Well-posedness and exponential stability of a coupled fluid–thermoelastic plate interaction model with second sound

In this paper, we investigate a coupled system modeled by fluid and thermoelastic plate, while the heat effects are modeled by the Cattaneo’s law giving rise to a “second sound” effect. We proved that the coupled system admits a unique global mild solution. Furthermore, we construct the second-order energy to control the term ‖∇θ‖L2(Γ0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Vert \nabla \theta \Vert _{L^2(\Gamma _0)} $$\end{document} so as to establish the exponential decay of the solutions.


Introduction
In this paper, we consider a coupled system of partial differential equations (PDEs) modeling the interaction of an incompressible fluid, which occupies a domain Ω ⊂ R 3 bounded by the (solid) walls of the container Γ 1 and a horizontal boundary Γ 0 on which a thin thermoelastic plate including the second sound effect is placed (see Fig. 1). Mathematically, the model is formulated as follows. The sufficiently smooth boundary of Ω is denoted here as Γ =Γ 0 ∪Γ 1 , where Γ 0 and Γ 1 are nonempty and Γ 0 ∩ Γ 1 = ∅. And we suppose that the boundary Σ 0 of Γ 0 is nonempty and smooth enough.
The dynamics of the fluid are described by the 3D linear Navier-Stokes equation while the thin thermoelastic plate dynamics are governed by Kirchhoff's equation and are subjected to second sound effects described by the heat equation where u, ϕ are the velocity vector fields of the fluid and the transversal displacement of the plate, respectively, and θ, q are the temperature and the heat flux vector, respectively. Here, μ > 0 is the viscosity of the fluid, while "rotational inertia parameter" γ > 0 is proportional to the thickness of the plate and constant τ 0 > 0 represents the relaxation time, describing the time lag of the heat flux response to the temperature gradient. p − ν · ∂u ∂ν is the viscous shear stress exerted by the fluid on the plate and where ν is the outer normal vector to Γ 0 . Equation (1.2) 3 was proposed by Cattaneo in [10] to correct the paradox of transient propagation of thermal disturbances predicted by Fourier's heat conduction theory (τ 0 = 0). Moreover, formula (1.2) 3 is the most obvious and simplest generalization of Fourier's law, which produces a finite velocity of propagation. From (1.2) 2 and (1.2) 3 , we can get the equation which is a hyperbolic equation that predicts a finite velocity 1 τ0 of heat propagation. We note here the necessity of enforcing that both the plate initial displacement and velocity have zero average, which explains the choice of the structural finite energy space component of H below. Recalling the boundary condition (1.1) 4 and the fact that the normal vector ν is on Γ 0 , then by Green's formula we get the following compatibility condition Γ0 ϕ t dΓ 0 = Γ0 u · ν dΓ 0 = 0 for t ∈ (0, ∞). (1.3) This condition fulfills when Γ0 ϕ dΓ 0 = const for t ∈ (0, ∞), which can be interpreted as the preservation of fluid volume. We assume that the edge of the plate is given by (1.4) and the temperature at the boundary takes θ = 0 on (0, ∞) × Σ 0 . (1.5) We assume that the system is subject to usual initial conditions System (1.2) may represent different models for Kirchhoff type thermoelastic plate equations, with (γ > 0) or without (γ = 0) inertial term, either with Fourier's law (τ 0 = 0) of heat conduction or with Cattaneo's law (τ 0 > 0). For (1.2) 1 , if γ = 0, then it corresponds to Euler-Bernoulli beam, while if γ > 0, then it is a Kirchhoff's model. For (1.2) 3 , if τ 0 = 0, then it becomes a parabolic equation, while if τ 0 > 0, then it is a hyperbolic equation.

