Regularity of boundary traces for a fluid-solid interaction model

We consider a mathematical model for the interactions of an elastic body fully immersed in a viscous, incompressible fluid. The corresponding composite PDE system comprises a linearized Navier-Stokes system and a dynamic system of elasticity; the coupling takes place on the interface between the two regions occupied by the fluid and the solid, respectively. We specifically study the regularity of boundary traces (on the interface) for the fluid velocity field. The obtained trace regularity theory for the fluid component of the system -- of interest in its own right -- establishes, in addition, solvability of the associated optimal (quadratic) control problems on a finite time interval, along with well-posedness of the corresponding operator Differential Riccati equations. These results complement the recent advances in the PDE analysis and control of the Stokes-Lam\'e system.


Introduction
The description of the interactions between a fluid flow and an elastic structure via an interface is a classical topic in continuum mechanics. Motivated by an impressive variety of fields of application, such as, e.g., cell biology, biomedical engineering, ultrasound imaging, in the past decade fluid-structure interaction models have received a renewed attention. Numerical and experimental studies have pursued the derivation of appropriate PDE models for specific physical problems, along with the understanding of the corresponding dynamics; see, e.g., [26] and [25]. On the other hand, basic mathematical questions such as well-posedness (in a natural functional setting) of the systems of coupled partial differential equations (PDE) which result from the modeling of these interactions, have been explored and answered only recently.
Let us recall that a major technical issue which arises in the PDE analysis of fluid-solid interaction models is the apparent discrepancy between the regularity of boundary traces for the fluid and the elastic (hyperbolic) components on the interface, with the latter not defined a priori by means of standard trace theory. Different solutions have been proposed to cope with the mismatch between parabolic and hyperbolic regularity. For instance, in [9] a regularizing term is included in the elastic equation, thereby ensuring the soughtafter boundary regularity, as a consequence of the improved interior regularity; the works [13] and [15] consider instead very smooth data, thus obtaining only local (in time) existence. In contrast with previous work, the study of [6] achieves-at least in the case of static interface-well-posedness of an established nonlinear fluid-structure interaction model in the natural energy space, along with the exceptional yet intrinsic regularity of the boundary traces of the (elastic) hyperbolic component of the system on the interface. (To the authors' best knowledge, a similar well-posedness result for the more challenging case of moving interface is, at present, lacking.) The composite PDE system under consideration, which is problem (2.1) in the next section, is a linearized version of the nonlinear system studied in [6,7]; from a modeling point of view, the linear evolution is consistent with the case of very slow motion of the fluid. The boundary value problem comprises a linearized Navier-Stokes system (often referred to as Oseen equations) for the velocity field u of the fluid flow and a dynamic system of elasticity for the displacement w of the solid. The interaction between the two means takes place at an interface between the two regions occupied by the fluid and the solid, respectively. We consider the case in which the motion of the solid is entirely due to infinitesimal displacements, and therefore the fluid-solid interface, later denoted by Γ s , is stationary. (Yet, although the displacement u is small, the velocity u t is not, resulting in the interface condition u = w t .) While the PDE problem (2.1) is present in the literature since the late sixties (see [23]), its well-posedness was not shown until [6], whose theory encompasses the present model. In the case of the (linear) Stokes-Lamé system, the contemporary work of [3,4,5] provides not only well-posedness, but also a careful stability analysis. A very nice overview of recent (theoretical as well as numerical) contributions to this topic is found in [14], whose introduction also provides insight into modeling aspects; see the references therein.
The present study focuses on the regularity of boundary traces for the velocity field of the fluid flow u (and u t ) on the interface Γ s . Although the obtained regularity properties are of interest in their own right, they are also critical in the study of the optimal boundary control problems (with quadratic functionals) associated with the fluid-solid interaction (2.1), in the presence of active controls on the interface Γ s , thus constituting its main motivation as well as application.
