HEAT–STRUCTURE INTERACTION WITH VISCOELASTIC DAMPING: ANALYTICITY WITH SHARP ANALYTIC SECTOR, EXPONENTIAL DECAY, FRACTIONAL POWERS

. We consider a heat–structure interaction model where the structure is subject to viscoelastic (strong) damping. This is a preliminary step toward the study of a ﬂuid–structure interaction model where the heat equation is replaced by the linear version of the Navier–Stokes equation as it arises in applications. We prove four main results: analyticity of the corresponding contraction semigroup (which cannot follow by perturbation); sharp location of the spectrum of its generator, which does not have compact resolvent, and has the point λ = − 1 in its continuous spectrum; exponential decay of the semigroup with sharp decay rate; ﬁnally, a characterization of the domains of fractional power related to the generator.

1. Introduction and statement of main result.
1.1. Introduction. We proceed to describe the canonical heat-structure PDE model of the present paper: its distinctive feature with respect to past literature on truly fluid-structure models such as [2,3], [4] - [10], [30,35] (involving actually the dynamic Stokes problems in place of the heat equation) is that it possesses a viscoelastic ('strong') damping term of the structure. It is a first study which is preliminarily focused on the boundary (interface) homogeneous problem because of space constraints. It is followed [40] by the ultimate case of interest, where the heat-structure PDE system is subject to a boundary control acting at the interface between the two media: the heat component and the structure component. The control may be either of Neumann-type or of Dirichlet-type, depending on the specific interface condition on which it acts, see (4) below. For this boundary non-homogeneous case, we study in [40] the corresponding (optimal) boundary → interior regularity, from which solution to corresponding optimal control problems, as well as min-max game theory problems will follow as a specialization of the abstract PDE-based theory in [31]. In turn, this project is the first step toward the more realistic fluid-structure PDE model which has the more challenging dynamic Stokes equation in place of the n-dimensional heat equation [32, p. 121], [19]. It will be treated in a subsequent publication. Such model arises in biological applications [12,13]. In the present work, we shall consider the structure immersed in the fluid, while leaving to other efforts the study of the fluid within a structure (such as blood running inside arteries).
The above references study the fluid-structure model (with dynamic Stoke equation involving pressure) with 'static' interface (justified in [19] to be appropriate under the assumption of small, rapid oscillations of the structure). For well-posedness result in the challenging case of moving interface, we quote [22,25].
Thus, throughout, Ω f ⊆ R n , n = 2 or 3, 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 s ∪ Γ f , Γ s ∩ Γ f = ∅, with each boundary piece being sufficiently smooth. Moreover, the geometry Ω s , immersed within Ω f , will be the domain on which the structural component evolves with time. As configured then, the coupling between the two distinct fluid and elastic dynamics occurs across boundary interface Γ s = ∂Ω s ; see Figure 1. In addition, the unit normal vector ν(x) will be directed away from Ω f ; thus on Γ s , toward Ω s . (This specification of the direction of ν will influence the computations to be done below.)

Fig. 1: The Fluid-Structure Interaction
On this geometry in Figure 1, we thus consider the following fluid-structure PDE model in solution variables u = [u 1 (t, x), u 2 (t, x),. . . , u n (t, x)] (the heat component here replacing the usual velocity field), and w = [w 1 (t, x), w 2 (t, x), . . . , w n (t, x)] (the structural displacement field): (IC) [w(0, ·), w t (0, ·), u(0, ·)] = [w 0 , w 1 , The constant b in (1b) will take up either the value b = 0, or else the value b = 1, as explained below. Accordingly, the space of well-posedness is taken to be the finite energy space for the variable [w, w t , u]. (We are using the common notation H s ≡ [H s ] n .) H b is a Hilbert space with the following norm inducing inner product, where (f, g) Ω ≡ Ω fḡ dΩ:  (3b) the first line for b = 0, the second for b = 1. In (2a), the space H 1 (Ω s )/R = H 1 (Ω s )/const is endowed with the gradient norm.
