Existence and uniqueness result for a fluid–structure–interaction evolution problem in an unbounded 2D channel

In an unbounded 2D channel, we consider the vertical displacement of a rectangular obstacle in a regime of small flux for the incoming flow field, modelling the interaction between the cross-section of the deck of a suspension bridge and the wind. We prove an existence and uniqueness result for a fluid–structure-interaction evolution problem set in this channel, where at infinity the velocity field of the fluid has a Poiseuille flow profile. We introduce a suitable definition of weak solutions and we make use of a penalty method. In order to prevent the obstacle from going excessively far from the equilibrium position and colliding with the boundary of the channel, we introduce a strong force in the differential equation governing the motion of the rigid body and we find a unique global-in-time solution.


Introduction and main result
Suspension bridges may experience several types of instability phenomena, which affect more or less critically each component of the structure. Among all components, the deck is the most sensitive part. When analyzing the dynamic response of the bridge to the wind from the engineering point of view, we observe that the bridge may suffer from a variety of problems: one degree and two degrees of freedom instability, buffeting and vortex shedding. Two degrees of freedom instability, also known as flutter instability, occurs when the vertical and the torsional motion of the deck synchronize so that aerodynamic forces introduce energy into the system. However, this only occurs after reaching a critical value of the incoming wind velocity (see [9,14,28]). Thus, the vertical and torsional displacements are decoupled in a regime of small oscillations. We set up our model precisely under the hypothesis of a small flux for the incoming flow field. We consider the interaction between an obstacle and a fluid in a 2D unbounded channel, where the flow is of Poiseuille type at infinity; we analyze a fluid-structure problem by allowing the obstacle to move in a vertical translation. In a regime of strong incoming flux, it would be necessary to consider the full coupled vertical-torsional motion.
We follow the two-dimensional fluid-structure-interaction evolution problem introduced in [4]. We label as B = [−d, d] × [−δ, δ] a rectangular rigid body representing the 2D cross-section of the deck of a suspension bridge; we denote by m > 0 the mass of the body. Without loss of generality, we can assume d to be equal to 1, by using the length of the body as reference length scale. Such body is free to move vertically inside an infinitely long 2D channel tracks the position of the body after the vertical translation. Notice that, when |h| = L − δ, the obstacle collides with Γ. Due to the motion of the rigid body, the domain occupied by the fluid Ω h depends itself on h, which explains the subscript, and it is given by: (1.1) T (u, p)n t∈ (0, T ), (1.2) to which we associate the initial conditions u(x, 0) = u 0 , h(0) = h 0 , h (0) = k 0 .
Here u : Ω h(t) × (0, T ) → R 2 and p : Ω h(t) × (0, T ) → R are respectively the unknown velocity vector field and the unknown scalar pressure field, whilê n denotes the outward normal to ∂Ω h , thus directed in the interior of ∂B h . The constant λ in q measures the magnitude of the Poiseuille flow, which is prescribed, thus λ is given. The motion of the body is governed by the ODE in (1. 2): f (h) is an elastic smooth restoring force, which we suppose to be f ∈ C 1 (−L + δ, L − δ), and T (u, p) = −pI + 2μD(u) is the fluid Cauchy stress tensor, with I the 2 × 2-identity matrix and D(u) = 1 2 (∇u + ∇u T ), thus the right hand side of the ODE corresponds to the fluid lift. We also assume that f (h) satisfies some further conditions, given later in (1.4), which translate the assumption of f being a strong force, preventing the obstacle from colliding with the boundary of the channel Γ. From the physical view point indeed, f resumes the action of three kind of forces acting on the deck of a suspension bridge, as also explained in [4,Introduction]: the upward restoring force due to the elastic action of both the hangers and the sustaining cables, the weight of the deck acting downwards and the elastic resistance to deformations of the whole deck, preventing the obstacle to go too far from its equilibrium horizontal position (see also [14]). Our model indeed loses its physical meaning in case of collisions; on the other hand, collisions would be purely virtual in the physical framework at consideration, because we can not expect the bridge to be subjected to very large deformations. This justifies the limit in the assumption in (1.4); we emphasize that, in this context, we are not interested in choosing the optimal hypothesis on f ensuring the absence of collisions. As in [26], we suppose that the initial velocity u 0 admits the following representation where ζ(x 1 ) is a smooth cutoff function such that ζ(τ ) = 0, if |τ | < 2, ζ(τ ) = 1 if |τ | > 3.
In [4], the authors prove that for the stationary version of (1.2) the equilibrium position of the obstacle is perfectly symmetric under smallness assumption on the imposed flow rate magnitude of the Poiseuille flow; actually, 39 Page 4 of 38 C. Patriarca NoDEA they are able to prove their result with an assumption on f weaker than (1.4).
