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1 SIAM J. MATH. ANAL. Vol. 49, No. 5, pp c 27 Society for Industrial and Applied Mathematics Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see ANALYSIS OF TRANSIENT ACOUSTIC-ELASTIC INTERACTION IN AN UNBOUNDED STRUCTURE YIXIAN GAO, PEIJUN LI, AND BO ZHANG Abstract. Consider the wave propagation in a two-layered medium consisting of a homogeneous compressible air or fluid on top of a homogeneous isotropic elastic solid in three dimensions. The interface between the two layers is assumed to be an unbounded rough surface. This work presents the first mathematical analysis for the transient acoustic-elastic interaction problem in such an unbounded structure. Using an exact transparent boundary condition and suitable interface conditions, we study an initial boundary value problem for the coupling of the acoustic and elastic wave equations. The well-posedness and stability are established for the reduced problem. Moreover, a priori estimates with explicit dependence on the time are obtained for the acoustic pressure and the elastic displacement. The problem addressed is sufficiently general to have potential use in applications. Key words. acoustic wave equation, elastic wave equation, unbounded rough surface, time domain, stability, a priori estimates AMS subject classifications. 78A46, 65C3 DOI..37/6M9326. Introduction. Consider a two-layered medium which consists of a homogeneous compressible air or fluid on top of a homogeneous isotropic elastic solid. The interface between air/fluid and solid is assumed to be an unbounded rough surface, which refers to a nonlocal perturbation of an infinite plane surface such that the whole surface lies within a finite distance of the original plane. As a source located in the solid layer, the external force generates an elastic wave, which propagates towards the interface and further excites an acoustic wave in the air/fluid layer. The wave propagation leads to an air/fluid-solid interaction problem with an unbounded interface separating the acoustic and elastic waves which are coupled on the interface through two interface conditions. The first kinematic interface condition is imposed to ensure that the normal velocity of the air/fluid on one side of the boundary matches the accelerated velocity of the solid on the other side. The second one is the dynamic condition which results from the balance of forces on two sides of the interface. The problem addressed above is sufficiently general to have potential use in applications. For example, the model problem can be used to describe the seismic wave propagation in the air/fluid-solid medium due to the excitation of an earthquake source which is located in the crust between the lithosphere and the mantle of the Earth. Mathematically, there are two major challenges for this acoustic-elastic interaction problem: the time dependence and the unbounded rough interface. Received by the editors August 6, 26; accepted for publication in revised form July 24, 27; published electronically October 2, Funding: The first author was supported in part by the NSFC grants 5765, 677, National Research program of China Grant 23CB834, JLSTDP JH and FRFCU2427FZ5. The second author was supported in part by the NSF grant DMS-538. The third author was partially supported by the NSFC grants and School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 324, China gaoyx643@nenu.edu.cn. Department of Mathematics, Purdue University, West Lafayette, IN 4797 lipeijun@math. purdue.edu. LSEC and Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 9, China b.zhang@amt.ac.cn. 395

2 3952 YIXIAN GAO, PEIJUN LI, AND BO ZHANG Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see This problem falls into the class of unbounded rough surface scattering problems, which have been of great interest to physicists, engineers, and applied mathematicians for many years. These problems arise from diverse scientific areas such as optics, acoustics, electromagnetics, and radar techniques [,, 32, 38, 4]. In particular, the elastic wave scattering by unbounded interfaces has significant applications in geophysics and seismology. For instance, the problem of elastic pulse transmission and reflection through the Earth is fundamental to the investigation of earthquakes and the utility of controlled explosions in search for oil and ore bodies [4, 5, 35]. The unbounded rough surface scattering problems are quite challenging due to unbounded structures. The usual Sommerfeld for acoustic waves, Kupradze-Sommerfeld for elastic waves, or Silver Müller for electromagnetic waves radiation condition is no longer valid [2, 43]. The typical Fredholm alternative argument is not applicable either, due to the lack of compactness results. For the time-harmonic problems, we refer to [3, 4, 5, 25, 27] for some mathematical studies on the two-dimensional Helmholtz equation and [7, 29, 3] for the three-dimensional Maxwell equations. Despite many studies conducted so far, it is still unclear what the least restrictive conditions are for those physical parameters to assure the well-posedness of the wave equations in unbounded structures. The time-domain scattering problems have recently attracted considerable attention due to their capability of capturing wideband signals and modeling more general material and nonlinearity [6, 24, 26, 33, 4]. These attractive features motivate us to tune our focus from seeking the best possible conditions for those physical parameters to the time-domain problems. Comparing them with the time-harmonic problems, the time-domain problems are less studied due to the additional challenge of the temporal dependence. The analysis can be found in [7, 9, 39] for the time-domain acoustic and electromagnetic obstacle scattering problems. We refer to [28] and [6] for the analysis of the time-dependent electromagnetic scattering from an open cavity and a periodic structure, respectively. The acoustic-elastic interaction problems have received much attention in both the mathematical and engineering communities [9,, 8, 2, 2, 3]. Many approaches have been attempted to solve numerically the time-domain problems such as coupling of the boundary element and finite element with different time quadratures [2, 22, 34, 36, 3]. There are also some numerical studies on the inverse problems arising from the fluid-solid interaction such as reconstruction of surfaces of periodic structures or obstacles [23, 42]. However, rigorous mathematical study, especially the stability, is very rare at present. This work presents the first mathematical analysis for the time-domain acousticelastic interaction problem in an unbounded structure. The problem is reformulated as an initial boundary value problem by adopting an exact transparent boundary condition TBC. Using the Laplace transform and energy method, we show that the reduced variational problem has a unique weak solution in the frequency domain. Meanwhile, we obtain the stability estimate to show the existence of the solution in the time domain. In addition, we achieve a priori estimates with explicit dependence on the time for the pressure of the acoustic wave and the displacement of the elastic wave by considering directly the time-domain variational problem and taking special test functions. The paper is organized as follows. In section 2, we introduce the model equations and interface conditions for the acoustic-elastic interaction problem. The time-domain TBC is presented and some trace results are proved. Section 3 is devoted to the analysis of the reduced problem, where the well-posedness and stability are addressed

