ANALYSIS OF A LINEAR FLUID-STRUCTURE INTERACTION PROBLEM. Q. Du. M.D. Gunzburger. L.S. Hou. J. Lee. (Communicated by Jianhong Wu)

Transcription

1 DISCRETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS Volume 9, Number3, May23 pp ANALYSIS OF A LINEAR FLUID-STRUCTURE INTERACTION PROBLEM Q. Du Department of Mathematics Penn State University, State College, PA 1682, USA M.D. Gunzburger School of Computational Science Information Technology Florida State University; Tallahassee FL 3236, USA L.S. Hou Department of Mathematics Iowa State University; Ames, IA 511, USA J. Lee Department of Mathematics Carnegie Mellon University, Pittsburgh, PA 15213, USA (Communicated by Jianhong Wu Abstract. A time-dependent system modeling the interaction between a Stokes fluid an elastic structure is studied. A divergence-free weak formulation is introduced which does not involve the fluid pressure field. The existence uniqueness of a weak solution is proved. Strong energy estimates are derived under additional assumptions on the data. The existence of an L 2 integrable pressure field is established after the verification of an inf-sup condition. 1. Introduction. Fluid-structure interaction problems have been extensively studied in the past continue to be the focus of much attention today. We classify a number of different types of mathematical models for fluid-structure interactions into the following three categories. Elementary fluid. The fluid motion is governed by equations for a potential function, e.g., the Laplace equation or the wave equation. In [25], a coupled system of a potential equation a wave equation was considered. Elementary fluids interacting with a rigid cavity or a moving wall were studied in [15] with an elastic solid in [3]. Inviscid fluid. The fluid motion is governed by inviscid fluid models, e.g., the Euler equations. Interactions between linearized inviscid fluids elastic solids were analyzed in [1, 27]. An algorithm applicable to an inviscid nonlinear fluid coupled with rigid walls was given in [2]. Viscous fluid. The fluid motion is governed by viscous, incompressible or compressible fluid models, e.g., the Stokes or Navier-Stokes equations. There is an extensive literature on linearized viscous fluids coupled with solids. Solids modeled 1991 Mathematics Subject Classification. 35Q3, 76D7, 73C2, 73K12. Key words phrases. fluid-structure interaction, incompressible fluid, linear elasticity. The work of Q. Du was supported in part by NSF grant DMS ; that of M. Gunzburger was supported in part by NSF grant DMS

2 634 Q. DU, M.D. GUNZBURGER, L.S. HOU, J. LEE by plate equations or shell equations were treated in [11, 12, 14, 23]. The Stokes equations coupled with a beam equation was analyzed in [18]. In [6, 22], interactions between a linearized viscous fluid elastic solids were studied; [4] discussed interactions with rigid walls. There also is a vast literature on fluid-structure interactions for which the fluid is modeled by nonlinear viscous fluid models. Rigid body motions of solids in a nonlinear viscous fluid were studied in [5, 7, 2, 19, 21]. In [13], the Navier-Stokes equations coupled to the plate equations were studied. The work of [5, 8, 1, 28, 29] treated interactions between nonlinear viscous fluids elastic solids. It should be noted that the majority of the references cited use solid models of lower spatial dimensions, e.g., one-dimensional beams interacting with twodimensional fluids or two-dimensional plates interacting with three-dimensional fluids. Rigorous mathematical results are rare for fluid-solid interaction problems in which both the fluid the solid occupy true spatial domains. Eigenmodes of the coupled system (1.1 (1.2 were studied in [29]. In [6], the homogenization of a mathematical model for the Stokes equation coupled to the equations of linear elasticity was considered. Both existence of a solution numerical experiments for a problem in which a nonlinear viscous fluids is coupled to elastic solids in one dimension were discussed in [8]. For a numerical algorithm for solving interaction problems of elastic body motions in a fluid flow, see [1]. In this paper, we consider the interaction of a linear, viscous fluid with elastic body motions in a two or three dimensional bounded domain. The specific fluidstructure interaction model is described as follows. We assume the fluid solid occupy the open, Lipschitz domains Ω 1 R d Ω 2 R d, respectively, where d = 2 or 3 is the space dimensions. We denote by Ω the entire fluid-solid region under consideration, i.e., Ω is the interior of Ω 1 Ω 2. Let Γ = Ω 1 Ω 2 denote the interface between the fluid solid let Γ 1 = Ω 1 \ Γ Γ 2 = Ω 2 \ Γ respectively denote the parts of the fluid solid boundaries excluding the interface Γ. For obvious reasons, we assume that meas(γ 1 Γ 2. In the fluid region Ω 1, we consider the Stokes system ρ 1 v t + p µ 1 ( v + v T =ρ 1 f 1 in (,T Ω 1 v = in (,T Ω 1 (1.1 v = on (,T Γ 1, v t= = v in Ω 1, where v denotes the fluid velocity, p the fluid pressure, f 1 the given body force per unit mass, ρ 1 µ 1 the constant fluid density viscosity, v the given initial velocity, T>the terminal time. In the solid region, we consider the equations of linear elasticity { ρ2 u tt µ 2 ( u + u T λ 2 ( u =ρ 2 f 2 in (,T Ω 2 (1.2 u = on (,T Γ 2, u t= = u in Ω 2, u t t= = u 1 in Ω 2, where u denotes the displacement of the solid, f 2 the given loading force per unit mass, µ 2 λ 2 the Lamé constants, ρ 2 the constant solid density u u 1 the given initial data. Across the fixed interface Γ between the fluid solid, the velocity stress vector are continuous. Thus, we have u t = v on Γ (1.3