The main purpose of our work is to prove the existence and the uniqueness of a weak solution for the fluid-structure-interaction evolution problem (1.2). In the literature several definitions of weak solutions have been adopted, as well as several techniques to find such solutions (see, e.g., [6,8] for the 3D case, [23,27] for the 2D case). In order to prove existence we adopt a penalty method devised in a paper by Fujita and Sauer, see [10]; this technique was later exploited by Conca, San Martin and Tucsnak in [6] in order to prove existence of solutions for a boundary problem modelling the motion of a rigid ball in a viscous fluid occupying a bounded domain. Among the several aforementioned techniques, we choose to apply the method by [6], because it enables to work with a non-global weak formulation, in the sense that the terms concerning the fluid sub-problem and those concerning the rigid body sub-problem of (1.2), although coupled, remain distinguished. On the other hand, both the method introduced in [8,23,27] would require working with global quantities and a global weak formulation, in the sense that the integrals would be defined on the whole domain A; in this context, the approach by [6] is more convenient because of the presence of the strong force f in the ordinary differential equation governing the motion of the obstacle, which makes it non-trivial to define a global weak form for the original coupled frame. The main difference with the problem considered in [6] lies in the fact that the domain occupied by both the rigid body and the fluid in (1.2) is an unbounded channel where a non-zero velocity field is imposed at infinity; this requires building a solenoidal extension of the Poiseuille flow. Moreover, besides the less affecting difference of a rectangular shaped obstacle, the introduction of the strong force f in the ordinary differential equation governing the motion of the obstacle in (1.2) allows to obtain solutions with a global character in time, as well as uniqueness of such solutions: indeed, this force prevents the obstacle from colliding with the boundary of the channel.
Collisions are a delicate issue in the context of fluid-structure-interaction problems, especially because different phenomena occur in the context of strong or weak solutions. For strong solutions, in some particular 2D cases, collisions are excluded (see [21,22,30]). However, this no-collision result appears physically unrealistic both at the macroscopic and microscopic scales, as pointed out in [19,20] where it is suggested that the flaw lies in the modelling. If we move to the realm of weak solutions, it is not known in general whether collisions do occur in finite time or not, although, in some particular cases, where the contact surfaces are regular enough, they might be excluded, see [30,Theorem 3.1]. However weak solutions exist globally even if collisions occur in finite time, because they are characterized by vanishing relative velocity; this result holds under regularity hypothesis on the boundary of the body and on the fluid domain, see [27,30]. In [31] the author was able to construct at least two weak solutions admitting collision in finite time and, in order to guarantee uniqueness of weak solutions, one then needs to exclude collisions as pointed out in [16]. In particular, while uniqueness of strong solutions is a known fact both in two and three-dimensions, and both for no-slip case and the slip case (see [1,7,18,32]), in [16] the authors proved for the first time a result of uniqueness for weak solutions for a two-dimensional fluid-rigid body system and in [5] the author proved it for the slip case. Here, we adopt a different strategy in order to explore existence and uniqueness of weak solutions, we make use of the strong force f , as it is done in [4].
Our main result states the existence and uniqueness of weak solutions to problem (1.2).
Moreover, let |h 0 | < L − δ and u 0 satisfying (1. 3) be such that u 0 ·n = k 0ê2 ·n on ∂B h0 . Then, problem (1.2) admits a unique weak solution (u, h), defined in a suitable sense, for any T < ∞. Moreover the energy of (u, h) is bounded.
Theorem 1.1 deserves some comments. First, we emphasize that, since (1.2) is a fluid-structure-interaction problem, one needs to give a suitable definition of weak solutions which takes into account the presence of the moving obstacle. Because of such difficulty, we adopt a definition of weak solutions which is a compromise between the one given in [13] and the one in [6]; weak solutions are defined in Definition 2.3 after transforming problem (1.2) into an equivalent problem, as we shall see in Sect. 2. Finally, the global-in-time property of solutions is ensured because of the energy estimate associated to problem (1.2), which guarantees that no collision occurs between the obstacle and the boundary of the channel. Such energy estimate will be made explicit in Theorem 2.6 for the equivalent problem. In particular, as in the case of the Navier-Stokes equations in a 2D domain with cylindrical outlets to infinity, we merely obtain an inequality, differently from what one usually obtains in a bounded 2D domain: see [26].
The rest of the paper is devoted to proving Theorem 1.1 and it is organized as follows. In Sect. 2 we present some notations and preliminary results which are essential to apply the penalty method and are useful throughout the paper. Then we reformulate problem (1.2) in a reference frame attached to the obstacle; this produces the equivalent problem, (2.7)-(2.8). At the end of Sect. 2, Theorem 2.6 states the existence and uniqueness of solutions for the equivalent problem (2.7)-(2.8). Thus, Theorem 1.1 is proven once we develop the proof of such statement, which is addressed in the subsequent sections. In Sect. 3, we introduce an auxiliary problem, at the core of the penalty method, for which we prove existence of solutions with the Faedo-Galerkin procedure. In Sect. 4, after some additional results, we conclude the proof of the existence part of Theorem 2.6 and, consequently, the existence part of the main result of the work, Theorem 1.1. Section 5 is devoted to proving uniqueness of solutions to problem (2.7)-(2.8), which concludes the proof of Theorem 2.6 and thus also of Theorem 1.1. The main difficulty of proving uniqueness is that we cannot 39 Page 6 of 38 C. Patriarca NoDEA simply take the difference between two weak solutions of problem (2.7)-(2.8) because, since the fluid domain has moving boundaries, those solutions are not defined on the same domain, thus we will adopt a suitable change of variables to solve such issue, following the work in [16]. As we already mentioned, if we moved to the regime of strong incoming flux, we would have to consider an obstacle characterized by a fully-coupled vertical-torsional motion. As a step towards this direction, it appears interesting to look at the second model introduced in [4] where the obstacle is immersed in the same channel but it is only free to rotate around a fixed pin. In particular, in Sect. 6 we make some remarks about this second fluid-structure-interaction evolution problem: we highlight how this is a significantly different problem with respect to the case of vertical translation both from the physical and mathematical point of view.