3 TRANSIENT ACOUSTIC-ELASTIC INTERACTION 3953 Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see air or fluid crust mantle Fig.. Problem geometry of the acoustic-elastic interaction in an unbounded structure. in both the frequency and time domains. We conclude the paper with some remarks in section Problem formulation. In this section, we define some notation, introduce the model equations, and present an initial boundary value problem for the acousticelastic scattering in an air/fluid-solid medium. 2.. Problem geometry. As shown in Figure, we consider an active source which is embedded in an elastic solid medium. It models an earthquake focus located in the crust which lies between the lithosphere and the rigid mantle of the Earth. Due to the excitation of the source, an elastic wave is generated in the solid and propagates through to the medium of the air/fluid. Clearly, this process leads to the air/fluidsolid interaction problem with the scattering interface separating the domains where the acoustic and elastic waves travel. Let r = x, y R 2 and x = x, y, z R 3. Denote by Γ f = {x R 3 : z = fr} the surface separating the air/fluid and the solid, where f is assumed to be a W, R 2 function. Let Γ g = {x R 3 : z = gr} be the surface separating the crust and the mantle, where g is an L R 2 function satisfying gr fr, r R 2. We assume that the open space Ω + f = {x R3 : z > fr} is filled with a homogeneous compressible air or a compressible inviscid fluid with the constant density ρ. The space Ω 2 = {x R 3 : gr < z < fr} is assumed to be occupied by a homogeneous isotropic linear elastic solid which is characterized by the constant mass density ρ 2 and Lamé parameters µ, λ. Define an artificial planar surface Γ h = {x R 3 : z = h}, where h > sup r R 2 fr is a constant. Let Ω = {x R 3 : fr < z < h} and Ω = Ω Ω Acoustic wave equation. The acoustic wave field in air/fluid is governed by the conservation and the dynamics equations in the time domain: px, t = ρ t vx, t, c 2 ρ vx, t = t px, t, x Ω + f, t >, where p is the pressure, v is the velocity, and the constants ρ > and c > are the density and sound speed, respectively. Eliminating the velocity v from, we obtain the acoustic wave equation for the pressure p: px, t c 2 2 t px, t =, x Ω + f, t >. The homogeneous initial conditions are considered as causality: Ω Ω 2 p t= =, t p t= = in Ω + f. Γ h Γ f Γ g

4 3954 YIXIAN GAO, PEIJUN LI, AND BO ZHANG Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see Elastic wave equation. For the solid, the elastic wave field in a homogeneous isotropic solid material satisfies the linear time-domain elasticity equation, 2 σux, t ρ 2 2 t ux, t = jx, t, x Ω 2, t >, where u = u, u 2, u 3 is the displacement vector, ρ 2 > is the density of the elastic solid material, jx, t is the source which models the earthquake focus and is assumed to have a compact support contained in Ω 2, T for any T >, and the symmetric stress tensor σu is given by the generalized Hooke s law 3 σu = 2µEu + λtr Eu I, Eu = 2 u + u. Here µ, λ are the Lamé parameters satisfying µ >, λ+µ >, I R 3 3 is the identity matrix, treu is the trace of the matrix Eu, and u is the displacement gradient tensor given by u = xu y u z u x u 2 y u 2 z u 2. x u 3 y u 3 z u 3 Substituting 3 into 2, we obtain the time-domain Navier equation for the displacement u: 4 µ ux, t + λ + µ ux, t ρ 2 2 t ux, t = jx, t, x Ω 2, t >. By assuming that the mantle is rigid, we have u = on Γ g, t >. The homogeneous initial conditions are also imposed for the elastic wave equation: u t= =, t u t= = in Ω Interface conditions. To couple the acoustic wave equation in the air/fluid and the elastic wave equation in the solid, the kinematic interface condition is imposed to ensure the continuity of the normal component of the velocity on Γ f : 5 n vx, t = n t ux, t, x Γ f, t >, where n is the unit normal on Γ f pointing from Ω 2 to Ω. Noting, we have from 5 that n p = n p = ρ n 2 t u on Γ f, t >. In addition, the dynamic interface condition is required to ensure the continuity of traction: pn = σu n on Γ f, t >, where σu n denotes the multiplication of the stress tensor σu with the normal vector n.

