Existence of time-periodic strong solutions to a fluid-structure system

We study a nonlinear coupled fluid-structure system modelling the blood flow through arteries. The fluid is described by the incompressible Navier-Stokes equations in a 2D rectangular domain where the upper part depends on a structure satisfying a damped Euler-Bernoulli beam equation. The system is driven by time-periodic source terms on the inflow and outflow boundaries. We prove the existence of time-periodic strong solutions for this problem under smallness assumptions for the source terms.


Introduction
In this paper we are interested in the existence of time-periodic solutions for a fluid-structure system involving the incompressible Navier-Stokes equations coupled with a damped Euler-Bernoulli beam equation located on a part of the fluid domain boundary. This system can be used to model the blood flow through human arteries and serves as a benchmark problem for FSI solvers in hemodynamics. When the system is driven by periodic source terms, related for example to the periodic heartbeat, we expect a periodic response of the system. In this article, we prove the existence of time-periodic solutions for the fluid-structure system subject to small periodic impulses on the inflow and outflow boundaries. The study of this fluid-structure model in a periodic framework seems to be new. and Γ d = Γ s ∪ Γ i ∪ Γ b . Let T > 0 be a period of the system, the domain of the fluid at the time 0 ≤ t ≤ T is denoted by Ω η(t) and depends on the displacement of the beam η : Γ s × (0, T ) → (−1, +∞). More precisely Ω η(t) = {(x, y) ∈ R 2 | x ∈ (0, L), 0 < y < 1 + η(x, t)}, Γ η(t) = {(x, y) ∈ R 2 | x ∈ (0, L), y = 1 + η(x, t)}.
The fluid-structure system (1.1) has been investigated with different conditions on the inflow and outflow boundaries: (DBC) homogeneous Dirichlet boundary conditions. (PBC) periodic boundary conditions. (PrBC) pressure boundary conditions. For (DBC), the existence of strong solutions is proved in [2,18,31]. The first result, stated in [2], is the existence of local-in-time strong solutions for small data. This result is then improved in [31], where the stabilization process directly implies the existence of strong solutions, on an arbitrary time interval [0, T ] with T > 0, for small data. Finally, in [18], the existence of strong solutions for small data and of local-in-time strong solutions without smallness assumptions on the initial data is proved. As specified in [5], the strategy developed in [18] works for zero (or small) initial beam displacement. This difficulty, purely nonlinear, was solved in [5] and more recently in [12].
For (PBC), the existence of global strong solutions without smallness assumptions on the initial data is proved in [11]. For a wide range of beam equations, depending on the positivity of the coefficients β, γ, α, the existence of local-in-time strong solutions without smallness assumptions is proved in [11,12].
The third case (PrBC) is introduced in [27] where the existence of weak solutions is proved. We investigated in [5] the existence of local-in-time strong solutions without smallness assumptions on the initial data, which includes non-small initial beam displacement, and the existence of strong solution on [0, T ] with T > 0 for small data.
Here we are interested in the existence of time-periodic strong solutions. The term 'strong solutions' is related to the spacial regularity of the solution, which is typically, for the fluid, H 2 . In the semigroup terminology of evolution equations, the solutions considered in [2,5,11,12,18,31] correspond to strict solutions in L 2 (see Definition 14 in the appendix). Motivated by the stabilization of (1.1) in a neighbourhood of a periodic solution, we prove the existence of a time-periodic strict solution in C 0 for (1.1) with Hölder regularity in time. Our result can be directly adapted for the boundary conditions (DBC)-(PBC)-(PrBC). The Dirichlet boundary condition on the inflow is motivated, once again, by stabilization purpose.
Let us describe the general strategy to construct a periodic solution for (1.1). First, we perform a change of variables mapping the moving domain Ω η(t) into the fixed domain Ω. We then linearize and we rewrite the coupled system as an abstract evolution equation driven by an unbounded operator (A, D(A)) in Section 2. We prove that (A, D(A)) is the infinitesimal generator of an analytic semigroup and that its resolvent is compact. At this stage we use the abstract results developed in the appendix to ensure the existence of a time-periodic solution for the linear system. Finally, we study the nonlinear system in Section 3 with a fixed point argument in the space of periodic functions. The main theorem of this paper, where the notation ♯ denotes time-periodic functions, can be formulated as follows.
