# Existence and Uniqueness of Strong Solutions to a Bi-dimensional Fluid-Structure Interaction System

Céline Grandmont, Lamis Sabbagh

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Céline Grandmont, Lamis Sabbagh. Existence and Uniqueness of Strong Solutions to a Bi-dimensional Fluid-Structure Interaction System. 2024. hal-04624252

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# Existence and Uniqueness of Strong Solutions to a Bi-dimensional
Fluid-Structure Interaction System

Céline Grandmont<sup>2,3,1</sup>, Lamis Sabbagh<sup>1</sup>

<sup>1</sup>Département de Mathématique, Université Libre de Bruxelles,
Boulevard du Triomphe, 1050 Bruxelles, Belgium

<sup>2</sup> LJLL, Sorbonne Université, CNRS, 4 Place Jussieu, 75005 Paris, France

<sup>3</sup> Centre Inria de Paris, 48, rue Barrault, 75647 Paris cedex, France

June 25, 2024

## Abstract

We study an unsteady non-linear fluid-structure interaction problem where a two-dimensional viscous incompressible Newtonian fluid and a mono-dimensional elastic structure, located on one part of the fluid domain boundary, interacts. The fluid motion is modelled by the bi-dimensional incompressible Navier-Stokes equations set in an unknown domain which depends on the structure's displacement. We consider longitudinal as well as transversal structure displacement. We assume that the longitudinal displacement of the structure satisfies a wave equation whereas the transversal displacement follows a beam equation with inertia of rotation. The fluid and structure systems are coupled through a kinematic condition which corresponds to a no-slip condition at the fluid-structure interface and the fluid exerts a force on the elastic structure. We prove the existence and uniqueness of strong solution to the considered problem with no gap between the initial conditions regularity and the ones obtained in positive time. To our knowledge, this is the first result regarding existence and uniqueness of strong solutions for fluid-beam interaction problem in the unsteady case taking into account both the transversal displacement and the longitudinal structure displacement.

**Keywords.** Fluid-structure interaction, Navier-Stokes system, beam equations, strong solutions.

## 1 Introduction

In this paper, we study a fluid-structure interaction problem involving a two-dimensional viscous incompressible Newtonian homogeneous fluid and a moving deformable structure located on one part of the fluid domain boundary. The model that we consider herein can be viewed as a toy model to describe blood flow in arteries [11, 36]. We suppose that the fluid is occupying a 2D container whose reference configuration is $\overline{\mathcal{F}}_0 = \cup_{k \in \mathbb{Z}} (\overline{\Omega}_0 + kLe_1)$ where $\Omega_0 = (0, L) \times (0, 1)$ denotes the reference periodic cell as we are looking for $L$-periodic in $x$ solutions. The lower boundary $\mathbb{R} \times \{0\} = \cup_{k \in \mathbb{Z}} (\overline{\Gamma}_0 + kLe_1)$, with $\Gamma_0 = (0, L) \times \{0\}$ is assumed to be fixed and the reference upper boundary $\cup_{k \in \mathbb{Z}} (\overline{\Gamma}_e + kLe_1)$, with $\Gamma_e = (0, L) \times \{1\}$ is an elastic mono-dimensional structure, that can move in the longitudinal direction and in the transversal one. Let us denote by $(x, t) \mapsto (\zeta(x, t), \eta(x, t))^T$ the displacement of the structure at time $t$. The displacement of the structure modifies the fluid domain and hence the fluid domain is a time
# Existence and Uniqueness of Strong Solutions to a Bi-dimensional
Fluid-Structure Interaction System

Céline Grandmont<sup>2,3,1</sup>, Lamis Sabbagh<sup>1</sup>

<sup>1</sup>Département de Mathématique, Université Libre de Bruxelles,
Boulevard du Triomphe, 1050 Bruxelles, Belgium

<sup>2</sup> LJLL, Sorbonne Université, CNRS, 4 Place Jussieu, 75005 Paris, France

<sup>3</sup> Centre Inria de Paris, 48, rue Barrault, 75647 Paris cedex, France

June 25, 2024

## Abstract

We study an unsteady non-linear fluid-structure interaction problem where a two-dimensional viscous incompressible Newtonian fluid and a mono-dimensional elastic structure, located on one part of the fluid domain boundary, interacts. The fluid motion is modelled by the bi-dimensional incompressible Navier-Stokes equations set in an unknown domain which depends on the structure's displacement. We consider longitudinal as well as transversal structure displacement. We assume that the longitudinal displacement of the structure satisfies a wave equation whereas the transversal displacement follows a beam equation with inertia of rotation. The fluid and structure systems are coupled through a kinematic condition which corresponds to a no-slip condition at the fluid-structure interface and the fluid exerts a force on the elastic structure. We prove the existence and uniqueness of strong solution to the considered problem with no gap between the initial conditions regularity and the ones obtained in positive time. To our knowledge, this is the first result regarding existence and uniqueness of strong solutions for fluid-beam interaction problem in the unsteady case taking into account both the transversal displacement and the longitudinal structure displacement.

