The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework

In this paper, we study a fluid-structure interaction model of Stokes-wave equation coupling system with Kelvin-Voigt type of damping. We show that this damped coupling system generates an analyticity semigroup and thus the semigroup solution, which also satisfies variational framework of weak solution, decays to zero at exponential rate.


1.
Introduction. Fluid-structure interaction model has been intensively studied for the last decade because of its wide applications to engineering, physics, biology and biomedicine sciences. The model in this paper describes the motion of a solid inside a fluid. The mathematical challenge of this model stems from the coupling of the hyperbolic component (elastic system) and the parabolic component (fluid). The dismatch of the parabolic-hyperbolic regularity at the interface raises difficulty in the establishment of well-posedness of the solutions in suitable functional spaces [14,8,9,24,25]. Furthermore, the parabolic-hyperbolic coupling is also the main obstacle in the study of the stability and stabilization of the fluid-structure system. As indicated in [27], [26], the dissipative propagation from the parabolic component is of weak type, thus can not overcome the conservative hyperbolic component to yield uniform stability for the overall evolution if the system is undamped. Therefore, a damping has to be added if one expects to get uniform stabilization for the system. The damping can be added on fluid-structure interface as a boundary control for the stabilization of the solid. Past literature on this topic includes [3,4,28] and the references therein. It can also be included directly as the structural damping in the hyperbolic component, the recent papers [30,40,5] work in this direction. This paper is a subsequent work of [30,40]. The distinct feature of [30,40] is that the model couples a heat equation with a viscoelastic damped wave equation, in which the damping is of Kelvin-Voigt type. The physical importance of this model is that the solid (inside the fluid) described in this system exhibits viscoelastic property, that is, gradual deformation and recovery when it is subject to loading and unloading [36]. In fact, such viscoelastic property presents in a large group of materials -polymer plastics, almost all biological materials, and etc. Thus, our model encompasses practical applications in engineering and biomedical science. From the mathematical point of view, compared to [41], where an undamped heat-wave coupling can not achieve uniform exponential decay under any geometric condition, [30] shows that the coupling of heat equation and Kelvin-Voigts type damped wave equation with suitable trace formulation on the interface yields very strong dissipative property for the overall system: the semigroup generated by the system is analytic. As Stokes equation and heat equation share many similarities, specially, the semigroup generated by Stokes operator also exhibits analyticity in certain function spaces, [16,19]. It is natural to raise the question: Does the Stokeswave coupling with Kelvin-Voigt damping still generate an analytic semigroup? In this paper, we address this problem under variational framework of the system. [11] has shown that the semigroup generated from the viscoelastic damped wave equation is indeed analytic. Thus the two components in our model are both analytic. However, the coupling formulation within the system makes the proof of overall analyticity challenging. It is not with the range of relatively bounded perturbation, thus we can not apply any perturbation theories for analytic semigroup in hope of obtaining the analyticity of the overall coupling system. The estimates need to be developed from scratch. The second difficulty comes from the presence of the pressure in the Stokes equation. The standard method of treating pressure term is Leray's projection in variational framework. But in our Stokes-wave coupling, pressure also involves in the boundary condition of the interface of the coupling system. Thus it persists in the structure and may potentially contribute to the singularities of the coupling system. Handling this needs introduction of new multipliers and new weights in the energy calculations. To the best of our knowledge, the present paper is the first one that considers the presence of pressure within the framework of analytic semigroup in the context of fluid-structure coupling system.
In the related literature, the Westervelt type wave equation introduced in [13,22,23] also includes a Kelvin-Voigt-damping-like term in the hyperbolic equation. In [33], a detailed theoretical approach using the tool of maximal L p regularity [15], [21] shows that the global solution of this Westervelt equation exponentially decays to zero. In addition, [34,35,10] explore the full Navier-Stokes fluid coupled with the viscoelastic damped structures using numerical methods: the ALE(Arbitrary Lagrangian-Eulerian) mapping and Lie splitting scheme. The existence of weak solution is obtained, the stability of the weak solution is also studied. Let Ω be a bounded domain in R n , (n = 2, 3), Ω is partitioned into two subsets Ω f and Ω s , with Ω s being surrounded by Ω f , as shown in figure 1. The figure describes the scenario of a structure Ω s fully immersed in a fluid Ω f . The boundary of Ω f is Γ f ∪ Γ s , and the boundary of Ω s is Γ s . Thus Γ s is the interface where the interactions between the fluid and the structure take place.