Literature
The motion of elastic solids in incompressible viscous fluids is ubiquitous in nature, which has attracted extensive attention from engineers, biologists and mathematical researchers. Mathematically, such a motion is described as a PDEs system that couples parabolic and hyperbolic phases, the latter of which may lead to a loss of regularity. For the model of similar system as (1.1)-(1.2) but without thermal effect and rotational force (γ = 0), the well-posedness and exponential stability has been established in [3,11,12] and more recently in [8,25,33,34]. In [12], Coutand and Shkoller proved the existence of local strong solutions for small enough data with high regularity assumptions. In [3], Avalos and Clark presented the well-posedness and numerical approximation results through a nonstandard variational formulation. Lu [25] studied the uniform stability to a non-trivial equilibrium of a nonlinear fluid-structure interaction model. Avalos and Bucci [2] firstly showed that the system is stable even for γ > 0, but with rational decay rate. A recent study by Shen, Wang and Feng [37] on small displacement oscillation fluid-structure interaction systems should also be mentioned. For other findings on fluid-elastic structural systems, we refer the reader to the brief review in the introduction of [17].
For the case of a fluid enclosed in a solid membrane or interacting with a plate, such as blood flowing in a vessel, Muha andČanić [27][28][29][30] studied the existence of weak solutions to a series of different fluidstructure interaction problems. Later, Trifunović and Wang [39] used a hybrid approximation scheme to deal with a 3D incompressible fluid coupled with a nonlinear plate equation and considered an interaction problem between a viscous fluid and a thermoelastic plate in [40].
Recently, by making use of appropriate multipliers in the time-space domain, Peralta [32] exhibited the exponential decay of the system similar to (1.1)-(1.6) whose heat conduction is modeled with Fourier's law, and the temperature at the boundary takes where λ 1 , λ 2 ≥ 0 and λ 1 + λ 2 > 0. For the thermoelasticity system that without the Navier-Stokes equation, there is a large number of work describing the elastic behavior of hyperbolic equation/system coupled with heat conduction. Hanni, Djebabla and Tatar [13] considered a one-dimensional full von Kármán system coupled to a heat equation simulated by the expected dissipative effects of heat transfer governed by Cattaneo's law and proved well-posed and exponentially stable results for the system. In [15], Fernández Sare and Racke investigated the stability of Timoshenko systems coupled with heat conduction equations, and proved that, the resulting systems with Fourier's law of heat conduction are exponentially stable, but the coupling with Cattaneo's law leads to not exponentially stability, which means that the exponential stability of solutions in thermoelasticity depends on the constitutive laws. But there is a gap in [15]. Santos, Almeida Júnior and Muñoz Rivera in [35] corrected the gap and introduced a new number χ 0 that characterizes the exponential decay and proved that the semigroup exponential decay holds if and only if χ 0 = 0; otherwise, it lacks exponential stability. In this case, the semigroup decays to t − 1 2 and the rate is optimal. For the thermoelastic plate models with Cattaneo's law, Fernández Sare and Muñoz Rivera [14] gave the exponential stability by semigroup techniques when the system is clamped with hinged boundary conditions using ϕ| Σ0 = Δϕ| Σ0 = 0 for γ > 0 and pointed out that the system is polynomial stability in the case γ = 0, see also [31].

Challenges encountered and novelty
In this paper, our purpose is to study the fluid-structure interaction models (1.1)-(1.6) when the heat flux is modeled by the Cattaneo's law (1.2) 3 . We shall show that the fluid-thermoelastic plate interaction model with Cattaneo's law is exponentially stable if the inertia rotational coefficient γ is strictly positive. The replacement of Fourier's law by Cattaneo's law in this paper such that the term ∇θ L 2 (Γ0) is absent from the dissipative term and we cannot control the energy integral any more. To overcome this difficulty, we construct the second-order energy so that the term ∇θ L 2 (Γ0) can be controlled by the terms q L 2 (Γ0) and q t L 2 (Γ0) . We note that the similar second-order energy methods were presented in [1,9,23,24,31,36] for other systems, but in this paper we need to develop some new estimates to obtain the desired exponential decay results.

