Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction

A coupled parabolic-hyperbolic system of partial differential equations modeling the interaction of a structure submerged in a fluid is studied. The system being considered incorporates delays in the interaction on the interface between the fluid and the solid. We study the stability properties of the interaction model under suitable assumptions between the competing strengths of the delays and the feedback controls.


1.
Introduction. This paper is devoted to the study of stability properties of a coupled linear parabolic-hyperbolic PDE system with delay in the interaction. It addresses asymptotic, rational and exponential stability. The system being considered is a simplified model describing the interaction of an elastic body that is completely submerged in a fluid. Delay between the interaction on the interface is being considered.
First, let us set-up the notation and the geometrical configuration of the problem. Let Ω s be a bounded smooth domain in R d occupied by the structure. The relevant physical scenarios are d = 2 or d = 3, however, we shall consider the general case where d ≥ 2 in the analysis. Denote the boundary of Ω s by Γ s . Let Ω f ⊂ R d be the region occupied by the heat component. We also assume that that Ω f is sufficiently smooth, and that its boundary consists of two parts ∂Ω f = Γ s ∪ Γ f where Γ f = Γ f and Γ s = Γ s have no common points. This means that the interface Γ s between the solid and the fluid does not touch the part Γ f of the boundary of Ω f . Let u(t, x) = (u 1 (t, x), . . . , u d (t, x)) and w(t, x) = (w 1 (t, x), . . . , w d (t, x)) be the velocity and displacement of the heat and structure at time t and position x, respectively. Then a linear model describing the dynamics of the above system is given by the coupled heat-wave system in Ω s .
We shall make the convention that the unit normal vector ν is outward to Ω s on Γ s and is outward to Ω f on Γ f . In particular, ν on Γ s will be inward with respect to fluid domain Ω f . The terms F and G represent feedback controls that will be specified below.
In the case of small but rapid oscillations, the assumption that the interface Γ s is stationary is reasonable. The system (1) is based on the one studied in [5], in the cases where there is no delay (τ f = τ s = 0) and no feedbacks (F = G = 0). The boundary conditions in (1) on Γ s are obtained by matching the velocities and stresses of the fluid and solid components. However, in the current problem, these are not equal. The velocity of the structure serves as a source term at the boundary for the heat equation, and its effect is not instantaneous but a delay takes place. In this case, the constant τ s > 0 represents the extent of the delay while the constant α > 0 signifies the strength of the delayed-velocity term. On the other hand, the normal stress of the fluid on the interface enters as a source term for the wave equation and delay is also being considered, with τ f > 0 measuring the extent of delay while β > 0 denotes the strength of the delayed-stress term. Such delay may occur if for instance there is a small boundary layer on the interface that impedes the instantaneous interaction between the fluid and the solid.
The fluid-structure model (1) without delay and several variants both in the linear and nonlinear settings have been studied extensively for the past years. This includes the case where instead of the heat equation for the fluid component, the Navier-Stokes equation takes place, and the Lamé equation is used instead of the wave equation for the solid component. We refer the reader to [2,3,4,5,6,7,9,10,15,19,20,29,30] and the references therein for the analysis of such variations. These investigations include the well-posedness, regularity and stability of the interaction models. With regards to a fluid-structure model with delay only on the interior feedback for the structure, see [34].
Without the feedback controls and in the case where α = β and F = G = 0 one can easily check that the system is dissipative, that is, the energy is decreasing. The source of dissipation is from the diffusion of the fluid. In this work, we consider the following feedback controls where γ and δ are positive constants. To achieve a well-posed system, the initial histories for the stress and velocity of the fluid and the solid on the interface, respectively, shall be specified In Section 2, the well-posedness of (1)-(3) based on semigroup methods will be discussed.
We would like to point out that Neumann-type boundary feedbacks in terms of the solid component have been considered in [20], whereas in the present paper we consider a Neumann-type boundary feedback in terms of the fluid component. Technically this will serve as a regularization in terms of the Neumann trace for the fluid component. In effect, this has an advantage when one needs to define the appropriate function spaces for the delay variables. These particular feedbacks turns the system (1) into a coupled parabolic-hyperbolic system with Robin-Neumann boundary conditions on the interface.
It is well-known that delays have a destabilizing effect [12,13]. This is due to the fact that delay induces a transport phenomenon in the system that creates oscillations which can lead to instability. The goal is then to determine the relationship among the parameters α, β, γ and δ for which the energy associated with the above model decays to zero asymptotically, or better, to have uniform exponential decay rates. Under the assumption αβ ≤ γδ, we show in Section 3 that the system is asymptotically stable using spectral methods and a generalized Lax-Milgram Lemma. If αβ < γδ then we prove in Section 4 that the energy decays exponentially to zero through the energy method, following the approach given in [6,21,22,23,24].