The dynamics of this fluid-structure system is described by the following equation system: In (1.1), u is a vector function representing the velocity field of the fluid. p is a scalar function representing the fluid pressure. w and w t are both vectors and represents the displacement and velocity of the structure respectively. ν is the unit outward normal vector on Γ s with respect to the region Ω s . For simplicity, the constant b equals either 0 or 1.
In addition, we assume the interface Γ s is static. This assumption is appropriate if we further assume that the structure component presents the property of small and rapid oscillations [18].
Throughout the paper we will consider the dynamic system in the following energy space The inner product in H is defined as In addition, we define the following space: It should be noted that in the rest part of this paper, we will omit the exponent n in (H 1 (Ω f )) n , (L 2 (Ω f )) n , (H 1 (Ω s )) n and (L 2 (Ω s )) n , and denote these spaces as H 1 (Ω f ), L 2 (Ω f ), H 1 (Ω s ) and L 2 (Ω s ) for simplicity. The energy for the system is defined as follows: and 2. Main Theorem. We first define the weak solution for (1.1) as follows: Definition 2.1. Let (u 0 , w 0 , w 1 ) ∈ H and T > 0. A triple (u, w, w t ) ∈ L ∞ (0, T ; H) is a weak solution for system (1.1) if • (u(0), w(0), w t (0)) = (u 0 , w 0 , w 1 ) • (u, w, w t ) satisfies the variational form for a.e. t ∈ (0, T ) for every φ ∈ V and ψ ∈ H 1 (Ω s ).
We now define the notation ρ(w, z) = ∇w + ∇z, and the operator A : Under these definitions, the weak solution in Definition 2.1 can be rewritten as with domain of A as follows: The domain is defined as such for the need in the proof of analyticity in section 4. Under the definition of A in (2.6) with its domain in (2.7), (2.3)-(2.5) can be written into an abstract evolution as To this end, it should be noted that in (2.6) the term bw when b = 1 can be seen as a bounded perturbation to the case b = 0 and thus does not alter the well-posedness and analyticity conclusions on the abstract evolution (2.8), [17], and therefore does not change these conclusions on the original system (1.1). Hence forward, we only work on the case b = 0. All the Main Theorems (theorems presented in this section) in this paper are under the assumption that b = 0, but they also hold for the case b = 1.
with domain of A unchanged as (2.7). And the consequently, the energy E(t) of the evolution system becomes We first develop the following Theorem: Theorem 2.1. The operator A defined in (2.9) and (2.7) generates a strongly continuous semigroup of contraction on the Hilbert space H. Thus, the solution (u, w, w t ) for the system (2.8) can be represented as In addition, the energy E(t) defined in (2.10) satisfies The subsequent theorems are the main theorems of this paper.
Theorem 2.2. For A defined in (2.9) and (2.7), its spectrum σ(A) is contained in the following region: where S r=1 (x 0 ) is the closed disk centered at x 0 = (−1, 0) with radius r = 1, S r0 is the open circle centered at origin with small radius r 0 > 0. The whole imaginary axis is in the resolvent of A, that is iR ⊂ ρ(A). In addition, there is a τ 0 > 0, such that for τ ∈ R and |τ | > τ 0 , the resolvent R(iτ, A) satisfies the estimates: Thus, the s.c. contraction semigroup generated by A is analytic on H.
Theorem 2.3. Theorem 2.1 and Theorem 2.2 yield that the energy of the evolution system defined in (2.10) satisfies exponential decay rate: There exist M > 0, δ > 0, such that Thus, the semigroup solution, which is also the weak solution under Definition 2.1, decays to zero at uniform exponential rate.
Remark 2.2. In our model, we describe the motion of the structure by wave equation with viscoelastic damping. This can be generalized to elastodynamic equation with Lamé coefficient with the damping term changing accordingly (Indeed, as one can see this damping is of Kelvin-Voigt type). Using similar approach presented in this paper we expect the above main theorems will still hold when wave equation is replaced by the following elastodynamic equation. That is, the following system generates an analytic semigroup and its weak solution decays to 0 at uniform exponential rate where σ(w) is the stress tensor and σ(w) = 2µε(w) + λtr(ε(w))I with ε(w) being strain tensor and λ > 0, µ > 0 being the Lamé coefficient.
Our approach in proving the contraction and analyticity of the semigroup generated by the system (1.1) is based on the variational method. The strong convergence of the solution (uniformly decays to 0) allows to adapt this variational approach to Galerkin Approximation of the system (1.1), thus, making the construction of finite element numerical scheme of system (1.1) a feasible option. And indeed, developing a Galerkin finite element numerical scheme for system (1.1) is in the plan of our future study of this fluid-structure interaction model.

3.
Proof of Theorem 2.1. We now verify the assumptions of Lumer-Philips Theorem, which yields exactly Theorem 2.1. That means we are going to prove (i) A is dissipative; (ii) λI − A is maximal for some λ > 0. In the following proof, we set λ = 1.