**Keywords.** Fluid-structure interaction, Navier-Stokes system, beam equations, strong solutions.

# 1 Introduction

In this paper, we study a fluid-structure interaction problem involving a two-dimensional viscous incompressible Newtonian homogeneous fluid and a moving deformable structure located on one part of the fluid domain boundary. The model that we consider herein can be viewed as a toy model to describe blood flow in arteries [11, 36]. We suppose that the fluid is occupying a 2D container whose reference configuration is $\overline{\mathcal{F}}_0 = \cup_{k \in \mathbb{Z}} (\overline{\Omega}_0 + kLe_1)$ where $\Omega_0 = (0, L) \times (0, 1)$ denotes the reference periodic cell as we are looking for $L$-periodic in $x$ solutions. The lower boundary $\mathbb{R} \times \{0\} = \cup_{k \in \mathbb{Z}} (\overline{\Gamma}_0 + kLe_1)$, with $\Gamma_0 = (0, L) \times \{0\}$ is assumed to be fixed and the reference upper boundary $\cup_{k \in \mathbb{Z}} (\overline{\Gamma}_e + kLe_1)$, with $\Gamma_e = (0, L) \times \{1\}$ is an elastic mono-dimensional structure, that can move in the longitudinal direction and in the transversal one. Let us denote by $(x, t) \mapsto (\zeta(x, t), \eta(x, t))^T$ the displacement of the structure at time $t$. The displacement of the structure modifies the fluid domain and hence the fluid domain is a time
dependent unknown in the system. Let us denote by $\Gamma_{\zeta(t),\eta(t)} = \{(x + \zeta(x, t), 1 + \eta(x, t)), x \in [0, L] + kLe_1, k \in \mathbb{Z}\}$ the deformed upper boundary. We then denote by $\mathcal{F}_{\zeta(t),\eta(t)} \subset \mathbb{R}^2$ the fluid domain at time $t > 0$ enclosed between $\Gamma_{\zeta(t),\eta(t)}$ and $\mathbb{R} \times \{0\}$. We assume that the motion of the fluid is described by the incompressible Navier-Stokes equations in $F_T := \bigcup_{t \in (0,T)} \mathcal{F}_{\zeta(t),\eta(t)} \times \{t\}$, with $T > 0$. Thus the fluid velocity $u$ and the fluid pressure $p$ satisfy in $F_T$

$$ \rho_f(\partial_t u + (u \cdot \nabla)u) - \operatorname{div} \sigma(u, p) = 0, \quad (1.1) $$

$$ \nabla \cdot u = 0, \quad (1.2) $$

with $\rho_f > 0$ the fluid density, and where $\sigma$ denotes the stress tensor of the fluid which is defined by

$$ \sigma(u, p) = -pI_2 + 2\mu D(u), $$

where $\mu$ denotes the viscosity of the fluid, $I_2$ is the identity matrix and $D(u) = \frac{1}{2}(\nabla u + \nabla u^T)$ is deformation tensor ($A^T$ denotes the transpose of the matrix $A$).

Furthermore the longitudinal structure displacement $\zeta$ solves a wave equation in the reference structure cell $(0, L)$, that writes

$$ \rho_s \partial_{tt} \zeta - \beta \partial_{xx} \zeta = \phi(u, p, \zeta, \eta) \cdot e_1, \quad \text{in } (0, L) \times (0, T), \quad (1.3) $$

whereas the transversal displacement $\eta$ satisfies a linear beam equation with inertia of rotation

$$ \rho_s \partial_{tt} \eta - \gamma \partial_{xxtt} \eta + \alpha \partial_{xxxx} \eta = \phi(u, p, \zeta, \eta) \cdot e_2, \quad \text{in } (0, L) \times (0, T), \quad (1.4) $$