Our result will be valid for positive delay parameters. A direct application of the Cauchy-Schwarz inequality will give us a sufficient condition for stability, however this is not optimal. The results will be based on the positivity of a quadratic form induced by the boundary terms and the corresponding result obtained by merely applying the Cauchy-Schwarz inequality is just a specific case. The said quadratic form is a special case of the one considered in [37] for coupled wave equations in the entire space with delay. It is worth noting that the condition αβ ≤ γδ for the stability of our problem coincides with the one given in [28] in the case of a coupled ordinary differential equations with delay.
Under the critical case, the dissipation terms induced by the feedbacks are cancelled, however the system still posseses dissipation due to the diffusion of the fluid. Without delays and feedbacks and with α = β = 1, the rational stability of the system (1) has been proved in [5] using a resolvent-based approach. This particular method relies on establishing a polynomial or algebraic growth of the resolvents on the imaginary axis. The corresponding decay estimate follows from a theorem in [11]. The same strategy has been also applied to a fluid-structure interaction model in [2]. Other relevant references for the rational decay rates of coupled heat-wave systems without delay are provided in [5]. We would like to point out that the decay rate O(t − 1 3 ) obtained in the present case when delay is incorporated is weaker than the one obtained based on the original model without delay. This is due to the additional terms in the interface, which are only square integrable. In the absence of delay, a decay rate O(t − 1 2 ) was established in [5] and recently improved to O(t −1 ) using a microlocal analysis argument in [8]. Nevertheless, we will show that the system is asymptotically stable under the critical case and with an additional geometric condition, the decay rate O(t − 1 3 ) will be established in Section 5. Further related problems will be mentioned in Section 6.
2. The evolution system. In this section, we recast (1)-(3) as a first order evolution system on a certain Hilbert space and prove its well-posedness through semigroup theory. The regularity of solutions for smooth and compatible data will be provided using elliptic regularity. Before we proceed with the formulation, we first recall in the following subsection the traces for a graph space. We follow the usual notations H k (Ω) and L p (Ω) for the Sobolev and Lebesgue spaces, respectively. For simplicity, the product of m copies of a Banach space X will be denoted by the same notation X instead of X m .
2.1. Traces for a graph space. It is known that if a certain function satisfies an elliptic problem, then a generalize boundary trace for that function can be defined, see [26,Chapter 2]. In the following discussion, we take the formulation in [38, pp. 432-433] . Endowed with the inner product associated with this norm, W(Ω f ) is a Hilbert space. For a given φ ∈ H 1 2 (Γ s ), we extend it by zero to ∂Ω f and obtain an element in H 1 2 (∂Ω f ) and denote this extension by the same notation. Recall that the trace map γ 0 : .
Consequently, ∂u ∂ν | Γs ∈ H − 1 2 (Γ s ) and the operator u → ∂u ∂ν | Γs is a bounded linear operator from the graph space W(Ω f ) into H − 1 2 (Γ s ). If u ∈ H 2 (Ω f ) then this definition coincides with the usual first order trace of u on Γ s . Indeed, this can be seen immediately from Green's identity. According to the definition it follows easily that for every u ∈ W(Ω f ) and ϕ ∈ H 1 (Ω f ) with ϕ = 0 on Γ f . In the succeeding sections, the expression Γ s will be removed in the notation for the trace for convenience.

Semigroup formulation.
Keeping track of the memory terms, we introduce the following auxiliary state variables for (t, θ, x) ∈ (0, ∞) × (0, 1) × Γ s corresponding to the delay terms in the stress of the fluid and velocity of the structure on the interface. Notice that y satisfies the following uncoupled transport system on the bounded interval (0, 1) with parameter whereỹ 0 (θ, x) = y 0 (−θτ f , x). On the other hand, z satisfies a similar transport system with parameter wherez 0 (θ, x) = z 0 (−θτ s , x). We recast (1)-(3) as a first order system in the state variables (u, w, v, y, z), where v = w t . Here, the wave equation is formulated as a first order system in terms of the displacement and velocity. Consider the following Hilbert space as our state space . Due to different factors in the delay and feedback terms, this space will be equipped with a weighted inner product. For where a = (a 1 , a 2 , a 3 , a 4 ) is a quadruple consisting of positive constants that will be specified below. Here, the dot represents either the inner product in C d or C d×d where it is applicable. The norm associated with this inner product will be denoted by · X,a . Let , see for instance [17, p. 286]. From the above discussion, we already know that the traces ∂u ∂ν and ∂w ∂ν both exist as elements in H − 1 2 (Γ s ). According to the definition of D(A), we have ∂u ∂ν = 1 γ (u − αz(1)) ∈ L 2 (Γ s ) since γ > 0. This means that the condition y(0) = ∂u ∂ν is meaningful under the assumption that y ∈ H 1 (0, 1; L 2 (Γ s )). Likewise, we also have ∂w ∂ν = βy(1) − δv ∈ L 2 (Γ s ).