Preliminaries: an equivalent formulation
Problem (1.2) is set in a 2D unbounded channel with a prescribed non-zero velocity field at infinity. We begin by capturing the flow at large |x 1 |: we construct a suitable extension of the Poiseuille velocity profile at infinity by using similar arguments to the classical procedure by Ladyzhenskaya [25] (see also [2]). Let b 3 be the following function, defined over (−L, L), . We consider a partition of the channel A as We define ζ 1 to be a smooth cutoff function acting in the horizontal direction such that Then, let ε 0 > 0. We take ζ 2 to be a twice continously differentiable cutoff function acting in the vertical direction obeying the inequalities with some constants C > 0. Then we take The above construction enables us to state: , q as in (1.2) and let ε 0 > 0 small be arbitrarily fixed. Let is as in (2.1), and ζ(x) as in (2.3). Then the vector field s(x) is such that , and supp(s) = A\{|x 1 | < 2 ∧ |x 2 | < L − ε 0 }, and, for every ε 0 > 0, there hold the estimates where c 1 , c 2 > 0 depend on L and λ.
As a first step of the penalty method implemented in [6], we change problem (1.2) into an equivalent problem by considering a frame attached to the rigid body, whose origin coincides with its center of mass. Thus we set The domain of the fluid in the new reference frameΩ shall also be partitioned We emphasize that the obstacle is now fixed, while the domain occupied by both the fluid and the rigid body, A h(t) = B ∪ ∂B ∪Ω(t), changes with time.
Then, we notice that Thus, we obtain the following problem (see also [12]): where v 0 (y) = u 0 (y + h 0ê2 ) with u 0 as in (1.3). Notice that all derivatives appearing in this problem are now taken with respect to the new variable y = (y 1 , y 2 ). The Poiseuille flow at infinity is obtained by transporting q as in (1.2) to the new reference frame: The motion of the rectangular body is governed by: The original problem (1.2) is equivalent to (2.7)-(2.8), because we simply adopted a change in the system of coordinates. Thus Theorem 1.1 is proven if we prove existence of solutions to (2.7)-(2.8). We look for solutions to the problem (2.7)- where a is the solenoidal extension of the Poiseuille flow in the new reference frame, strongly depending on h and on the choice of ε 0 in Lemma 2.1: The function a enjoys the same properties of s stated in Lemma 2.1 once we substituted q and A with their counterparts in the new reference frame,q and A h , and we partitioned A h similarly to what we did in (2.2). Assume that whereε > 0 small is arbitrarily fixed. Then, the functionv solves the following problem: whereĝ := μ Δa − (a · ∇) a. (2.11) Notice thatv → 0 as |y 1 | → ∞ andv 0 ∈ L 2 (Ω(0)) is such thatv 0 ·n = k 0ê2 ·n on ∂B. We also point out that supp(ĝ) The vertical translation of the obstacle h responds to As already mentioned, in order to prove our main result we make use of a procedure which is similar to the one adopted in [6]: problem (2.10)-(2.12) is set in a region with moving boundaries,Ω(t), which makes it impossible to apply the Faedo-Galerkin approximation with the standard functional spaces of hydrodynamic evolutionary problems. The idea exploited in [6] is that of a penalty method and it was first elaborated in the paper by Fujita and Sauer (see [10]). The crucial idea of the method implies introducing an auxiliary fixed, infinite domainÃ given by: Fig. 2 for the new configuration). Notice that the vertical translation ofΩ(t) insideÃ is confined: dist (∂Ã,Γ(t)) ≥ δ. This is an obvious consequence of impenetrability of bodies but we will later prove that this inequality actually holds strictly, thus the obstacle never collides with the boundary of the channel. Inside the auxiliary fixed domainÃ, we can naturally extend the velocity field at infinityq(y) outside A h(t) for every h(t) through its definition in (2.7), see Fig. 2.