where $\rho_s$, $\alpha$, $\beta$, and $\gamma$ are positive real constants, and $(e_1, e_2)$ is the canonical basis of $\mathbb{R}^2$. These beam equations can be derived from three dimensional elasticity thanks to an asymptotic analysis and, in particular, in such analysis the term $-\gamma \partial_{xxtt} \eta$, which models the inertia of rotation, appears (see [37], in the case of a plate). We emphasize that the equations modelling the displacement of structure are written in a reference configuration whereas the fluid equations are written in Eulerian coordinates and consequently in an unknown non cylindrical domain $F_T$. The fluid and structure equations are coupled through the forcing term $\phi(u, p, h)$ in the right hand side of (1.3) and (1.4). It corresponds to the trace of normal component of the fluid stress tensor, multiplied by the element length, $\sigma(u, p) n \cdot d\ell$ (where $n$ is the unit normal vector directed outward to $\Gamma_{\zeta(t),\eta(t)}$) that is further transported in the reference configuration. Precisely, $\phi$ reads as

$$ \phi(u, p, \zeta, \eta)(x, t) = -\sigma(u, p)(x + \zeta(x, t), 1 + \eta(x, t), t) \cdot (-\partial_x \eta(x, t)e_1 + (1 + \partial_x \zeta(x, t))e_2), \quad \text{on } (0, L) \times (0, T). \quad (1.5) $$

We underline that the vector $-\partial_x \eta(x, t)e_1 + (1 + \partial_x \zeta(x, t))e_2$ is a non unitary normal vector to the elastic the interface $\Gamma_{\zeta(t),\eta(t)}$. The fluid and the structure are also coupled by the no-slip boundary condition at the interface:

$$ u(x + \zeta(x, t), 1 + \eta(x, t), t) = \partial_t \zeta(x, t)e_1 + \partial_t \eta(x, t)e_2, \quad (x, t) \in (0, L) \times (0, T). \quad (1.6) $$

We complement our system with

i. L-periodicity w.r.t $x$ for the fluid velocity and pressure and for the structure displacement;
dependent unknown in the system. Let us denote by $\Gamma_{\zeta(t),\eta(t)} = \{(x + \zeta(x, t), 1 + \eta(x, t)), x \in [0, L] + kLe_1, k \in \mathbb{Z}\}$ the deformed upper boundary. We then denote by $\mathcal{F}_{\zeta(t),\eta(t)} \subset \mathbb{R}^2$ the fluid domain at time $t > 0$ enclosed between $\Gamma_{\zeta(t),\eta(t)}$ and $\mathbb{R} \times \{0\}$. We assume that the motion of the fluid is described by the incompressible Navier-Stokes equations in $F_T := \bigcup_{t \in (0,T)} \mathcal{F}_{\zeta(t),\eta(t)} \times \{t\}$, with $T > 0$. Thus the fluid velocity $u$ and the fluid pressure $p$ satisfy in $F_T$

$$ \rho_f(\partial_t u + (u \cdot \nabla)u) - \operatorname{div} \sigma(u, p) = 0, \qquad (1.1) $$

$$ \nabla \cdot u = 0, \qquad (1.2) $$

with $\rho_f > 0$ the fluid density, and where $\sigma$ denotes the stress tensor of the fluid which is defined by

$$ \sigma(u, p) = -pI_2 + 2\mu D(u), $$

where $\mu$ denotes the viscosity of the fluid, $I_2$ is the identity matrix and $D(u) = \frac{1}{2}(\nabla u + \nabla u^T)$ is deformation tensor ($A^T$ denotes the transpose of the matrix $A$).

Furthermore the longitudinal structure displacement $\zeta$ solves a wave equation in the reference structure cell $(0, L)$, that writes

$$ \rho_s \partial_{tt} \zeta - \beta \partial_{xx} \zeta = \phi(u, p, \zeta, \eta) \cdot e_1, \quad \text{in } (0, L) \times (0, T), \qquad (1.3) $$

whereas the transversal displacement $\eta$ satisfies a linear beam equation with inertia of rotation

$$ \rho_s \partial_{tt} \eta - \gamma \partial_{xxtt} \eta + \alpha \partial_{xxxx} \eta = \phi(u, p, \zeta, \eta) \cdot e_2, \quad \text{in } (0, L) \times (0, T), \qquad (1.4) $$

where $\rho_s$, $\alpha$, $\beta$, and $\gamma$ are positive real constants, and $(e_1, e_2)$ is the canonical basis of $\mathbb{R}^2$. These beam equations can be derived from three dimensional elasticity thanks to an asymptotic analysis and, in particular, in such analysis the term $-\gamma \partial_{xxtt} \eta$, which models the inertia of rotation, appears (see [37], in the case of a plate). We emphasize that the equations modelling the displacement of structure are written in a reference configuration whereas the fluid equations are written in Eulerian coordinates and consequently in an unknown non cylindrical domain $F_T$. The fluid and structure equations are coupled through the forcing term $\phi(u, p, h)$ in the right hand side of (1.3) and (1.4). It corresponds to the trace of normal component of the fluid stress tensor, multiplied by the element length, $\sigma(u, p) n \cdot d\ell$ (where $n$ is the unit normal vector directed outward to $\Gamma_{\zeta(t),\eta(t)}$) that is further transported in the reference configuration. Precisely, $\phi$ reads as

