HEAT–VISCOELASTIC PLATE INTERACTION: ANALYTICITY, SPECTRAL ANALYSIS, EXPONENTIAL DECAY

. We consider a heat-plate interaction model where the 2-dimen-sional plate is subject to viscoelastic (strong) damping. Coupling occurs at the interface between the two media, where each components evolves. In this paper, we apply “low”, physically hinged boundary interface conditions, which involve the bending moment operator for the plate. We prove three main results: analyticity of the corresponding contraction semigroup on the natural energy space; sharp location of the spectrum of its generator, which does not have compact resolvent, and has the point λ = − 1 /ρ in its continuous spectrum; exponential decay of the semigroup with sharp decay rate. Here analyticity cannot follow by perturbation.


1.
Introduction. Fluid-structure interactions models in the physical dimensions d = 2, 3, have been the object of mathematical studies in the past several years, particularly since the appearance of [16] which in turn followed [29, p.121]. In this setting, the typical configuration sees a structure -modeled by a d-dimensional wave-type equation -surrounded by a fluid -modeled by the d-dimensional dynamic Stokes equation involving pressure. Thus, the overall system provides a physical illustration of hyperbolic-parabolic coupling. Moreover, coupling takes place at the common interface of the two media where the individual equations evolve and is given by differential operators. In a preliminary step, the structure was taken at first to have static interface, a case justified to be appropriate under the assumption of small, rapid oscillations of the structure [16]; see [1], [2]- [8] [9], [10], [20], [21], [23], [26], [31] for a certainly non-exhaustive list of works. These studies have provided considerable insight in the dynamical properties of the overall coupled system. Moreover, they have revealed the mathematical challenges imposed by the interface (boundary) coupling with mismatched dynamics, parabolic traces versus hyperbolic traces. This preliminary work has thus proved beneficial for subsequent Throughout, Ω f ∈ R 2 will denote the bounded domain on which the heat component of the coupled PDE system evolves. Its boundary will be denoted here as ∂Ω f = Γ f ∪Γ s , Γ f ∩Γ s = ∅, with each boundary piece being sufficiently smooth. Moreover, the geometry Ω s , also 2-dimensional, immersed within Ω f , will be the plate domain on which the structure component evolves with time. The coupling between the two distinct fluid (heat) and elastic dynamics occurs across the boundary interface Γ s = ∂Ω s , see Figure 1 in the previous page.
In addition, the unit normal vector ν(x) will be directed away from Ω f on Γ f , and away from Ω s on Γ s . This choice will allow us to use the setting of [27,Chapter 3], regarding the plate on Ω s , and quote from this reference the relevant formulas.

ROBERTO TRIGGIANI AND JING ZHANG
The computational proof is relegated to Appendix A.
On the space H introduce the following bounded, symmetric operator Then, one may verify the following properties, as in [28, Section 1]: (i) and T A is self-adjoint with domain D(A). and then λ is an eigenvalue of A * with corresponding eigenvector (T e). And conversely.
The proof is given in section 3.
Our main results are as follows: Theorem 2.3. (i) For any fixed 0 < ρ ≤ 1, the generator A in (2.6) and (2.10) of the s.c.contraction semigroup e At asserted by Theorem 2.2 has no spectrum on the imaginary axis, and satisfies the following resolvent condition: there is a constant C > 0 such that Hence, the s.c. contraction semigroup e At is analytic on the finite energy space H, t > 0, [27,Thm 3E.3,p 334].
(ii) Complementing (2.23) we have that the resolvent R(·, A) is uniformly bounded on the imaginary axis for any fixed 0 < ρ ≤ 1: there is a constant C > 0 such that Hence, the s.c. contraction analytic semigroup e At is uniformly exponentially stable on H for any fixed 0 < ρ ≤ 1. There exist constants M ≥ 1, δ > 0, such that [33] e At L(H) ≤ M e −δt , ∀t > 0. (2.25) For an estimate of δ, see r 0 in Theorem 2.2(ii) as well as k in Proposition 4.2.
See Proposition 4.3 for ρ = 0. The proof is given in Section 5.
Remark 2.4. Once estimate (2.23) is established, then the proof in [27, p335] show that in fact, for a suitable M > 0, we have where one may take the angle θ 1 , 0 < θ 1 < π 2 , such that tan( C the constant in (2.23), for an arbitrary fixed constant 0 < κ < 1. We seek the 'largest' possible angle θ 1 < π 2 , at least after moving the vertex of the triangular sector in a nearby point. In our case, this nearby point will be , 0} the angle θ 1 will be arbitrarily close to π 2 . This is contained in the next result, which makes more precise Theorem 2.3 (i) from a spectral analysis viewpoint.
The proof is given in Section 6.