5 TRANSIENT ACOUSTIC-ELASTIC INTERACTION 3955 Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see Laplace transform and some functional spaces. We first introduce some properties of the Laplace transform. For any s = s + is 2 with s >, s 2 R, define ŭs to be the Laplace transform of the function ut, i.e., ŭs = L us = It follows from the integration by parts that t e st utdt. uτdτ = L s ŭs, where L is the inverse Laplace transform. It is easy to verify from the inverse Laplace transform that ut = F e st L us + is 2, where F denotes the inverse Fourier transform with respect to s 2. Recall the Plancherel or Parseval identity for the Laplace transform cf. [8, 2.46]: 6 ŭs vsds 2 = e 2st ut vtdt 2π s > σ, where ŭ = L u, v = L v, and σ is the abscissa of convergence for the Laplace transform of u and v. Hereafter, the expression a b or a b stands for a Cb or a Cb, where C is a positive constant and its specific value is not required but should be always clear from the context. The following lemma cf. [37, Theorem 43.] is an analogue of the Paley Wiener Schwarz theorem for the Fourier transform of the distributions with compact supports in the case of the Laplace transform. Lemma 2.. Let hs be a holomorphic function in the half-plane Re s > σ and be valued in the Banach space E. The following two conditions are equivalent:. there is a distribution h D +E whose Laplace transform is equal to hs; 2. there is a real σ with σ σ < and an integer m such that for all complex numbers s with Res > σ, we have hs E + s m, where D +E is the space of distributions on the real line which vanish identically in the open negative half-line. Next we introduce some Sobolev spaces. For any u L 2 Γ h, which is identified as L 2 R 2, we denote by û the Fourier transform of u: ûξ = ure ir ξ dr, 2π R 2 where ξ = ξ, ξ 2 R 2. For any α R, define { } H α Γ h = H α R 2 = u L 2 R 2 : + ξ 2 α û 2 dξ <. R 2 It is clear to note that the dual space associated with H α Γ h is the space H α Γ h with respect to the scalar product in L 2 R 2 defined by u, v Γh = ur vrdr = Γ h ûξ ˆvξdξ. R 2

6 3956 YIXIAN GAO, PEIJUN LI, AND BO ZHANG Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see Define with the associated image norm H /2 Γ f = {u : Γ f C : u, f H /2 R 2 } u H /2 Γ f = ur, fr 2 dr + R 2 R 2 R2 ur, fr ur 2, fr 2 2 r r 2 3 dr dr 2 /2. Denote by H k Ω = {u L 2 Ω : D α u L 2 Ω for all α k} the standard Sobolev space of square integrable functions with the order of derivatives up to k N. Let HΓ g Ω = {u H Ω : u = on Γ g }. Let HΓ g Ω 3 and H /2 Γ f 3 be the Cartesian product spaces equipped with the corresponding 2-norms of HΓ g Ω and H /2 Γ f, respectively. For any u = u, u 2, u 3 H Γ g Ω 2 3, define It is easy to verify that 3 u L 2 Ω = u j 2 dx Ω 2 j= /2 u 2 L 2 Ω u 2 L 2 Ω 2 u 2 H Ω 2 3. The following trace results can be easily proved by using the definitions. The proofs are omitted for simplicity. Lemma 2.2. There exists a positive constant C such that u H /2 Γ f C u H Ω u H Ω. Lemma 2.3. There exists a positive constant C such that u H /2 Γ f 3 C u H Ω 2 3 u H Γ g Ω Transparent boundary condition. In this subsection, we will introduce an exact time-domain TBC to formulate the acoustic-elastic wave interaction problem into the following coupled initial boundary value problem: p c 2 t 2 p = in Ω, t >, µ u + λ + µ u ρ 2 t 2 u = j in Ω 2, t >, p t= = t p t= =, u t= = t u t= = in Ω, 7 n p = ρ n t 2 u, pn = σu n on Γ f, t >, ν p = T p on Γ h, t >, u = on Γ g, t >, where ν =,, is the unit normal vector on Γ h pointing from Ω to Ω + h = {x R 3 : z > h}, and T is the time-domain TBC operator on Γ h. In what follows, we shall derive the formulation of the operator T and show some of its properties. Let px, s = L p and ŭx, s = L u be the Laplace transform of px, t and ux, t with respect to t, respectively. Recall that L t p = s p, s p,, L 2 t p = s 2 p, s sp, t p,, L t u = sŭ, s u,, L 2 t u = s 2 ŭ, s su, t u,..

7 TRANSIENT ACOUSTIC-ELASTIC INTERACTION 3957 Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see Taking the Laplace transform of 7 and using the initial conditions, we obtain the acoustic-elastic wave interaction problem in the s-domain: 8a p s2 c 2 p = in Ω, 8b µ ŭ + λ + µ ŭ ρ 2 s 2 ŭ = j in Ω 2, 8c 8d 8e n p = ρ s 2 n ŭ, pn = σŭ n on Γ f, ν p = B p on Γ h, ŭ = on Γ g. where j = L j, B is the Dirichlet-to-Neumann DtN operator on Γ h in the s-domain and satisfies T = L B L. In order to deduce the TBC, we consider the Helmholtz equation with a complex wavenumber, 9 p s2 c 2 p = in Ω+ h. Taking the Fourier transform of 9 with respect to r yields { d 2 ˆ pξ,z s dz 2 2 c + ξ 2 ˆ pξ, z =, z > h, 2 ˆ pξ, z = ˆ pξ, h, z = h. Solving and using the bounded outgoing wave condition, we get where ˆ pξ, z = ˆ pξ, he βξz h, z > h, β 2 ξ = s2 c 2 + ξ 2 with Reβξ >. Thus we obtain the solution of 9, 2 pr, z = ˆ pξ, he βξz h e iξ r dξ. R 2 Taking the normal derivative of 2 on Γ h, we have ν pr, h = βξˆ pξ, he iξ r dξ. R 2 For any function u defined on Γ h, we defined the DtN operator 3 Bu r = βξûξe iξ r dξ. R 2 Lemma 2.4. Let s = s + is 2, s σ >, s 2 R. The DtN operator Bs : H /2 Γ h H /2 Γ h is continuous, i.e., Bsu H /2 Γ h Cσ s u H /2 Γ h u H /2 Γ h, where Cσ = max{c, σ }.