3 FLUID-STRUCTURE INTERACTION 635 µ 2 ( u + u T n 2 + λ 2 ( un 2 = pn 1 µ 1 ( v + v T n 1 on Γ, (1.4 where n i is the outward-pointing unit normal vector along Ω i, i =1, 2. Some remarks about the use of a fixed fluid-solid interface Γ are in order. The general motion of a solid body immersed in a fluid involves rigid body motions superimposed with displacements caused by the stresses strains induced in the solid by the loads resulting from the interaction with the fluid motion. Even if the the latter are infinitesimal in size, the rigid motion of the solid by itself would result in a moving fluid-solid interface. Certainly, if the solid motion involves large stress-induced displacements, the fluid-solid interface is not stationary. Thus, the general case is that of a moving fluid-solid interface which, in fact, is generally not known must be found as part of the solution process. At the other extreme is the case for which, first, there is no rigid motion or that motion is purely translational with constant speed one attaches the coordinate system to the solid,, second, the solid undergoes only infinitesimal elastic displacements. Then, since the motion of the fluid-solid interface is wholly determined by the elastic displacement, one may assume that that interface is stationary. Moreover, it is usually the case that the velocity in the solid is also infinitesimal so that that the interface condition v = u t simply reduces to v =, i.e., the classical no-slip condition, the fluid motion decouples from that of the solid. Once the fluid motion is determined, the other interface condition provides an inhomogeneous traction boundary condition which may be used to determine the elastic motion of the solid. In between the two extremes, i.e., between having a moving interface a fixed interface with uncoupled fluid motion, is the case considered here. We begin as in the second case: the motion of the solid is wholly due to infinitesimal displacements. Again, we may then assume that the fluid-solid interface is stationary. However, we operate in the case that although the displacement u is small, the velocity u t is not. Thus, we cannot impose the no-slip condition on the fluid velocity must retain the interface condition v = u t, but the latter may be imposed along a fixed boundary. We note that the case of having O(1 solid velocities even though the solid displacements are of infinitesimal size is of practical interest. For example, this setting arises in the high frequency, small displacement oscillation of elastic structures. Our model may be justified by stard asymptotic analysis techniques (see, e.g., [16] the details are omitted. The plan of the paper is as follows. In Section 2, working in a divergence-free setting, we first introduce some notations define a weak formulation for the model fluid-structure interaction problem. We next formulate an auxiliary, equivalent parabolic type problem, define its Galerkin approximations derive a priori estimates for the Galerkin sequence. By passing to the limit in the Galerkin approximations, we prove the existence uniqueness of a solution for the auxiliary problem. The existence uniqueness of the velocity v displacement u for the fluid-structure interaction problem follows directly from the existence of a unique solution for the auxiliary problem. In Section 3, we prove a regularity result for the weak solution under additional assumptions. We then verify an inf-sup condition establish the existence of an L 2 integrable pressure. In the end of this paper we give some concluding remarks regarding forthcoming work on the model studied in this paper on extensions into more general models.

4 636 Q. DU, M.D. GUNZBURGER, L.S. HOU, J. LEE 2. The existence of a weak solution Notations. Throughout this paper, C denotes a positive constant, depending on the domains Ω, Ω 1 Ω 2, whose meaning value changes with context. H s (D, s R, denotes the stard Sobolev space of order s with respect to the set D equipped with the stard norm s,d. Vector-valued Sobolev spaces are denoted by H s (D, with norms still denoted by s,d. H 1 (D denotes the space of functions belonging to H 1 (D that vanish on the boundary D of D; H 1 (D denotes the vector-valued counterpart. We will use the following L 2 inner product notations on scalar vector-valued L 2 spaces: [p, q] D = pq dd p, q L 2 (D, [u, v] D = u v dd u, v L 2 (D, D where the spatial set D is Ω or Γ or Ω i,fori =1, 2. We introduce the function spaces X i =[H 1 (Ω] Ωi with the norm Xi = 1,Ωi, i =1, 2, V 1 = {v H 1 (Ω 1 : div v =} with the norm V1 = 1,Ω1, Ψ = {η H 1 (Ω : div η =inω 1 } with the norm 1,Ω, Φ = the closure of Ψ for the norm induced by the inner product [[, ]], where [[, ]] denotes the weighted L 2 inner product [[ξ, η]] = [ρ 1 ξ, η] Ω1 +[ρ 2 ξ, η] Ω2 ξ, η L 2 (Ω. (2.1 Clearly, the induced norm on Φ is equivalent to,ω Φ preserves the divergence-free property in Ω 1. We denote by, the duality pairing between Ψ Ψ that is generated from the weighted L 2 inner product [[, ]]. The norm on the dual space Ψ is defined in the conventional manner: g Ψ = sup g, η η Ψ, η 1,Ω 1 g Ψ. We define the bilinear forms a 1 [u, v] = 1 µ 1 ( u + u T :( v + v T dω u, v X 1, 2 Ω 1 b[v,q]= q v dω Ω 1 v X 1, q L 2 (Ω 1 { µ2 } a 2 [u, v] = Ω 2 2 ( u + ut :( v + v T +λ 2 ( u( v dω u, v X 2. It is evident that the forms a 1 [, ], a 2 [, ] b[, ] are continuous, i.e., there exist positive constants K 1, K 2 K b such that a 1 [u, v] K 1 u 1,Ω1 v 1,Ω1 u, v X 1, a 2 [u, v] K 2 u 1,Ω2 v 1,Ω2 u, v X 2, b[v,q] K b v 1,Ω1 q,ω1 v X 1, q L 2 (Ω 1. D

5 FLUID-STRUCTURE INTERACTION 637 Also, it can be verified with the help of Korn s inequalities [26, p.31, p.12] that for i =1, 2, a i [η, η] k i η 2 1,Ω i η X i, if meas(γ i (2.2 [η, η] Ωi + a i [η, η] k i η 2 1,Ω i η X i, if meas(γ i =. ( A divergence-free weak formulation. We are now prepared to introduce a weak formulation for (1.1-(1.4. We assume that the fluid force f 1, the elastic force f 2 the initial data v, u u 1 satisfy { f1 L 2 (,T; L 2 (Ω 1, f 2 L 2 (,T; L 2 (Ω 2, u X 2, (2.4 v X 1, div v =inω 1, u 1 X 2, v Γ = u 1 Γ. A desired weak formulation for (1.1 (1.2 can be derived by multiplying the Stokes elasticity equations by an η H 1 (Ω performing integration by parts; this yields ρ 1 [v t, η] Ω1 + b[η,p]+a 1 [v, η]+ρ 2 [u tt, η] Ω2 + a 2 [u, η] (2.5 = ρ 1 [f 1, η] Ω1 + ρ 2 [f 2, η] Ω2 η H 1 (Ω, a.e. t [,T]. The choice of a global test function η H 1 (Ω instead of two independent test functions on the two subdomains allows us to incorporate the stress interface condition (1.4 into the weak formulation. The weak form (2.5 requires v t L 2 (,T; L 2 (Ω 1, u t L 2 (,T; X 2, u tt L 2 (,T; L 2 (Ω 2, p L 2 (,T; L 2 (Ω 1. In general, such regularity conditions are too strong to assume for a weak solution. Analogous to the stard practice [3] of defining a weak formulation for the Stokes equation with respect to divergence-free function spaces, we introduce the following divergence-free weak formulation for (1.1-(1.4: seek a pair (v, u L 2 (,T; X 1 L 2 (,T; X 2 satisfying div v =inω 1, d (ρ 1 [v, η] Ω1 + ρ 2 [ t u, η] Ω2 + a 1 [v, η]+a 2 [u, η] dt (2.6 = ρ 1 [f 1, η] Ω1 + ρ 2 [f 2, η] Ω2 η Ψ, v t= = v, u t= = u, u t t= = u 1 (2.7 v(s Γ ds = u(t Γ u Γ a.e. t, (2.8 where (2.6 holds in the sense of distributions on (,T (see [3]. Equations (2.6 (2.8 will be the primary weak formulation we study in this paper. Note that the pressure does not appear in (2.6 (2.8. A weak formulation involving the pressure will be considered in Section 3.2. The natural interface condition (1.4 is built into the weak form (2.5 (2.6. In the divergence-free weak formulation, we enforce the essential interface condition (1.3 in the weak sense (2.8.