We report three facts, that we will later use. First, an estimate for the L 2 -norm of the gradient ofq in each one-dimensional section of the domaiñ A: (2.14) Then, we give the following partition ofÃ: Finally, we comment on some properties which can be derived from the assumptions of f in (2.12) being a strong force, given in (1.4  The last condition (1.4) may be interpreted in the spirit of [17]. In [17], the author considers systems of the general type x + ∇V (x) = 0, with x ∈ R n , where the potential V (x) associated to a conservative dynamical system, such as a n-body system, is assumed to be C 2 everywhere except at a closed non empty set S at which it has infinitely deep wells, i.e. V (x) → −∞ as x → S; then, the system is said to satisfy the strong force condition if and only if there exists a neighborhood N of S and In a n-body system the singularities correspond to collisions of the masses and the strong force condition allows to avoid them (see also [3]). In our case, in the terminology of [17], f plays the role of V and the ODE in (2.12) satisfies the strong force assumption, with the singularity exhibited at |h| = L − δ. Now, we seek a rigorous definition of weak solutions to (2.7)-(2.8). We introduce some classical functional spaces from mathematical fluid dynamics (see [33] for instance): We emphasize that sinceÃ is bounded in the vertical direction, there holds the Poincaré inequality, which makes H 1 0 (Ã) an Hilbert space with respect to the scalar product (u, v) . The presence of the obstacle B requires introducing some further spaces: Finally, the weak formulation of problem (2.7)-(2.8) will exploit the following spaces, which may be defined for any given function h(t). In particular, for every t Then, we introduce the standard trilinear form: We are ready to state and prove the following proposition. Let us denote by ·, · the duality pairing between V and V .
Proof. Consider the problem satisfied by (v, h), (2.10)-(2.12). In order to obtain (2.20), we choose a test couple (φ, l) ∈ C 1 ([0, T ], V h(t) ) such that φ(·, T ) = l(T ) = 0. We multiply the first equation in (2.10) by φ and integrate by parts onQ T =Ω(t) × [0, T ]. All terms may be treated in a standard manner (see, e.g., [11]). Though, a particular attention must be devoted to the diffusive and pressure terms. Indeed, we temporally move the term μΔa appearing inĝ in (2.10) on the left-hand side and we get: Thus, given ψ in (2.19) and we obtain the weak formulation (2.20).
These tools enable us to define weak solutions of (2.7)-(2.8). This definition is a compromise between the definition given in [6, Definition 1] and the one given in [13, Definition 3.1], although the extension in [13] is constructed so as to isolate the obstacle.
where a = a h is the extension in (2.9) depending on some ε 0 = ε 0 (v 0 , h 0 , k 0 , T ) > 0, it satisfies the following requirements: ensures that no collision occurs between the obstacle and the boundary of the channel as there exists a separation strip of size ε 0 > 0 for all t ∈ [0, T ]. This makes the definition of weak solution consistent, since it also allows to build the solenoidal extension a = a h in (2.9) precisely by choosing such ε 0 > 0. As already mentioned, we will prove that this requirement is satisfied by making use of the strong force assumption satisfied by f , given in (1.4). On the other hand, it is worth mentioning that, one could prove the no-collisions result without adding the strong force assumption, at least in the case of an obstacle purely translating in the vertical direction, because the contact surfaces are of the class C ∞ (see [20,30]). However, as soon as one allows the obstacle to rotate, this result does not hold anymore because the contact surfaces are merely Lipschitz continuous.
Finally, we provide an estimate on the norm ofĝ, defined as in (2.11), on the auxiliary domainÃ, which we will exploit later. (2.11), a(y) as in (2.9) for some ε 0 > 0 and A 0 as in (2.15). Thenĝ ∈ V (Ã), the dual space of V (Ã), and

Lemma 2.5. Letĝ be defined as in
Proof. Multiplyĝ by ϕ ∈ V (Ã) and integrate by parts overÃ. Since ϕ = 0 on ∂Ã, we obtain We proceed similarly to [15,Lemma 4.1] and we exploit the partition (2.15) together with the properties of a once we naturally extended this function to the whole fixed domainÃ through its definition (2.9). In particular a =q iñ A 1 ∪Ã 2 . Thus we divide the first term in (2.23): ∇a : ∇ϕ dy and we remark that as ϕ carries no flux being divergence-free onÃ. For the same reason, also the integral overÃ 2 vanishes and we obtain, by applying the Hölder inequality, Since a =q inÃ 1 ∪Ã 2 and since (q · ∇)q ≡ 0, we have that, again by the Hölder inequality and the property of the trilinear form (see [33, Chapter 2, Thus we obtain from which the thesis of the lemma follows. The following theorem states existence and uniqueness of weak solutions to problem (2.7)-(2.8).
We observe that one may expect the energy inequality in Theorem 2.6 to be an equality as for the case of the Navier-Stokes equations in a bounded 2D domain, but so far it is not known whether we should expect an equality also in the case of a 2D domain with cylindrical outlets to infinity (see [26]). Since the original problem (1.2) is equivalent to problem (2.7)-(2.8), the proof of Theorem 1.1 is completed if we prove Theorem 2.6. We emphasize that Theorem 2.6 has a stronger statement than Theorem 1.1, because it guarantees the continuity of the (unique) solution in a suitable sense. The proof of the existence part is developed in Sects. 3 and 4 and, as previously mentioned, it takes advantage of a penalty method. Uniqueness is proven in Sect. 5: there, we will consider two weak solutions of problem (2.7)-(2.8), (v 1 , h 1 ) and (v 2 , h 2 ), in the sense of Definition 2.3. Given the extensions of the Poiseuille flow a 1 = a h1 and a 2 = a h2 defined as in (2.9), one can putv 1 = v 1 − a 1 andv 2 = v 2 − a 2 and do the computations on (v 1 , h 1 ) and (v 2 , h 2 ). As we already pointed out in the Introduction, we are not allowed to take the difference between the equation in weak form satisfied by the two solutions, (2.20), becausev 1 andv 2 are not defined on the same domain: the definition of the functional spaces, (2.18), depends on h 1 , h 2 , and the weak formulation (2.20) is set on two different domains,Ω 1 (t) andΩ 2 (t). To solve such issue, we follow the procedure devised in [16]; here, the authors build a map ψ t projectingΩ 2 (t) onΩ 1 (t), which allows to define a change of variables, so that they can introduce a solenoidal velocity vector fieldv 2 , the pullback ofv 2 by such map, onΩ 1 (t). As a consequence, one can define (2.26) Then, one proceeds standardly, obtaining the equation satisfied by (w,ĥ), providing some proper estimates, and finally one can conclude by applying a Grönwall's inequality.