8 3958 YIXIAN GAO, PEIJUN LI, AND BO ZHANG Proof. For any u H /2 Γ h, it follows from 3 and that Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see Bu 2 H /2 Γ h R = + ξ 2 /2 βξûξ 2 dξ 2 = + ξ 2 /2 + ξ 2 βξ 2 ûξ 2 dξ R 2 Cσ 2 s 2 u 2 H /2 Γ h, where we have used 2. to derive the explicit bound of βξ 2 : βξ 2 = s 2 c 2 + ξ 2 s 2 c 2 + ξ 2 which completes the proof. Lemma 2.5. We have s 2 c 2 + σ 2 ξ 2 Cσ 2 s 2 + ξ 2, Re s Bsu, u Γh u H /2 Γ h, s = s + is 2, s >, s 2 R. Proof. A simple calculation yields that s Bu, u Γh = s βξ ûξ 2 dξ = R 2 R 2 sβξ s 2 ûξ 2 dξ. Let βξ = a + ib, s = s + is 2 with a >, s >. Taking the real part of the above equation gives Re s s a + s 2 b 4 Bu, u Γh = R s 2 ûξ 2 dξ. 2 Recalling β 2 ξ = s2 c + ξ 2, we have 2 5 Substituting 5 into 4 yields a 2 b 2 = s2 s 2 2 c 2 + ξ 2, ab = s s 2 c 2. Re s Bu, u Γh = which completes the proof. R 2 s 2 as + s a s 2 2 c 2 ûξ 2 dξ, Using the DtN operator 3, we can get the following TBC for the acoustic pressure in the s-domain: 6 ν p = B p on Γ h. Using Lemmas 2. and 2.4, we may take the inverse Laplace transform of 6 and obtain the TBC in the time domain, ν p = T p on Γ h.

9 TRANSIENT ACOUSTIC-ELASTIC INTERACTION 3959 Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see 3. The reduced problem. In this section, we present the main results of this paper, which include the well-posedness and stability of the scattering problem and related a priori estimates. 3.. Well-posedness in the s-domain. Consider the reduced problem 8a 8d in the s-domain. Multiplying 8a and 8b by the complex conjugate of a test function q H Ω and a test function v HΓ g Ω 2 3, respectively, using the integration by parts and boundary conditions, which include the TBC condition 8d, the kinematic and dynamic interface conditions 8c, and the rigid boundary condition 8e, we arrive at the variational problem: to find p, ŭ H Ω HΓ g Ω 2 3 such that 7 and 8 Ω 2 Ω s p q + s c 2 p q dx s B p, q Γh ρ s n ŭ qdγ = q H Ω Γ f s µ ŭ : v + λ + µ ŭ v + ρ 2sŭ v dx + pn vdγ = s Ω 2 s j vdx v HΓ g Ω 2, Γ f where A : B = tr AB is the Frobenius inner product of square matrices A and B. We multiply 8 by ρ s 2 and add the obtained result to 7 to obtain an equivalent variational problem: to find p, ŭ H Ω HΓ g Ω 2 3 such that 9 a p, ŭ; q, v = ρ s j vdx Ω 2 q, v H Ω HΓ g Ω 2 3, where the sesquilinear form 2 a p, ŭ; q, v = Ω s p q + s c 2 p q dx + ρ s µ ŭ : v Ω 2 + λ + µ ŭ v + ρ ρ 2 s s 2 ŭ v dx s B p, q Γh + ρ Γ f s pn v s qn ŭ dγ. Theorem 3.. The variational problem 9 has a unique weak solution p, ŭ H Ω H Γ g Ω 2 3, which satisfies 2 22 p L 2 Ω 3 + s p L 2 Ω j L 2 Ω 2 3, ŭ L 2 Ω ŭ L 2 Ω 2 + sŭ L 2 Ω 2 3 s j L 2 Ω 2 3.

10 396 YIXIAN GAO, PEIJUN LI, AND BO ZHANG Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see Proof. We have from the Cauchy Schwarz inequality and Lemmas 2.2, 2.3, and 2.4 that a p, ŭ; q, v s p L 2 Ω 3 q L 2 Ω 3 + s c 2 p L 2 Ω q L 2 Ω + ρ s µ ŭ L 2 Ω 3 3 v 2 L 2 Ω λ + µ ŭ L2 Ω 2 v L2 Ω 2 + ρ ρ 2 s 3 ŭ L 2 Ω 3 v 2 L 2 Ω s B p H /2 Γ h q H /2 Γ h + ρ s p L 2 Γ f n v L 2 Γ f + q L 2 Γ f n ŭ L 2 Γ f p H Ω q H Ω + ŭ H Ω 2 3 v H Ω p H /2 Γ h q H /2 Γ h + p H /2 Γ f v H /2 Γ f 3 + q H /2 Γ f ŭ H /2 Γ f 3 p H Ω q H Ω + ŭ H Ω 2 3 v H Ω p H Ω q H Ω + p H Ω v H Ω q H Ω ŭ H Ω 2 3, which shows that the sesquilinear form is bounded. Letting q, v = p, ŭ in 2 yields a p, ŭ; p, ŭ = s p 2 + s c 2 p 2 dx + ρ s µ ŭ : ŭ + λ + µ ŭ 2 Ω 2 23 Ω + ρ ρ 2 s s 2 ŭ 2 dx s B p, p Γh + ρ Γ f s pn ŭ s pn ŭ dγ. Taking the real part of 23 and using Lemma 2.5, we obtain s Rea p, ŭ; p, ŭ = Ω s 2 p 2 + s c 2 p 2 dx + ρ s µ ŭ 2 L 2 Ω λ + µ ŭ 2 L 2 Ω 2 +ρ ρ 2 s s 2 ŭ 2 L 2 Ω 2 3 Re s B p, p Γh s s 2 p 2 L 2 Ω + 3 s p 2 L 2 Ω 24 + s ŭ 2 L 2 Ω ŭ 2 L 2 Ω + 2 sŭ 2 L 2 Ω 2. 3 It follows from the Lax Milgram theorem that the variational problem 9 has a unique weak solution p, ŭ H Ω H Γ g Ω 2 3. Moreover, we have from 9 that 25 a p, ŭ; p, ŭ s s j L2 Ω 2 3 sŭ L 2 Ω 2 3. Combing 24 and 25 completes the proof Well-posedness in the time domain. We now consider the reduced problem in the time-domain: 26a p c 2 2 t p = in Ω, t >, 26b 26c 26d 26e 26f µ u + λ + µ u ρ 2 t 2 u = j in Ω 2, t >, p t= = t p t= =, u t= = t u t= = in Ω, n p = ρ n 2 t u, pn = σu n on Γ f, t >, ν p = T p on Γ h, t >, u = on Γ g, t >.