Notation
In this subsection, we introduce the relevant spaces and operators necessary in the abstract formulation in system (1.1)-(1.6) as in [32]. For the fluid component, we define the state space as 3 : div u = 0 in Ω and u · ν = 0 on Γ 1 } and endow it with the L 2 -norm. By trace theory, we know that if u ∈ [L 2 (Ω)] 3 and div u ∈ L 2 (Ω), then ). One is also applicable for the generalized trace u · ν on Γ 0 and Γ 1 . Moreover, we define the space 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. Next, we define the bi-Laplace and Laplace operators on various domains. Define the bi-Laplacian operator A : . It is known that A is positive self-adjoint on L 2 (Γ 0 ) and hence the fractional power A    where D(A D ) = H 2 (Γ 0 ) ∩ H 1 0 (Γ 0 ). It is easy to see that the norm · D(AD) := A D · L 2 (Γ0) on D(A D ) is equivalent to the induced norm of D(A D ) after using the Poincaré inequality and standard elliptic result. Moreover, the operator A D is a positive self-adjoint operator on L 2 (Γ 0 ) and there holds (θ,θ) D(AD) = (Δθ, Δθ) L 2 (Γ0) .

So the operator
exists. Given γ > 0, the operator P γ : is also a positive self-adjoint operator with P We extend the domain from D(A D ) to H 1 0 (Γ 0 ), and a typical extension procedure shows that P γ becomes Indeed, if the space H 1 0 (Γ 0 ) is endowed with the norm P 1 2 γ · L 2 (Γ0) and we combine the fact that H −1 (Γ 0 ) is the dual of H 1 0 (Γ 0 ) with respect to the pivot space L 2 (Γ 0 ), then P γ becomes a unitary operator, see [42,Corollary 3.4.6].
With this consideration, we shall take the state space which is a Hilbert space subject to the inner product

Plan of the paper
The paper is organized as follows. Section 2 gives the abstract system and preliminary results. The wellposedness of the model will be established in Sect. 3. In Sect. 4, we introduce the notion of modified multipliers. We use modified multipliers and the second-order energy to prove the uniform exponential decay of the system in Sect. 5.

Elimination of the pressure
The elimination of the pressure term p of system (1.1)-(1.6) depends on the following observation: For each time t, the pressure p(x, t) solves the elliptic boundary value problem (BVP) Compared with the literature [32], for the pressure harmonic equation, the influence of the two kinds of heat conduction on the pressure at the boundary is the same as the heat conduction, while compared with system without heat conduction [3], heat transfer affects the normal pressure gradient at the boundary Γ 0 . The trick in this case to eliminate the pressure p by constructing a suitable elliptic problem in p was introduced in [6] by Triggiani in 2007, and used systematically by these authors in several subsequent papers including [3,4] (see also [41]). Because of the elliptic BVP above with mixed Neumann-and Robintype boundary conditions, see [3,4,6], we give the following mixed Neumann-Robin maps R 0 and R 1 , compare with [3], Then by elliptic regularity results in [22, p. 152], the maps satisfy for s ∈ R and i = 0, 1. Due to the compatibility condition (1.3), we need to further introduce functions on Γ 0 that satisfy zero average. LetL for every s ≥ 0. It is easy to see thatĤ s 0 (Γ 0 ) is a closed subspace of H s 0 (Γ 0 ). At last, we solve the pressure p of BVP (2.1) in regard to R 0 , R 1 as Well-posedness and exponential stability of a coupled fluid Page 7 of 33 132 If we introduce the following maps where γ 1 is the first-order trace, then the expression in (2.3) can be represented as (2.4)

The fluid-plate generator
With respect to the above setting, the PDEs system given in (1.1)-(1.6) can be written as an abstract with the domain D(A) given by

Known results of energy estimates
To deal with the related estimates of A −1 D θ which will be defined later, we recall a result about the estimate of div q in [26].
Hence, we get the desired result. Peralta [32] has used the following important theorem due to Temam to eliminate the challenge caused by p − ν · ∂u ∂ν in the plate equation when he uses the multiplier method. The advantage of this theorem is that an operator S is used to connect the regularity between the boundary Γ 0 and the interior of the region Ω.