It was shown in [21] that the fluid-structure interaction model under consideration yields a suitable ('singular') estimate for the corresponding abstract evolution, which plays a major role in the study of the associated optimal control problems. Indeed, in light of the regularity result obtained in [21,Theorem 5.1], the optimal control theory in [17,19,20] applies, ensuring a feedback control law with a bounded gain operator (defined on the state space), along with well-posed Riccati equations. This, however, holds true assuming that the functional penalizes the quadratic energy of the solid component in a weaker topology than the actual energy level (this is a consequence of the fact that the singular estimate holds in the norm of a function space H −α which is larger than the energy space).
A first attempt to remove the constraint on the structure of the quadratic cost functional-which corresponds to a constraint on the observation operator-has been successfully carried out in [12] in the case of the simpler Stokes-Lamé system. The PDE analysis performed in [12] shows indeed that the singular estimate in [21,Theorem 5.1] does not hold for α = 0 and hence that the optimal boundary control theory of [17,20] is inapplicable to the case of natural cost functionals such as the integral of the quadratic energy of the system. Nonetheless, [12] provides novel regularity results which ensure solvability of the associated optimal control problems, with well-posed Riccati equations, according to the distinct (optimal control) theory contained in [2].
In the present paper we show that the complex of boundary regularity results established in [12] for the uncontrolled Stokes-Lamé system may be extended to more general fluid-structure interaction models comprising a linearized Navier-Stokes (rather than the Stokes) system for a slow, viscous, incompressible fluid (see (2.1) below). As we shall see, the regularity of boundary traces for the fluid component of the system will follow combining the exceptional boundary regularity of the hyperbolic component of the system (cf. Lemma 2.7), the regularizing effect of the parabolic component, and interpolation techniques. The same factors are behind the proof of the specific regularity estimates recently derived for diverse composite PDE systems, including thermoelastic systems ( [11,1]) and acoustic-structure interaction models ( [10]).
An outline of the paper follows below. In Section 2 we introduce the coupled PDE system, along with the natural functional setting where well-posedness holds true, in both the variational and semigroup sense. The statement of the present work's main contribution, namely Theorem 2.6, which collects several boundary regularity results for the fluid velocity field of problem (2.1), is given here. Section 2.4 contains some preliminary trace regularity results which are crucially instrumental to the proof of Theorem 2.6. Section 3 is entirely devoted to the proof of all the assertions of Theorem 2.6. Application of the obtained regularity theory to the associated optimal control problems is discussed in Section 4, which concludes the paper.

The mathematical model, statement of the main result
In this section we introduce the PDE system under investigation. We preliminarly recall from [6] the well-posedeness and regularity theory which constitutes the basis of our analysis. Theorem 2.6 contains the statement of the novel regularity results established for the fluid component of the system.

The PDE problem
Let Ω f and Ω s be the open smooth (three-dimensional) domains occupied by the fluid and the solid, respectively. The boundary of Ω s represents the interface between the fluid and the solid, and is denoted by Γ s = ∂Ω s ; Γ f is the outer boundary of Ω f , that is Γ f = ∂Ω f \ ∂Ω s . It is assumed that the motion of the solid is entirely due to infinitesimal displacements, and hence that the interface Γ s is stationary. The PDE system under investigation comprises a linearized Navier-Stokes system for the fluid velocity u in Ω f , and an elastic equation for the displacement w of the solid in Ω s . The coupling takes place on the interface Γ s ; the case of small, but rapid oscillations of the elastic body is considered, which yields the interface condition u = w t on Γ s in lieu of the no-slip boundary condition u = 0. (For a detailed discussion of modeling issues see, e.g. [14].) Thus, if p denotes the fluid pressure, the pair (u, w, p) satisfies In the above PDE model ǫ and σ denote the strain tensor and the elastic stress tensor, respectively, that are where λ, µ > 0 are the Lamé constants and δ ij is the Kronecker symbol. The (uncoupled) linear system for the fluid velocity field, often referred to as Oseen equations, is the linearization of the Navier-Stokes equations about a given (time-independent) smooth vector valued function v such that div v = 0; see, e.g., [16]. Thus, the operator L is defined as follows:

Functional setting, well-posedness
The energy space for the PDE problem (2.1) is where H is defined as follows: where the divergence is meant in a distributional sense. P H will denote the Leray's projection operator from L 2 (Ω f ) onto H; then, L = P H L. In addition, V denotes the space defined as follows: The following distinct inner products will be used throughout the paper: The space V is topologized with respect to the inner product given by the corresponding (induced) norm | · | 1,f is equivalent to the usual H 1 (Ω f ) norm, in view of Korn inequality and the Poincaré inequality. The norm · H r (D) in the Sobolev space H r (D) will be denoted by | · | r,D throughout.