Theorem 1. Fix θ ∈ (0, 1) and T > 0. There exists R > 0 such that, for all T -periodic source terms satisfying ;H 1/2 (Γo)) ≤ R, the system (1.1) admits a T -periodic strict solution (u, p, η) belonging to (after a change of variables mapping Ω η(t) into Ω) The functional spaces are introduced in Section 1.2. In the appendix we present existence results for time-periodic abstract evolution equation. For a periodic evolution equation with T > 0, the existence of a solution is related to the spectral criteria 1 ∈ ρ(S(T )) where (S(t)) t≥0 is the semigroup associated with A. This simple criteria follows from the Duhamel formula and is well known. It is stated, for example, in [8,7] for T -periodic mild solutions and in [21,22] for strict solutions in C 0 with Hölder regularity in time (and for time-dependent operator A(t)). Our approach, however, specifies the different regularities that we can expect on the periodic solution, depending on the source term f . We also provide explicit conditions on the pair (A, T ) to ensure that the spectral criteria is satisfied. Remark that the previous results always assume that A is the infinitesimal generator of an analytic semigroup. For abstract periodic evolution equations with weaker assumptions on A we refer to [4].
Let us conclude this introduction with a brief history on the existence of time-periodic solutions for the Navier-Stokes equations. This question was initially considered in 1960s in [13,29,30,32]. In particular, in [13,29,30], the authors obtained a periodic weak solution by considering a fixed point of the Poincaré map which takes an initial value and provides the state of the corresponding initial-value problem at time T . The existence of strong solutions for small data is proved in [14] in 3D and without size rectriction in [33] in 2D. For more recent results with non-homogeneous boundary conditions see [15,26]. The existence theory for the periodic Navier-Stokes equations in bounded domain is now as developed as the existence theory for the initial value problem. For unbounded domain the question is still delicate and was investigated, with zero boundary conditions at the infinity, in [9,10,16,23,24,34]. For further references on the existence of periodic solutions for the Navier-Stokes equations we refer to [17].
The method developed in this article corresponds to the Poincaré map approach, applied on the whole coupled fluid-structure system. Note that the periodic solution obtained for the Navier-Stokes equations is usually unique. Here the free boundary makes the analysis of the uniqueness more complicated. For instance, we cannot considered the difference of two periodic solutions in their respective time-dependent domains, which may be different. The difference has to be taken after a change of variables mapping both periodic solutions in the same domain. In that case, energy estimates are difficult to obtain due to the higher order 'geometrical' nonlinear terms. The uniqueness question remains an open question in our work.
1.1. Equivalent system in a reference configuration. To fix the domain we perform the following change of variables (1.2) We study the linear periodic system associated to (1.3) in Section 2.3-2.5. The existence of time-periodic solution for (1.3) is established in Section 3 with a fixed point procedure.

Function spaces.
To deal with the mixed boundary conditions introduce the spaces and the orthogonal decomposition of L 2 (Ω) = L 2 (Ω, R 2 ) be the so-called Leray projector associated with this decomposition. If u belongs to L 2 (Ω) then Πu = u − ∇p u − ∇q u where p u and q u are solutions to the following elliptic equations Throughout this article the functions and spaces with vector values are written with a bold typography.
For example H 2 (Ω) = H 2 (Ω, R 2 ). Using the notations in [19,Theorem 11.7], we introduce the space H Odd and even symmetries preserve the H k -regularity for functions in H k 0 (Γ s ) with k = 1, 2, thus, by interpolation, the H 3/2 -regularity is also preserved for functions in H 3/2 00 (Γ s ). This property is used in [5] to handle the pressure boundary condition.