6 638 Q. DU, M.D. GUNZBURGER, L.S. HOU, J. LEE 2.3. An auxiliary weak formulation its Galerkin approximations. Introducing auxiliary functions { { v in Ω1 v in Ω 1 ξ = ξ = (2.9 u t in Ω 2 u 1 in Ω 2, we see that (2.6 (2.8 is equivalent to the auxiliary problem t ξ, η + a 1 [ξ, η]+a 2 [ ξ(s ds, η] (2.1 = ρ 1 [f 1, η] Ω1 + ρ 2 [f 2, η] Ω2 a 2 [u, η] η Ψ, a.e. t [,T], ξ( = ξ in Ψ (2.11 ( ( Γ Γ ξ(s Ω1 ds = ξ(s Ω2 ds a.e. t. (2.12 Here we recall that the space Ψ introduced in Section 2.1 is divergence-free in Ω 1 so that the term b[η,p] vanishes. Under assumption (2.4, ξ defined by (2.9 obviously satisfies ξ Ψ. The initial condition (2.11 is equivalent to ξ(, η =[[ξ, η]] η Ψ. The precise meaning of the initial condition requires certain continuity of ξ in t will be made clear in subsequent discussions. To prove the existence of a solution for (2.1 (2.12, we employ adapt the widely-used Galerkin approach (see, e.g., [9] [24]. Let {ψ j } j=1 be a basis for the space Ψ. In particular, we choose {ψ j } j=1 to be the complete set of eigenfunctions for the eigenvalue problem ψ Ψ, [ ψ, η] Ω = λ[[ψ, η]] η Ψ, (2.13 where the weighted L 2 (Ω inner product [[, ]] is defined by (2.1. Moreover, we assume that {ψ j } j=1 is orthonormalized with respect to the H1 (Ω-inner product [, ] Ω. Then, from equation (2.13 we also have that {ψ j } j=1 is orthogonal with respect to the weighted L 2 (Ω inner product [[, ]]. Let Ψ m = span {ψ 1,...,ψ m }. A Galerkin approximation ξ m C 1 ([,T]; Ψ m of the form ξ m = m j=1 g (m j (tψ j (x (2.14 is defined as the solution of ρ 1 [ t ξ m (t, η] Ω1 + ρ 2 [ t ξ m (t, η] Ω2 + a 1 [ξ m (t, η]+a 2 [ ξ m(s ds, η] (2.15 =[[f(t, η]] a 2 [u, η]+a 1 [ξ m ( v, η] η Ψ m,t [,T] [[ξ m (, η]] = [[ξ, η]] η Ψ m, (2.16 where f L 2 (,T; L 2 (Ω is defined by { f1 in Ω 1 f = f 2 in Ω 2. Note that ξ m defined as in (2.14 trivially satisfies (ξ m (s Ω1 Γ ds = (ξ m (s Ω2 Γ ds t [,T]. (2.17

7 FLUID-STRUCTURE INTERACTION 639 Remark: Comparing (2.15 (2.1, the term a 1 [ξ m ( v, η] appears to be redundant. Indeed, it will be shown in Lemma 2.2 that ξ m ( ξ 1,Ω sothat a 1 [ξ m ( v, η] asm.theterma 1 [ξ m ( v, η] is added into (2.15 for technical reasons connected with the derivation of strong a priori estimates in Section 3.1. We can write (2.15 (2.16 as an equivalent system of first-order, linear ordinary differential equations (ODEs for {g (m j } m j=1 : m ( [[ψ j, ψ i ]] d dt g(m j (t+a 1 [ψ j, ψ i ] g (m j (t+a 2 [ψ j, ψ i ] j=1 =[[f(t, ψ i ]] a 2 [u, ψ i ]+a 1 [ξ m ( v, ψ i ] g (m j (s ds i =1,...,m, t [,T] (2.18 m [[ψ j, ψ i ]] g (m j ( = [[ξ, ψ i ]] i =1,...,m. (2.19 j=1 The following theorem states the existence of a solution for (2.18 (2.19, or equivalently, (2.15 (2.16. Theorem 2.1. Assume that f 1, v, f 2, u u 1 satisfy (2.4. Then, for each integer m>, there exists a unique set of functions {g (m j } m j=1 C1 [,T] which satisfy (2.18 (2.19. Proof: Setting h (m j (t = g(m j (s ds for j =1,...,m, we see that (2.18 (2.19 is equivalent to the following ODE initial value problem: m [[ψ j, ψ i ]] d m dt g(m (t+ j=1 dt h(m i j j=1 a 1 [ψ j, ψ i ] g (m j (t+ m j=1 = c i (t i =1,...,m, d (t =g (m i (t i =1,...,m, m [[ψ j, ψ i ]] g (m j ( = [[ξ, ψ i ]] i =1,...,m, j=1 h (m i ( =, i =1,...,m, j=1 a 2 [ψ j, ψ i ] h (m j (t where m c i (t =ρ 1 [f 1 (t, ψ i ] Ω1 +ρ 2 [f 2 (t, ψ i ] Ω2 a 2 [u, ψ i ]+ g (m j (a 1 [ψ j, ψ i ] a 1 [v, ψ i ]. The matrix {[[ψ i, ψ j ]]} m i,j=1 is positive definite as the function set {ψ 1,...,ψ m } is linearly independent. Thus, using stard theories for (constant coefficient systems of linear, first-order ODEs we see that the above ODE system has a unique C 1 solution (g (m 1,...,g m (m,h (m 1,...,h (m m on[,t]. Upon eliminating h (m j in the system, we arrive at the assertion of the theorem. Theorem 2.1 immediately yields the existence of the Galerkin approximate solutions {ξ m } satisfying (2.15 (2.16. We proceed to derive a priori estimates for {ξ m }.

