Uniform exponential stability of a fluid-plate interaction model due to thermal effects

We consider a coupled fluid-thermoelastic plate interaction model. The fluid velocity is modeled by the linearized 3D Navier-Stokes equation while the plate dynamics is described by a thermoelastic Kirchoff system. By eliminating the pressure term, the system is reformulated as an abstract evolution problem and its well-posedness is proved by semigroup methods. The dissipation in the system is due to the diffusion of the fluid and heat components. Uniform stability of the coupled system is established through multipliers and the energy method. The multipliers used for thermoelastic plate models in the literature are modified in accordance to the applicability of a certain Stokes map.


1.
Introduction. Consider an incompressible fluid occupying a bounded domain Ω ⊂ R 3 with sufficiently smooth boundary Γ. Suppose that part of the boundary is enclosed by a solid wall while the remaining part is enclosed by a thin elastic plate. Let Γ 1 and Γ 0 be the regions where the solid wall and the plate are located, respectively. Here, Γ 0 and Γ 1 are nonempty, Γ 1 ∪ Γ 0 = Γ and Γ 1 ∩ Γ 0 = ∅. In this paper, we assume that the boundary Σ 0 of Γ 0 is nonempty and smooth enough. The fluid is modeled by the linearized 3D Navier-Stokes equation (1.1) In (1.1), u is the velocity vector field of the fluid, µ > 0 is the viscosity of the fluid, ϕ is the transversal displacement of the plate and ∆ Ω is the Laplace operator in Ω. The coupling between the fluid and the plate is attained by matching the corresponding velocities on Γ 0 . For the problem (1.1) to be well-posed one must Thus, the average plate velocity should be zero for all times. Suppose that the transversal displacement ϕ of the plate is described by Kirchoff's equation and is subjected to thermal effects given by the heat equation where α, β, η > 0, γ, σ ≥ 0 and ∆ Γ0 is the Laplace-Beltrami operator on Γ 0 considered as a Riemannian manifold with boundary. The constant γ is proportional to the thickness of the plate. The case γ = 0, that is, plate's thickness is negligible, corresponds to the Euler-Bernoulli beam while γ > 0 is the Kirchoff's model. Without fluid interaction, it is well-known that the case γ > 0 is of parabolic-type at least for certain boundary conditions [17,22]. More precisely, the corresponding system generates an analytic semigroup and hence a fortiori the uniform stability of the solutions. On the other hand, if γ = 0 then the system is of mixed hyperbolic-parabolic type. We consider the case where the edge of the plate is clamped 4) and the temperature at the boundary satisfies ηλ 1 ∂θ ∂ν + λ 2 θ = 0 on (0, ∞) × Σ 0 , (1.5) where λ 1 , λ 2 ≥ 0 and λ 1 + λ 2 > 0. The case λ 1 , λ 2 > 0 is Newton's law of cooling and the cases λ 1 = 0 and λ 2 = 0 mean that the temperature and the temperature flux at the boundary are zero, respectively. Finally, we supply the following initial data u(0) = u 0 in Ω, (1.6) ϕ(0) = ϕ 0 , ϕ t (0) = ϕ 1 , θ(0) = θ 0 in Γ 0 . (1.7) All throughout this paper we will assume that γ > 0 and σ + λ 2 > 0, and for the sake of simplicity we set without loss of generality that α = β = η = µ = 1. We will use the same notation for the Laplace operators in Ω and Γ 0 , that is, ∆ Ω and ∆ Γ0 will be both denoted by ∆.