A penalized problem
After changing problem (1.2) into the equivalent problem (2.10)-(2.12), the second step of the penalty method exactly implies penalizing (2.20), which is the weak formulation of problem (2.10)-(2.12). We denote by E h the complementary domain of A h inÃ, E h =Ã\A h and we introduce its characteristic function χ E h . We emphasize that the functions belonging to W(Ã) differ from those belonging to W h precisely because their support might be also in E h . The penalty method eliminates the difficulty induced by the time dependent domain by allowing to solve the problem in the fixed domainÃ, so that classical methods can be applied, by introducing a penalization term which takes care of the remainder in E h . We extendv 0 by zero outside A h(0) , while a(y) is naturally extended outside A h(t) to the whole fixed domainÃ through its definition (2.9) for every t ∈ [0, T ] and we solve the following problem: Let n ≥ 1 be fixed. Find We remark that the trilinear forms in (3.1) are defined as in (2.19), where the integral is now onÃ\B, but with a little abuse of notation we still use ψ as a label function. The existence of a solution to the penalized problem is proven in the following proposition: Proof. Since the domainÃ is fixed, we can apply the Faedo-Galerkin procedure. The space V(Ã) is a separable Hilbert space, hence we can choose a sequence given by a countable set of couples where the coefficients c iN are determined by the following first-order integro- The initial conditions are given by the orthogonal projections in H(Ã) of (v 0 , k 0 ) onto the space spanned by {(w i , 1)} N i=1 , which we call (v 0,N , k 0,N ). This system has a solution defined on some interval [0, t N ], provided that there exists ε 0 > 0 such that In this way, we are allowed to build the function a as in (2.9), by choosing the same ε 0 . We shall see that this condition is indeed always guaranteed for any t ∈ [0, T ] with T < ∞, because of the presence of f in (2.8). We will prove this claim at the end of the proof of this proposition. Our aim is finding an apriori estimate for the approximate solution  (v N , H N ). To this end, we multiply (3.3) by c jN (t) and add the equations for j = 1, . . . , N: where F is defined in (2.16). Our aim is finding an a priori estimate for the approximate solution (v N , H N ). To this end, we start by estimating the righthand side of (3.5). The first trilinear form may be bounded exploiting the Hölder inequality and the Young inequality where we used the fact that the Poincaré constant in the domainÃ is π 2 /(4L− 2δ) 2 . For what concerns the second trilinear form, we exploit again the definition of a, as we did when proving Lemma 2.5. Since a(y) is equal to 0 on the obstacle B, we write (H Nê2 · ∇)a ·v N dy 2 dy 1 . Then Consider the partition (2.15). The first term is treated as follows, by using (2.14): The third integral in (3.6) can be treated analogously, so that we obtain:

For what concerns the term in the regionÃ
Thus, we finally obtain that: where we used the Young inequality and the fact that ∇v N 2 L 2 (Ã1) . Then, we apply again the Young inequality, after the Schwarz inequality, to provide a bound for the first term on the right-hand side in (3.5): since Lemma 2.5 guarantees thatĝ ∈ V (Ã\B). Thus, by reordering Eq. (3.5), once we have plugged these estimates and used the following fact sincev N is a divergence free vector field vanishing on ∂Ã, we obtain (3.7) From (3.7), by integrating between 0 and t, we deduce that Since F (h N ) ≥ 0 by (2.17), invoking Grönwall's lemma, we obtain for any with

Thus, we finally get
for all t ∈ [0, T ], where the right hand-side is bounded if T < ∞. In particular, the solution exists globally in time provided that condition (3.4) holds, which we still need to prove; such condition in the original inertial reference system translates the condition of absence of collisions occurring between the rectangular obstacle and the boundary of the channel. Estimate (3.8) implies the existence of a couple (v, h ) ∈ L ∞ (0, T ; H(Ã))∩ L 2 (0, T ; V(Ã)) and of a subsequence, which we denote by (v N , h N ), such that, as N → ∞: the latter holding because of the compact embedding of W 1,∞ (0, T ; R) into C 0 ([0, T ]; R). Moreover, through classical methods (see [6,Section 3] and [33, Chapter 3, Section 3]), in view of (3.8) together with the fact that f (h N ) can be thought to be a bounded function as long as the rigid obstacle does not touch the boundary of the channel, one can prove a further convergence property (up to the extraction of a subsequence): for any compact setO ⊂Ã.
By [6, Lemma 1] we also have that for any compact set O ⊂Ã.