Remark 2.1. Note that all the Sobolev spaces H r related to u and w are actually (H r ) 3 : the exponent is omitted just for simplicity of notation.
Let us recall from [6] the definition of weak solutions to the (uncontrolled) PDE system (2.1).
Existence of weak (global) solutions to the PDE problem (2.1) follows by a more general-because pertaining to the actual nonlinear model, which comprises the Navier-Stokes system-result established in [6].

Remark 2.3.
It is important to emphasise that the regularity of boundary traces of the elastic component required by Definition 2.2 does not follow from the interior regularity (u, w, w t ) ∈ C([0, T ], H × H 1 (Ω s ) × L 2 (Ω s )) via standard trace theory. This is an independent regularity result, valid for any solution corresponding to initial data in the finite energy space; see the statement of Theorem 2.4 below.
Theorem 2.4 (Existence of weak solutions, [6]). For any initial datum (u 0 , w 0 , z 0 ) ∈ Y and any T > 0, there exists a weak solution (u, w, w t ) to system (2.1), along with the following trace regularity for the elastic component of the system: It is important to recall from [6] that by properly defining the overall dynamics operator A, the PDE system (2.1) admits as well a semigroup representation. Preliminarly, one needs to introduce the fluid dynamic operator A : and the (Neumann) map N : L 2 (Γ s ) → H defined as follows: Then, by using the operators introduced above, the initial/boundary value problem (2.1) becomes the following abstract differential system in the variable y := (u, w, w t ) where It was shown in [6, Proposition 3.1] that the operator A : A major achievement of the study performed in [6] is that it gives, in addition, that any semigroup solution turns out to be as well a weak solution of system (2.1). In particular, it satisfies the variational formulation (2.4), along with the intrinsic boundary regularity of the stresses on the interface. A different semigroup set-up, which does not require-at the level of (semigroup) generalized solutions-the present boundary regularity of the normal stresses, has been developed in [3]; see also [4,5].
Remark 2.5. We finally recall from [21] that the operator A defined by (2.5) may be considered as acting on H with domain D( Readily A : D(A) ⊂ H → H is a self-adjoint, negative operator and therefore it generates an analytic semigroup e At on H. Then, the fractional powers of −A are well defined. A well known result relating (−A) α and the semigroup generated by A is the pointwise (in time) estimate, true for any 0 < α ≤ 1: (2.10) (To simplify the notation, we shall write A α , rather than (−A) α , throughout; similarly, later A α will mean (−A) α .)

New results: regularity of boundary traces for the fluid component
The main contribution of this work is a set of regularity results for the traces of the fluid velocity field u (and u t ) on the interface between the fluid and the solid regions, collected in Theorem 2.6 below. While these regularity estimates are of intrinsic interest, they additionally play a major role in the study of the quadratic optimal control problems associated with the PDE system under investigation, in the presence of control actions on the interface. This fact will be briefly discussed in Section 4.
Theorem 2.6 (Main result: boundary regularity for the fluid component). Let y(t) = (u(t), w(t), w t (t)) be the solution to the coupled PDE system (2.1) corresponding to an initial state y 0 = (u 0 , w 0 , z 0 ). The following assertions pertaining to the regularity of boundary traces for the fluid velocity field u (and u t ) are valid.
for arbitrarily small h > 0. More precisely, u admits a decomposition

12)
where u 1 satisfies a pointwise (in time) "singular estimate", namely there exists a positive constant C T such that
(2.18) 5 that for simplicity of notation we write A α instead of (−A) α .) It is this latter formulation, more appropriate in the present context, to require some adjustment of the arguments used for the proof of Lemma 5.2 in [21]. The technical details are the same as the ones in [6,21], if one takes into account that the solutions considered are generated by the semigroup flow. However, for the reader's convenience the proof of Lemma 2.7 follows below.