Formally differentiating the energy E and using the differential equations and boundary conditions in (1.1)-(1.5) we obtain where κ = 0 if λ 1 = 0, λ 2 /λ 1 if λ 1 > 0. (1.8) This means that the dissipation of the system is due to the diffusion in the fluid and heat components. Without thermal effects and with hinged boundary conditions ϕ| Σ0 = ∆ϕ| Σ0 = 0 for the plate, the approximate controllability of the associated system was established by Lions and Zuazua [21] using duality and variational techniques. The system is somewhat similar to those presented in [11,18]. In the absence of rotational forces (γ = 0) and thermal effects, the well-posedness and exponential stability of the system has been established in [1,3,10] and in [9] accounting only for longitudinal displacement of the plate. To show the stability property, the authors in [1] used the frequency domain approach which is done by showing the uniform boundedness of the resolvents on the imaginary axis. On the other hand, the authors of [10] make use of an appropriate Lyapunov functional. It has been shown by Avalos and Bucci [2] that the system is stable even for γ > 0, but now the decay rate is rational. Recent related works for fluid-structure interaction models incorporating viscoelasticity can be found in [28] and [30].
In this work, we will exhibit the exponential decay of the energy for the solutions of the system (1.1)-(1.7) using appropriate multipliers in the time-domain space. One hindrance in using the multiplier method is on how to eliminate the terms arising from p − ν · ∂u ∂ν in the plate equation. We can view the latter term as an (unbounded) feedback interconnection of the fluid and plate components. In fact, using the Agmon-Douglis-Nirenberg Theorem and a standard trace estimate one can majorize the L 2 -norm of p − ν · ∂u ∂ν in terms of the H 3 2 -norm of ϕ t , an estimate incompatible to the natural state space associated with ϕ t . Alternatively, we shall eliminate this term using the properties of the Stokes map introduced by Chuesov and Ryzhkova [10], which is an improvement of the classical results for the Stokes equation, see [27] for example. However, a problem arises in applying this map, it only applies to functions in Γ 0 that have zero average. To circumvent this problem, we shall enforce the multipliers to have zero average through mollifiers.
Although both the fluid and plate domains are assumed to be smooth, our results can be eventually extended to certain non-smooth geometries, for example those that satisfy the criteria in [2]. Such configurations include canonical polyhedra and cylinders that are utilized in numerical analysis and simulations, for instance, in finite element methods. This paper will be organized as follows. In Section 2, we present the suitable function spaces for which system (1.1)-(1.7) will be studied. The semigroup wellposedness of the model will be established in Section 3. In Section 4, we introduce the notion of modified multipliers. Finally, using appropriate multipliers along with their modified versions, we prove the uniform exponential stability of the system in Section 5.
2. Spaces and operators for the abstract formulation. In this section we introduce the relevant spaces and operators necessary in the abstract formulation and analysis of the system (1.1)-(1.7). For the fluid component, the state space is given by H = {u ∈ [L 2 (Ω)] 3 : div u = 0 in Ω and u · ν = 0 on Γ 1 } and it is endowed with the L 2 -norm. Recall that an element u ∈ [L 2 (Ω)] 3 with L 2 -distributional divergence admits a generalized trace u · ν ∈ H − 1 2 (Γ). In fact, if is equipped with the graph norm then u → u · ν ∈ L(L 2 div (Ω), H − 1 2 (Γ)). One can also localize the generalized trace u · ν on Γ 0 and Γ 1 . Consider the space 3 : div u = 0 in Ω and u = 0 on Γ 1 } with the norm ∇ · [L 2 (Ω)] 3×3 , which is equivalent to the full norm in [H 1 (Ω)] 3 according to the Poincaré inequality. If C ∞ σ,Γ0 (Ω) is the space of all infinitely differentiable vector-valued functions on Ω that are divergence-free and vanish on a neighborhood of Γ 1 , then H and V are the completions of C ∞ σ,Γ0 (Ω) with respect to the L 2 -norm and H 1 -norm, respectively.