8 64 Q. DU, M.D. GUNZBURGER, L.S. HOU, J. LEE Lemma 2.2. Let P m denote the weighted L 2 (Ω projection from L 2 (Ω onto Ψ m, i.e., for every η L 2 (Ω, [[P m η, z]] = [[η, z]] z Ψ m. (2.2 Then, P m η,ω η,ω η L 2 (Ω, (2.21 P m η 1,Ω η 1,Ω η Ψ, (2.22 P m η η 1,Ω as m η Ψ (2.23 P m η η,ω as m η Φ. (2.24 Proof: Setting z = P m η in (2.2 applying the Cauchy-Schwarz inequality we easily obtain (2.21. Next, we prove (2.22 (2.23. Let η Ψ be given. Since {ψ i } i=1 forms a basis for Ψ, we can write η = i=1 α iψ i ; moreover, we have that m 1,Ω m α i ψ i η 1,Ω α i ψ i η asm. (2.25 1,Ω i=1 i=1 Using the orthogonality of {ψ i } with respect to the [[, ]] inner product we find P m η = m i=1 α iψ i. Hence, (2.22 (2.23 follows from (2.25. It remains to prove (2.24. Let η Φ be given. Using the triangle inequality (2.21 we have P m η η,ω η η ɛ,ω + η ɛ P m η,ω + P m (η ɛ η,ω ( η η ɛ,ω + η ɛ P m η,ω η ɛ Ψ. From relations (2.23 (2.26 as well as the denseness of Ψ in Φ for the L 2 (Ω norm we obtain (2.24. Theorem 2.3. Assume that f 1, v, f 2, u u 1 satisfy (2.4. Then, for each integer m >, there exists a function ξ m C 1 ([,T]; Ψ m of the form (2.14 which satisfies (2.15 (2.17. Moreover, ξ m (t 2,Ω + ξ m 2 L 2 (,T ;H 1 (Ω + 1 ξ m(s ds 2 H 1 (Ω 2 Ce CT( (2.27 f 2 L 2 (,T ;L 2 (Ω + v 2 1,Ω 2 + u 2 1,Ω 1 for all t [,T], ξ m 2 L 2 (,T ;Ψ Ce CT( f 2 L 2 (,T ;L 2 (Ω + v 2 1,Ω 2 + u 2 1,Ω 1, (2.28 t [ξ m Ω1 ] 2 L 2 (,T ;V 1 Ce CT( f 2 L 2 (,T ;L 2 (Ω + v 2 1,Ω 2 + u 2 1,Ω 1, t [ξ m Ω2 ] 2 L 2 (,T ;H 1 (Ω 2 Ce CT( f 2 L 2 (,T ;L 2 (Ω + v 2 1,Ω 2 + u 2 1,Ω 1. (2.29 (2.3

9 FLUID-STRUCTURE INTERACTION 641 Proof: The existence of ξ m follows directly from Theorem 2.1 so we only need to prove the a priori estimates (2.27 (2.3. Setting η = ξ m (t in (2.15 using the weighted L 2 (Ω inner product notation, we have [[ξ m(t, ξ m (t]] + a 1 [ξ m (t, ξ m (t] + a 2 [ ξ m(s ds, ξ m (t] =[[f(t, ξ m (t]] a 2 [u, ξ m (t] + a 1 [ξ m ( v, ξ m (t] which can be rewritten as 1 d 2 dt [[ξ m(t, ξ m (t]] + a 1 [ξ m (t, ξ m (t] + 1 d 2 dt a 2[ ξ m(s ds, ξ m(s ds] =[[f(t, ξ m (t]] a 2 [u, ξ m (t] + a 1 [ξ m ( v, ξ m (t] C( f(t 2,Ω + ξ m ( v 2 1,Ω ξ m(t 2,Ω a 2 [u, ξ m (t] a 1[ξ m (t, ξ m (t]. Integrating the last relation in t noting that (Lemma 2.2 we are led to ξ m ( 2 1,Ω = P m ξ 2 1,Ω ξ 2 1,Ω = v 2 1,Ω 1 [[ξ m (t, ξ m (t]] + a 1 [ξ m (s, ξ m (s] ds + a 2 [ ξ m(s ds, ξ m(s ds] C ( ξ m ( 2,Ω + f 2 L 2 (,T ;L 2 (Ω + u 2 1,Ω2 + T ξ m ( v 2 1,Ω1 + ξ m (s 2,Ω ds a 2 [u, ξ m(s ds] C ( ξ m ( 2,Ω + f 2 L 2 (,T ;L 2 (Ω + u 2 1,Ω2 + T v 2 1,Ω1 + T u 1 2 1,Ω2 so that + ξ m (t 2,Ω + ξ m (s 2,Ω ds a 2[ ξ m(s ds, ξ m(s ds] a 1 [ξ m (s, ξ m (s] ds + a 2 [ ξ m(s ds, ξ m(s ds] ξ m (s 2,Ω ds +C ( f 2 L 2 (,T ;L 2 (Ω + T v 2 1,Ω1 + u 2 1,Ω2 + T u 1 2 1,Ω2. (2.31 Dropping the second third terms on the left side of (2.31 then applying the following version of Gronwall s inequality: we deduce if h(t C 1 + C 2 h(s ds, then h(t C 1 C 2 e C2t, (2.32 ξ m (t 2,Ω Ce CT( f 2 L 2 (,T ;L 2 (Ω + v 2 1,Ω 1 + u 2 1,Ω 2. (2.33