Main results
We now state the first main result of this paper, which gives the global existence of mild solution.
The second main result of this paper provides the uniform exponential stability of system (1.1)-(1.6).
We refer to Sect. 1.3 for the definitions of H (1.10) and V (1.7).

Semigroup well-posedness
This section is devoted to prove the semigroup well-posedness of the coupled problem given in (1.1)-(1.6).
Proof. (Proof of Theorem 3.1) Our proof depends on the application of Lumer-Phillips theorem in reflexive Banach spaces which says that if we can prove that the operator A is dissipative and that 0 belongs to the resolvent set ρ(A), then we can show that the operator is a generator of a strongly continuous semigroup of contractions.
Step 1 (Dissipativity of A) Firstly, the density of D(A) in H is obvious and then we only need to show the dissipativity of A in H. The proof process is similar to [32]. For convenience, we repeat it here. Let According to Green's formula, we have since div u = 0 on Ω and using the boundary conditions u = ϕ 2 ν on Γ 0 and u = 0 on Γ 1 . Moreover, from (1.8) and using ϕ 2 = 0 on Σ 0 , we have For the heat component, it holds that On the other hand, we have Combining (4.1)-(4.4) and getting the real part, we yield Step 2 (0 ∈ ρ(A)) We know that 0 ∈ ρ(A) ⇔ A is invertible. This is equivalent to prove that the following two properties are satisfied: where the constant C > 0 is independent of (u, ϕ 1 , ϕ 2 , θ, q) and (u * , ϕ * 1 , ϕ * 2 , θ * , q * ). To the end of solving (4.5), we divide the system into three subproblem and utilize known regularity results to obtain the existence and uniqueness, respectively. For given data (u * , ϕ * 1 , ϕ * 2 , θ * , q * ) ∈ H, the u-subproblem of problem (4.5) can be written as: The solution of the plate component of (4.5) will be denoted below as 8) and the temperature component of (4.5) solves Notice that we can obtain the solution of (4.7) and (4.9) and then use the solution to solve (4.8 Applying the elliptic theory, temperature subproblem (4.9) has a unique solution θ ∈ H 1 (Γ 0 ) ∩ L 2 (Γ 0 ) and it has ) after using Poincaré inequality. To proceed, we invoke the embedding L 2 (Γ 0 ) ⊂ H −1 (Γ 0 ) and Lemma 2.1 and obtain For plate subproblem (4.8), similar to the process in [32], we have −Δθ+p−ν · ∂u ∂ν −(I −γΔ)ϕ * 2 ∈ H −1 (Γ 0 ) according to the regularity of the solution u, p and θ. In virtue of elliptic theory, the subproblem exists a unique solution [22, p.152]. But we do not know whether ϕ 1 is average zero or not. Following the method in [7], we can make sure that ϕ 1 satisfies the zero-average property by choosing a suitable constant p * .
The above theorem gives that the operator A can raise the regularity. According to the standard semigroup theory, as in [32] we can further get the additional regularity of the strong solutions of (2.5), when U 0 ∈ D(A). These regularity properties will justify the calculations that will be provided in the succeeding sections.
Similar to (4.10), one has the estimate Thus, This plays a crucial role in the derivation of the energy estimates below.

Auxiliary results for exponential decay
We will use the modified multiplier M ρ f to meet the condition of zero average. This idea was proposed by Seidman [20], see more details in [16,19]. The modified multiplier makes an important role to derive exponential decay of the energy for the solutions of the system, and we will utilize the Stokes map S in Theorem 2.1 to eliminate the terms arising from p − ν · ∂ν ∂ν . Fix a smooth cutoff function ρ ∈ C ∞ 0 (Γ 0 ) such that ρ ≥ 0 in Γ 0 and Γ0 ρ dΓ 0 = 1.

