Rational decay rates for a PDE heat--structure interaction: Afrequency domain approach

In this paper, we consider a simplified version of a fluid--structure PDE 
model 
---in fact, a heat--structure interaction PDE-model. It is intended to be a first step toward a more realistic fluid--structure PDE model which has been of longstanding interest within the mathematical and biological sciences [33, p. 121], [17], [19]. This physically more sound and mathematically more challenging model will be treated in [13]. The simplified model replaces the linear dynamic Stokes equation with a linear $n$-dimensional heat equation (heat--structure interaction). The entire dynamics manifests both hyperbolic and parabolic features. Our main result is as follows: Given smooth initial data---i.e., data in the domain of the associated semigroup generator---the corresponding solutions decay at the rate $o( t^{-\frac{1}{2}}) $ 
(see Theorem 1.3 below). The basis of our proof is the recently derived resolvent criterion in [15]. In order to apply it, however, suitable PDE-estimates need to be established for each component by also making critical use of the interface conditions. A companion paper [6] will sharpen Lemma 5.8 of the present work by use of a lengthy and technical microlocal argument as in [26,29,30,31], to obtain the optimal value $\alpha =1$; hence, the optimal decay rate $o(t^{-1})$. See Remarks 1.2,1.3.

1. Introduction and statement of main result. Introduction. We proceed to describe the canonical heat-structure PDE model of the present paper. This 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 in (1.1a) below [33, p. 121], [19]. It will be treated in a subsequent publication [13]. The presence of the pressure is responsible for significant additional mathematical challenges already at the level of establishing semigroup well posedness [7], [10], [4]. For the present problem of rational decay, the present treatment of a simplified model offers a strategy, a template to be followed also in the original fluid-structure model. Due to the presence of the pressure, however, serious additional mathematical challenges need to be overcome which require a lengthy 234 GEORGE AVALOS AND ROBERTO TRIGGIANI technical treatment [13]. Such reference yields the same decay o(t − 1 2 ) as in the present paper. 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 , and so 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 heat-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 fluid velocity field), and w = [w 1 (t, x), w 2 (t, x), . . . , w n (t, x)] (the structural displacement field): Semigroup well-posedness. As was done in [5], [7]- [11], [4] for a fluidstructure system in which Stokes flow is used to describe the fluid component of the dynamics, one can provide a non-trivial semigroup formulation so as to describe the time-evolving PDE model (1.1a)-(1.1f). In fact (as a very special case of the above references, as no pressure term appears now), one can define a modeling generator A : H → H as follows: 3) The structural component w 0 satisfies ∆w 0 ∈ L 2 (Ω s ). In consequence, elliptic theory provides that ∂w0 ∂ν Γs is well-defined as an element of [H − 1 2 (Γ s )] n ; see e.g., [22, p. 71 (1.5) (A.6) The components [u 0 , w 0 ] obey the following relation on the boundary interface Γ s : In regard to well-posedness, one can proceed as in [7]-or as in [4] and [9], these being inf-sup Babuska-Brezzi approaches to well-posedness-to establish the following: (1.7) in fact, maximal dissipative, and thus it generates a contraction C 0 -semigroup e At t≥0 on H. Thus, given (1.8) Moreover, the fluid component satisfies the additional regularity, u ∈ L 2 (0, T ; H 1 (Ω f )).
(iii) The resolvent operator R(λ, A), λ ∈ C + is not compact on the state space H. More precisely, the component of The semigroup e At is strongly stable on H: e At x → 0 as t → +∞, ∀ x ∈ H. Remark 1.1. If one replaces the interface condition (1.1e) by the following dissipative condition: ∂u ∂ν = ∂w ∂ν − w t in (0, T ) × Γ S , then the corresponding problem still generates a s.c. semigroup e A d t which, moreover, is now uniformly (exponentially) stable in L(H): The counterpart of this uniform stabilization result for the definitely more challenging cases of actual fluid-structure or Stokes-Lamé models is proved by PDE methods in [8], [12], respectively, without assumed geometrical conditions on Ω s .
Statement of main result. Part (ii) of Theorem 1.1 allows for the inference of strong decay of the fluid-structure model (1.1a)-(1.1f), as asserted in Part (iv). This can be done by invoking [2], [35], or else [16], see [7] and also [10]; while the Nagy-Foias-Foguel approach [32] fails because of the lack of compactness inferred in part (iii). The main topic of the present paper deals rather with the more advanced notion of rational decay of the semigroup e At (which readily then implies its strong stability, as D(A) is dense in H). To achieve this result, one may seek to pursue either an analysis in the t-domain by using suitable energy methods [36], [40, Thm. 6.2, p. 694], or else an analysis in the λ-domain. In this paper we will follow the second approach. More precisely, our main result of rational decay for solutions of (1.1a-f) will ultimately invoke the following operator-theoretic (and sharp) recent result in [15]. [15].) Let {T (t)} t≥0 be a bounded C 0 -semigroup on a Hilbert space H with generator A such that iR ⊂ ρ(A). Then for fixed α > 0 the following are equivalent: (1.10) In order to apply Theorem 1.2 to problem (1.1a-f), we shall need to establish suitable PDE-estimates for each component of the system, by also making critical use of the interface conditions (1.1d-f). In the present paper, we shall succeed to apply Theorem 1.2 with α = 2 to our original semigroup e At of Theorem 1.1. [The exponent α = 2 is not optimal, see Remark 1.3 below.] As a consequence, we will have the following result of rational decay. The main result of this paper is: , the corresponding solution [w, w t , u] of (1.1a)-(1.1f ) obeys the following decay rate for large time t > 0:  , t > 0, t → ∞. (1.11)