10 642 Q. DU, M.D. GUNZBURGER, L.S. HOU, J. LEE The estimates (2.33 (2.31 yield: for all t [,T], ξ m (s 2,Ω 1 ds + a 1 [ξ m (s, ξ m (s] ds Ce CT( (2.34 f 2 L 2 (,T ;L 2 (Ω + v 2 1,Ω 1 + u 2 1,Ω 2 ξ m(s ds 2,Ω 2 + a 2 [ ξ m(s ds, ξ m(s ds] Ce CT( f 2 L 2 (,T ;L 2 (Ω + v 2 1,Ω 1 + u 2 1,Ω 2. (2.35 Combining (2.33 (2.35 utilizing the coercivity conditions (2.2 or (2.3 we arrive at the desired energy estimate (2.27. We next prove (2.28. For each given η Ψ with η 1,Ω 1, we write η as η = P m η +(η P m η where P m is defined by (2.2. Since ξ m(t Ψ m,wehave [[ξ m(t, η]] = [[ξ m(t,p m η]] + [[ξ m(t, η P m η]] = [[ξ m(t,p m η]]. From the last equation (2.15 we obtain [[ξ m(t, η]] = [[ξ m(t,p m η]] =[[f(t,p m η]] a 1 [ξ m (t,p m η] a 2 [u,p m η] a 2 [ ξ m(s ds, P m η]+a 1 [ξ m ( v,p m η] ( C f(t,ω + ξ m (t 1,Ω1 + ξ m(s ds 1,Ω2 ( u 1,Ω2 + u 1 1,Ω2 + v 1,Ω1 P m η 1,Ω. Thus, (2.28 follows from (2.36, (2.27 the fact that P m η 1,Ω η 1,Ω 1 (see Lemma 2.2. To prove (2.29, we arbitrarily choose a v V 1 with v 1,Ω1 1. We then define an η Ψ by η Ω1 = v η Ω2 =. Evidently we have that η 1,Ω = v 1,Ω1 1 [ξ m(t, v] Ω1 =[ξ m(t, η] Ω C ξ m(t Ψ. The last estimate together with (2.28 implies (2.29. We may prove (2.3 analogously Existence uniqueness of a weak solution. By passing to the limit in the Galerkin approximations, we may prove the existence of a solution ξ for the auxiliary problem (2.1 (2.11 which in turn directly yields the existence uniqueness of a solution to the fluid-structure interaction problem. Note that as in the Galerkin approximations, we consider in this subsection the existence of an auxiliary weak solution on the space Ψ so that the weak formulation does not contain the pressure term. Theorem 2.4. Assume that f 1, v, f 2, u u 1 satisfy (2.4. Then, there exists auniqueξ L 2 (,T; L 2 (Ω H 1 (,T; Ψ which satisfies ξ Ω1 L 2 (,T; X 1, div ξ Ω1 =, ξ(s Ω2 ds L (,T; X 2 (2.37

11 FLUID-STRUCTURE INTERACTION 643 (2.1 (2.12. Moreover, ξ(t 2,Ω + ξ 2 L 2 (,T ;H 1 (Ω + 1 ξ(s ds 2 H 1 (Ω 2 Ce CT( (2.38 f 2 L 2 (,T ;L 2 (Ω + u 2 1,Ω 2 + v 2 1,Ω 1 for all t [,T], ξ 2 L 2 (,T ;Ψ Ce CT( f 2 L 2 (,T ;L 2 (Ω + u 2 1,Ω 2 + v 2 1,Ω 1, (2.39 t [ξ Ω1 ] 2 L 2 (,T ;V 1 Ce CT( f 2 L 2 (,T ;L 2 (Ω + u 2 1,Ω 2 + v 2 1,Ω 1, t [ξ Ω2 ] 2 L 2 (,T ;H 1 (Ω 2 Ce CT( f 2 L 2 (,T ;L 2 (Ω + u 2 1,Ω 2 + v 2 1,Ω 1. (2.4 (2.41 Proof: Letξ m of the form (2.14 be a solution of (2.15 (2.16. Using the initial condition the energy estimates (2.27 (2.28 we may extract subsequence of {ξ m }, still denoted by {ξ m }, such that ξ m ξ in L (,T; L 2 (Ω, ξ m ξ in L 2 (,T; L 2 (Ω, ξ m Ω1 ξ Ω1 in L 2 (,T; H 1 (Ω 1, [ξ m (s] Ω2 ds [ξ(s] Ω2 ds in L (,T; H 1 (Ω 2 t ξ m t ξ in L 2 (,T; Ψ for some ξ L (,T; L 2 (Ω H 1 (,T; Ψ satisfying (2.37. These weak limits also imply [ξ m (s Ω1 ] Γ ds [ξ(s Ω1 ] Γ ds in L 2 (,T; H 1/2 (Γ [ξ m (s Ω2 ] Γ ds [ξ(s Ω2 ] Γ ds in L 2 (,T; H 1/2 (Γ. Passing to the limit in (2.17, we obtain (2.12. Also, passing to the limit in (2.27 (2.3 yields (2.38 (2.41. To prove ξ satisfies (2.1, we proceed as follows. We fix an integer N choose a function η C 1 ([,T]; Ψ havingtheform N η = d j (tψ j. (2.42 For each m>n, we integrate (2.15 with respect to t to obtain ρ 1 [ t ξ m, η] Ω1 + ρ 2 [ t ξ m, η] Ω2 + a 1 [ξ m, η]+a 2 [ t ξ m(s ds, η] dt = ρ 1 [f 1, η] Ω1 + ρ 2 [f 2, η] Ω2 a 2 [u, η]+a 1 [ξ m ( v, η] dt. j=1 (2.43

12 644 Q. DU, M.D. GUNZBURGER, L.S. HOU, J. LEE By passing to the limit as m, we find t ξ, η + a 1 [ξ, η+a 2 [ t ξ(s ds, η] dt = ρ 1 [f 1, η] Ω1 + ρ 2 [f 2, η] Ω2 a 2 [u, η] dt. (2.44 Here we have used the fact that (see Lemma 2.2 ξ m ( v 1,Ω1 = P m ξ ξ 1,Ω1 P m ξ ξ 1,Ω as m Equality (2.44 then holds for all η L 2 (,T; Ψ, since functions of the form (2.42 are dense in L 2 (,T; Ψ. In particular, (2.44 implies (2.1. To verify the initial condition (2.11, we first note that the regularity ξ L 2 (,T; Ψ t ξ L 2 (,T; Ψ implies that ξ C([,T]; Ψ. On the one h, by choosing an η C 1 ([,T]; Ψ in (2.44 integrating by parts we have ξ, t η + a 1 [ξ, η]+a 2 [ t ξ m(s ds, η] dt (2.45 = [[f, η]] a 2 [u, η] dt + ξ(, η(. On the other h, from (2.43 we deduce ρ 1 [ξ m, t η] Ω1 ρ 2 [ξ m, t η] Ω2 + a 1 [ξ m, η]+a 2 [ t ξ m(s ds, η] dt = [[f, η]] a 2 [u, η]+a 1 [ξ m ( v, η] dt +[[ξ m (, η(]]. Passing to the limit in (2.46 yields ξ, t η + a 1 [ξ, η]+a 2 [ t ξ(s ds, η] dt = ρ 1 [f 1, η] Ω1 + ρ 2 [f 2, η] Ω2 a 2 [u, η] dt +[[ξ, η(]]. (2.46 (2.47 Comparing (2.45 (2.47 we obtain ξ( ξ, η( = η( Ψ, (2.48 which is precisely (2.11 as η( Ψ is arbitrary. It remains to prove the uniqueness. The following proof is adapted from [9, pp ]. Assume ξ ξ are two solutions of (2.1 (2.11. Setting θ = ξ ξ we have that θ L 2 (,T; Φ, θ L 2 (,T; Ψ, θ, η + a 1 [θ, η]+a 2 [ θ(s ds, η] = η Ψ, a.e. t, (2.49 θ( =. (2.5 We define ζ = θ(s ds. Then ζ L 2 (,T; Ψ, ζ L 2 (,T; Φ, ζ L 2 (,T; Ψ, ζ, η + a 1 [ζ, η]+a 2 [ζ, η] = η Ψ, a.e. t, (2.51 ζ( = ζ ( =, (2.52 where ( ( denotes the first second time derivatives, respectively. We note that from the regularity ζ L 2 (,T; Ψ H 1 (,T; Ψ [3, p.176, Lemma