Theorem 2.1. Let α, β, γ, δ > 0. Suppose that there exists a = (a 1 , a 2 , a 3 , a 4 ) ∈ R 4 + such that the quadratic form Q a defined by is nonnegative definite. Then the operator A generates a strongly continuous semigroup of contractions on X. In particular, for every Y 0 ∈ X the Cauchy problem If γδ > αβ then for every T > 0 it holds that A sufficient condition for the nonnegativity of the quadratic form Q a is αβ ≤ γδ. The proof of this remark is provided in the Appendix.
Proof of Theorem 2.1. We shall apply the Lumer-Phillips theorem in reflexive Banach spaces, see [16,Corollary III.3.20]. The first step is to check the dissipativity of A. For this, take an arbitrary element Y = (u, w, v, y, z) ∈ D(A). Employing Green's identity with respect to the fluid domain and using the boundary conditions u = 0 on Γ f and u = γ ∂u ∂ν + αz(1) on Γ s we have Re The negative sign on the boundary terms is due to the convention that ν on Γ s is inward to the fluid domain Ω f . Likewise, using Green's identity with respect to the structural domain and the boundary condition ∂w ∂ν = βy(1) − δv give us For the delay variable y we integrate by parts, take the real part and apply the condition y(0) = ∂u ∂ν to obtain In a similar way, for the delay variable z we have using z(0) = v Multiplying (10)-(12) by constants a 1 , a 2 , a 3 , a 4 , respectively, and then using the Cauchy-Schwarz inequality to the last terms on the right hand sides of (10) and (11), it is not difficult to see that we have where Q a is the quadratic form defined by (7). Consequently, it follows from (14) and the nonnegativity of Q a that A is dissipative. The next step is to prove that 0 lies in the resolvent set ρ(A) of A. This implies that A has a nonempty resolvent, and thus A is closed according to [38,Remark 2.2.4]. The fact that the resolvent set is open implies that λ ∈ ρ(A) for some λ > 0, which proves the range condition R(λI − A) = X in the Lumer-Phillips Theorem.
u satisfies an elliptic problem on Ω f with mixed Dirichlet-Robin boundary condi- and w satisfies an elliptic problem on Ω s with Neumann boundary condition on Γ s The boundary conditions on Γ s for (17) and (18) can be written, according to (16) and (15), respectively as follows By elliptic theory, the boundary value problem (17) admits a solution u ∈ H 1 Γ f (Ω f ) and since u * ∈ L 2 (Ω f ) we deduce that u ∈ W(Ω f ). Moreover, the boundary condition (19) implies that ∂u ∂ν ∈ L 2 (Γ s ) and thus y ∈ H 1 (0, 1; L 2 (Γ s )) where y is defined by (15). We can see immediately from (16) that z ∈ H 1 (0, 1; L 2 (Γ s )). On the other hand, the boundary value problem (18) has a unique solution w ∈ H 1 (Ω s ) and since v * + w ∈ L 2 (Ω s ) we have w ∈ W(Ω s ). Therefore there exists Y ∈ D(A) such that AY = Y * .
We show that there is a constant C > 0 independent of Y and Y * such that and this will prove that 0 ∈ ρ(A). Using a standard elliptic estimate, the trace theorem, Hölder's inequality and Fubini's theorem to the term involving z * , we obtain from (17) and (19) that By (19), the trace theorem once more and (22) we get Using (23) we can estimate the norm of y as follows Finally, from the equation v = w * , a basic elliptic estimate applied to the boundary value problem (18) and (20) we have the following estimates Combining (22)- (27) proves (21). This completes the proof that A is the generator of a strongly continuous semigroup of contractions on X.
for all t > 0 and the component u of the corresponding semigroup solution Y (t) = e tA Y 0 satisfies u = 0 on (0, ∞) × Γ f and according to (14) we have for every T > 0. This estimate implies that u ∈ L 2 (0, ∞; H 1 Γ f (Ω f )), hence (8). The corresponding result for data in X follows immediately from the density of D(A) in X. Finally, (9) is a consequence of the assumption that αβ < γδ, which implies that Q a is positive definite (see Theorem 7.1 below) and the fact that they satisfy the onedimensional transport equations (4) and (5) with parameter x ∈ Γ s , respectively. Such regularity can be proved by following the arguments as in [36].