For the boundary condition on the inflow, we use the results developed in [25] for elliptic equations in a dihedron. In our case, the angle between Γ i and Γ s is equal to π 2 . If ω (resp. g) denotes the boundary condition on Γ i (resp. Γ s,0 ), the Laplace and Stokes equations possess solutions with H 2 -regularity near C 0,1 = (0, 1) provided that the data are regular enough and that the compatibility conditions ω(C 0,1 ) = g(C 0,1 ) is satisfied. To ensure these conditions, the non-homogeneous boundary condition on The energy space associated with (1.5) is The regularity result for (1.5) is similar to [5,Theorem 5.4] and we define the Stokes operator ( We also introduce the space V s (Ω) = {u ∈ H s (Ω) | div u = 0} for s ≥ 0. To describe the Dirichlet boundary condition on Γ s set For space-time dependent functions we use the notations introduced in [20]:

Linear system
2.1. Stokes system with non-homogeneous mixed boundary conditions. In this section we consider the Stokes system λu − ν∆u + ∇p = f , div u = 0 in Ω, The following lemmas provide suitable lifting of the non-homogeneous Dirichlet boundary conditions on Γ s and Γ i .
Proof. The idea to solve (2.2) is to use a Stokes system with Dirichlet boundary conditions on an extended domain. We set Ω e = (0, 2L) Thanks to the properties of the space H 3/2 00 (Γ s ) with respect to symmetries, the functionĝ is in H 3/2 00 (Γ s,e ). Moreover, it has a zero average by construction. Consider the Stokes system This system admits a unique solution (v, q) ∈ H 2 (Ω e ) × H 1 (Ω e ) (see for example [25]; note that one could not find w directly by solving (2.3) on Ω, since g does not necessarily have a zero average on Γ s , contrary toĝ on Γ s,e . We introduce the function v s (x, y) : The function v s ∈ H 2 (Ω e ) still satisfies andv := v+vs 2 verifiesv 2 (L, y) = 0 for all y ∈ (0, 1). The restriction to Ω ofv is solution to (2.2). The linearity of the mapping g → w is obvious from the construction above, and its continuity (that is, an estimate w H 2 (Ω) ≤ C g H 3/2 00 (Γs) ) follows from the classical estimates for the Stokes system with Dirichlet boundary conditions.
Proof. Once again we construct w by solving a Stokes system with Dirichlet boundary conditions. First, we have to compensate the non-zero average of ω · n on Γ i . Consider the function where ϕ ∈ C ∞ 0 (Γ s ) satisfies Γs ϕ = 0. Consider then the system Using [25], we obtain a solution (v, q) ∈ H 2 (Ω) × H 1 (Ω) to this system. Finally Once again, the linearity of Φ i : ω → w is trivial by construction, and its continuity follows from the classical estimates for the Stokes equations with Dirichlet boundary conditions, and from the construction of ω − .
We can now specify the regularity results for (2.1).
The H 2 -regularity of v in a neighbourhood of Γ i is well known for Stokes with homogeneous Dirichlet conditions. The lower order term λv does not impact the regularity of the system and can be dealt with a bootstrap argument. The regularity on a neighbourhood of Γ o is proved in [5,Theorem 5.4]. Hence, (v, p) ∈ H 2 (Ω) × H 1 (Ω), and thus (u, p) ∈ H 2 (Ω) × H 1 (Ω) with the desired estimates.
We introduce the lifting operators: where (w 2 , ρ 2 ) is the solution to (2.1) with (f , g, ω) = (0, 0, ω) and λ = 0. • L Γo ∈ L(H 1/2 (Γ o ), H 1 (Ω)) a continuous lifting operator. In order to express the pressure, we also consider the operators: Lemma 5. The operator N s can be extended as follows: Proof. The first result is obtained by duality. The second follows from interpolation techniques.
To prepare the matrix formulation of the fluid-structure system, we recast the Stokes system in terms of Πu and (I − Π)u.
Proof. Remark that u − L 1 (g) belongs to D(A s ) and In The space H 2 0 (Γ s ) is equipped with the inner product 2.3. Semigroup formulation of the linear fluid-structure system. Consider a period T > 0. Set θ ∈ (0, 1) and .
consider the following linear system For a scalar function η defined on Γ s we use the notation L 1 (η) = L 1 (ηe 2 ). We look for a solution to (2.8) under the form (u, p, η) = (v, q, η) We then apply Theorem 6 to conclude.
Owing to Theorem 8, System (2.9) can be recast in terms of (Πv, η, η t ): and A is the unbounded operator in H defined by with ∆ s = ∂ xx .