11 TRANSIENT ACOUSTIC-ELASTIC INTERACTION 396 To show the well-posedness of the reduced problem 26, we assume Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see 27 j H, T ; L 2 Ω 2 3. Theorem 3.2. The initial boundary value problem 26 has a unique solution p, u which satisfies px, t L 2, T ; H Ω H, T ; L 2 Ω, ux, t L 2, T ; H Γ g Ω 2 3 H, T ; L 2 Ω 2 3 and the stability estimates t p L2 Ω + p L2 Ω max t [,T ] max t [,T ] t j L,T ; L 2 Ω 2 3, t u L2 Ω u L 2 Ω 2 + u L2 Ω t j L,T ; L 2 Ω 2 3. Proof. For the air/fluid pressure p, we have T p 2 L 2 Ω + tp 2 3 L 2 Ω dt T e 2st T p 2 L 2 Ω 3 + tp 2 L 2 Ω dt T = e 2sT e 2st p 2 L 2 Ω + tp 2 3 L 2 Ω dt e 2st p 2 L 2 Ω + tp 2 2 L 2 Ω dt. Similarly, we have for the elastic displacement u that T t u 2 L 2 Ω u 2 L 2 Ω u 2 L 2 Ω 2 dt e 2st t u 2 L 2 Ω u 2 L 2 Ω u 2 L 2 Ω 2 Hence it suffices to estimate the integrals and e 2st p 2 L 2 Ω 3 + tp 2 L 2 Ω dt e 2st t u 2 L 2 Ω u 2 L 2 Ω u 2 L 2 Ω 2 dt. dt. Taking the Laplace transform of 26, we obtain the reduced acoustic-elastic interaction problem in the s-domain 8. It follows from Theorem 3. that p and ŭ satisfy the stability estimates 2 and 22, respectively. It follows from [37, Lemma 44.] that p and ŭ are holomorphic functions of s on the half-plane Res > σ. Hence we have from Lemma 2. that the inverse Laplace transform of p and ŭ exists and is supported in [,.

12 3962 YIXIAN GAO, PEIJUN LI, AND BO ZHANG Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see Using the Parseval identity 6, the assumptions 27, and the stability estimate 2, we have e 2st p 2 L 2 Ω + tp 2 3 L 2 Ω dt = 2π s 2 s j 2 L 2 Ω 2 3ds 2 = s 2 s 2 T which shows that e 2st t j 2 L 2 Ω 2 3dt, p 2 L 2 Ω + 3 s p 2 L 2 Ω ds 2 L t j 2 L 2 Ω 2 3ds 2 px, t L 2, T ; H Ω H, T ; L 2 Ω. Since ŭ = L u = F e st u, where F is the Fourier transform in s 2, we have similarly from the Parseval identity 6 and the stability estimate 22 that which shows that e 2st t u 2 L 2 Ω u 2 L 2 Ω u 2 L 2 Ω 2 T s 2 e 2st j 2 L 2 Ω 3dt, 2 u L 2, T ; H Γ g Ω 2 3 H, T ; L 2 Ω 2 3. dt Next we show the stability estimates. Let p be the extension of p with respect to t in R such that p = outside the interval [, t]. By the Parseval identity 6 and Lemma 2.5, we get Re t = Re = 2π Γ h e 2st T p, t p Γh dt = Re t e 2st T p t pdtdr = 2π s 2 Re s B p, p Γh ds 2, which yields after taking s that 3 Re t e 2st Γ h T p t pdrdt Γ h T p t pdrdt. Re B p, s p Γh ds 2 Taking the partial derivative of 26b 26d and 26f with respect to t, we get µ t u + λ + µ t u ρ 2 t 2 t u = t j in Ω 2, t >, t u t= = in Ω 2, 3 t 2 u t= = ρ 2 µ u + λ + µ u j t= = in Ω 2, t p n = t σu n = σ t u n on Γ f, t >, t u = on Γ g, t >.