(3.10) The convergence results in (3.9) together with (3.10), where we choose as O the support of w j , us to pass to the limit in the system satisfied by (v N and to obtain that (v, h) in (3.9) satisfies (3.1), as well asv(0) =v 0 in the distributional sense, by exploiting classical arguments (see for instance [33, Chapter 3, Section 3]). To prove that (v, h ) is almost everywhere equal to a function continuous from [0, T ] into H(Ã), one can easily proceed as in [33, Chapter 3, Theorem 3.1] to obtain that (v , h ) ∈ L 2 (0, T ; V (Ã)) and then apply [33, Chapter 3, Lemma 1.2], by exploiting the fact that {V(Ã), H(Ã), V (Ã)} is an Hilbert triplet. Moreover, the functionv obeys the energy inequality (3.2), which is a natural consequence of the convergence results that we have just proved.
The proof of Proposition 3.1 is complete once we prove the following lemma, which, combined with the last convergence result in (3.9), allows to conclude on the global-in-time character of the solution.
From the energy estimate (3.8) (which is uniform with respect to N ), (3.11) and the estimates (2.4), we obtain that there exist a positive constant C 0 , depending on the initial data h 0 , k 0 ,v 0 , and two positive constants C 1 , C 2 depending on the geometrical and physical parameters, i.e. m, λ, L, δ, μ, such that Integrating by parts the right-hand side of (3.12), we obtain that This immediately yields a contradiction, since r > 0. From the above argument, for all T < ∞ it must be that |h N (t)| < L − δ when t ∈ [0, T ]. Since h N ∈ C 0 ([0, T ]; R) and [0, T ] is compact, the thesis of the lemma follows.

Proof of the main result: existence
We already mentioned how proving Theorem 1.1 is equivalent to proving Theorem 2.6. The idea of the proof of the first part of Theorem 2.6 is exploiting the result of existence for the penalized problem, given in Proposition 3.1. Indeed, the next step of the penalty method implies passing to the limit in (3.1) with respect to n. Again, we will follow the procedure by [6], highlighting the differences when it is necessary. Let us label the weak solution to (3.1) making the dependence on n explicit as (v n , h n ); of course E h also depends on n, thus it may be relabelled as E hn This solution satisfies the energy estimate (3.2), where we should also make explicit the dependence on n. Let us state and prove two consequences of this fact. The first consequence is natural: there exists a subsequence, which we denote again by (v n , h n ) such that where the latter is due to the compact embedding of W 1,∞ (0, T ; R) onto C 0 ([0, T ]; R).
The second consequence is proven in the following lemma. Then, we use the Hölder inequality: Now, we introduce, for any η > 0, given O h(t) any bounded set such that that is Q i,η contains the couple (y, t) such that y belongs to the interior boundary strip of O h . The sets in (4.6) depend on the choice of O h , but from now on we will omit the subscript O h not to make the notation heavy. In order to develop the proof of Theorem 2.6, we need to first prove an auxiliary result, i.e. thatv n is relatively compact in L 2 (Q), which implies the existence of a subsequence, still labelled asv n , satisfying the following strong convergence resultv n →v in L 2 (Q). (4.7) The procedure implemented in [6] in order to prove such convergence result implies exploiting an Aubin-Lions type lemma: more precisely, one wants to where H(D) can be characterized as follows since D is a Lipschitz open bounded set (see [33,Theorem 1.4]) We associate to both Z(D) and M (D) the following scalar product: which makes both of them Hilbert spaces. Then, we define the projector P (D), We will now prove two technical lemmas, starting by a useful property of such projector.

Lemma 4.2. There exists a positive constant C such that
Proof. We follow step by step the proof of [6, Lemma 4]. We emphasize that even if D is merely a Lipschitz domain this does not compromise the validity of the proof.
Then we prove the second technical lemma needed to guarantee (4.7).
Proof. We introduce an auxiliary function space: Any element in F (D) can be extended by 0 inÃ\B, so we can consider F (D) ⊆ V(Ã). We pick as a test function φ in (3.1) φ = a(t)ϕ, with ϕ ∈ F (D) and a(t) ∈ D(α, β) to obtain provided that n is large enough. Since F (D) ⊆ Z(D), we may rewrite (4.8) as From (4.9), we have that (4.10) Then, we can prove that the right-hand side of (4.10) can be bounded. Indeed, by exploiting classical estimates (see [33, Chapter 3, Section 3.3]), Lemma 2.5, the energy estimate (2.24), we obtain that where M 1 is a constant depending on T, μ, λ, the geometry of the problem and the initial conditions. Also because f (h n ) can be thought to be a bounded function as long as the rigid obstacle does not touch the boundary of the channel, which has been proven. Bounds (4.11) and (4.12) imply that This inequality, together with the fact that U n is a bounded set in  (α, β)).
Finally, one can give the following lemma: Proof. In order to prove this lemma, one can follow precisely the procedure given in [6, Theorem 3], once we declare the following notations: given s ∈ N, define α 0 , α 1 , . . . , α s , real numbers, and the sets Ω 1 , Ω 2 , . . . , Ω s such that Then we denote With the help of such definitions, we follow step by step the proof of [6, Theorem 3], which is divided in two parts. We can now proceed to the proof of Theorem 2.6. The objective is proving that the limit (v, h) in (4.1) is a solution to the original problem, thus it satisfies (2.21)-(2.22)-(2.20).