Next we prove the regularity of solutions with initial data in D(A 2 ). The following theorem will be utilized in establishing the stability of (1)-(3) in Section 4.
and (u, w, v, y, z) = e tA Y 0 be the semigroup solution of (6). For every T > τ s it holds that and for every T > τ f + τ s we have , H 1 (Ω s )) follows from the equation w t = v and w(0) ∈ H 1 (Ω s ). Therefore, by trace theory we have for every T > τ s , and hence it follows that u ∈ L 2 (τ s , T ; H 2 (Ω f )) by elliptic regularity. On the other hand, we have ∆w = v t + w ∈ C([0, T ], H 1 (Ω s )) and for every T > τ f + τ s , and therefore w ∈ L 2 (τ f + τ s , T ; H 2 (Ω s )) by applying elliptic regularity once more.
3. Spectral properties and asymptotic stability. In the absence of delays and feedback controls in the coupled system (1), the corresponding semigroup generator lacks the compactness of its resolvents. However, the projections of the resolvents onto the product of the state spaces corresponding to the fluid and structure velocities are compact, see [3] for the details. The goal of this section is to show that the spectrum of the generator A corresponding to the problem (1)- (3) consists of eigenvalues only. Our strategy is to rewrite the resolvent equation for a given fixed element of X in variational form and then apply a generalized Lax-Milgram argument.
The method we employ has been utilized for wave equations with viscoelastic surface [14] and for a fluid-structure interaction model with delay in the interior feedback control for the structural component [34]. The corresponding results in these works do not consider the whole spectrum and the analysis tackles only the part of spectrum neglecting the negative real axis. This is due to the fact that the boundary condition on the interface for the equation corresponding to the fluid is of Dirichlet type. This essential boundary condition entails to incorporate the compatibility condition in the definition of the space for the variational formulation. In this direction, one needs to formulate the variational form in terms of the velocities of the fluid and the structure. For our problem, due to the Neumann-type feedback on Γ s we basically have boundary value problems that have either Neumann or Robin boundary conditions on the interface. As a result of these natural boundary conditions, we can formulate the variational equation in terms of the velocity of the fluid and the displacement of the structure, and from this we can cover the whole spectrum of the generator in the analysis. The first step is to rewrite the resolvent equation in variational form.
if and only if there exists a pair (u, for every (ϕ, and F λ, Proof. First, let us suppose that the equation (28) holds. Notice that (28) is equivalent to a system where u is the solution of the boundary value problem v and w satisfies the system while the delay variables y and z satisfy respectively the following ordinary differential equations with parameter Applying the variation of parameters formula to (34) and (35) and using the equation v = λw − w * we obtain From (37), the boundary condition for u on Γ s in (32) becomes Consequently, from (36) and (38) the boundary condition for w can be written as Taking the inner product in L 2 (Ω f ) of the first equation in (32) with ϕ and then applying Green's identity we have Using (38), the boundary integral in (40) can be written as With regards to the elliptic equation for w in (33), we take the inner product in L 2 (Ω s ) with ψ ∈ H 1 (Ω s ), use v = λw − w * and apply Green's identity to obtain Ωs v * · ψ dx = (λ 2 + 1) From (39) the boundary term in the above equation can be expressed as Taking the sum of (40) and (42), substituting the boundary terms according to (41) and (43), and then rearranging the terms yield the variational equation (29). Conversely, suppose that there exists (u, . We define v = λw − w * and z by (37). By definition we have v ∈ H 1 (Ω s ), z ∈ H 1 (0, 1; L 2 (Γ s )) and z satisfies the differential equation (34). By taking ψ = 0 in (29) and using the definition of z, we can see that u is the weak solution of the boundary value problem (32). Thus, in particular we have ∂u ∂ν ∈ L 2 (Γ s ). Now define y according to (36), which satisfies (34) and y ∈ H 1 (0, 1; L 2 (Γ s )). Taking ϕ = 0 in (29) and then using (38) and the definition of z, we can see that w satisfies the boundary value problem in (33). These observations imply that Y = (u, w, v, y, z) ∈ D(A) and it satisfies equation (28).
To characterize the spectrum of A, we need the following combination of the Lax-Milgram Lemma and the Fredholm alternative. For the proof we refer the reader to [14] or [35].
has either a unique solution u ∈ H 1 for all F ∈ H 1 or has a nontrivial solution for F = 0.