2.4. Analyticity of A. The unbounded operator A has already been studied, with small variations related to the boundary conditions, in [31,5].

Theorem 9. The operator (A, D(A)) is the infinitesimal generator of an analytic semigroup on H.
Moreover, the resolvent of (A, D(A)) is compact.
The Stokes system can then be solved, and we find (u, p) ∈ H 2 (Ω) × H 1 (Ω) solution to (2.14) 1−4 such that In the appendix, we prove the existence of time-periodic solutions for abstract evolution equations y ′ (t) = Ay(t) + f (t) under the assumption (A.4). This assumption is a restriction on the period T of the system depending on the eigenvalues of A lying on the imaginary axis. Here, this condition does not restrict the choice of T as we are able to prove that all the non-zero eigenvalues of A have a negative real part. Indeed, let λ ∈ C be a non-zero eigenvalues of A and (Πu, η 1 , η 2 ) ∈ D(A) be an associated eigenvector. The system  Combining the previous energy estimates we obtain Taking the real part of the previous identity we deduce that Re λ < 0. It is easily verified that 0 ∈ σ p (A) (recall that σ p (A) = σ(A) as the resolvent of A is compact) and we can apply Theorem 20 to solve the linear system (2.12) without restriction on the period T . Let W be the set defined by . The regularity space for the beam is denoted by Moreover (u(0), η(0), η t (0)) is given by (2.16)

Nonlinear system
In this section we prove the existence of classical solutions for the nonlinear system (1.3) using a fixed point argument. Without additional source terms in the model, here given through the inflow and outflow boundary conditions, the solution obtained with the fixed point procedure is the null solution. Hence, in what follows, the pair (ω 1 , ω 2 ) is assumed to be non trivial, eventually small enough, and represent the 'impulse' of the system. The period T of (ω 1 , ω 2 ) determines the period of the whole system.
Theorem 12. There exists R * > 0 and µ * > 0 such that for all where C L is the constant involved in (2.16), system (1.3) admits a unique solution (u, p, η) in the ball B(R * , µ * ).
Proof. Let R 1 > 0 and µ * > 1. In order to ensure that the map F is well defined from B(R * , µ * ) into itself (with R * to be defined) we control the estimate on (1 + η) −1 C([0,T ]×Γs) with the parameter R 1 . Precisely, for all (u, p, η) ∈ B(R 1 , µ * ), the following estimate holds with C ∞ > 0 a positive constant. Then we choose R 2 < µ * −1 C∞µ * and for all (u, p, η) ∈ B(R 2 , µ * ) the following estimate holds The linear estimate 2.16 implies that, for all (u, p, η) ∈ B(R 2 , µ * ), We choose 0 < R * < R 2 such that C(µ * )Q(R * ) < min( 1 2CL , 1 2 ). Finally choose (ω 1 , ω 2 ) such that At this point we proved that F is well defined from B(R * , µ * ) into itself. Moreover, using (3.1), the Lipschitz estimate In what follows we assume that the pair (A, T ) satisfies the assumption: where {ib j } 0≤j≤NA denote the non zero eigenvalues of A on the imaginary axis iR Remark that the assumptions A generates an analytic semigroup and has a compact resolvent directly imply that N A is a finite number.
A.1. Hilbert case. In this section we obtain a simple criteria to ensure that the problem (A.2) admits a unique strict solution in L 2 (0, T ; H). The assumption that A has a compact resolvent implies (see [28,Theorem 3.3] and recall that S(t) is analytic and thus differentiable, which implies the continuity for the uniform operator topology for t > 0) that S(t) is a compact semigroup. Hence σ(S(T )) = σ p (S(T )) and the spectral mapping theorem e T σp(A) = σ p (S(T )), coupled with the assumption (A.4), shows that 1 ∈ ρ(S(T )). Thus Hence we have proved the following theorem. Using the regularization properties of analytic semigroup for t > 0, that is S(t)z ∈ D(A n ) for all n ≥ 1 and z ∈ H, we can prove that the regularity of the solution solely depends on the source term f . Hence the previous result can be improved when f is more regular. We introduce the space H r = [D(A n+1 ), D(A n )] 1−α with r = n + α, n ≥ 0 an integer and 0 ≤ α ≤ 1 a real number.