) by means of the implicit function theorem.
Note that, in the Dirichlet case, the fact that the operator −A D : D(A D ) → X D defined by (3.3) has a strictly negative spectral bound (Lemma 3.2) was crucial in order to show that the linearized equation (3.1) has maximal regularity on R + , see the proof of Theorem 3.3.
In the Neumann case, due to the zero eigenvalue of However, if we restrict ourselves to finite time intervals J = (0, T ), then we do not necessarily need to use − N ,0 . In case of finite time intervals, we use − N . As a consequence, we will prove global well-posedness of (1.6) only if the data u 0 , u 1 , u 2 and g, h have zero mean, whereas local well-posedness holds also for data with nonzero mean.  ∈ (1, ∞). Here, for where u 0 , u 1 , u 2 : → R and g, h : J × → R are the given initial and boundary data, respectively. We say that the data (g, h, u 0 , u 1 , u 2 ) are compatible, if ∂ ν u 0 | = Vol. 16 (2016) Optimal regularity and exponential stability 965 g| t=0 , ∂ ν u 0 | = h| t=0 , ∂ ν u 1 | = g t | t=0 and, if p > 3, also ∂ ν u 1 | = h t | t=0 , ∂ ν u 2 | = g tt | t=0 in the sense of traces.
Analogously to the Dirichlet case, we first represent (4.1) with g = h = 0 as an abstract evolution equation of the form introducing the Banach space X N = D(( N ) 2 ) × D( N ) × L p ( ) and defining the coefficient operator A N : D(A N ) → X N via  Proof. The result can be proved similarly to Proposition 3.1. For some α > 0, consider The operator A N 1 : D(A N 1 ) → X N has maximal L p -regularity on R + . This is seen as in the proof of Proposition 3.1 by Again, for F ∈ L p (R + ; X N ) one solves stepwise the equations above, starting with the last one. For the second equation, we need higher regularity for the heat equation provided by Lemma B.5. Since A N 2 : X N → X N is bounded, there exists some ν > 0 966 R. Brunnhuber and S. Meyer J. Evol. Equ.
such that ν + A N 1 + A N 2 = ν + A N has maximal L p -regularity on R + . Maximal Proof. Since ν + A N has maximal L p -regularity on R + , multiplication of (4.2) with e −νt shows that A N has maximal L p -regularity on bounded intervals J = (0, T ). Moreover, likewise to Lemma 3.