Moreover, the following estimates hold true.
Proof.  .21) involves instead initial data whose regularity is 'above' the energy level, and it is thus expressed in terms of the domains of fractional powers of the generator A (rather than fractional Sobolev spaces as in [21]). In order to establish (2.21) we appeal, as in [21], to interpolation methods. Accordingly, it will suffice to prove (2.21) for α = 0 and α = 1. The regularity result corresponding to α = 0 has been shown in [6, Theorem 3.3]; thus, it remains to analyze the case α = 1. Note that (2.21) with α = 1 provides H 1 (Σ s )-regularity of the component σ(w 2 ) in terms of Ay 0 H + f H 1 (Σs) . As in [21] and [6], we exploit the fact that w 2 = χw, where χ is supported in the microlocal hyperbolic sector; see [6,Section 5]. This means that the tangential boundary regularity is implied by the time regularity alone. In other words, H 1 (Σ s )-regularity is equivalent to H 1 (0, T ; L 2 (Γ s )) for functions supported in the hyperbolic sector; see [6, Section 5] for more details.
The sharp results of Lemma 2.7 yield the following improved boundary regularity for the fluid velocity field. Lemma 2.8. Consider the PDE problem (2.1), and take initial data y 0 = (u 0 , w 0 , z 0 ) ∈ D(A α ), 0 ≤ α < 1 4 . Then, for any T < ∞ we have u| Γs ∈ H α (Σ s ) and the following estimate holds true: The proof of this Lemma is identical with the one given in [21, Lemma 5.3], after taking into consideration the estimates derived in Lemma 2.7.

Proof of the main result
This section is devoted to the proof of Theorem 2.6. As we shall see, the regularity of boundary traces for the fluid component of the system will follow combining the exceptional boundary regularity of the hyperbolic component of the system (Lemma 2.7), the smoothing properties of the (fluid) analytic semigroup, and the theory of interpolation spaces.
To begin with, let us recall from [21] the basic regularity of the Neumann map, which will used repeatedly in the proof of our main result. This Lemma is well known in the context of classical elliptic equations, where N is the associated Green (Neumann) map. The same result has been proved for the abstract operators (defined by (2.5) and (2.6)) which arise in the PDE problem under investigation; see [21,Proposition 4.3]. For the reader's convenience we recall few details.
The composition N * A (N * being the L 2 -adjoint of N ) is computed from Since u| Γs is bounded on V = D(A 1/2 ), the above identity can be extended by density to all of V , even more, to D(A 1/4+δ ). This gives the identification in part (ii) of the Lemma.   In the remainder of this section we will use freely the above identifications.
Proof of Theorem 2.6. Our starting point is the equation satisfied by u(·), namely u t = (A − L)u + AN σ(w) · ν, whose evolution/integral equivalent is  1. Since by Lemma 3.1 we know that u| Γs = −N * Au, the splitting (3.2) yields as well respectively. The term u 1 yields a pretty straightforward regularity estimate. Indeed, combining the (analytic) estimates (2.10) with the regularity of the Neumann map recalled in Lemma 3.1, (ii), we obtain i.e. the "singular" estimate (2.13) holds true. In particular, Thus, to establish (2.11), it remains to demonstrate that the boundary trace of the second component u 2 on Γ s is well defined and satisfies (2.14). Taking account of (3.3), since the regularity of U 2 has been dealt with in [12], we will be done once we show We consider the convolution integral z(t) = t 0 e A(t−s) Lu(s) ds, and recall from [6, Theorem 3.2] the additional property u ∈ L 2 (0, T ; V ), which implies Lu ∈ L 2 (0, T ; L 2 (Ω f )) and Lu = P H Lu ∈ L 2 (0, T ; H). Thus, by parabolic regularity theory we deduce z ∈ L 2 (0, T ; D(A)) ∩ H 1 (0, T ; H); therefore, by interpolation, ∀α ∈ (0, 1) .