We start by proving that (2.20) is satisfied. In other words, we pass to the limit in (3.1). Take (φ, l) ∈ C 1 ([0, T ]; W h(t) ). Since there holds the last convergence result in (4.1), there exists n 0 ∈ N such that ∀n ≥ n 0 , and the penalty term in (3.1) vanishes for all n ≥ n 0 , by applying Lemma 4.1, where K is the compact support of φ. The convergence results (4.1), (4.7) (where as O h defining Q in (4.6) we take the compact support of φ) together with the density of The results of convergence (4.1) prove that (v, h) ∈ L 2 (0, T ; V(Ã)) ∩ L ∞ (0, T ; H(Ã)). By contradiction, let us suppose that supp(v) invades E h . Then, we can find K ⊂Ã such that supp(v) ⊂ K. However, from (4.7) and Lemma 4.1, we get a contradiction; thus in particular we obtain that (v, h) satisfies (2.21)-(2.22). Moreover, with analogous considerations as those that we did when proving Proposition 3.1, we obtain the continuity property of (v, h) expressed in the statement of Theorem 2.6. Finally, the energy estimate (2.24) simply follows from taking the limit as n → ∞ in (3.2); that explains the expression for α(s) in (2.25). As a consequence of (3.2), we also obtain the global character in time of the solution and the existence of some ε 0 > 0 such that |h(t)| ≤ L − δ − ε 0 for all t ∈ [0, T ], by proceeding precisely as in Lemma 3.2.

Proof of the main result: uniqueness
We begin by stating the following regularity property on a solution given by Theorem 2.6.
Let us consider two weak solutions of the equivalent problem (2.7)-(2.8), (v 1 , h 1 ) and (v 2 , h 2 ), in the sense of Definition 2.3, with the same initial conditions, where v 1 is defined on the fluid domainΩ 1 (t) while v 2 is defined oñ Ω 2 (t). Let ε 0 > 0 be such that min t∈[0,T ] (dist(B, ∂Ω i )) ≥ ε 0 for i = 1, 2; the existence of such ε 0 comes from Theorem 2.6. Finally, let ζ(y 1 , y 2 ) be a smooth cutoff function equal to 0 in a ε 0 /4 neighbourhood of ∂B and to 1 for any (t, y) such that dist(y, ∂B) ≥ ε 0 /2. Then, for each of the two solutions, we define a solenoidal velocity vector field We introduce a further domainΩ 0 , which serves as reference configuration, corresponding to the initial condition (v 0 , h 0 ). Then, we build the deformation mappings of such domain respectively intoΩ 1 andΩ 2 , X i : [0, T ]×Ω 0 →Ω i (t), i = 1, 2 as the flow associated to (5.3): Notice that, since ∇ · V i = 0, X i is volume preserving. More precisely, taking y = (y 1 , y 2 ) ∈Ω 0 , The mapping X i is a smooth function of V i . In particular, for some C > 0 For each t ∈ [0, T ] we define the volume preserving diffeomorphisms y)).
Given the extensions of the Poiseuille flow associated to each of the two solutions, a 1 = a h1 and a 2 = a h2 , defined as in (2.9), we putv 1 = v 1 − a 1 and v 2 = v 2 − a 2 . For any given y = (y 1 , y 2 ) ∈Ω 1 (t), we introduce the function v 2 = ∇ψ t (y) ·v 2 (t, ϕ t (y)), 39 Page 28 of 38 C. Patriarca NoDEA the pullback ofv 2 by map ϕ t in (5.5). Since a 2 = 0 near the obstacle B, because of (2.9) and the properties of s as in Lemma 2.1, we obtain that the pullback of a 2 corresponds to a 2 = ∇ψ t (y) · a 2 (ϕ t (y)) = a 2 (y 1 , which implies that the solenoidal extension a 1 and a 2 are equal after the change of variables. Thus, from now on, a 1 = a 2 = a. We remark thatv 2 mantains the property of being solenoidal since ϕ t is volume preserving. The weak formulation satisfied by (v 1 , h 1 ) can be obtained from (2.20), after rewriting the equation by integrating by parts the two first terms. For a.e. t ∈ [0, T ], there holds, for every (φ(t), l(t)) ∈ V h1 such that φ(·, T ) = l(T ) = 0, We refer to [16,Section 3.2] for the explicit computation of the partial derivatives ofv 2 in terms of those ofv 2 , so as to obtain that the equation satisfied by v 2 reads as for every (φ(t), l(t)) ∈ V h1 such that φ(·, T ) = l(T ) = 0, where, using Einstein's summation convention and omitting the index t in ψ t and ϕ t , . Now, let (w,ĥ) be defined as in (2.26). Then, taking the difference of the weak formulations satisfied byv 1 and v 2 , one has Then we take (φ, l) = (w, h ) and we obtain w, a, w) + ψ(ĥ ê 2 , a, w).