237
Remark 1.2. To our knowledge e.g., [37], Theorem 1.2 has so far been employed to obtain rational decay of solutions only in the case of some 1-dimensional, single (uncoupled) PDEs, using also a spectral analysis, in particular a Riesz-basis property available for such 1-d models, e.g., [1]. The present paper appears to be the first one where such λ-analysis ultimately based on Theorem 1.2 is successful in the case of complicated multi-dimensional systems of two strongly coupled (at the interface) PDEs of different type. Thus, the analysis provided here may serve as a template for other complicated multi-dimensional systems of coupled PDEs.
In contrast, the literature on polynomial/rational decay is mostly based on a t-domain analysis, by using energy methods. A precursor paper which we wish to recall is paper [36], reported also in [40, Thm. 6.2, p. 695]. More recent works include [38], [39], etc. However, a first relevant reference to our present paper in [45], where a t-analysis for the same heat-wave system (1.1a-f) produces the rational decay (1.9) with α = 6, i.e., with rate (t − 1 6 ) [45, Eqn. (7.3), Thm. 11, p. 88]; that is, " 1 3 ," worse than in the present Theorem 1.3. This result of [45] is improved to α = 1 − 2 , > 0, ie with rate (t −(1− ) ) in [20], again by using a time-domain approach. Remark 1.3. As noted above the statement of Theorem 1.3, our result with α = 2 is not optimal. The optimal parameter is, in fact, α = 1, yielding therefore the optimal decay rate (t −1 ) instead of (t − 1 2 ) in (1.10). The treatment of the present paper is optimal up until the analysis culminating with Lemma 5.7. This is noted in Remark 5.1. To obtain the optimal parameter one needs to revisit estimate (5.38) of Lemma 5.8 involving the "last term" of the analysis ∂w ∂ν , ∂f3 ∂ν Γs . In a subsequent paper [6]-which is built upon the present one up to Lemma 5.7-we shall privde the noted improvement to α = 1. This, however, will require performing a lengthy technical microlocal analysis argument as in [26,29,30,31]. Space restrictions do not permit it to include this argument here. We do not believe that it is possible to obtain α = 1 unless microlocal estimates are used in estimating the term in Lemma 5.8.

The resolvent equation and orientation of the proof of Theorem
From the definition of the domain for (1.4), and as noted for the Stokes-wave system of [7, p. 28], the relation (2.1) gives the explicit relations: Orientation of the proof of Theorem 1. 3.
∈ H as above in (2.1), we introduce the notation Our goal and guiding idea in proving Theorem 1.3 will be as follows: We seek the lowest positive number α and a positive number σ, with 0 ≤ σ ≤ 2α, such that (2.6) Indeed, once the estimate (2.6) is established in Proposition 6.2 below-by (6.5), with α = 2 and σ = 1 2 -the proof of Theorem 1.3 will be completed via the following argument.
3. An auxillary system and corresponding static energy identity. We start by introducing the "Dirichlet" map on Ω s [27, p. 181], [34]: Following past strategies employed in the theory of boundary control-see e.g., [42], [25,26,27,28]-we seek to homogenize the v 1 -boundary value problem (2.3ab). To this end, we introduce a new variable so that it satisfies the following two homogeneous boundary value problems: A bound on ω * will be given in (4.3) below. As such, we have by elliptic theory (see [34]) that with continuous dependence on the data. (Note that, in contrast, the solution of (2.3a-b) satisfies only v 1 ∈ H 1 (Ω s ) and ∂v 1 /∂ν ∈ H − 1 2 (Γ s ).) Thus, via componentwise application of Proposition A.1 of the Appendix, the following energy identities are suitable for the ω-problem (3.3a-b), rather than the v 1 -problem (2.3a-b). To this end, we recall the following conventional notation [41].
Convention. ∇ω = n × n matrix whose i th column is ∇ω i ; for an n-vector h, then h · ∇ω = n × n matrix whose i th column is h · ∇ω i . If A = (a jk ) n j,k=1 , B = (b jk ) n j,k=1 are n × n matrices, then we set A · B = j,k=1 a jk b jk , while AB is written for the usual matrix product.
semidefinite Jacobian matrix H(x), then the solution of the homogeneous problem on the LHS of (3.3a-b) satisfies the following identities:  We next specify vector field h( where the interior terms (IT) are given by Ωs ω * · ωdΩ s + Re We note that all the terms on the RHS of (3.9) and (3.10a) are well defined as L 2 -inner products, as ω ∈ H 2 (Ω s ), while the last integral term on the RHS of (3.10b) is to be interpreted as a duality pairing between ω ∈ H 1 0 (Ω s ) and h·∇ω * ∈ H −1 (Ω s ), as ω * ∈ L 2 (Ω s ) by (3.4).
The estimate (3.13) for |(IT)| is sufficient for the time being; it will be taken to completion in Section 4, Eqn. (4.8). The heart of the matter in this work is Section 5, which deals with an appropriate estimation of ∂ω ∂ν L 2 (Γs) .