$$ \phi(u, p, \zeta, \eta)(x, t) = -\sigma(u, p)(x + \zeta(x, t), 1 + \eta(x, t), t) \cdot (-\partial_x \eta(x, t)e_1 + (1 + \partial_x \zeta(x, t))e_2), \quad \text{on } (0, L) \times (0, T). \qquad (1.5) $$

We underline that the vector $-\partial_x \eta(x, t)e_1 + (1 + \partial_x \zeta(x, t))e_2$ is a non unitary normal vector to the elastic the interface $\Gamma_{\zeta(t),\eta(t)}$. The fluid and the structure are also coupled by the no-slip boundary condition at the interface:

$$ u(x + \zeta(x, t), 1 + \eta(x, t), t) = \partial_t \zeta(x, t)e_1 + \partial_t \eta(x, t)e_2, \quad (x, t) \in (0, L) \times (0, T). \qquad (1.6) $$

We complement our system with

i. L-periodicity w.r.t $x$ for the fluid velocity and pressure and for the structure displacement;
ii. no-slip boundary condition on the bottom of the fluid container:

$$u(., 0, t) = 0. \qquad (1.7)$$

To close the problem herein, we impose initial conditions at $t = 0$:

$$\zeta(x, 0) = \zeta_0(x), \quad x \in (0, L), \qquad (1.8)$$

$$\partial_t \zeta(x, 0) = \zeta_1(x), \quad x \in (0, L), \qquad (1.9)$$

$$\eta(x, 0) = \eta_0(x), \quad x \in (0, L), \qquad (1.10)$$

$$\partial_t \eta(x, 0) = \eta_1(x), \quad x \in (0, L), \qquad (1.11)$$

$$u(x, y, 0) = u_0(x, y), \quad \text{in } \mathcal{F}_{\zeta_0, \eta_0}. \qquad (1.12)$$

The initial displacement and velocities are assumed to be L-periodic in $x$. The purpose of this work is to show the existence of a unique local-in-time strong solution to the system (1.1)-(1.7) completed with initial conditions (1.8)-(1.12).

Most of the literature on the well-posedness of 2D/1D or 3D/2D fluid-structure coupled problems is done ignoring the longitudinal displacement of the structure. Indeed, these studies tackle the fluid-structure interaction problem consisting of the incompressible Navier-Stokes equations (1.1)-(1.2) describing the fluid behaviour and an elastic beam, plate or shell in flexion. In the two-dimensional case either the wave equation (1.3) or the beam equation (1.4) or the damped beam equation have been considered to describe the transversal structure displacement as the structure is only allowed to move along the normal direction of the reference configuration. In dimension three, the case of plate or shell in flexion have also been considered. In this respect, existence of weak solutions as long as the structure does not touch the bottom of the fluid cavity is established in [14] in the case the transversal structure displacement satisfies a beam equation with an additional damping term. In this case existence beyond contact has also been derived in [13]. We also mention [25,32] in 2D and [26] in 3D for the case of a plate equation without any additional damping or inertia of rotation terms. The case of Koiter shell is also considered in [33] and in [18] for generalized Newtonian flow. The uniqueness of weak solution is not an easy issue, even with small initial data and it is still an open question. However, a weak-strong uniqueness result is established in [38] in case where the structure satisfies a wave equation.

Several works have also been devoted to study of the existence and uniqueness of strong solutions to similar problems but still neglecting the longitudinal displacement of the structure. In particular, existence and uniqueness of strong solutions in two-dimension was established in the case the transversal structure displacement satisfies a wave equation with additional damping term in [31] and without damping in [4, 28]. The method used in [28] is based on a regularization arguments and a priori estimates whereas the strategy of proof in [4] consists on showing that the semi-group associated to the linearized problem is analytic. In both strategy the damping coming from the fluid is crucial. Moreover when considering the beam equation (1.4) the presence of a damping or inertia rotation term seems to be important to obtain existence and uniqueness of strong solutions in the case where the initial fluid velocity is in $H^1$ (see [12, 28, 30]), in particular with no gap between the regularity in space of the initial conditions and the one obtained for the solution [28]. Note however a recent preprint [34] requiring more regular initial data.
ii. no-slip boundary condition on the bottom of the fluid container:

$$u(., 0, t) = 0. \qquad (1.7)$$

To close the problem herein, we impose initial conditions at $t = 0$:

$$\zeta(x, 0) = \zeta_0(x), \quad x \in (0, L), \qquad (1.8)$$

$$\partial_t \zeta(x, 0) = \zeta_1(x), \quad x \in (0, L), \qquad (1.9)$$

$$\eta(x, 0) = \eta_0(x), \quad x \in (0, L), \qquad (1.10)$$

$$\partial_t \eta(x, 0) = \eta_1(x), \quad x \in (0, L), \qquad (1.11)$$

$$u(x, y, 0) = u_0(x, y), \quad \text{in } \mathcal{F}_{\zeta_0, \eta_0}. \qquad (1.12)$$

The initial displacement and velocities are assumed to be L-periodic in $x$. The purpose of this work is to show the existence of a unique local-in-time strong solution to the system (1.1)-(1.7) completed with initial conditions (1.8)-(1.12).

Most of the literature on the well-posedness of 2D/1D or 3D/2D fluid-structure coupled problems is done ignoring the longitudinal displacement of the structure. Indeed, these studies tackle the fluid-structure interaction problem consisting of the incompressible Navier-Stokes equations (1.1)-(1.2) describing the fluid behaviour and an elastic beam, plate or shell in flexion. In the two-dimensional case either the wave equation (1.3) or the beam equation (1.4) or the damped beam equation have been considered to describe the transversal structure displacement as the structure is only allowed to move along the normal direction of the reference configuration. In dimension three, the case of plate or shell in flexion have also been considered. In this respect, existence of weak solutions as long as the structure does not touch the bottom of the fluid cavity is established in [14] in the case the transversal structure displacement satisfies a beam equation with an additional damping term. In this case existence beyond contact has also been derived in [13]. We also mention [25,32] in 2D and [26] in 3D for the case of a plate equation without any additional damping or inertia of rotation terms. The case of Koiter shell is also considered in [33] and in [18] for generalized Newtonian flow. The uniqueness of weak solution is not an easy issue, even with small initial data and it is still an open question. However, a weak-strong uniqueness result is established in [38] in case where the structure satisfies a wave equation.

Several works have also been devoted to study of the existence and uniqueness of strong solutions to similar problems but still neglecting the longitudinal displacement of the structure. In particular, existence and uniqueness of strong solutions in two-dimension was established in the case the transversal structure displacement satisfies a wave equation with additional damping term in [31] and without damping in [4, 28]. The method used in [28] is based on a regularization arguments and a priori estimates whereas the strategy of proof in [4] consists on showing that the semi-group associated to the linearized problem is analytic. In both strategy the damping coming from the fluid is crucial. Moreover when considering the beam equation (1.4) the presence of a damping or inertia rotation term seems to be important to obtain existence and uniqueness of strong solutions in the case where the initial fluid velocity is in $H^1$ (see [12, 28, 30]), in particular with no gap between the regularity in space of the initial conditions and the one obtained for the solution [28]. Note however a recent preprint [34] requiring more regular initial data.
All these results are local-in-time except [27] where the authors proved that the solution of the fluid-damped beam problem is global, by showing that no contact occurs between the structure and the bottom of the fluid cavity. Without any additional term (damping or inertia of rotation), existence and uniqueness of strong solutions to the fluid-beam interaction problem is studied in [2,3] by studying the underlying semi-group associated to the linearized coupled problem and [34] by energy estimates for more regular initial data. In particular, it has been proven that the associated semi-group operator is not analytic in this case, but only of Gevrey class unlike the damped beam or wave equation case. Existence and uniqueness of strong solutions of the fluid-beam problem in the $L^p - L^q$ setting is obtained in [19].

All the results mentioned up to now deals with the fluid-structure problem by ignoring the longitudinal displacement of the structure. An existence result of strong solution considering longitudinal displacement of the fluid-beam problem can be found in [23] in the steady state.

One of the main difficulties when the longitudinal displacement is taken into account is that the incompressibility condition (1.2) together with the boundary conditions (1.6)-(1.7) and the L-periodicity of the fluid velocity $u$ with respect to first space variable $x$ imply that the following volume preserving constraint should be satisfied

$$ \int_{0}^{L} (-\partial_{t}\zeta(x, t)\partial_{x}\eta(x, t) + \partial_{t}\eta(x, t)(1 + \partial_{x}\zeta(x, t))) dx = 0. \quad (1.13) $$