Thus, using [33, Chapter 3, Lemma 1.2] and the fact that (w ,ĥ ) ∈ L 2 (0, T ; V h1 ) from the properties of weak solutions to problem (2.7)-(2.8), there holds Now, we estimate the right hand side of the above inequality, starting from the trilinear forms. For what concerns the second term, we exploit Lemma 5.1 and the Young inequality: The third term can be estimated analogously to what we did in the proof of Proposition 3.1, through the Hölder inequality, the Poincaré inequality in the domainΩ 1 (t) and the Young inequality: Then, we consider the domainΩ 1 (t) to be partitioned as in (2.6) and we recall that the function a enjoys the same properties of s stated in Lemma 2.1 once we substituted q and A h withq as in (2.7) (where, instead of h, we consider h 1 ) and A h1 . The last term on the right hand side of (5.6) is bounded following the reasoning developed in the proof of Proposition 3.1 for the terms in (3.6). Given In order to estimate the first term on the right-hand side of Eq. (5.6), following [16], we divide f into pieces We have the following estimates, where we use (5.4).

Some remarks on torsional movements
If we aim at developing the analysis of the interaction between wind and suspension bridges in the hypothesis of strong incoming flux, we need to consider also the torsional movement of the deck. Hence, we refer to the second model introduced in [4]. This section is devoted to discussing some remarks about such a problem. In particular, we will briefly present the problem and illustrate the mathematical differences that the diverse physical behaviour of the obstacle entails: we focus on the reasons why we do not expect the proof used in the case of translation to work also in this case, because the same technique will generate an infinite energy term.
In this case, as already mentioned, the obstacle B is free to rotate around a fixed pin placed at its center of mass; this imposes a constraint on the dimensions of the channel and of the obstacle L 2 > δ 2 + d 2 , because we assume that the obstacle may a priori reach a vertical position. We denote by J the inertia of the body and by θ the angle of rotation with respect to the horizontal. The position of the body is tracked by The fluid-structure-interaction evolution problem that we analyze is: x ⊥ · T (u, p) n t ∈ (0, T ), (6.2) to which we associate the initial conditions u(x, 0) = u 0 , θ(0) = θ 0 , θ (0) = θ 1 . The rectangular rigid body B is free to rotate driven by the action of both an elastic angular restoring force g(θ) and the torque imposed by the fluid stress. The force g plays the role which was played by f in the case of pure translation, thus it resumes the action of the same three forces acting on the deck of a suspension bridge (see Introduction): in this case the action of the hangers and cables is symmetric, which makes g an odd function, and it is not as strong as the action of resistance to deformations of the whole deck. In particular, we assume that g(θ) ∈ C 1 (− π 2 , π 2 ) satisfies the following conditions We emphasize that the third hypothesis in (6.3) is needed in order to prevent the obstacle to reach a vertical position; if we imagine the obstacle to representing the cross section of the deck of a bridge, this assumption seems physically reasonable. We point out that (6.3) compared to (1.4) is not a strong force assumption, but it is weaker, as collisions are already ruled out by the geometric configuration. The difference in the physics, and consequently in the mathematics of this problem with respect to the case of an obstacle vertically translating is revealed when we introduce the change of variables allowing to write the equations of motion in a frame attached to B, whose coordinates are labelled as (y 1 , y 2 ): R y = x, (6.4) where R is defined as in (6.1). The fluid-structure-interaction evolution problem in the new rotating reference frame is obtained consequently, similarly to what is done in [12,29]. We just highlight that the Poiseuille flow at infinity after the change of variables (6.4) assumes the following expression, for each value of θ: q(y) = cos[θ λ L 2 − (y 1 sin θ + y 2 cos θ) 2 ], − sin[θ λ L 2 − (y 1 sin θ + y 2 cos θ) 2 ] . (6.5) The auxiliary fixed domain that we labelledÃ in Sect. 2 is given by: and it corresponds to R 2 ; indeed, the channel is unbounded and a priori it may cover the whole plane when rotating. Here lies the essential difference in the physical framework of this problem with respect to the previous one; in the case of an obstacle purely vertically translating, the movement of the channel was still confined to a region bounded in one direction. Figure 4 represents the new configuration. The auxiliary domainÃ corresponding to the whole plane R 2 also reveals the difference in the mathematical framework. Indeed, the velocity field at infinityq defined in (6.5) in the new reference frame is supposed to be extended outside the rotating channel, which we label A θ(t) , to the whole R 2 if one wants to apply the penalty method, as we did in Sect. 2. Thus, the Poiseuille flow would be extended to be a parabolic branch diverging to infinity: see Fig. 4, where we represented such extension only for the horizontal position not to burden the picture. This implies that the solenoidal extension now captures a flow to which it is associated an infinite energy, thus compromising the possibility to prove existence of a weak solution to the penalized problem and, consequently, also to the original problem, through the penalty method. On the other hand, as already mentioned in the Introduction, besides the penalty method, different methods have been conceived to find existence of solutions (see [8,23,27]), which would allow handling rotation without falling into the aforementioned difficulties. However, they require building global quantities and a global weak formulation, where the integrals must be defined on the whole domain A; as we are not dealing with a simple Newton law in (6.2) due to the presence of g(θ), such methods are not of trivial application.