13 FLUID-STRUCTURE INTERACTION ] we deduce that ζ C([,T]; Φ; indeed, the spaces Ψ, Φ Ψ satisfy that Ψ Φ Ψ that the duality pairing between Ψ Ψ is generated from the weighted L 2 inner product on Φ. We arbitrarily fix a t (,T] define { t ζ(τ dτ, η(t = t t [, t], t [ t, T ]. Then η(t Ψ for every t [,T] equation (2.51 yields t ( ζ, η + a 1 [ζ, η]+a 2 [ζ, η] dt =. Since ζ ( = η( t =, we obtain after integrating by parts in the first two terms that t ( [[ζ, η ]] a 1 [ζ, η ] a 2 [η, η] dt =. Noting that η (t = ζ(t fort [, t] weareledto t 1 d ( t [[ζ, ζ]] a 2 [η, η] dt = a 1 [ζ, ζ] dt 2 dt so that 1 2 [[ζ( t, ζ( t]] + 1 t 2 a 2[η(, η(] = a 1 [ζ, ζ] dt. This implies that ζ( t =. As t is arbitrary, we deduce that ζ = in [,T] Ω. Hence, ξ ξ = ζ = in [,T] Ω. Remark: In the proof for uniqueness, it is not known whether ξ(t ξ(t belong to Ψ for almost every t. Thus we cannot simply set η = θ(t in (2.49. If ξ is the solution of (2.1 (2.12 that is guaranteed to exist by Theorem 2.4, then by setting v = ξ Ω1 u = u + ξ(s Ω 2 ds we immediately obtain the existence of a weak solution (v, u for the divergence-free formulation of the fluidstructure interaction problem. Theorem 2.5. Assume that f 1, v, f 2, u u 1 satisfy (2.4. Then, there exists auniquepair(v, u L 2 (,T; X 1 L 2 (,T; X 2 which satisfies (2.6 (2.8, where (2.6 holds in the sense of distributions on (,T. Moreover, v(t 2,Ω + u t(t 2,Ω + v 2 L 2 (,T ;H 1 (Ω + 1 u(t 2 H 1 (Ω 2 Ce CT( (2.53 f 2 L 2 (,T ;L 2 (Ω + u 2 1,Ω 2 + v 2 1,Ω 1 for all t [,T] v t 2 L 2 (,T ;V 1 + u tt 2 L 2 (,T ;H 1 (Ω 2 Ce CT ( f 2 L 2 (,T ;L 2 (Ω + u 2 1,Ω 2 + v 2 1,Ω 1. (2.54

14 646 Q. DU, M.D. GUNZBURGER, L.S. HOU, J. LEE 3. The existence of a strong solution an L 2 pressure. Under additional regularity compatibility assumptions on the data, we may prove stronger energy estimates for the Galerkin solutions {ξ m }. Such a priori estimates will allow us to derive regularity results for the solution of (2.1 (2.11 to show the existence of a corresponding pressure field Strong energy estimates. Theorem 3.1. Assume that f 1, v, f 2, u u 1 satisfy (2.4 t f i L 2 (,T; L 2 (Ω i, i=1, 2, (3.1 v H 2 (Ω 1, u 1 H 2 (Ω 2, u H 2 (Ω 2. Assume further that there exists a p H 1 (Ω 1 such that ( p n 1 µ 1 ( v + v T Γ n 1 (3.2 = (µ 2 ( u + u T Γ n 2 +(λ 2 + µ 2 (div u n 2 where n i denotes the outward-pointing normal along Ω i. Then, for each integer m>, the solution ξ m of (2.15 (2.16 satisfies the estimate ξ m(t 2,Ω + ξ m 2 L 2 (,T ;H 1 (Ω + 1 ξ m(s ds 2 H 1 (Ω 2 Ce CT( (3.3 f 2 H 1 (,T ;L 2 (Ω + u 2 2,Ω 2 + v 2 2,Ω 1 + p 2 1,Ω 1 for all t [,T]. Proof: Defining ζ m = t ξ m differentiating (2.15 we obtain that for each t [,T], ρ 1 [ t ζ m, η] Ω1 + ρ 2 [ t ζ m, η] Ω2 + a 1 [ζ m, η]+a 2 [ ζ m(s ds, η] (3.4 = ρ 1 [ t f 1, η] Ω1 + ρ 2 [ t f 2, η] Ω2 a 2 [ξ m (, η] η Ψ m. Setting η = ζ m (t in (2.15 repeating the steps for the derivation of (2.31 we have ζ m (t 2,Ω + a 1 [ζ m (t, ζ m (t] dt + a 2 [ ζ m(s ds, ζ m(s ds] (3.5 C ( ζ m ( 2,Ω + t f 2 L 2 (,T ;L 2 (Ω + ξ m( 2 1,Ω2 so that using Gronwall s inequality (2.32 (2.16 we deduce ζ m (t 2,Ω Ce CT( ξ m( 2,Ω 1 + t f 2 L 2 (,T ;L 2 (Ω + v 2 1,Ω 1 + u 1 2 1,Ω 2. (3.6 The term ξ m( 2,Ω 1 can be estimated as follows. Evaluating (2.15 at t = then setting η = ξ m( using (2.16 the divergence-free property of Ψ m,we deduce [[ξ m(, ξ m(]] = [[f(, ξ m(]] a 2 [u, ξ m(] a 1 [v, ξ m(] b[ξ m(,p ] =[[f(, ξ m(]] + [ u + (div u, ξ m(] Ω2 +[ v p, ξ m(] Ω1 ( + µ 2 ( u + u T n 2 (λ 2 + µ 2 (div u n 2 Γ +p n 1 ( v + v T n 1 ξ m( dγ