13 TRANSIENT ACOUSTIC-ELASTIC INTERACTION 3963 For any < t < T, consider the energy function Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see where and E t = e t + e 2 t, e t = c tp 2 L 2 Ω + p 2 L 2 Ω 3 e 2 t = ρ ρ 2 /2 2 t u 2 L 2 Ω ρ λ + µ /2 t u 2 L 2 Ω 2 It is easy to note that 32 E t E = t E τdτ = + ρ µ /2 t u 2 L 2 Ω t e τ + e 2τ dτ. It follows from 26a, 26c 26e, and the integration by parts that 33 t e τdτ = 2Re t t Ω c 2 2 t p t p + t p p dxdτ = 2Re p t p + t p p dxdτ Ω t = 2Re p t p + t p p dxdτ Ω t t + 2Re T p t pdrdτ 2Re n p t pdγdτ Γ h Γ f t t = 2Re T p t pdrdτ + 2Re ρ n t 2 u t pdγdτ. Γ h Γ f Similarly, we have from 3 and the integration by parts that t t e 2τdτ = ρ 2Re ρ2 t t 2 u t 2 ū + λ + µ 2 t u t ū Ω 2 + µ t 2 u : t ū dxdτ t = ρ 2Re µ t u + λ + µ t u t j t 2 ū Ω 2 + λ + µ t 2 u t ū + µ t 2 u : t ū dxdτ t = ρ Re µ t u : t 2 ū λ + µ tu t 2 ū Ω 2 + λ + µ t 2 u t ū + µ t 2 u : t ū dxdτ t t 2Reρ t j t 2 ūdxdτ + 2Reρ σ t u n t 2 ūdγdτ Ω 2 Γ f t t 34 = 2Reρ t j t 2 ūdxdτ 2Reρ t pn t 2 ūdγdτ. Ω 2 Γ f

14 3964 YIXIAN GAO, PEIJUN LI, AND BO ZHANG Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see Since E =, combining and 3 gives t t E t = 2Re T p t pdrdτ 2Reρ t j t 2 ūdxdτ Γ h Ω 2 t 2Reρ t j t 2 ūdxdτ Ω 2 2ρ max t [,T ] 2 t u L 2 Ω 3 2 tj L,T ;L 2 Ω 2 3. It follows from Young s inequality that t p L 2 Ω + p L 2 Ω t u L 2 Ω 3 max t [,T ] + t u L 2 Ω 2 + t u L 2 Ω t j L,T ;L 2 Ω 2 3, which shows the stability estimate 28. For the elastic displacement u, we can also obtain t 2 u 2 L 2 Ω2 + tu 2 3 L 2 Ω2 + tu 2 L 2 Ω max t [,T ] For any function v with v, =, we have 36 v, t t t j 2 L,T ;L 2 Ω 2 3. t v, τ dτ T max t [,T ] tv, which holds for any norm. Combining gives t u 2 L 2 Ω2 + 3 u 2 L 2 Ω2 + u 2 L 2 Ω2 3 3 t j 2 L,T ;L 2 Ω 2 3, max t [,T ] which shows the estimate A priori estimates. In what follows, we derive a priori stability estimates for the air/fluid pressure p and the displacement u with a minimum regularity requirement for the data and an explicit dependence on the time. We shall consider the elastic wave equation for t u in order to match the interface conditions when deducing the stability estimates. Taking the partial derivative of 26b 26e and 26f with respect to t, we obtain a new reduced problem: 37 p c 2 t 2 p = in Ω, t >, ν p = T p on Γ h, t >, n p = ρ n t 2 u on Γ f, t >, p t= = t p t= = in Ω µ t u + λ + µ t u ρ 2 t 2 t u = t j in Ω 2, t >, t u t= = in Ω 2, t 2 u t= = ρ 2 µ u + λ + µ u j t= = in Ω 2, t pn = t σu n = σ t u n on Γ f, t >, t u = on Γ g, t >.

15 TRANSIENT ACOUSTIC-ELASTIC INTERACTION 3965 Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see The variational problems of 37 are to find p, u H Ω HΓ g Ω 2 3 for all t > such that Ω c 2 2 t p qdx = p qdx + T p qdr n p qdγ Ω Γ h Γ f 38 = p qdx + T p qdr + ρ n t 2 u qdγ, q H Ω Ω Γ h Γ f and ρ 2 t 2 t u vdx = µ t u : v + λ + µ t u v dx Ω 2 Ω 2 t j vdx + σ t u n vdγ Ω 2 Γ f = µ t u : v + λ + µ t u v + t j v dx Ω 2 39 t pn vdγ, v HΓ g Ω 2 3. Γ f To show the stability of the solution, we follow the argument in [37] but with a careful study of the TBC. The following lemma is useful for the subsequent analysis. Lemma 3.3. Given ξ and p H Ω, we have Re Γ h ξ t T p, τdτ p, tdtdr. Proof. Let p be the extension of p with respect to t in R such that p = outside the interval [, ξ]. We obtain from the Parseval identity 6 and Lemma 2.5 that Re Γ h ξ = Re = Re = Re = 2π = 2π e 2st τ Γ h Γ h Γ h where we have used the fact that T p, τdτ p, tdtdr e 2st t e 2st t T p, τdτ p, tdtdr L B L p, τdτ p, tdtdr e 2st L s B L p, t p, t dtdr Re s B p, s p, sdrds 2 Γ h Re s B p, p Γh ds 2, t p, τdτ = L s p, s. The proof is completed after taking the limit s.

16 3966 YIXIAN GAO, PEIJUN LI, AND BO ZHANG Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see Theorem 3.4. Let p, u H Ω H Γ g Ω 2 3 be the solution of Given t j L, T ; L 2 Ω 2 3 for any T >, we have p L,T ; L 2 Ω T t j L,T ; L 2 Ω 2 3, u L,T ;L 2 Ω 2 3 T 2 t j L,T ; L 2 Ω 2 3 p L2,T ; L 2 Ω T 3/2 t j L,T ; L 2 Ω 2 3, u L 2,T ;L 2 Ω 2 3 T 5/2 t j L,T ; L 2 Ω 2 3. Proof. Let < θ < T and define an auxiliary function It is clear to note that 44 ψ x, t = θ t ψ x, θ =, For any φx, t L 2, θ; L 2 Ω, we have 45 θ φx, t ψ x, tdt = px, τdτ, x Ω, t θ. t ψ x, t = px, t. θ t φx, τdτ px, tdt. Indeed, we have from the integration by parts and 44 that θ φx, t ψ x, tdt = = = = θ θ θ t θ t θ φx, t θ t px, τdτd φx, ςdς px, τdτ t t px, τdτ φx, ςdς θ t θ + t φx, τdτ px, tdt. dt φx, ςdς px, tdt Next, we take the test function q = ψ in 38 and get Ω c 2 2 t p ψ dx = p ψ dx + T p ψ dr Ω Γ h 46 + ρ n t 2 u ψ dγ. Γ f It follows from 44 and the initial conditions 26c that Re θ Ω c 2 2 t p ψ dxdt = Re = Re Ω θ Ω c 2 = 2 c p, θ 2 L 2 Ω. t t c 2 p ψ + t p p dtdx t p ψ θ + 2 p 2 θ dx