We emphasize that this constraint is non-linear contrary to the case where only the transversal displacement of the structure is taken into account [28]. In the latter case, (1.13) reduces to

$$ \frac{d}{dt} \int_{0}^{L} \eta(x, t) dx = 0. \quad (1.14) $$

Thus, with null mean value initial deformation $\eta_0$, (1.14) implies that $\int_0^L \eta(x, t)dx = 0$, $\forall t \ge 0$, which can easily be satisfied. In our case the volume preserving constraint (1.13) couples both beam equations (1.3)-(1.4). This volume preserving constraint fixes the pressure of the fluid $p$ in a unique way. So the pressure of the fluid $p$ in problem (1.1)-(1.7) is not defined up to a constant, but is uniquely defined thanks to the constraint (1.13). Precisely we can split the pressure into

$$ p = p_0 + c, \quad (1.15) $$

where one imposes

$$ \int_{\Omega_{\zeta(t), \eta(t)}} p_0 dxdy = 0, \quad (1.16) $$

for any deformed unit cell $\Omega_{\zeta(t), \eta(t)}$, such that $\bar{F}_{\zeta(t), \eta(t)} = \cup_{k \in \mathbb{Z}} (\bar{\Omega}_{\zeta(t), \eta(t)} + kLe_1)$. The average pressure, $t \mapsto c(t)$, is the Lagrange multiplier associated to the constraint (1.13). To find the expression of $c$, remembering the expression of the force applied by the fluid on the structure (1.5), we may multiply equations (1.3) and (1.4) by $-\partial_x \eta$ and $(1 + \partial_x \zeta)$ respectively, then the resulting equations are integrated with respect to $x$ on $(0, L)$, and the resulting identities are added. Doing so, we obtain by using the periodic boundary conditions of the structure that

$$ c(t) = - \frac{1}{\|(-\partial_x \eta, 1 + \partial_x \zeta)\|_{L^2(0,L)}^2} \left( \rho_s \int_0^L \left( \partial_{tt}\zeta \partial_x \eta - \partial_{tt}\eta (1 + \partial_x \zeta) \right) dx + \beta \int_0^L \partial_x \zeta \partial_{xx} \eta dx \right. \\ \left. + \alpha \int_0^L \partial_{xxx}\eta \partial_{xx}\zeta dx + \gamma \int_0^L \partial_{xtt}\eta \partial_{xx}\zeta dx + \int_0^L \phi(u, p_0, \zeta, \eta) \cdot (-\partial_x \eta, 1 + \partial_x \zeta) dx \right), $$
All these results are local-in-time except [27] where the authors proved that the solution of the fluid-damped beam problem is global, by showing that no contact occurs between the structure and the bottom of the fluid cavity. Without any additional term (damping or inertia of rotation), existence and uniqueness of strong solutions to the fluid-beam interaction problem is studied in [2,3] by studying the underlying semi-group associated to the linearized coupled problem and [34] by energy estimates for more regular initial data. In particular, it has been proven that the associated semi-group operator is not analytic in this case, but only of Gevrey class unlike the damped beam or wave equation case. Existence and uniqueness of strong solutions of the fluid-beam problem in the $L^p - L^q$ setting is obtained in [19].

All the results mentioned up to now deals with the fluid-structure problem by ignoring the longitudinal displacement of the structure. An existence result of strong solution considering longitudinal displacement of the fluid-beam problem can be found in [23] in the steady state.

One of the main difficulties when the longitudinal displacement is taken into account is that the incompressibility condition (1.2) together with the boundary conditions (1.6)-(1.7) and the L-periodicity of the fluid velocity $u$ with respect to first space variable $x$ imply that the following volume preserving constraint should be satisfied

$$ \int_{0}^{L} (-\partial_{t}\zeta(x, t)\partial_{x}\eta(x, t) + \partial_{t}\eta(x, t)(1 + \partial_{x}\zeta(x, t))) dx = 0. \qquad (1.13) $$

We emphasize that this constraint is non-linear contrary to the case where only the transversal displacement of the structure is taken into account [28]. In the latter case, (1.13) reduces to

$$ \frac{d}{dt} \int_{0}^{L} \eta(x, t) dx = 0. \qquad (1.14) $$

Thus, with null mean value initial deformation $\eta_0$, (1.14) implies that $\int_{0}^{L} \eta(x, t)dx = 0$, $\forall t \ge 0$, which can easily be satisfied. In our case the volume preserving constraint (1.13) couples both beam equations (1.3)-(1.4). This volume preserving constraint fixes the pressure of the fluid $p$ in a unique way. So the pressure of the fluid $p$ in problem (1.1)-(1.7) is not defined up to a constant, but is uniquely defined thanks to the constraint (1.13). Precisely we can split the pressure into