In the following theorem, we will use the weighted trace inequality: for every ε > 0 there exists a constant C ε > 0 such that where Ω is a sufficiently smooth bounded domain in R d . This inequality follows from the moment trace inequality for some constant C > 0 and Young's inequality |ab| ≤ εa 2 + C ε b 2 for some C ε > 0 and for every ε > 0 and a, b ∈ R. We denote by σ p (A) the point spectrum of a closed linear operator A. Proof. Let λ ∈ C and denote byã λ the sesquilinear form corresponding to the boundary terms in a λ , that is,ã λ : Taking ϕ = u and ψ = w and then applying the Cauchy-Schwarz inequality we obtain where the positive constants k 1 (λ) and k 2 (λ) are given by Choose the constant ε(λ) > 0 small enough so that 1 − k i (λ)ε(λ) > 0 for i = 1, 2. Let us split the sesquilinear form a λ into a λ = a λ,1 + a λ,2 where a λ,1 : Here C ε(λ) > 0 is the constant in (44) corresponding to ε(λ) > 0. One can immediately see that a λ,1 and a λ,2 are bounded. Now, we show that a λ,1 is coercive. Utilizing (44), (46) and recalling the choice of ε(λ), we can see that and Lemma 3.2 we either have λ in the resolvent set of A or in its point spectrum. This is equivalent to the conclusion of the lemma. Now we prove that A does not have purely imaginary eigenvalues. Proof. We already know that 0 ∈ ρ(A). Let r be a nonzero real number and Y := (u, w, v, y, z) ∈ D(A) be such that AY = irY . This is equivalent to the system (32)- (35) with Y * = 0 and λ = ir. From the dissipativity inequality (14) and the fact that Q a is nonnegative definite we have Thus u is constant and from the boundary condition on Γ f it follows that u must be zero in Ω f . From (36) we obtain y = 0 in (0, 1) × Γ s . The equation for w in (33) since v = irw. Multiplying (47) by w, integrating over Ω s and then using Green's identity we have From the imaginary part in the above equation, we can see that w = 0 on Γ s and consequently ∂w ∂ν = 0 according to the boundary condition in (47). By elliptic regularity we have w ∈ H 2 (Ω s ) ∩ H 1 0 (Ω s ) and therefore w = 0 in Ω s from the unique continuation theorem for elliptic operators, see [ Thus v = irw = 0 in Ω s and from (37) we have z = 0 in (0, 1) × Γ s . Therefore Y = 0 so that ir is not an eigenvalue of A and from Lemma 3.3 it follows that ir lies in the resolvent set of A. Thus, the imaginary axis lies in the resolvent set of A.
Theorem 3.5. Under the condition γδ ≥ αβ, we have e tA Y 0 X,a → 0 in X as t → ∞ for every Y 0 ∈ X.
In the succeeding sections we improve this theorem by providing explicit decay rates under additional conditions on the parameters α, β, γ and δ.

4.
Uniform exponential stability for the case αβ < γδ. The goal of this section is to prove the exponential decay of the energy for the solutions of the system (1)-(3) for the case αβ < γδ. From inequality (14), it can be seen that the dissipation is due to the diffusion of the fluid, the normal stress of the fluid component and velocity of the solid on the interface. The latter boundary dissipation for the wave equation is enough to obtain exponential decay, and this can be achieved through multipliers. We follow the methodology presented in [6] for the current problem.
First, we recall the following energy identities for the wave operator = ∂ tt − ∆. These are obtained by using the multipliers ϕ div η and η · ∇ϕ, respectively. We refer to [21,22,23,24] for their proofs in the scalar version and to [6] in the vector version.
Theorem 4.1. Let f ∈ L 2 (0, T ; L 2 (Ω)) and w ∈ L 2 (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) be a solution of the wave equation w = f . If w t , ∂w ∂ν ∈ L 2 (0, T ; L 2 (∂Ω)), then for every σ ∈ (0, T ) we have ∂w ∂τ ∈ L 2 (σ, T − σ; L 2 (∂Ω)). Furthermore, for every ε ∈ (0, 1 2 ] there exists a constant C σ,ε,T > 0 such that The estimate provided in [22] involves the L 2 (0, T ; H 1 2 +ε (Ω))-norm instead of the H For t ≥ 0, we define the energy of the solution to (1)-(3) associated with data in D(A) by and the corresponding dissipation term by Proof. The given assumption implies that the quadratic form Q a is positive definite. Hence, it follows from (14) and Theorem 7.1 that for some C > 0, where Y (t) = e tA X 0 and X 0 ∈ D(A). On the other hand, using the Cauchy-Schwarz inequality and similar calculations as in the proof of (14), we have the estimate from below From this inequality and (118), it can be seen that there is a constant c > 0 such that E (t) ≥ −cD(t). This completes the proof of the lemma.