1.2. The non-homogeneous case. As already noted, a companion publication [40] deals with the 'boundary' (interface) non-homogeneous problem, whereby the homogeneous boundary condition (B.C.) in (1e), respectively (1d), will be replaced by the following boundary non-homogeneous conditions, respectively: Here, g is a control function of Neumann-type, respectively, of Dirichlet-type. Writing each problem in the abstract formẋ = Ax + B N g (Neumann case) oṙ x = Ax + B D g (Dirichlet case), with explicit operators B N and B D , reference [40] shows two main preliminary results: (1) that A − 1 2 B N is a bounded operator from functions at the interface measured in the L 2 -norm to the energy space H 0 ; (2) that A −1 B D is a bounded operator from functions at the interface measured this time in the H 1 2 -norm to the energy space H 0 . In the much more challenging Neumann case, the proof that A − 1 2 B N ∈ L(L 2 (Γ s ), H 0 ) requires explicit knowledge of the D((−A) 1 2 ), or at least of a suitable subspace thereof [41]. In Theorem 1.5 below, we give a characterization in fact of D((−A) θ ), 0 < θ < 1. Henceforth, we shall restrict the present paper to the homogeneous case g = 0, while referring to [40] for the case g = 0.
1.3. Abstract model for the free dynamics (1a-f ) (i.e. g = 0 in (4).) The operator A. We rewrite problem (1a-e) as a first-order equation where we introduce the operator A : for {v 1 , v 2 , h} ∈ D(A). A description of D(A) is as follows (we write it explicitly only in the case b = 0, which is the one to which we shall restrict below): We now justify (7a-b). Minimal preliminary conditions for {v 1 , v 2 , h} ∈ D(A) on H b=0 are: as desired, by elliptic theory; hence |∇h| ∈ L 2 (Ω f ). Moreover, the claimed Neumann boundary regularity ∂h ∂ν | Γs ∈ H − 1 2 (Γ s ) follows from the identity with ∆h = f ∈ L 2 (Ω f ), test function ϕ ∈ H 1 (Ω f ), ϕ| Γ f ≡ 0, so that the last two interior integral terms in (9) are well defined, and then so is the boundary term on Γ s with ϕ ∈ H 1 2 (Γ s ). By surjectivity of the trace [34], ϕ runs over all of H 1 2 (Γ s ). Then ∂h ∂ν ∈ H − 1 2 (Γ s ), as desired.
Remark 1.1. The above description of D(A) in (7a-b) shows that the point {v 1 , v 2 , h} ∈ D(A) enjoys a smoothing of regularity by one Sobolev unit-from L 2 ( · ) to H 1 ( · ) -but only of the coordinates v 2 and h, with respect to the original finite energy state space H 0 in (2a). In contrast, the first coordinate v 1 experiences no smoothing: it is in H 1 (Ω s ), the first coordinate component of the space H 0 . This amounts to the fact that A has non-compact resolvent R(λ, A) on H 0 . Consistently, we shall see below in Proposition 2.4 that the point λ = −1 belongs to the continuous spectrum of A : −1 ∈ σ c (A). Henceforth, it will be convenient to focus on the case b = 0, as the case b = 1 is a cosmetic variation of it. The space H b=0 will be henceforth denoted simply by H 0 .

1.4.
Main results of the free dynamics (1a-f ), case b = 0. Our first main result in the homogeneous case (g = 0) is that the operator A in (6)-(7a-b) generates a s.c. contraction semigroup on the energy space H 0 which moreover is analytic (holomorphic) with maximal sector of analyticity, modulo a finite translation. More precisely, the location of the spectrum σ(A) of A in peculiar: conservatively, it is contained in an infinite key-shaped set K: is the open desk centered at the point x 0 = {−1, 0} and of radius 1, and S r0 (0) is the open disk centered at the origin of radius r 0 > 0. In addition, we shall see that the spectrum σ(A) of the operator A, which does not have compact resolvent, contains the point −1 in the continuous spectrum. We refer to Theorem 1.4 and Proposition 2.3. Similar results hold for the adjoint A * of A to be characterized next.
The H 0 -adjoint of the operator A defined in (6) (for b = 0) and (7) is given by with D(A * ) described as follows (compare with D(A) in (7a-b)): The computational proof is given in Appendix A.