2.5.
A physically more attractive case where the B.C.
In this subsection we briefly deal with the same model (2.1), except that the B.C. in (2.1d) is replaced with the following arguably more physical B.C.: Our conclusion will be that the corresponding generator is still dissipative, however in a different equivalent, yet more complicated, topology. More precisely, for f, g ∈ H 2 (Ω s ), let us introduce the following (well known) symmetric positive bilinear form [27, (3C.71), p310], still for 0 < µ < 1: Then our new energy spaceĤ will be topologized with the inner product instead of (2.4). On the first component space, the inner product (∆v 1 , ∆ v 1 ) Ωs in (2.4) is topologically equivalent to the inner product defined by the form (Ω s ), as called for by our model with B.C. w| Γs = 0 in (2.1c). To justify our statement, in addition to the bending boundary operator B 1 in (2.2), we need to introduce the shear forces boundary operator B 2 , [27, (3C.12), p.300, see also (3C.50) and (3C.54) in terms of only normal and tangential derivatives], Furthermore, we need to recall the following alternative Green formula [27,Proposition C.12,p310]. For f ∈ H 4 (Ω s ), g ∈ H 2 (Ω s ), we have (2.29). We can now state and proof our claimed desired result: Theorem 2.5. Let A be the same operator A as in (2.6)-(2.8) except for incorporating the B.C.
instead of (2.7c). Then A is dissipative on the space H topologized by (2.30). More precisely, we have The Proof is given in Appendix B. We note that the inner product generated by the form a(v 1 , v 1 ) in (2.29) is much more complicated than the inner product (∆v 1 , ∆ v 1 ) in (2.4) 3. Proof of Theorem 2.2.
. From (2.6), recalling the topology (2.4), compute by Green's second theorem on the plate and the Green's first theorem on the heat. We recall that the normal vector ν is outward with respect to Ω s on Γ s and as v 2 | Γs = 0 by (2.7a) and h| Γs = 0 by (2.8).
4. Spectral (Eigenvalues) analysis. We next present an eigenvalues analysis of the operator A. This will be much extended in Section 6, in proving Theorem 2.4, which, in particular, provides information on the entire spectrum of A. The proof here is more direct and much simpler than in Section 6. Specializing the treatment of Section 6, one obtains a different but parallel proof of Proposition 4.1 below, See Remark 6.1. Then where σ p (·) = point spectrum, σ r (·) = residual spectrum. In particular, Proof. We provide a proof for A. The proof for A * is similar, mutatis mutandis, or we simply invoke property (v) of T is Section 2.3. Regarding the last assertion in (4.2) for σ r (A), recall via [34, p282] that if λ = α + iω were in σ r (A), then λ = α + iω would be in σ p (A * ), which is excluded under the LHS of (4.2). Next, inner product the second relation in (4.4) with v 2 , apply Green's second theorem, recall v 2 | Γs = 0 from (2.7a), and obtain (4.6) Next, we recall v 2 = (α + iω)v 1 from (4.4), ∆(v 1 + ρv 2 ) = −ρ(1 − µ)B 1 v 2 + ∂h ∂ν on Γ s from (2.7c), and h = ∂v 2 ∂ν on Γ s from (2.8), to rewrite (4.6) as (4.7) We now sum up (4.5) and (4.7) and obtain after a cancellation of the boundary term ∂h ∂ν , h
Step 1. Given {v * 1 , v * 2 , h * } ∈ H and ω ∈ R\{0}, we first seek to solve the equation The analyticity condition (2.23): there is a constant C > 0 such that is equivalent (since AR(λ, A) = −I + λR(λ, A)) to showing: there is a constant C > 0 such that These constants will depend on 0 < ρ ≤ 1 fixed. In view of (5.2), condition (5.4) is in turn equivalent to showing the following estimate, recalling the topology of H in (2.4) (all norms are L 2 -norm in their respective domains): there is a constant C > 0 such that Below, we shall in fact establish a more precise inequality than (5.5): see the final estimate (5.20).
(ii) Proof of the uniform bound (2.24), i.e. exponential stability. By Theorem 2.2 (ii), 0 ∈ ρ(A). Hence there is r 0 > 0 sufficiently small such that where c depends on r 0 . This estimate, together with estimate (2.23) just established above with ω 0 arbitrarily small yields the uniform bound (2.24). Theorem 2.3 is proved.
Step 1. Given {v * 1 , v * 2 , h * } ∈ H, constants α < 0 and ω ∈ R\{0}, we seek via (2.6) to solve the equation in terms of {v 1 , v 2 , h} ∈ D(A) uniquely with bounded inverse. We have again via 3) The sought-after (analyticity) condition (2.27) is therefore explicitly rewritten as follows via (6.3): there exists a constant C > 0 such that, recalling the topology (2.4) for all λ = α + iω in ρ(A)\K ρ ; that is, outside the set K ρ defined in (2.20) (Figure 2) This estimate (6.4) -that gives analyticity and a much more precise description of the spectrum location for A over (2.26a), which is a consequence of (2.23) -is proved in the present section. The proof is a more delicate extension of the proof of Theorem 2.3 of Section 5.
Orientation (with reference to Figure 4) In the proof of statement (6.4) given below, we shall proceed as follows, after obtaining the basic identities (i) We start with an arbitrarily small positive number r 1 > 0, which we hold fixed throughout. (ii) Henceforth, the points {α, ω}, α < 0, will be further constrained to satisfy the preliminary constraint: |α| > r 1 arbitrarily small; that is, to lie in the half plane α < −r 1 < 0. See below (6.14). This allows one to select a number ε 1 > 0 such that 1 + ε 1 < |α| r 1 .