15 FLUID-STRUCTURE INTERACTION 647 so that upon substituting (3.2 into the last estimate we have ( [[ξ m(, ξ m(]] C f( 2 L 2 (Ω + v 2 2,Ω+ p 2 1,Ω 1 + u 2 2,Ω [[ξ m(, ξ m(]]. This last relation the estimate f( 2 L 2 (Ω ( f C 2 L 2 (,T ;L 2 (Ω + tf 2 L 2 (,T ;L 2 (Ω lead us to ( [[ξ m(, ξ m(]] C v 2 2,Ω + p 2 1,Ω 1 + u 2 2,Ω + f 2 H 1 (,T ;L 2 (Ω. From (3.5, (3.6 the last inequality we obtain (3.3. Remark: The term a 1 [ξ m ( v, η] is added to the Galerkin approximation (2.15 to provide the cancellation of a 1 [ξ m (, η] in the estimation of [[ξ m(, ξ m(]]. This extra term vanishes in the limit as m thus does not affect the continuous weak form. As an immediate consequence of Theorems we obtain the following strong a priori estimates for the solution to (2.6 (2.8. Theorem 3.2. Assume the hypotheses of Theorem 3.1 hold. Then, the solution (v, u to (2.6 (2.8 satisfies the estimate t v(t 2,Ω 1 + tt u(t 2,Ω 2 + t v 2 L 2 (,T ;X 1 + tu(t 2 1,Ω 2 Ce CT( f 2 H 1 (,T ;L 2 (Ω + u 2 2,Ω 2 + v 2 2,Ω 1 + p 2 1,Ω 1. ( The existence of a pressure. With the existence of a strong solution guaranteed by Theorem 3.2, it is now possible for us to establish the existence of an L 2 integrable pressure. To find a pressure that satisfies (2.5, we need to show the existence of a p L 2 (,T; L 2 (Ω 1 such that b[η,p]= ρ 1 [v t, η] Ω1 a 1 [v, η] ρ 2 [u tt, η] Ω2 a 2 [u, η] (3.8 +ρ 1 [f 1, η] Ω1 + ρ 2 [f 2, η] Ω2 η H 1 (Ω, a.e. t. Using the auxiliary formulation (2.1, we see that this is equivalent to showing there exists a p L 2 (,T; L 2 (Ω 1 which satisfies ρ 1 [ξ t, η] Ω1 + ρ 2 [ξ t, η] Ω2 + b[η,p]+a 1 [ξ, η]+a 2 [ ξ(s ds, η] (3.9 = ρ 1 [f 1, η] Ω1 + ρ 2 [f 2, η] Ω2 a 2 [u, η] η H 1 (Ω, a.e. t. By virtue of [17, p.58, Lemma 4.1], the proof of the existence of a pressure p for (3.9 is reduced to showing the following inf-sup condition for the pair of function spaces {H 1 (Ω,Q 1 }, where Q 1 = L 2 (Ω 1 : inf sup q Q 1 η H 1 (Ω b[η,q] η 1,Ω q,ω1 C. (3.1 The proof of this inf-sup condition follows exactly that of [1, Lemma 3.1], though the inf-sup condition established in that Lemma is for a slightly different pair of function spaces. Theorem 3.3. The inf-sup condition (3.1 holds.

16 648 Q. DU, M.D. GUNZBURGER, L.S. HOU, J. LEE Proof: Obviously, it suffices to show that q Q 1, an η Ψ such that div η Ω1 = q η 1,Ω C q,ω1. (3.11 Let q Q 1 be given. We define q L 2 (Ω = {s L 2 (Ω Ω sdω=} by q Ω 1 = q q Ω2 = (1/ Ω 2 Ω 1 qdx. By virtue of [17, p.24, Corollary 2.4], there exists a unique η H 1 (Ω such that div η = q, η 1,Ω C q,ω C q,ω1. This proves (3.11, which in turn implies (3.1. From [17, p.58, Lemma 4.1] Theorems 2.5, we readily deduce the following result. Theorem 3.4. Assume the hypotheses of Theorem 3.1 hold. Then, there exists a unique triplet (v,p,u which possesses the regularity v L (,T; L 2 (Ω 1 L 2 (,T; X 1, u L (,T; X 2, p L 2 (,T; L 2 (Ω 1 v t L (,T; L 2 (Ω 1 L 2 (,T; X 1, u t L (,T; X 2, u tt L (,T; L 2 (Ω 2 satisfies equations (2.5, (2.7, (2.8 b[v,q]= q L 2 (Ω 1, a.e. t. Moreover, the estimates (2.53, (3.7 hold. p L 2 (,T ;L 2 (Ω 1 Ce CT( f 2 H 1 (,T ;L 2 (Ω + u 2 2,Ω 2 + v 2 2,Ω 1 + p 2 1,Ω 1 For a strong solution in the sense of Theorem 3.4, the velocity interface condition holds in the strong sense v Γ = u t Γ in L 2 (,T; H 1/2 (Γ v Γ = u t Γ in H 1/2 (Γ, a.e. t. Also,itcanbeinferredthat v C([,T]; X 1, u C([,T]; X 2 t u C([,T]; L 2 (Ω 2 so that the initial conditions (2.7 hold in the respective strong senses. Remark. The inf-sup condition (3.1 is equivalent to the following inf-sup condition for the space pair {X 1,Q 1 }: b[v,q] inf sup C. (3.12 q Q 1 v X 1 v 1,Ω1 q,ω1 To see this, let the extension operator E : X 1 H 1 (Ω be defined as follows: for every z X 1,(Ez Ω1 = z (Ez Ω2 = z where z X 2 is the solution of [ z, w] Ω = w H 1 (Ω 2, z Γ2 =, z Γ = z Γ. (3.13 (For each given z X 1, the definition of X 1 ensures that there exists an η H 1 (Ω such that η Γ = z Γ so that (3.13 possesses a unique solution z X 2. Stard elliptic estimates (e.g., [17, p.12] yields Ez 1,Ω C (Ez Ω1 1,Ω1 + (Ez Ω2 1,Ω2 C( z 1,Ω1 + z 1/2,Γ C z 1,Ω1