Next, we define the bi-Laplace and Laplace operators on various domains. Define the bi-Laplacian operator A : It is known that A is a positive self-adjoint operator on L 2 (Γ 0 ) and hence the fractional powers A α of A are well-defined on suitable domains for α ∈ R. In particular, D(A

Moreover, A can be extended to a bounded linear operator
A ∈ L(D(A . By the Poincaré inequality and standard elliptic results it can be seen that the norm · D(A D ) := A D · L 2 (Γ0) on D(A D ) is equivalent to the induced norm of D(A D ) as a subspace of H 2 (Γ 0 ). The operator A D is a positive selfadjoint operator on L 2 (Γ 0 ) and Therefore, the pseudo-differential operator A −1 D is regularizing. Given γ > 0, the operator P γ : is also positive self-adjoint operator with square root P . A typical extension procedure shows that P γ admits an extension P γ ∈ L(H 1 0 (Γ 0 ), H −1 (Γ 0 )) which has a bounded inverse. In fact, if we equipped the space H 1 0 (Γ 0 ) with the norm P to the pivot space L 2 (Γ 0 ), then P γ becomes a unitary operator, see [29,Corollary 3.4.6].
where κ is the constant defined in (1.8). Because σ + λ 2 > 0, it follows that the norm induced by this inner product is equivalent to norm of H 1 (Γ 0 ).
It will be needed to rewrite the heat equation in terms of the Dirichlet Laplacian so as to apply the operator A −1 D . For this reason we define the Dirichlet map D The notations γ 0 and γ 1 for the zero and first order traces will be utilized. The Robin map A R can now be expressed in terms of the operators A D , D and γ 0 as follows In stuyding the system (1.1)-(1.7), the pressure term p will be eliminated as in [3,4,7]. This is done by observing that the pressure satisfies an elliptic boundary value problem with mixed Neumann and Robin type-boundary conditions. The suitable mixed Neumann-Robin maps R 0 and R 1 are as follows, compare with [2,3], The classic elliptic regularity results in [20, p. 152] give us for s ∈ R and i = 0, 1. According to the compatibility condition (1.2), one also need to introduce functions on Γ 0 that have average zero. Let viewed as a subspace of L 2 (Γ 0 ). For every s ≥ 0 we define In dealing with the energy estimates, we will frequently use the following result in [10,Proposition 2.2]. This has been shown under the hypothesis that Γ 0 is flat. However, the proof can be adapted to a curved Γ 0 . In fact, the flatness of Γ 0 was not explicitly used in the proof.
Theorem 2.1. Let S be the Stokes map defined in the following way We can think of the Stokes map S as a lifting operator for functions in Γ 0 to functions in Ω. This information will allow us to eliminate the terms involving p − ν · ∂u ∂ν in the thermoelastic system (1.3) when using the multiplier method. However, the drawback in applying the Stokes map is that the multiplier should have zero average. This means that we need to modify the multipliers introduced in the literature in order for them to be applicable in the present case. To do this, we subtract the original multiplier by a suitable regularized version with respect to space.
3. Abstract formulation and main results. As noted in the earlier works [7,8], one cannot apply the usual Leray projection method to deal with the Stokes equation (1.1) due to the Neumann-type boundary condition on Γ 0 . An alternative and novel approach presented in these papers to solve this problem is to eliminate the pressure by writing it in terms of the fluid velocity and the state variables associated with the structure. The same idea has been used to compute numerical approximations for the Stokes equation using pressure-matrix methods, see [26, Section 9.6.1] and the references therein. The procedure will also enable us to recover the pressure for the strong solutions of the system. In our case, we need to express p as a function of the fluid velocity u, the plate displacement ϕ and temperature θ. With this representation it is possible to express the system (1.1)-(1.7) in an abstract form and semigroup methods are amenable to establish its well-posedness.

3.1.
Resolution of the pressure. For smooth solutions it can be checked that for each time t the pressure is a harmonic function in Ω with mixed Neumann-Robin boundary conditions See [3] or the proof of Theorem 3.1 below for the verification of this claim. Introducing the following maps γ ∆θ), the pressure term can be written in terms of (u, ϕ, θ) as then the maps G i are well-defined and p given by (3.1) lies in H 1 (Ω).
