Remark on an elastic plate interacting with a gas in a semi-infinite tube: periodic solutions

We consider a conservative system consisting of an elastic plate interacting with a gas filling a semi-infinite tube. The plate is placed on the bottom of the tube. The dynamics of the gas velocity potential is governed by the linear wave equation. The plate displacement satisfies the linear Kirchhoff equation. We show that this system possesses an infinite number of periodic solutions with the frequencies tending to infinity. This means that the well-known property of decaying of local wave energy in tube domains does not hold for the system considered.

Recently there was great interest in the study of long-time dynamics of elastic plates interacting with a flow of gas (see, e.g., [3,4,6,8,9,10,11,13,14] and the literature cited in those sources). The corresponding model has the form where φ(x 1 , x 2 , x 3 ; t) solves the problem in R 3 (2) * e-mail: chueshov@karazin.ua Here above Ω is a bounded smooth domain in R 2 identified with {(x 1 ; x 2 ; 0) : (x 1 ; x 2 ) ∈ Ω} ⊂ R 3 + . The term f (u) describes a nonlinear force which can be (a) von Karman type (like as in [6]), or (b) Berger type (like as in [1]), or even (c) generated by some Nemytskii operator (see, e.g., [5,7]). The unknown function u = u(x 1 , x 2 ; t) measures the transverse displacement of the plate at the point (x 1 ; x 2 ) and time t. The boundary conditions for u means that the plate is hinged on its edge. The function φ(x, t) = φ(x 1 , x 2 , x 3 ; t) is velocity potential of the gas filing the domain R 3 + . Here we deal with interaction of a plate with a gas flow moving with the speed U in the direction of the axis x 1 . The aerodynamical pressure of the gas on the plate is given by the term p(x, t) = ν(φ t + U φ x1 )| x3=0 , the parameter ν > 0 characterizes the intensity of the interaction between the gas and the plate. The transverse displacement u(x, t) has influence on the gas via boundary condition in (2).
For recent surveys of mathematical and applied aspects of the model above we refer to [3,4]. Here we only mention the convergence results in the subsonic case (0 ≤ U < 1) which were established in [13,14] (see also [6] for related facts) and state stabilization of solutions to stationary states of the system when t → ∞ under some conditions concerning initial data of the gas velocity potential. The corresponding argument requires positivity of the damping parameter γ and involves a gradient-type structure of the system in the case considered.
In this relation the question (see [2] and [6, p.694]) arises whether it is possible to obtain a similar stabilization result in the absence (γ = 0) of the internal damping in the plate. This conjecture is based on the well-known property of the local energy decay for the wave equation in R 3 and some other unbounded domains (see also Proposition 2.1 below).
Our main goal in this note is to show that unboundness of the wave domain O is not sufficient to guarantee stabilization of solutions to equilibria. For this we consider a linear plate model without any damping (γ = 0), coupled to the flow via matching velocities. The parameter U is taken to be zero. We show that in this scenario, periodic solutions may exist. In the case of a specific tubular domain they are explicitly constructed. Whether the same result holds for nonlinear plate is an open question. However, the model indicates the necessity of introducing mechanical damping in the plate model if one expects a strong convergence to equilibria of the full flow-structure system.

Model
Let Ω be a smooth bounded domain in R 2 . We consider the following problem We denote by φ| x3=0 the trace of a function φ( The is velocity potential of the gas filing the tube O + . The pressure of the gas on the plate is given by the term p( In the case of bounded domains O systems like (3) and (4 was studied by many authors (see the discussion and the references in [6] and [12]). Below we use the notation H s (D) for the Sobolev space of order s on a domain D in R d , d = 2, 3.
We start with the following assertion.
Then there exists a unique couple {u; φ} of function and solving (3) and (4) in the sense of distributions. Moreover this solution satisfies the energy preservation law of the form where we use the notations Proof. We can apply well-known general result reported in [12], see also [6,Chapter 6], where nonlinear versions of similar problems are discussed. However we can also give more direct argument based on some symmetry of this linear problem and involving the variables separation. We sketch the corresponding argument below. Let {e k } be an orthonormal basis in L 2 (Ω) consisting of eigenvectors of the problem ∆w + λw = 0, w| ∂Ω = 0, and 0 < λ 1 ≤ λ 2 ≤ ... the corresponding eigenvalues. We are looking for solutions to problem (3) and (4) in the following form where u k (t) satisfies the equation It is easy to show that for each k problem (8) and (9) has a unique solution (φ k (t), u k (t)) for which we have the following energy balance relation where the energy E k of the k-mode has the form This observations allow us to obtain appropriate a priori estimates and conclude the proof by the standard compactness method.