17 FLUID-STRUCTURE INTERACTION 649 for all z X 1. Then, for every q Q 1 we have sup η H 1 (Ω b[η,q] sup q,ω1 η 1,Ω z X 1 b[ez,q] C sup z X 1 b[ez,q] q,ω1 Ez 1,Ω b[z,q] = C sup q,ω1 z 1,Ω1 z X 1 q,ω1 z 1,Ω1 b[η,q] b[η,q] b[z,q] sup sup sup ; η H 1 (Ω q,ω1 η 1,Ω η H 1 (Ω q,ω1 η 1,Ω1 z X 1 q,ω1 z 1,Ω1 in other words, (3.1 (3.12 are indeed equivalent. The inf-sup condition (3.12 is useful for recovering the pressure field from the fluid equations. As a side remark, we point out that this inf-sup condition is also useful for analyzing the Stokes problems with mixed velocity/stress boundary conditions. 4. Concluding remarks. We have demonstrated the existence uniqueness of weak (fluid velocity solid displacement solutions of a fluid-structure interaction problem involving a linear, viscous, incompressible fluid an elastic solid. We have considered the particular case of a solid that undergoes only infinitesimal elastic displacements but whose velocity is large enough so that the fluid structure remain fully coupled. Under additional smoothness assumptions on the data, we have also demonstrated the existence of an L 2 integrable fluid pressure. In forthcoming work, we will consider finite element approximations of our model fluid-structure interaction problem, in future work, we will consider the extension of our results to nonlinear fluid models such as the Navier-Stokes equations to optimization control problems involving the interactions between fluids structures. REFERENCES [1] A. Bermúdez, R. Durán R. Rodríguez (1998, Finite element analysis of compressible incompressible fluid-solid systems. Math. Comp., 67, pp [2] F. Blom (1998, A monolithical fluid-structure interaction algorithm applied to the piston problem. Comput. Methods Appl. Mech. Engrg., 167, pp [3] J. Boujot (1987, Mathematical formulation of fluid-structure interaction problems. RAIRO Modél. Math. Anal. Numér., 21, pp [4] C. Conca M. Durán (1995, A numerical study of a spectral problem in solid-fluid type structures. Numer. Methods Partial Differential Equations, 11, pp [5] C.Conca,J.Martín J. Tucsnak (1999, Motion of a rigid body in a viscous fluid. C. R. Acad. Sci. Paris Sér. I Math. 328, pp [6] S. Dasser (1995, A Penalization method for the homogenization of a mixed fluid-structure problem. C. R. Acad. Sci. Paris Sér. I Math., 32, pp [7] B. Desjardins M. Esteban (2, On weak solutions for fluid-rigid structure interaction: compressible incompressible models. Comm. Partial Differential Equations, 25, pp [8] D. Errate, M. Esteban Y. Maday (1994, Couplage fluid-structure. Un modéle simplifié en dimension 1, C. R. Acad. Sci. Paris Sér. I Math., 318, pp [9] L. Evans (1998, Partial Differential Equations, American Mathematical Society, Providence, Rhode Isl. [1] C. Farhat, M. Lesoinne P. LeTallec (1998, Load motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: momentum energy conservation, optimal discretization application to aeroelasticity. Comput. Methods Appl. Mech. Engrg. 157, pp [11] F. Flori P. Orenga (2, Fluid-structure interaction: analysis of a 3-D compressible model. Ann. Inst. H. Poincaré Anal. Non Linéaire, 17, pp

18 65 Q. DU, M.D. GUNZBURGER, L.S. HOU, J. LEE [12] F. Flori P. Orenga (1998, On a fluid-structure interaction problem. Trends in Applications of Mathematics to Mechanics (Nice, pp [13] F. Flori P. Orenga (1999, Analysis of a nonlinear fluid-structure interaction problem in velocity-displacement formulation. Nonlinear Anal., 35, pp [14] F. Flori P. Orenga (1999, On a nonlinear fluid-structure interaction problem defined on a domain depending on time. Nonlinear Anal. 38, pp [15] L. Gastaldi (1996, Mixed finite element methods in fluid structure system. Numer. Math., 74, pp [16] J.-F. Gerbeau B. Perthame (21, Derivation of viscous Saint-Venant system for laminar shallow water: numerical validation. Discrete Continuous Dynamical Systems - Series B, 1, pp [17] V. Girault P. Raviart (1986, Finite Element Methods for Navier-Stokes Equations, Springer, Berlin. [18] C. Grmont (1998, Existence uniqueness for a two-dimensional steady-state fluidstructure interaction problem. C. R. Acad. Sci. Paris Sér. I Math., 326, pp [19] C. Grmont Y. Maday (1998, Existence of solutions for a two-dimensional unsteady fluid-structure interaction problem. C. R. Acad. Sci. Paris Sér. I Math., 326, pp [2] C. Grmont Y. Maday (2, Existence for an unsteady fluid-structure interaction problem. M 2 ANMath.Model.Numer.Anal., 34, pp [21] M. Gunzburger, H.-C. Lee G. Seregin (2, Global existence of weak solutions for incompressible viscous flow around a moving rigid body in three dimensions; J. Math. Fluid Mech., 2, pp [22] G. Hsiao, R. Kleinman G. Roach (2, Weak solutions of fluid-solid interaction problems. Math. Nachr., 218, pp [23] P. LeTallec S. Mani (2, Numerical analysis of a linearized fluid-structure interaction problem. Numer. Math., 87, pp [24] C. Liu J. Shen (21, On liquid crystal flows with free-slip boundary conditions. Discrete Continuous Dynamical Systems, 7, pp [25] S. Micu E. Zuazua (1998, Asymptotics for the spectrum of a fluid/structure hybrid system arising in the control of noise. SIAM J. Math. Anal., 29, pp [26] P. Panagiotopoulos (1985, Inequality Problems in Mechanics Applications, Birkháuser, Boston. [27] R. Rodríguez J. Solomin (1996, The order of convergence of eigenfrequences in finite element approximations of fluid-structure interaction problems. Math. Comp., 65, pp [28] M. Rumpf (1998, On equilibria in the interaction of fluids elastic solids. Theory of the Navier-Stokes equations, Ser. Adv. Math. Appl. Sci., 47, pp [29] R. Schulkes (1992, Interactions of an elastic solid with a viscous fluid: eigenmode analysis. J. Comput. Phys., 1, pp [3] R. Temam (21, Navier-Stokes Equations, AMS Chelsea Publishing, Providence, Rhode Isl. Received March 22; revised July 22; final version December 22. address: qdu@math.psu.edu address: gunzburg@csit.fsu.edu address: hou@math.iastate.edu address: jeehyunl@rew.cmu.edu