$$ p = p_0 + c, \qquad (1.15) $$

where one imposes

$$ \int_{\Omega_{\zeta(t), \eta(t)}} p_0 dxdy = 0, \qquad (1.16) $$

for any deformed unit cell $\Omega_{\zeta(t), \eta(t)}$, such that $\bar{F}_{\zeta(t), \eta(t)} = \cup_{k \in \mathbb{Z}}(\bar{\Omega}_{\zeta(t), \eta(t)} + kLe_1)$. The average pressure, $t \mapsto c(t)$, is the Lagrange multiplier associated to the constraint (1.13). To find the expression of $c$, remembering the expression of the force applied by the fluid on the structure (1.5), we may multiply equations (1.3) and (1.4) by $-\partial_x \eta$ and $(1 + \partial_x \zeta)$ respectively, then the resulting equations are integrated with respect to $x$ on $(0, L)$, and the resulting identities are added. Doing so, we obtain by using the periodic boundary conditions of the structure that

$$ c(t) = - \frac{1}{\|(-\partial_x \eta, 1 + \partial_x \zeta)\|_{L^2(0,L)}^2} \left( \rho_s \int_0^L \left( \partial_{tt}\zeta \partial_x \eta - \partial_{tt}\eta (1 + \partial_x \zeta) \right) dx + \beta \int_0^L \partial_x \zeta \partial_{xx} \eta dx \right. \\ \left. + \alpha \int_0^L \partial_{xxx}\eta \partial_{xx}\zeta dx + \gamma \int_0^L \partial_{xtt}\eta \partial_{xx}\zeta dx + \int_0^L \phi(u, p_0, \zeta, \eta) \cdot (-\partial_x \eta, 1 + \partial_x \zeta) dx \right), $$
and hence

$$c(t) = - \frac{1}{\|(-\partial_x \eta, 1 + \partial_x \zeta)\|_{L^2(0,L)}^2} \left( \rho_s \frac{d}{dt} \int_0^L \left( \partial_t \zeta \partial_x \eta - \partial_t \eta (1 + \partial_x \zeta) \right) dx + \rho_s \int_0^L \left( -\partial_t \zeta \partial_{tx} \eta + \partial_t \eta \partial_{tx} \zeta \right) dx \right. \\ \left. + \beta \int_0^L \partial_x \zeta \partial_{xx} \eta dx + \alpha \int_0^L \partial_{xxx} \eta \partial_{xx} \zeta dx + \gamma \int_0^L \partial_{xtt} \eta \partial_{xx} \zeta dx + \int_0^L \phi(u, p_0, \zeta, \eta) \cdot (-\partial_x \eta, 1 + \partial_x \zeta) dx \right).$$

Thanks to the constraint (1.13), we get that

$$c(t) = - \frac{1}{\|(-\partial_x \eta, 1 + \partial_x \zeta)\|_{L^2(0,L)}^2} \left( \rho_s \int_0^L \left( -\partial_t \zeta \partial_{tx} \eta + \partial_t \eta \partial_{tx} \zeta \right) dx + \beta \int_0^L \partial_x \zeta \partial_{xx} \eta dx \right. \\ \left. + \alpha \int_0^L \partial_{xxx} \eta \partial_{xx} \zeta dx + \gamma \int_0^L \partial_{xtt} \eta \partial_{xx} \zeta dx + \int_0^L \phi(u, p_0, \zeta, \eta) \cdot (-\partial_x \eta, 1 + \partial_x \zeta) dx \right). \quad (1.17)$$

Reciprocally, given c as in (1.17) we can recover

$$\frac{d}{dt} \int_0^L \left( -\partial_t \zeta \partial_x \eta + \partial_t \eta (1 + \partial_x \zeta) \right) dx = 0,$$

and thus we recover the volume preserving constraint (1.13) if it is satisfied at the initial time.

Note that this additional constraint is not required when Neumann boundary conditions on inlet and outlet lateral boundaries are imposed. These conditions have been considered in [32] for the existence of weak solutions (by considering the total pressure instead of the pressure to obtain energy estimates) or in [12] in the 2D case, [16] in the 3D case both for existence of a unique strong solution for a damped beam or shell in flexion respectively. Studying existence of strong solutions of the problem considered here by replacing the periodic boundary conditions with more realistic Neumann boundary conditions presents additional difficulties since, in particular, one needs to derive elliptic estimates for the Stokes problem in a domain with corners, low regularity and mixed boundary conditions. In [12], and [16] such result are obtained for damped beam or shell in flexion (whose dynamic involves a fourth order operator in space and an additional damping term).