Define the map
for every s ≥ 0 and 1 ≤ p ≤ ∞. And we know that M ρ has the properties as follows:

The first-level estimates
We denote the first-order energy of the system by Differentiating the energy E 1 (t), we have  32]) Assume that T > 0, ε > 0 and f ∈ L 2 (0, T ; H 1 0 (Γ 0 )). Then for ∀U 0 ∈ D(A), the velocity of the fluid u and the pressure p satisfy the estimate

4)
for some constants C ρ > 0 and C ε,ρ > 0 independent of T, u and f .
Proof. Integrating by parts in time, using Green's identities, the boundary conditions and the fact that ∇θ · ∇(I ρ f ) dΓ 0 dt. According to the regularizing property of I ρ , see (5.1), we have the following estimates for some constant C = C ρ,Γ0 > 0. Using Young's inequality in (5.11) and then applying the estimates (5.12), we obtain the estimate of the lemma.

The second-level estimates
In order to obtain the inequalities about E 2 (t), we differentiate the full system (1.1)-(1.6) in time to To control the spatial derivatives of θ, we introduce the second-level energy Again, we get the second-level energy identity with the corresponding dissipation ZAMP Well-posedness and exponential stability of a coupled fluid Page 17 of 33 132 Lemma 5.3. Assume that T > 0, ε > 0 and f ∈ L 2 (0, T ; H 1 0 (Γ 0 )). Then for ∀ U 1 ∈ D(A), u t and p t satisfy the estimate for some constants C ρ > 0 and C ε,ρ > 0 independent of T, u and f .
Proof. We multiply both sides of (5.13) 5 by I ρ and integrate by parts in time to yield after using Green's identities together with the boundary conditions and the fact that I ρ f vanishes on Σ 0 . And according to (5.1), we have the following estimates for some constant C = C ρ,Γ0 > 0. For (5.20), we use Young's inequality and then apply the estimates in (5.21) along with (6.6), and we obtain the estimate of the lemma. Denote as the total energy, for which we shall prove a result on the exponential decay. Hence, we have We shall prove Theorem 3.2 by using suitable multipliers. To establish (3.2), we only need to derive the energy estimate Indeed, from the fact that the energy is decreasing, (6.3) gives Meanwhile, from (4.18), we get Then, we have Thus, if T > max{T * , 4C}, then where δ T = 2C T −2C ∈ (0, 1). To prove (6.3), we need to prove First, according to (4.21) and Poincaré inequality we have Moreover, we multiply both the sides of (1.2) 2 by θ t and integrate over space to obtain According to the Hölder and Young's inequalities together with Lemma 2.1, we get Furthermore, integrating over time we have Thus, there are four terms: Δϕ Γ0 , Δϕ t Γ0 , ∇ϕ t Γ0 , ∇ϕ tt Γ0 remained to be proved so that we can derive the exponential decay.

Hidden trace regularity estimates
We next obtain the following trace regularity results about ϕ and ϕ t as in [5], which are differ from standard Sobolev space trace theory, but play a key role in our estimate of the uniform decay.
where C = C ρ,γ,Γ0 > 0 is independent of U 0 and D is the function defined in (6.2).