3.2. Abstract formulation. As noted in the literature, we shall also need to factor the constant functions associated with the state space for displacement of the plate. Indeed, by integrating the compatibility condition (1.2) from 0 to t one obtains With this consideration, we shall take the state space which is a Hilbert space under the inner product consisting of all elements (u, ϕ 1 , ϕ 2 , θ) ∈ H satisfying all of the following conditions: . The system can now be written as an abstract Cauchy problem on H with the state variable U = (u, ϕ 1 , ϕ 2 , θ) and initial data U 0 = (u 0 , ϕ 0 , ϕ 1 , θ 0 ). In the succeeding theorem, we show that A is the generator of a strongly continuous semigroup of contractions on H by invoking the well-known Lumer-Philipps Theorem in reflexive Banach spaces. This requires to prove the dissipativity of A and the range condition R(λI −A) = H for some λ > 0. Without thermal effects and flat Γ 0 this range condition was demonstrated in [3] using a nonstandard variational mixed method, specifically, the Babuska-Brezzi Theorem. This mixed formulation can be utilized in the numerical analysis for the approximations of steady-state solutions. However, in our case it is enough to prove the range condition by showing that 0 ∈ ρ(A). Indeed, if this is the case then from the fact that the resolvent set ρ(A) is open, there exists λ > 0 such that λ ∈ ρ(A) and thus R(λI − A) = H. The latter method has its advantage. The system of equations that are needed to solve are now weakly coupled and one may solve them successively.
Theorem 3.1. The operator A generates a strongly continuous semigroup of contractions on H. In particular, for every initial data U 0 ∈ H the Cauchy problem admits a unique mild solution U ∈ C([0, ∞), H). Moreover, the components u and θ of the solution satisfy after using div u = 0 on Ω and the boundary conditions u = ϕ 2 ν on Γ 0 and u = 0 on Γ 1 . On the other hand, from the duality pairing (2.1) we have since ϕ 2 = 0 on Σ 0 . For the heat component it holds that Taking the sum of (3.4)-(3.6) and getting the real part yields and this shows that A is dissipative. Next, we prove that 0 lies in the resolvent set of A. Given (u * , ϕ * 1 , ϕ * 2 , θ * ) ∈ H we need to find a unique (u, for some constant C > 0 independent on (u, ϕ 1 , ϕ 2 , θ) and (u * , ϕ * 1 , ϕ * 2 , θ * ). The equation to be solved is equivalent in solving the Stokes equation  9) and the elliptic equation Therefore by trace theory we obtain from (3.11) Notice that p =p + p * , where p * is a constant, is still an admissible pressure for the Stokes problem and satisfies the stability estimate and hence (3.12) wherep is replaced by p.
In virtue of the Lax-Milgram lemma, the elliptic equation (3.10) has a unique solution θ ∈ D(A R ) and it satisfies (3.14) By standard elliptic theory, the biharmonic problem (3.9) admits a unique solution [20, p. 152] for example. However, there is no reason that ϕ 1 necessarily has average zero. This will be done, following the method in [8], by choosing the appropriate constant p * . Let We test equation (3.9) with ζ to obtain . Acording to the following equations Therefore we choose p * according to (3.17).