We are now in position to prove the exponential stability of the semigroup e tA .
Theorem 4.3. If γδ > αβ then the semigroup generated by A is uniformly exponentially stable, that is, there exist M ≥ 1 and σ > 0 such that e tA Y 0 X,a ≤ M e −σt Y 0 X,a for every t ≥ 0 and Y 0 ∈ X.
Proof. By the density of D(A 2 ) in X and strong continuity of the semigroup e tA , we may suppose that the initial data Y 0 = (u 0 , w 0 , v 0 , y 0 , z 0 ) lies in D(A 2 ). Let Y = (u, w, v, y, z) be the associated solution. The regularity of the corresponding components are provided in Theorem 2.2. From Lemma 4.2, for every T ≥ s ≥ 0 it follows that The goal is to prove that there exist T > 0 and a constant T ∈ (0, 1) such that E(T ) ≤ T E(0). Then according to standard results for semigroup theory, see for instance [33], we have exponential stability. By the linearity of the problem and the fact that the coefficients are real, we may suppose without loss of generality that the states are real-valued. The corresponding result can be obtained by separating the real and imaginary parts. We divide the arguments in several steps.
A similar procedure applied to the delay variable y provides us the following estimate for some constant C τ f ,a > 0 T s 1 0 Γs Step 2. Energy estimate for u. By the Poincaré inequality we have Next, we shall estimate the terms in the energy corresponding to the wave component. Let T > 2(τ f + τ s ) and τ f + τ s < σ < T − (τ f + τ s ). In the following, we establish that there exist a constant C η > 0 independent of T and a constant C ρ,σ,r,η,T > 0 such that for every r ∈ ( 1 2 , 1] we have Step 3. Energy estimates for w. We use the energy identities provided in Proposition 1, which is possible since w satisfies the necessary regularity condition according to Theorem 2.2. We choose a vector field η having a uniformly positive definite Jacobian, that is, for some some ρ > 0 we have Let us estimate each term on the right hand side of (49) where T and s are replaced by T − σ and σ, respectively. Since w = −w we have Applying the Cauchy-Schwarz inequality to the boundary terms in (49) one obtains The last integral term in (49) are estimated as follows The remaining term in (49) is estimated by using (48) and Young's inequality to obtain Using the estimates (56)-(59) in (49) with ϕ = w we have, after rearranging terms and using (55), It remains to estimate the velocity term w t , and this is the place where we use the energy equation (48). Let us choose the vector field η so that div η = 1, e.g. the radial vector field η(x) = x/d. We estimate the right hand side of (48) by applying the Cauchy-Schwarz inequality and then using estimate (60) with ϕ = w to get Taking the sum of (60) and (61), using the decomposition |∇w| 2 = | ∂w ∂ν | 2 + | ∂w ∂τ | 2 on Γ s , the tangential trace estimate in Theorem 4.1, and the boundary condition ∂w ∂ν (t) = β ∂u ∂ν (t − τ f ) − δw t (t) on Γ s we obtain the desired estimate (55) where r = 1 2 + ε.
Step 4. From (51)-(55) and the fact that the energy is decreasing we have Replacing T and s by T − σ and 0 in (50), respectively, we can estimate E(0) from above in terms of E(T −σ) and the dissipation term D. Using this in (62) we obtain for some constant C η independent of T . If we take σ > τ f + τ s and T > 2σ + C η and then use the fact that E(T ) ≤ E(T − σ) in (63) we obtain the energy estimate where r ∈ ( 1 2 , 1]. Step 5. Absorption of lower order term. In this step, we prove that the lower order term on the right hand side of (64) can be absorbed by the dissipation term. More precisely, we show that there exists C T > 0 such that This is done via a compactness-uniquess argument as in [6]. Suppose on the contrary that for each integer n > 0 there is a data Y 0n = (u 0n , w 0n , v 0n , y 0n , z 0n ) ∈ D(A) with where D n is the dissipation term associated with the solution Y n = (u n , w n , v n , y n , z n ) with initial data Y 0n = (u 0n , w 0n , v 0n , y 0n , z 0n ). By rescaling the initial data, we may suppose without loss of generality that for each n we have w n H r ((0,T )×Ωs) = 1.