4.
A preliminary estimate. The goal of this section is to establish the following basic estimate: The solution variables f and ω of (2.4a-b) and (3.3a-b), respectively, satisfy say for |β| ≥ 3:
Step 2. Applying Lemma 4.1(i) to the right-hand side of the relation (3.13) for the interior term (IT), we obtain the following corollary.
Step 4. To conclude the proof of Proposition 3, we add the term f 2 L 2 (Ω f ) + ∇ω 2 L 2 (Ωs) to both sides of the inequality (4.7). On the RHS of the resulting inequality, we invoke estimate (4.9) for ∇ω 2 , as well as the Poincaré Inequality so as to estimate f In this way, upon taking > 0 small enough, we obtain the sought-after inequality (4.1) of Proposition 4.1.
The next result will be invoked in Section 5 twice, in obtaining estimates (5.37) as well as (5.39).
5. An estimate for ∂ω ∂ν 2 L 2 (Γs) . With reference to estimate (4.1), the objective of the present section is to derive the following estimate: The solution variables f and ω of (2.4) and (3.3), respectively, satisfy the estimate, say for |β| ≥ √ 3: Proof of Theorem 5.1(i). Step 1. It will be critical in our analysis to split the solution of the fluid system (2.4a-b) into three components; i.e., where the f i satisfy the respective problems (here again, ∂Ω f = Γ s ∪ Γ s ): Step 2. The relevant information concerning the three problems in (5.4) are given next.
The following estimate holds (an improvement over a straightforward majorization obtained by Poincaré's Inequality): Proof. (a)(i) Using the f 1 -equation in (5.4) we have after invoking the fluid system (2.4). Applying Green's First Theorem to both sides of the relation, and using f 1 | ∂Ω f = 0 will then yield via Poincaré Inequality, which establishes (5.5) by using 2ab iβ . Hence (after also using Poincaré's Inequality via (5.4b)). But In addition, a direct elliptic estimate for the f 1 -problem in (5.4) is

DECAY RATES FOR A PDE HEAT-STRUCTURE INTERACTION 245
Substituting the estimate (5.12) into the right-hand side of (5.13), followed by the estimate (5.5), we then have for |β| ≥ 1, This is the estimate (5.6).
By a immediate application of elliptic and Sobolev Trace Theory, we also have the following lemmas.
Lemma 5.7. Regarding the fourth term on the right-hand side of (5.28), we have
Remark 5.1. Up to this point, that is, neglecting for the moment the third term ∂ω ∂ν , ∂f3 ∂ν Γs on the RHS of (5.28), then according to (2.6) in the orientation in Section 2, the analysis so far carried out would yield the value of the parameter in Theorem 1.2 to be α = 1 2 , as dictated by (5.29); while (5.33) and (5.34) would yield "α = 0." This is seen by substituting estimates (5.29), (5.33), (5.34) on the RHS of identity (5.28) [with the above third term neglected] and using the resulting estimate for ∂ω ∂ν L 2 (Γs) on the RHS of the basic estimate (4.1) of Proposition 4.1.
It is the estimate of the third so far 'neglected' term-the one in the subsequent Lemma 5.8-which is responsible for obtaining α = 2 in the sought-after estimate (2.6), when substituting the estimate for the RHS of (5.28) in (4.1). Lemma 5.8 below will be sharpened in [6] by use of a lengthy and technical microlocal argument as in [26,29,30,31] to ultimately obtain α = 1, the optimal value.
Lemma 5.8. Regarding the third term on the right-hand side of (5.28), we have which establishes (5.38).
This concludes the proof of Theorem 5.1.

6.
Completion of the proof of Theorem 1.3: From ω back to v 1 .
This concludes the proof of Theorem