Thus p = G(u, ϕ 1 , θ) and we obtain from the above discussions that 0 ∈ ρ(A). Therefore A generates a strongly continuous semigroup of contractions on X by the Lumer-Philipps Theorem. The additional regularity (3.3) for u and θ is a consequence of the identity  Indeed, for every t ∈ [0, ∞) we have where the first equation holds in [L 2 (Ω)] 3 . Hence one has the estimate , t ≥ 0. The continuity of p and u as mentioned in (3.21) follows from this estimate and the fact that u t ∈ C([0, ∞), [L 2 (Ω)] 3 ) and ϕ t ∈ C([0, ∞), H 2 0 (Ω)). If the initial data U 0 lies in D(A) then we have the following equations hold pointwise-in-time. If in addition, U 0 ∈ D(A 2 ) then (3.23) holds in L 2 (Γ 0 ) and The above regularity properties will justify the calculations that is provided in the succeeding section. We shall prove this theorem using suitable multipliers. To establish (3.26), it is enough to estimate the total energy on sufficiently large time intervals [0, T ] in terms of the final energy, the initial energy and the total dissipation on [0, T ]. In other words, we want to derive the energy estimate for some constants C > 0 (independent of T ) and T * ≥ 0, where D is the dissipation term given by Indeed, from (3.20) and the fact that the energy is decreasing, (3.27) implies Thus, if T > max(T * , 4C) then where δ T = 2C/(T − 2C) ∈ (0, 1). The exponential decay property (3.26) now follows from the latter inequality together with induction and the evolution property. We note that as in Avalos and Lasiecka [6], the decay rate in Theorem 3.2 is not uniform in γ > 0. However, in the case of γ = 0 one may proceed using the same techniques presented here.

Modification of multipliers.
As stated in the introduction, we will utilize suitable multipliers to derive the energy estimates required to obtain exponential stability of the system (1.1)-(1.7). First, we discuss a simple but efficient way of modifying the multipliers suitable to the Stokes map S in Theorem 2.1. Fix a smooth cut-off function ρ ∈ C ∞ 0 (Γ 0 ) such that ρ ≥ 0 in Γ 0 and Γ0 ρ dΓ 0 = 1. For a function f defined on (0, T ) × Γ 0 we set whenever the integral makes sense. Note that supp I ρ f (t, ·) ⊂ supp ρ and hence I ρ f (t, ·) is also compactly supported in Γ 0 . The operator I ρ is regularizing with respect to space in the sense that I ρ ∈ L(L p (0, T ; L 2 (Γ 0 )), L p (0, T ; H s 0 (Γ 0 ))) (4.1) for every s ≥ 0 and 1 ≤ p ≤ ∞, with operator norm independent of T and depending only on (p, s, ρ, Γ 0 ). Let us define the map In the subsequent discussions, if f is a specific multiplier then M ρ f is called the modified multiplier. According to the definition of M ρ f we have for suitable f . Therefore M ρ f is a suitable argument for the Stokes map S whenever f is sufficiently regular. This choice of multiplier M ρ f is somewhat similar to the one given by Haraux [12] for damped wave equations, see also [15].
for some constant C > 0 independent of f . Therefore Lemma 4.1. Let T > 0, ε > 0 and f ∈ L 2 (0, T ; H 1 0 (Γ 0 )) ∩ H 1 (0, T ; L 2 (Γ 0 )). Then for every initial data in D(A), the component u of the solution and the pressure p satisfy the estimate for some constants C ρ > 0 and C ε,ρ > 0 independent of T , u and f .
We will utilize this lemma with f being h · ∇ϕ, A −1 D θ or ϕ, where h is a smooth extension of ν to Γ 0 (see Remark 3 below for a short historical account regarding these multipliers). The additional terms obtained by enforcing the compatibility condition will be estimated with the help of the following lemma.
Proof. Integrating by parts in time, using Green's identities, the boundary conditions and the fact that According to the regularizing property of I ρ , see (4.1), we have the following estimates for some constant C = C ρ,Γ0 > 0. Using Young's inequality in (4.8) and then applying the estimates (4.9)-(4.11) we obtain the estimate of the lemma.

5.
Proof of uniform stability. First, we give a hidden trace regularity for the plate component that is similar to [6,14,19].