According to (64), (66) and (67), Y n (T ) is uniformly bounded in X, and consequently Y n (0) = Y 0n is uniformly bounded in X according to (50) with s = 0. Thus, up to a subsequence, we have Y n0 → Y 0 weakly in X for some Y 0 = (u 0 , w 0 , v 0 , y 0 , z 0 ) ∈ X. By the uniform boundedness of the adjoint semigroup on compact intervals, it follows that where Y (t) = (u(t), w(t), v(t), y(t), z(t)) = e tA Y 0 . In particular, from the last two parts in (70) we can see that w n is uniformly bounded in H 1 ((0, T ) × Ω s ). By compactness, there is a subsequence which we denote by the same indices such that w n → w strongly in H r ((0, T ) × Ω s ) for r ∈ ( 1 2 , 1), and by passing to the limit in (67) the limit satisfies w H r ((0,T )×Ωs) = 1.
Integrating by parts in time and space we infer that Passing to the limit n → ∞ and using (68)-(70), w 0n → w 0 weakly in H 1 (Ω s ) and v 0n → v 0 weakly in L 2 (Ω s ) we have, for every f ∈ L 2 ((0, T ) × Ω s ), In a similar fashion it can be shown that for every f ∈ L 2 ((0, T ) × Ω s ) we have where From (73) and (74) it follows that v is indeed the very weak solution of (72).
For sufficiently large T > 0, the solution of (72) is identically zero according to the Holmgren-uniqueness principle, see [25,Chapter I,Theorem 8.2]. Therefore w is constant with respect to t and it is a weak solution of the over-determined elliptic problem in Ω s , ∂w ∂ν = 0, w = 0, on Γ s .
Applying the unique-continuation condition for elliptic operators we infer that w = 0, contradicting (71). This completes the proof of (65).
Step 6. Let us finish the proof of the theorem. First, we obtain immediately from (64) and (65) that Moreover, according to Lemma 4.2 we have Combining the last two estimates leads us to E(T ) ≤ T E(0) where T = C T C −1 /(1 + C T C −1 ) ∈ (0, 1). This proves the desired inequality, which in turn implies the exponential stability of the semigroup generated by A.

5.
Rational stability under the critical case γδ = αβ. In the critical case γδ = αβ, the dissipation induced by the feedback controls is cancelled due to the delay in the interaction. However, we can see that the energy still dissipates through the diffusion of the fluid component. In this section, we show that this dissipative property leads to a rational decay rate for the interaction model, however, we are only able to prove this under an additional geometric condition. This geometrical condition has been utilized in several works and we refer to [20] in the context of fluid-structure interaction models. Our method relies on a resolvent-based approach, which has been used in fluidstructure models without delay in [2,5]. The method we shall employ here uses the ideas and techniques given in [5], where instead of the Dirichlet map we use the Robin map. The success of these methods is based on the following abstract result by Borichev and Tomilov [11].
Theorem 5.1. Let A be a generator of a bounded semigroup on a Hilbert space X such that iR ⊂ ρ(A) and let α > 0. Then there exist r 0 > 0 and C(r 0 ) > 0 such that (irI − A) −1 X ≤ C r0 |r| σ for every r ∈ R with |r| ≥ r 0 , where σ ∈ (0, ∞), if and only if there exist t 0 > 0 and C t0 > 0 such that To prove rational stability, we need the static version of the energy identity (49) in Proposition 1, compare [5]. For convenience, we state it in the following proposition.

Proposition 2.
Let Ω be a smooth bounded domain and suppose that ψ ∈ H 2 (Ω) is a vector-valued function satisfying the elliptic equation where f ∈ L 2 (Ω) and s ∈ R. If η ∈ [C 2 (Ω)] d is a vector field then we have We are now ready to state and prove the main result of this section.
Proof. We will establish the following inequality for every real number r with |r| ≥ r 0 for some r 0 > 1 and C = C(r 0 ) > 0, where Y = (u, v, w, y, z) ∈ D(A) and (irI − A)Y = Y * = (u * , w * , v * , y * , z * ). The latter equation is equivalent to system (32)- (35) where λ = ir. We note that to prove (77), it is enough to prove that . Applying Young's inequality, we have The last two inequalities, together with the assumption that r 0 > 1, imply (77) after taking square roots. Therefore, in the following we will show (78). This will be done in several steps.
Step 1. An auxiliary variable. Define the Robin map Rg = ϕ as follows We infer from the regularity theory for elliptic operators in [26,Chapter 2] that the linear operator R satisfies R ∈ L(H σ (Γ s ), H σ+ 3 2 (Ω s )) for every σ ∈ R, assuming that Ω s is sufficiently smooth. Let ϕ = R( ∂w ∂ν +w). The function ψ = w −ϕ satisfies the boundary value problem Notice that ψ ∈ H 2 (Ω s ). The right hand side of the first equation in (79) will be denoted by ψ * .