In particular, we may take α = 3 4 − δ to find we obtain where the inclusion follows by Sobolev embedding theorems in one dimension. Thus, the validity of (3.6) is ensured by (3.7). The analysis of the summand U 2 (t) critically relies on the sharp regularity theory for the elastic component of the system, as given by the first part of Lemma 2.7. Specifically, on the basis of it was shown in [12] that Consequently, combining (3.8) with (3.6) we obtain (2.14), as required. This concludes the proof of all assertions in item 1.
3. We turn our attention to the time derivative u t . In order to prove (2.16), we compute from where, initially, y 0 ∈ Y and v 2 reads as As V 1 is nothing but z t (z is defined in step 1), and according to parabolic regularity z ∈ H 1 (0, T ; H), we immediately obtain The more challenging analysis of V 2 yields, utilizing once again Lemma 2.7, for any 0 < δ 1 < 1 4 ; see [12] for all details. Combining (3.11)-(3.12) leads us to the estimate When y 0 ∈ D(A), we rewrite v 2 in a different fashion, namely On the other hand, with y 0 ∈ D(A) we have u 0 ∈ V so that Lu 0 ∈ H and in particular which is weaker than (3.15); consequently, Consider next the term Z 2 , and notice that y 0 ∈ D(A) implies, in particular, (w 0 , z 0 ) ∈ H 2 (Ω s ) × H 1 (Ω s ). By standard semigroup arguments and according to Lemma 2.7 we know that In light of the above result, it was shown in [12] that the convolution integral which gives On the other hand, still with y 0 ∈ D(A), one has just σ(w)·ν ∈ C([0, T ], H −1/2 (Γ s )). Rewrite the summand Z 22 (t) := Ae At N σ(w(0)) · ν in (3.14) as Z 22 (t) = A 1/2 e At A 1/2 N σ(w(0)) · ν , and use once more Lemma 3.1, along with the singular estimates pertaining to analytic semigroups, to find for any 0 < δ 2 < 1 2 , where q 2 ∈ (1, 2) depends on δ 2 ; more precisely, (3.20) We proceed now as in the conclusion of step (iii) in [12, Proof of Theorem 2.9]. Combining (3.19) with (3.13), gives by interpolation where q 2 is as in (3.20) and W is the interpolation space see [24,Chapter 1,Theorem 12.5]. Notice that by taking, for instance, δ 1 = δ 2 =: δ, one has s ≥ 1/2 provided that In this case v 2 takes values in H s (Ω f ) with s ≥ 1/2 and hence its trace on Γ s is well defined. Notice that, in view of the constraint (3.22), we need to require 0 < θ < 1 4 . Consequently, given any θ such that 0 < θ < 1 4 , choosing, e.g., δ = 1/4 − θ in view of (3.22), from (3.21) it follows . (3.23) It remains to establish the regularity of the first summand N * Av where ǫ is independent of θ and can be taken arbitrarily small. (Notice that the Sobolev exponent q 1 belongs to (1,2), as well.) Thus, in view of (3.23) and (3.24) we finally obtain , (3.25) which concludes the proof.

Application to the quadratic optimal control problem
In this section we briefly discuss the implications of the obtained trace regularity results in the study of the optimal control problem (with quadratic functionals) associated with the fluid-solid interaction model (2.1). Thus, we consider the initial/boundary value problem which includes a function g acting as a control on the interface Γ s . The PDE problem (4.1) readily corresponds to an abstract (linear) control system, that is where the (free dynamics) operator A is the one in ( is a consequence of AN being bounded from L 2 (Γ s ) into V ′ (yet unbounded from L 2 (Γ s ) into H); see [21] for more details. We associate to (4.2) a quadratic functional where g varies in L 2 (0, T ; U ), and initially R ∈ L(Y, Z) (R is called observation operator, Z is the observations space). Then, the (optimal) quadratic control problem over a finite horizon is stated as follows.
Problem (The optimal control problem). Given y 0 ∈ Y , seek a control function g ∈ L 2 (0, T ; U ) which minimizes the cost functional (4.4), where y(t) = y(t; y 0 , g) is the solution to (4.2) corresponding to g.