Dynamics
We start with the following assertion that shows a local energy decay in the case when the bottom Ω of the cylinder O + is rigid. This means that we consider the wave dynamics only. This dynamics is described by the following equations Proposition 2.1 (Local energy decay). Assume that φ 0 ∈ H 1 (O + ) and φ 1 ∈ L 2 (O + ). Then problem (10) has a unique variational solution φ which belongs to the class This this solution satisfies the energy preservation law of the form Moreover, we have decaying of φ as t → ∞ in the local energy norm, i.e., where E gas Proof. The existence and uniqueness of solutions to (10) is obvious (we can use the same idea as in Proposition 1.1, for instance). It is also clear that the energy relation is satisfied. Thus we only need to establish the property in (11). Extending the initial data as even functions in the variable x 3 on the whole x 3 -axis we can consider the wave equation in the domain with the Dirichlet boundary conditions on ∂O. Now we can separate variables as above an apply the same idea as in [15] to prove (11) for localized initial data φ 0 and φ 1 . In fact the article [15] contains exactly this statement for the case when Ω = (0, π)×(0, π). The method suggested in [15] relies on the presentation of the solution φ in the form where e mn (x 1 , x 2 ) = 2π −1 sin ma 1 sin nx 2 are solutions to the spectral problem (6) for Ω = (0, π) × (0, π) and φ mn (t, z) solves the equation ∂ tt φ mn − ∂ zz φ mn + (m 2 + n 2 )φ mn = 0, z ∈ R, t > 0, Establishing appropriate bounds for φ mn (see [15]) one can prove the desired result for Ω = (0, π)×(0, π). The calculations given in [15] can be easily extended to the case of general domains Ω.
Then using approximation procedure for initial data and the energy relation we can obtain the result for every pair ( The decay property of the local wave energy demonstrated in Proposition 2.1 is not valid for the coupled system in (3) and (4). More precisely, we show that problem (3) and (4)  such that the functions and where A k and B k are arbitrary real numbers, solve problem (3) and (4) with appropriate initial data. Each trajectory (ϕ k ; ϕ k t ; u k ; u k t ) is Lyapunov stable in the phase space Proof. Let us look for solutions to (8) and (9) of the form φ k (t, z) = e iωt e −αz , u k (t) = ae iωt with α > 0 and ω, a ∈ C. The substitution in (8) and (9) gives us the relations − ω 2 − α 2 + λ k = 0, α = iωa, a(−ω 2 + λ 2 k ) + iνω = 0. This implies that a = −iαω −1 and also One can see for every k there exists unique solution (ω 2 k , α k ) to (15). It is also easy to find that (ω 2 k , α k ) ∼ (λ k , νλ −1 k ) when k → +∞ in the sense of (12). This implies the structure of a solution written in (13) and (14).
Stability properties of solutions follow from the energy preservation law.
Theorem 2.2 shows that the elasticity of the bottom Ω of the cylinder O + destroy the local energy decay property which we observe in the case of rigid bottom (see Proposition 2.1).
We conclude this note with several open questions which, we believe, are important for understanding of long-time dynamics of flow-structure systems.
Open Questions: • Can we show that the minimal subspace in H containing all solutions (ϕ k ; ϕ k t ; u k ; u k t ) is asymptotically stable? Is this subspace a global minimal attractor? If not, what is a real candidate on the role of global attractor for (3) and (4)?
• What can we say about stability and spectral properties of the generators of C 0 semigroups generated by (3) and (4) and its dissipative perturbation? For instance, is it possible to stabilize the system by introducing internal damping in the plate component only?
These questions are important not only for linear dynamics, but also for nonlinear perturbations of (3) and (4).