17 TRANSIENT ACOUSTIC-ELASTIC INTERACTION 3967 Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see It is easy to verify that θ θ Re ρ n t 2 u ψ dγdt = Re ρ t n t u ψ + n t u p dtdγ Γ f Γ f = Re ρ n t u ψ θ θ dγ + Re ρ n t u pdγdt Γ f Γ f θ = Re ρ n t u pdγdt. Γ f Integrating 46 from t = to t = θ and taking the real parts yield 2 θ 2 p, θ c + Re p ψ dxdt L 2 Ω Ω = 2 2 p, θ c + 2 θ p, tdt dx L 2 Ω 2 Ω θ θ = Re T p, ψ Γh dt + Re ρ n t 2 u ψ dγdt Γ f θ θ 47 = Re T p, ψ Γh dt + Re ρ n t u pdγdt. Γ f We define another auxiliary function Clearly, we have 48 ψ 2 x, t = θ t ψ 2 x, θ =, τ ux, τdτ, x Ω 2, t θ < T. t ψ 2 x, t = t ux, t. Using a similar proof as that for 45, for any φx, t L 2, θ; L 2 Ω 2 2, we may show that θ θ t 49 φx, t ψ 2 x, tdt = φx, τdτ t ūx, tdt. Taking the test function v = ψ 2 in 39, we can get ρ 2 t 2 t u ψ 2 dx = µ t u : ψ 2 + λ + µ t u ψ 2 Ω 2 Ω t j ψ 2 dx Γ f t pn ψ 2 dγ. It follows from 48 and the initial condition in 37 that θ θ Re ρ 2 t 2 t u ψ 2 dxdt = Re ρ 2 t t 2 u ψ 2 + t 2 u t ū dtdx Ω 2 Ω 2 = Re t 2 u ψ 2 θ + 2 tu 2 θ dx Ω 2 ρ 2 = ρ 2 2 tu, θ 2 L 2 Ω 2 3

18 3968 YIXIAN GAO, PEIJUN LI, AND BO ZHANG Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see and Re θ t pn ψ 2 dγdt = Re Γ f Γ f θ t pn ψ2 + pn t ū dtdγ θ = Re pn ψ2 θ dγ + Re pn t ūdγdt Γ f Γ f θ = Re pn t ūdγdt. Γ f Integrating 5 from t = to t = θ and taking the real parts yields ρ θ 2 2 tu, θ 2 L 2 Ω + Re 2 3 µ t u, t : ψ 2, t Ω 2 + λ + µ t u, t ψ 2, t dxdt = ρ 2 2 tu, θ 2 L 2 Ω + 2 θ 2 µ 2 2 Ω 2 t u, tdt F 2 θ + λ + µ t u, tdt dx θ θ 5 = Re t j ψ 2 dxdt Re pn t ūdγdt, Ω 2 Γ f where θ t u, tdt 2 F := θ t u, tdt : θ t ū, tdt. Multiplying 5 by ρ and adding it to 47 give 2 2 p, θ c + 2 θ p, tdt dx + ρ ρ 2 L 2 Ω 2 Ω 2 tu, θ 2 L 2 Ω ρ 2 θ θ 2 µ 2 t u, tdt + λ + µ t u, tdt dx 52 Ω 2 θ θ = Re T p, ψ Γh dt + Re ρ n t u pdγdt Γ f θ θ Re ρ t j ψ 2 dxdt Re ρ pn t ūdγdt Ω 2 Γ f θ θ = Re T p, ψ Γh dt Re ρ t j ψ 2 dxdt. Ω 2 In what follows, we estimate the two terms on the right-hand side of 52 separately. Using Lemma 3.3 and 45, we obtain θ θ Re T p, ψ Γh dt = Re T p ψ drdt Γ h θ t 53 = Re T p, τdτ p, tdtdr. Γ h F