As in the classical Linear Quadratic (LQ) problem in a finite dimensional context, solvability of the optimal control problem defined above does not just mean existence of a (unique) minimizer g * . We aim to obtain the feedback representation of the optimal control, defined for any y 0 in the state space Y , along with well-posedness of the corresponding operator Riccati equations. On the other hand, it is well known that in the case of boundary control systems, such as the present (4.2), a major difficulty comes from the fact that the gain operator B * P (t) which occurrs in the feedback formula g * (t) = −B * P (t)y * (t) a.e. in [0, T ], may not be defined, or may be unbounded. Let us recall that owing to the well-known representation of the Riccati operator P (t) (involving the evolution map Φ(·, ·) associated with the optimal dynamics) it is immediately seen that boundedness of the operator B * P (t) will crucially depend upon the regularity properties of the operator Thus, a PDE intepretation of the composition B * e A * t is called for. This is obtainedfor a given control system (4.2)-by computing B * and by noting that e A * t y 0 solves the uncontrolled Cauchy problemŷ ′ = A * ŷ ,ŷ(0) = y 0 . These calculations eventually yield the distinct trace operator corresponding to B * e A * t . (This connection appeared clear since the former studies of the LQ-problem for simple hyperbolic equations; see, e.g., [18] and [8].) In the present case we obtain, in particular, B * e A * · y 0 = [N * A, 0, 0]ŷ = N * Aû = −û| Γs , (4.5) whereŷ = (û,ŵ,ŵ t ) solves the initial/boundary value problem corresponding to the afore-said adjoint system, which is almost identical with the PDE problem (2.1). This explains why a study of the boundary regularity of the fluid velocity field on the interface is necessitated. A situation where one can conclude that B * P (t) is bounded on the state space Y is when the pair (A, B) yields a 'singular' estimate for the operator e At B near t = 0; see [17], [19], and [20]. It was shown in [21] that the PDE problem (4.1) does give rise to a singular estimate; however, this holds true provided the observation operator R possesses a suitable smoothing property.
In contrast, in light of the regularity results established in Theorem 2.6, we are able to show that the abstract control system (4.2) corresponding to the initial/boundary value problem (4.1) falls in the more general class of systems introduced in [2]. Indeed, all the regularity results proved in Theorem 2.6-which yield relative assertions for a similar (dual) PDE problem-are the PDE counterpart of the control-theoretic properties (Hypotheses 2.1-2.2 therein) required to apply the theory of [2]. We do not discuss the proof, which follows the lines of [12, Proof of Theorem 2.6], but let us remark that while the regularity of the boundary traces of u on Γ s corresponds to the regularity of the operator B * e A * t , the obtained regularity for u t in (2.16) yields a sought-after and most challenging property for B * e A * t A * θ (that is condition (3.23) in [2], or (4.4) in [12]). We just note that the latter PDE interpretation follows from the following key observation: where (û,ŵ,ŵ t ) solves the same (adjoint) PDE problem as before, yet with initial state ξ 0 ≡ A * θ−1 y 0 ∈ D(A * 1−θ ) (while time derivatives are understood distributionally); the reader is referred to [12, Section 4.1] for more details. Thus, in order for all the hypotheses needed to apply [2, Theorem 2.3] to be fulfilled, we just need to assume the one which pertains to the observation operator R; namely, R * R ∈ L(D(A ǫ ), D(A * ǫ )) (4.6) for some ǫ ∈ (0, 1/4). Then, under (4.6) all the conclusions of [2, Theorem 2.3] follow.
In particular, one obtains the feedback synthesis of the optimal control for all y 0 in the finite energy space Y , with a gain operator which is densely defined, that is enough to get well-posedness of the corresponding differential Riccati equations.
Remark 4.1. We finally note that while a case of particular interest is R = I, the requirement (4.6) does not 'force' smoothing effects of the observation operator. Indeed, in the present case D(A * ǫ ) coincides with D(A ǫ ) for ǫ sufficiently small, and hence (4.6) is satisfied by any operator which 'maintains' regularity, such as the identity R = I; see [12, Remark 2.7] for a more detailed explanation. This answers the final question raised in [21, Remark 6.1].