Proof. By a density argument, we may suppose that the initial data lies in D(A 2 ). Let h ∈ [C 2 (Γ 0 )] 2 be a vector field such that h = ν on Σ 0 . We multiply the plate equation in (1.3) by the multiplier M ρ (h · ∇ϕ) and integrate over time and space to obtain According to Remark 1, h · ∇ϕ ∈ C 1 ([0, T ], H 1 0 (Γ 0 )) for each T > 0. Thus, we can apply Lemma 4.1 and Lemma 4.2. Using Lemma 4.1 with f = h · ∇ϕ and ε = 1 we have On the other hand, Lemma 4.2 with f = h · ∇ϕ and ε = 1 produces the estimate To estimate the remaining terms in (5.2) we shall proceed as in [6, pp. 168-169]. Indeed, integrating by parts and using the fact that ϕ = ϕ t = ∂ϕ ∂ν = ∂ϕt ∂ν = 0 and h = ν on Σ 0 we have We shall estimate each terms on the right hand side of (5.5).
The first three terms can be simply estimated using the Cauchy-Schwarz and Poincaré inequalities. For the fourth term, we use the divergence theorem to get where the remainder term R 1 (T ) can be estimated by Applying the divergence theorem once again together with |∇ϕ t | 2 = 0 and h · ν = 1 on Σ 0 we obtain Consequently, from (5.6)-(5.8) one acquires the estimate Let us estimate the fifth term on the right hand side of (5.5). Before we proceed, we note the following standard identities where H is the Jacobian of h, ∇ 2 ϕ is the Hessian of ϕ and Thus from (5.10)-(5.12) we have Using the fact that ∂ 2 u ∂τ 2 = ((∇ 2 ϕ)τ ) · τ = 0 on Γ 0 , where τ = (−ν 2 , ν 1 ), one can show through integration by parts that (see [6]) On the other hand, according to the divergence theorem and |∇ϕ| = 0 on Γ 0 The second term on the right hand side of (5.13) can be expressed using the divergence theorem as The last term on the right hand side of (5.5) can be estimated as The next step is to estimate the total kinetic energy of the plate component. Due to (3.25) it is enough to consider the total energy of ∇ϕ t . This is achieved by the modified multiplier M ρ (A −1 D θ). We recall that the pseudodifferential operator A −1 D is smoothing in the sense that This follows from the representation (2.3). Therefore from (5.19), (5.20) and Dγ 0 ∈ L(H 1 (Γ 0 )) Similarly, from (5.19) it holds that for every 0 < ε < γ 2 . Proof. The proof is similar as in the proof of the previous theorem, but now using the modified multiplier M ρ (A −1 D θ). Because θ ∈ C 1 ([0, T ], L 2 (Γ 0 )) for each T > 0 we have A −1 D θ ∈ C 1 ([0, T ], H 2 (Γ 0 )), and thus Lemma 4.1 and Lemma 4.2 are applicable.

GILBERT PERALTA
The estimate will be derived from the following identity    If J 1 is the second term on the right hand side of the equation (5.26) then using Young's inequality, (3.25), (5.19), and (5.27). Since A −1 D θ = 0 on Σ 0 we obtain from Green's identity |∇ϕ t | 2 dΓ 0 dt + 2ε 3 Solving for the integral term involving ∆ϕ in (5.32) and then invoking (5.33) and (5.34) proves the lemma.
Remark 3. We would like to point out that the multipliers ϕ and h · ∇ϕ are now standard and they were used in various plate equations and even for wave equations, e.g. [13,16] and the references therein. The multiplier A −1 D θ was first introduced by Avalos and Lasiecka [5,6] for certain thermoelastic plate systems. It was used later for certain thermoelastic von Karman plate models by Perla Menzala and Zuazua [23,24,25]. The idea of using the Stokes map S originated from the work of Chuesov and Ryzhkova [10]. It was used in deriving suitable estimates for a Lyapunov functional associated with a fluid-plate interaction model. for every 0 < ε < γ 2 . Choosing ε > 0 in such a way that 2ε < min( 1 4 , γ) we obtain from the above inequality the estimate