Dissipativity of A.
Proposition 3.1. The operator A defined in (2.9) and (2.7) satisfies Because w ∈ H 1 (Ω s ) and z ∈ H 1 (Ω s ), the last term Plug in this result, keep in mind u| Γs = z| Γs , we get  The dissipativity of operator A is proved.
Here and throughout this paper equation We write in the form of (3.8) for simplicity, when detailed calculations are involved, we use the variational form (3.12). Eliminate z, we get From elliptic theory, we know for u ∈ V , g ∈ H 1 (Ω s ), h ∈ L 2 (Ω s ), there exists w ∈ H 1 (Ω s ) satisfies the system (3.14) and (3.15): The solution w is continuously dependent on u, h, g. Thus we write it as w(u, h, g), and it satisfies the following inequality. Recall the boundary condition (3.17), the above equation becomes This yields, Linear theory shows that w(u, h, g) can be decomposed as w(u, h, g) = w(0, h, g)+ w(u, 0, 0). w(0, h, g) is independent from u, and elliptic theory guarantees that the solution w(0, h, g) exists. Thus, (3.13) can be written as Also reminding that f ∈ H, we have the right side of (3.22), −A(0, w hg , w hg − g) + f ∈ V . By now, to prove the maximality of I − A, we only need to prove that I + A : V → V defined by the left side of (3.22) is surjective from V to V . By [7], this is to show I + A is continuous, coercive and monotone in u. Thus, together with Poincare's inequality,

Continuity of I +
So I + A defined in (3.22) is bounded from V to V , together with the fact that it is also linear on V , we obtain its continuity from V to V .
So far, we have proved the continuity, coercivity and monotonicity of the operator I + A defined in (3.22). Thus the surjectivity of I + A is established, and therefore the maximality of the operator I −A is established. Combined with the dissipativity of A in Proposition 3.1. We have proved that A defined in (2.9) and (2.7) indeed generates a strongly continuous semigroup of contraction. Theorem 2.1 is thus proved. And it is natural to obtain the following corollary: Corollary 3.3. Let the initial condition (u 0 , w 0 , w 1 ) ∈ D(A), the semigroup solution y(t) = (u(t), w(t), w t (t)) obtained in Theorem 2.1 satisfis the following regularity: , w t (t)) also satisfies the variational formulation (2.1) and (2.2) in Definition 2.1. Thus, (u(t), w(t), w t (t)) is also a weak solution defined by Definition 2.1.
We want to prove the analyticity of A using the tool introduced in [29] and [32], which require to show that the imaginary axis is in the resolvent set. Specially, we need to show 0 is in the resolvent set.
Following Theorem 4.3, because a resolvent set is an open set, we have the following corollary: there exists a small r 0 > 0, such that on the complex plane, the open disk S r0 centered at the origin with radius r 0 is in the resolvent set of A.
To prove that the operator A generates an analytic semigroup on H. We will first establish the following theorem Theorem 4.5. For p ∈ L 2 ((0, T ); L 2 (Ω s )) satisfying system (1.1) obtained in Lemma 4.1, the resolvent operator R(λ, A) = (λ − A) −1 for the generator A defined in (2.9) and (2.7) satisfies the following estimates: is the closed disk centered at x 0 = (−1, 0) with radius r = 1.
Remark 5.1. Compare the above results with [41], we know the damping term ∆w t steers the system into exponential decay. An interesting question is what will happen if we replace ∆w t by small structural damping such as α∆w t , or ∆(αw t ), with 0 < α < 1. In fact, for any fixed α > 0, the proofs of contraction (Theorem 2.1) and analyticity (Theorem 2.2) of the generated semigroup still hold, thus, the new system also keep the uniform stability property. However, if we let α → 0, the semigroup is still contractive, but analyticity is lost (The proof of Theorem 4.3 runs into trouble), and uniform stability is lost too. However, in this scenario, from [27], we do have the solution (u, w, w t ) decays to zero in the energy space H.
Remark 5.2. Another interesting extension to our current problem is what happens if the structural damping −∆w t is replaced by fractional structural damping (−∆) α w t where 0 < α < 1. Following [11,31], when 1/2 ≤ α < 1, the structure component of the coupling model is analytic. Thus, the coupling is still analyticanalytic type. Under this scenario, we hope to see the overall evolution system remains analytic. This is still an open question that needs further investigation. For the situation 0 < α < 1/2, [12] indicates that the structure component generates a semigroup of Gevrey class δ, with δ > 1 2α . One can expect the Gevrey regularity to propagate through the interface. The regularity of the overall coupling under such situation (0 < α < 1/2) is also an interesting problem for further study.