On the space H 0 introduce the following bounded, symmetric operator Then, one may verify the following properties: then applying T on both sides and recalling T Ae = A * T e by (16) yields and then λ is an eigenvalue of A * with corresponding eigenvector (T e). And conversely, Theorem 1.2 (Generation by A and A * ). (i) The operator A defined by (6), (7) and its adjoint A * given by (11), (12) are dissipative: (15)), we have, complementing (17b): in the L 2 ( · )-norms of Ω s and Ω f .
in the L 2 (·)-norms of Ω s and Ω f . (iii) The operator A is boundedly invertible on H 0 : A −1 ∈ L(H 0 ). (A more precise result is given in Lemma 2.1 below.) Thus, there exists a disk S r0 of the complex plane C, centered at the origin and of suitably small radius r 0 > 0 such that S r0 ⊂ ρ(A), the resolvent set of A. Similarly for the operator A * .
(iv) Hence, A is maximal dissipative on H 0 . By the Lumer-Phillips Theorem, A generates a s.c.
(v) The same generation results hold also for A * .
The proof is given in Section 2.
(b) The interior regularity in C([0, T ]; ·) in (28)- (30) is an explicit restatement of (23). The interior regularity in L 2 (0, T ; ·) in (29), (30) is a consequence of identity (27), whereby then the Dirichlet boundary regularity of u| Γs in the LHS of (21) follows by trace theory. In turn, such result on u| Γs , used with the property that t 0 Γs ∂u ∂ν u dΓ s ds ∈ L 2 (0, T ) yields then the Neumann boundary regularity in (31). 1.5. Orientation on analyticity. A first main result of the present paper is Theorem 1.4 below that claims that the s.c. semigroup e At asserted by Theorem 1.2 is, in fact, analytic on the space H 0 , and in fact, with the triangular sector containing the spectrum σ(A) of its generator A that reduces itself to the infinite axis (−∞, −2) for Reλ < −2. A more precise statement on the location of the spectrum σ(A) of A is given in (10).
Analyticity per se is not surprising in view of the following motivating considerations.
A motivating result. (a) Analyticity. The following is a very special case of a much more general result (noted below) for which we refer to [15], [16], (see also [31, Appendix 3B of Chapter 3, pp 285-296], [17], [18]). These references solve and improve upon the conjectures posted in [14]. Let A be a positive, self-adjoint operator on the Hilbert space Y . On it, consider the following abstract equation The operator A is dissipative and with domain (33b) is closed and generates a s.c. contraction semigroup e At on the finite energy space E ≡ D(A 1 2 ) × Y , which moreover is analytic on E. Thus, the second-order dynamic (32) with strong 'structural' damping is parabolic-like. Indeed, [15], [16], (see also [31, Appendix 3B of Chapter 3, pp 285-296]) show the more general result, and more useful in application to mixed PDEs-problems, that analyticity holds true if in equation (32) the damping term Aẋ is replaced by Bẋ, where B is another positive self-adjoint operator (which needs not commute with A) which is comparable with A α , 1 2 ≤ α ≤ 1, in the sense of inner product: shows that the spectrum σ(A) of the operator A defined in (32)-(33b) (case α = 1) has the following features assuming that the positive self-adjoint operator A has compact resolvent on Y : The spectrum of A consists of two branches of eigenvalues λ +,− n : solutions of the algebraic equation λ 2 + µ n λ + µ n = 0, where {µ n } ∞ n=1 are the eigenvalues of the positive self-adjoint operator A : 0 < µ 1 < · · · < µ n → +∞. The branch λ − n −∞ monotonically. The branch λ + n −1 monotonically. Moreover, the point λ = −1 belongs to the continuous spectrum σ c (A) of the operator A. The operator A does not have compact resolvent on the finite energy space E, even though A has compact resolvent on Y .
By looking at the operator A in (6), the above abstract result for equation (33a) suggests, or makes one surmise, that the homogeneous problem (1a-e) is the coupling of 'two parabolic problems' and hence generates an analytic semigroup e At (A in (6), (7)) on the finite energy space H b in (2). Of course, the above considerations are purely indicative and qualitatively suggestive, as the Laplacian ∆ in (1b) has coupled, high-level, non-homogeneous interface boundary conditions which constitute the crux of the matter to be resolved before making the assertion of analyticity of problem (1). At any rate analyticity cannot follow by a perturbation argument.
We have already noted in Remark 1.1 that the operator A in (6), (7) does not have compact resolvent on the finite energy space H 0 . We shall see in Proposition 2.4 below that λ = −1 is a point in the continuous spectrum of the operator A: −1 ∈ σ c (A). This result coupled with the location of the spectrum of A anticipated that, qualitatively, the spectrum of A is like the spectrum of the operator A in (33a), with one branch of eigenvalues being negative and going to −∞, and the other branch going to the point −1 of the continuous spectrum.