19 TRANSIENT ACOUSTIC-ELASTIC INTERACTION 3969 Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see For t θ T, we have from 49 that 54 Re θ θ t ρ t j ψ 2 dxdt = ρ Re t j, τdτ t ū, tdtdx Ω 2 Ω 2 θ t = ρ Re t j, τ t ū, tdxdτdt Ω 2 θ t ρ t j, τ L 2 Ω 3dτ 2 t u, t L 2 Ω 2dt 2 θ θ ρ t j, t L2 Ω 2 3dt t u, t L2 Ω 2 3dt θ θ ρ t j, t L2 Ω 2 3dt t u, t L2 Ω 2 3dt. Substituting 53 and 54 into 52, we have for any θ [, T ] that p, θ c + ρ ρ 2 L 2 Ω 2 tu, θ 2 L 2 Ω p, θ c + 2 θ p, tdt dx + ρ ρ 2 L 2 Ω 2 Ω 2 tu, θ 2 L 2 Ω ρ 2 θ θ 2 µ t u, tdt + λ + µ t u, tdt dx 2 Ω 2 F θ θ ρ t j, t L 2 Ω 3dt 2 t u, t L 2 Ω 2dt 2. Taking the L norm with respect to θ on both sides of 55 yields p 2 L,T ; L 2 Ω + tu 2 L,T ; L 2 Ω 2 3 T t j L,T ; L 2 Ω 2 3 t u L,T ; L 2 Ω 2 3. Applying the Young inequality to the above inequality, we get 56 p 2 L,T ; L 2 Ω + tu 2 L,T ; L 2 Ω 2 3 T 2 t j 2 L,T ; L 2 Ω 2 3. It follows from the Cauchy Schwarz inequality that p L,T ;L 2 Ω p L,T ; L 2 Ω + t u L,T ; L 2 Ω 2 3 T t j L,T ; L 2 Ω 2 3, which gives the estimate 4. For the elastic displacement u, using Young s inequality again gives t u, t 2 L 2 Ω = 2 3 τ u, τ 2 L 2 Ω 3dτ ɛt u, 2 t 2 L 2 Ω + T 2 3 ɛ tu, t 2 L 2 Ω 2 3 ɛt u 2 L,T ;L 2 Ω T ɛ tu 2 L,T ; L 2 Ω 2 3,

20 397 YIXIAN GAO, PEIJUN LI, AND BO ZHANG which gives Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see u 2 L,T ;L 2 Ω 2 3 ɛt u 2 L,T ;L 2 Ω T ɛ tu 2 L,T ; L 2 Ω 2 3. Choosing ɛ = 2T, we have from 56 that u 2 L,T ;L 2 Ω 2 3 T 2 t u 2 L,T ; L 2 Ω 2 3 T 2 p 2 L,T ; L 2 Ω + tu 2 L,T ; L 2 Ω 2 3 T 4 t j 2 L,T ; L 2 Ω 2 3, which implies the estimate 4. The estimates 42 and 43 are simply the straightforward consequences of 4 and 4, respectively. The details are omitted and the proof is completed. 4. Conclusion. In this paper we have studied the time-domain acoustic-elastic interaction problem in an unbounded structure in three dimensions. The problem models the wave propagation in a two-layered medium consisting of the air/fluid and the solid due to an active source located in the solid. We reduce the scattering problem into an initial boundary value problem by using the exact TBC. We establish the wellposedness and the stability for the variational problem in the s-domain. In the time domain, we show that the reduced problem has a unique weak solution by using the energy method. The main ingredients of the proofs are the Laplace transform, the Lax Milgram theorem, and the Parseval identity. Moreover, we obtain a priori estimates with explicit time dependence for the quantities of acoustic wave pressure and elastic wave displacement by taking special test functions to the time-domain variational problem. Acknowledgment. The authors wish to thank the anonymous reviewers for their constructive suggestions and comments on improving the presentation of the paper. REFERENCES [] I. Abubakar, Scattering of plane elastic waves at rough surfaces. I, Math. Proc. Cambridge Philos. Soc., , pp [2] T. Arens and T. Hohage, On radiation conditions for rough surface scattering problems, IMA J. Appl. Math., 7 25, pp [3] S. N. Chandler-Wilde, E. Heinemeyer, and R. Potthast, Acoustic scattering by mildly rough unbounded surfaces in three dimensions, SIAM J. Appl. Math., 66 26, pp [4] S. N. Chandler-Wilde and P. Monk, Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal., 37 25, pp [5] S. N. Chandler-Wilde and B. Zhang, Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers, SIAM J. Math. Anal., 3 999, pp [6] Q. Chen and P. Monk, Discretization of the time domain CFIE for acoustic scattering problems using convolution quadrature, SIAM J. Math. Anal., 46 24, pp [7] Z. Chen and J.-C. Nédélec, On Maxwell s equations with the transparent boundary condition, J. Comput. Math., 26 28, pp [8] A. M. Cohen, Numerical Methods for Laplace Transform Inversion, Numer. Methods Algorithms 5, Springer, New York, 27. [9] A. G. Dallas, Analysis of a Limiting-Amplitude Problem in Acousto-Elastic Interactions, Technical report, NRL Report 972, Naval Research Laboratory, Washington, DC, 989. [] J. Donea, S. Giuliani, and J.-P. Halleux, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Comput. Methods Appl. Mech. Engrg., 33982, pp

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22 3972 YIXIAN GAO, PEIJUN LI, AND BO ZHANG Downloaded /7/7 to Redistribution subject to SIAM license or copyright; see [38] A. G. Voronovich, Non-classical approaches in the theory of wave scattering from rough surfaces, Mathematical and numerical aspects of wave propagation Golden, CO, 998, SIAM, Philadelphia, 998, pp [39] B. Wang and L. Wang, On L 2 -stability analysis of time-domain acoustic scattering problems with exact nonreflecting boundary conditions, J. Math. Study, 24, pp [4] L.-L. Wang, B. Wang, and X. Zhao, Fast and accurate computation of time-domain acoustic scattering problems with exact nonreflecting boundary conditions, SIAM J. Appl. Math., 72 22, pp [4] K. F. Warnick and W. C. Chew, Numerical simulation methods for rough surface scattering, Waves Random Media, 2, pp. R R3. [42] T. Yin, G. Hu, L. Xu, and B. Zhang, Near-field imaging of obstacles with the factorization method: Fluid-solid interaction, Inverse Problems, 32 26, 53. [43] B. Zhang and S. N. Chandler-Wilde, Acoustic scattering by an inhomogeneous layer on a rigid plate, SIAM J. Appl. Math., , pp