Step 2. We prove that there is a constant C > 0 such that for every |r| ≥ √ 6, Multiplying the first equation in (79) by ψ, integrating over Ω f and then using the generalized Green's identity we obtain (r 2 − 1) The assumption |r| ≥ √ 6 implies that (r 2 − 3)/2 ≥ r 2 /4 and applying this to the above inequality we obtain (80).
Step 3. The next step is to show that for every ε > 0 there exist constants C > 0 and C ε > 0 such that for every |r| ≥ 1. This is the place where we apply the energy identity in Proposition 2 where s = 1 − r 2 , f = ψ * and with the vector-field η in the statement of the theorem.
Step 4. The next step is to estimate the first two terms on the right hand side of (83). We show that there exists C > 0 such that for every |r| ≥ 1 there holds and as a consequence for every ε 1 > 0 there exists C ε1 > 0 such that for every |r| ≥ 1 First we prove (91). Rewrite the boundary conditions for u and w on Γ s given in (38) and (39) as follows Following [5], we defineũ ∈ H 1 (Ω f ) to be the solution of the following elliptic problem ∆ũ = ∆u + u * in Ω f ,ũ = 0 on ∂Ω f , which, by a standard elliptic estimate, satisfies the inequality Since iru = ∆ũ = div(∇ũ) we have |r| From this estimate and (95), together with interpolation, the Poincaré and Cauchy-Schwarz inequalities, we have By the continuity of the first-order trace operator (recall Section 1.1), the equation ∆u = iru − u * , the Poincaré inequality and (96) we obtain for |r| ≥ 1 Using the embedding L 2 (Γ s ) ⊂ H − 1 2 (Γ s ), trace theory and Poincaré inequality to majorize the norm of u in L 2 (Γ s ) by the norm of its gradient in L 2 (Ω f ) in (93) Employing the inequalities (97) and (98) in (94) we get The sum of (98) and (99) gives us (91), and this in turn implies the estimate by the continuity of the Robin map R. To estimate the norm of rψ in L 2 (Γ s ), we rewrite this norm, using the equation ψ = w − ϕ, as follows where the brackets denote the duality pairing between H − 1 2 (Γ s ) and H 1 2 (Γ s ). The first term on the right hand side of (101) can be estimated above by The second term of the said equation will be simply estimated as follows Using (102) and (103) in (101), taking the sum of the resulting estimate with (100) and finally using (98), we obtain (92) by virtue of the assumption that |r| ≥ 1.
Step 5. Now we combine the estimates provided in the previous steps to prove that there exist r 0 > √ 6 and a constant C > 0 such that for |r| ≥ r 0 there holds From (80), (83) and (92), it follows that if |r| ≥ √ 6 then where C is independent of r and ε. We choose the constants r 0 , ε and ε 1 so that the following inequalities are satisfied whenever |r| ≥ r 0 . For example, we take ε > 0 small enough so that 1 − Cε ≥ 1/2, and then choose ε 1 sufficiently small so that the second inequality in (106) is satisfied. The first inequality in (106) is satisfied for every |r| ≥ r 0 if we choose r 0 sufficiently large, for example one may take r 0 = max( √ 6, 2 √ C + C ε ε 1 ). Rearranging the terms in (105) and then using (106) yield (104).
Finally, we note that the above results can be adapted when there is no delay, that is, τ f =τ f = τ s =τ s = 0.
Theorem 7.1. Suppose that α, β, γ, δ > 0 satisfies αβ < γδ. Then the quadratic form Q a is positive definite for some a ∈ R 4 + . If αβ = γδ then Q a is nonnegative definite for some a ∈ R 4 + . Proof. The positive definiteness of the form Q a for some a ∈ R 4 + under the condition αβ < γδ has been already established in [37]. Thus, we only need to prove the second part. Following [37], we rewrite the quadratic form as follows If we choose the positive constants a 3 and a 4 to satisfy a 4 /a 3 = α 2 /γ 2 and then take a 1 = a 4 γ/α 2 , then the coefficient for u 2 in the above equation will become zero. On the other hand, by taking a 2 = a 3 δ/β 2 and using the fact that a 3 /a 4 = γ 2 /α 2 = β 2 /δ 2 , we can see that the coefficient of v 2 also vanishes. Therefore, with these choices for the constants a 1 , a 2 , a 3 , a 4 , the quadratic form is nonnegative definite provided that αβ = γδ.