Our first main result is contraction semigroup e At asserted by Theorem 1.2 satisfies the following resolvent condition with ω ∈ R. Hence, the s.c. semigroup e At is analytic on the finite energy space (ii) More precisely, the resolvent operator R(λ, A) = (λI − A) −1 of the generator A in (6), (7) (with b = 0), satisfies the following estimate (iv) Complementing (35) we have that the resolvent R( · , A) is uniformly bounded on the imaginary axis Hence, the s.c. analytic semigroup e At is uniformly exponentially stable on H 0 : there exist constants M ≥ 1, δ > 0, such that [38] e At L(H0) ≤ M e −δt , t ≥ 0.
For s.c. analytic semigroups such as e At , domains of fractional powers of (−A) are very important. For instance, a special subcase of the following result allows one to obtain optimal regularity results of the non-homogeneous mixed problem (1a-d) with forcing term g ∈ L 2 (0, T : L 2 (Γ s )) acting at the interface Γ s in the Neumann B.C. as in (4) (LHS).
A proof is given in Section 4. We note that the well-known sufficient condition of [33, Theorem 6.1, p238] implying D((−A)  (6), (7), with strongly coupled boundary conditions, as it arises in the original physical model. Such property of coincidence is usually referred to as Kato's problem. It is well-known that in 1961, T. Kato [23] showed that for a maximal accretive operator Q, one always has D(Q θ ) ≡ D(Q * θ ) for θ ∈ [0, 1/2). He also conjectured that the coincidence should also attain for θ = 1/2 for regularly accretive operators. One year later, J.L. Lions [32], page 240, provided a counterexample for θ = 1/2, by considering a first order differential operator Q 1 = d dx on L 2 (0, ∞), with boundary condition u(0) = 0. Then such accretive operator Q 1 has D(Q  (40)), define the sesquilinear form which is continuous on the space V defined in (40). The RHS of (41) is the opposite of the RHS of identity (45) below, which will be derived for Φ = {v 1 , v 2 , h} ∈ D(A) and Φ ∈ V . By density and continuity, we can extend the validity of identity (45) to all of Φ ∈ V as well, and thus write In particular, recalling (21), we can write for so that the sesquilinear form a(·, ·) is regular accretive [33, p233, footnote]. However, as noted above, the setting of [32] on bilinear forms and trace spaces cannot be invoked to obtain the characterizations (40).
Refer also to Remark 1.2.
(iii) If we specialize further and let e = [v 1 , v 2 , h] ∈ D(A) be a normalized (in H 0 ) eigenvector of A corresponding to the eigenvalue λ : Ae = λe, then λ = RHS of (46a); and λ need not be real.
In this section, we shall establish a resolvent estimate such as (78a) for all λ ∈ C\K, namely K being the infinite key-shaped set defined in (36b); whereby, moving the vertex of the triangular sector of analyticity to concide with the point x 0 = {−1, 0}, the corresponding angle θ 1 is arbitrarily close to π 2 (Fig 3.1). (b) Then (79) and S r0 ⊂ ρ(A) will imply (Section 3.2) that the real part of the spectrum σ(A) of A is confined inside the negative axis (−∞, −r 0 ]. The direct passage from (79) to (78) is exhibited in Remark 3.1 below. Moreover, our proof below, once specialized with Re λ = 0, λ = iω, ω ∈ R, will yield (through simplified computations in Remark 3.2 below) the establishment of inequality (77) for any ω 0 > 0. This result, combined with S r0 ⊂ ρ(A) will allow us to conclude that Then, (80) will imply [38] uniform stabilization of the analytic semigroup e At : there exists constants M ≥ 1, δ > 0 such that and hence that In conclusion, the present section establishes three results: (a) analyticity of the semigroup e At ; (b) location of the spectrum σ(A) of A, in Subsection 3.1; and (c) exponential stability (81a) in Subsection 3.2.
Remark 3.2 (Specialization to the case α = 0). We specialize the above computations to the case α = 0, λ = iω, to obtain: (a) The counterpart of identity (92) (real part) is which then yields the counterpart of estimate (94).
where (117b) can also be obtained through the argument in Remark 3.1 (with α = 0). Theorem 1.4(i) is established.