Lemma 6.2.
For ∀ U 1 ∈ H, the component ϕ t of the solution of the first-order derivative of (2.5) satisfies Δϕ t ∈ L 2 (0, T ; L 2 (Σ 0 )) and where C = C ρ,γ,Γ0 > 0 is independent of U 1 and D is the function defined in (6.2). Proof. Due to the density of D(A 2 ) in H, we suppose U 1 ∈ D(A 2 ). Let h ∈ C 2 (Γ 0 ) 2 be a vector field such that h = ν on Γ 0 . Multiplying the plate equation (5.13) 5 by M ρ (h · ∇ϕ) and integrating over time and space, we get According to Lemma 5.4, we choose f = h · ∇ϕ and ε = 1 to get the estimate To estimate the remaining terms in (6.10), we integrate by parts and use the fact that ϕ t = ϕ tt = ∂ϕt ∂ν = ∂ϕtt ∂ν = 0 and h = ν on Σ 0 to obtain ∇ϕ tt · ∇(h · ∇ϕ t ) dΓ 0 dt For the fourth term, we use the divergence theorem together with |∇ϕ tt | 2 = 0 and h · ν = 1 on Σ 0 to get (6.14) Thus, one acquires the estimate Using the divergence theorem, we can get According to the following standard identities where H is the Jacobian of h, ∇ 2 ϕ t is the Hessian of ϕ t and Well-posedness and exponential stability of a coupled fluid Page 23 of 33 132 From (6.18), we have Combining (6.17) with (6.19), we get Using the fact that ∂ 2 ut ∂τ 2 = ((∇ 2 ϕ t )τ )·τ = 0 on Γ 0 , where τ = (−ν 2 , ν 1 ), one can show through integration by parts that On the other hand, according to the divergence theorem and |∇ϕ| = 0 on Γ 0 , we have Thus, we have |Δϕ t | 2 dΓ 0 dt. (6.23) According to (5.2), the term about temperature on the right-hand side of (6.13) can be estimated as Combining (6.10)-(6.15), (6.23) with (6.24), we prove the theorem. |∇ϕ tt | 2 dΓ 0 dt. These two are achieved by the mod- , respectively, where the multiplier A −1 D (θ) and subsequently A −1 D (θ t ) were introduced by Avalos and Lasiecka in [5], and it is reproduced by Lasiecka and Triggiani in [21, P409]. We give the related estimates about the operator A −1 D . According to the definition of the operator and Notice that the heat equation (1.2) 2 can be rewritten in terms of the pseudo-differential operator Since q ∈ L 2 (Γ 0 ), recalling Lemma 2.1, we get ∇ · q ∈ H −1 (Γ 0 ), and then, Therefore, from (6.27) and (6.28) Similarly, it also holds that Moreover, note that the heat equation (5.13) 6 can be written as Similar to the inequality above, we have and for every 0 < ε < γ 2 . Proof. For this proof, which is similar to the one of [32, Lemma 5.2], we will use the multiplier M ρ (A −1 D θ), and then, we derive the following identity Because the estimates of A −1 D θ differ from the ones in [32, Lemma 5.2], we repeat the following process here. First, we estimate the right-hand side of (6.34). We use Lemma 5.1 with f = A −1 D θ, (6.29), (6.30) and (4.21) to derive Next, to derive the estimate of the left-hand side in (6.34), we will utilize the identity Finally, we estimate the remaining terms in (6.34) separately. First, from (6.27) we have (|ϕ t | 2 + γ|∇ϕ t | 2 ) dΓ 0 dt. Let J 1 be the second term on the right-hand side of the equation (6.37), we then use Young's inequality, Poincaré's inequality and (6.28) to get after using Green's identity. Therefore according to trace theory, Theorem 6.1, (6.25) and Young's inequality, we deduce with C being the constant in Theorem 6.1. The desired estimate follows from (6.6), (6.34)-(6.37) and (6.38)-(6.40), and we choose ε ∈ (0, γ 2 ) here. Page 27 of 33 132 Lemma 6.4. Assume that T > 0, then for ∀ U 1 ∈ D(A 2 ) we have the estimate Proof. Multiplying the plate equation (5.13) 5 by M ρ (A −1 D θ t ) and integrating over time and space, we derive the identity First, for the right-hand side of (6.41), using Lemma 5.3 with f = A −1 D θ t , (6.32), (6.33) and (4.21), we derive Next, according to Lemma 5.4 with f = A −1 D θ t , (6.30), (6.33), (4.21) and Poincaré's inequality we obtain
Now, from Lemma 6.3 and Lemma 6.5, we have for every 0 < ε < γ 2 . We further choose ε > 0 such that 2ε < min( 1 4 , γ) to obtain The estimates of remaining four terms are obtained, and thus, the proof of Theorem 3.2 is now completed.