The paper follows now standard steps when dealing with fluid-structure interaction problems. In Section 2, we introduce some functional spaces and we construct a suitable change of variables which enables us to write the considered free boundary problem into a problem set in the reference configuration that can be reformulated as a linear problem involving variable coefficients (as the initial displacement of structure is non-zero), and whose source terms are perturbations of the initial structure deformations. Note that, by doing so, we also linearized the non linear volume preserving constraint (1.13) and the right hand side of the divergence constraint has no zero mean, contrary to the case when only the transversal displacement is considered (in the present study this average is equal to the right hand side coming from the linearization of (1.13)). Next, we state our main result. We end this section by proving an elliptic result for a steady Stokes-like problem set in the reference configuration. In Section 3, we tackle the linear problem and we prove existence and uniqueness of strong solutions. We first study the linear problem with homogeneous divergence constraint and then we study the non-homogeneous case by lifting the divergence term using the elliptic result of the steady Stokes-like problem established in the previous section. Note that, here, in order to lift the divergence constraint one needs to both change the fluid unknowns and the structure unknowns, contrary to the case of only transversal motions. In the last section, we introduce a mapping for which a solution of the nonlinear coupled problem can be defined as a fixed point of this mapping. By applying
and hence

$$c(t) = -\frac{1}{\|(-\partial_x \eta, 1 + \partial_x \zeta)\|_{L^2(0,L)}^2} \left( \rho_s \frac{d}{dt} \int_0^L \left( \partial_t \zeta \partial_x \eta - \partial_t \eta (1 + \partial_x \zeta) \right) dx + \rho_s \int_0^L \left( -\partial_t \zeta \partial_{tx} \eta + \partial_t \eta \partial_{tx} \zeta \right) dx \right. \\ \left. + \beta \int_0^L \partial_x \zeta \partial_{xx} \eta dx + \alpha \int_0^L \partial_{xxx} \eta \partial_{xx} \zeta dx + \gamma \int_0^L \partial_{xtt} \eta \partial_{xx} \zeta dx + \int_0^L \phi(u, p_0, \zeta, \eta) \cdot (-\partial_x \eta, 1 + \partial_x \zeta) dx \right).$$

Thanks to the constraint (1.13), we get that

$$c(t) = -\frac{1}{\|(-\partial_x \eta, 1 + \partial_x \zeta)\|_{L^2(0,L)}^2} \left( \rho_s \int_0^L \left( -\partial_t \zeta \partial_{tx} \eta + \partial_t \eta \partial_{tx} \zeta \right) dx + \beta \int_0^L \partial_x \zeta \partial_{xx} \eta dx \right. \\ \left. + \alpha \int_0^L \partial_{xxx} \eta \partial_{xx} \zeta dx + \gamma \int_0^L \partial_{xtt} \eta \partial_{xx} \zeta dx + \int_0^L \phi(u, p_0, \zeta, \eta) \cdot (-\partial_x \eta, 1 + \partial_x \zeta) dx \right). \quad (1.17)$$

Reciprocally, given c as in (1.17) we can recover

$$\frac{d}{dt} \int_0^L \left( -\partial_t \zeta \partial_x \eta + \partial_t \eta (1 + \partial_x \zeta) \right) dx = 0,$$

and thus we recover the volume preserving constraint (1.13) if it is satisfied at the initial time.

Note that this additional constraint is not required when Neumann boundary conditions on inlet and outlet lateral boundaries are imposed. These conditions have been considered in [32] for the existence of weak solutions (by considering the total pressure instead of the pressure to obtain energy estimates) or in [12] in the 2D case, [16] in the 3D case both for existence of a unique strong solution for a damped beam or shell in flexion respectively. Studying existence of strong solutions of the problem considered here by replacing the periodic boundary conditions with more realistic Neumann boundary conditions presents additional difficulties since, in particular, one needs to derive elliptic estimates for the Stokes problem in a domain with corners, low regularity and mixed boundary conditions. In [12], and [16] such result are obtained for damped beam or shell in flexion (whose dynamic involves a fourth order operator in space and an additional damping term).

The paper follows now standard steps when dealing with fluid-structure interaction problems. In Section 2, we introduce some functional spaces and we construct a suitable change of variables which enables us to write the considered free boundary problem into a problem set in the reference configuration that can be reformulated as a linear problem involving variable coefficients (as the initial displacement of structure is non-zero), and whose source terms are perturbations of the initial structure deformations. Note that, by doing so, we also linearized the non linear volume preserving constraint (1.13) and the right hand side of the divergence constraint has no zero mean, contrary to the case when only the transversal displacement is considered (in the present study this average is equal to the right hand side coming from the linearization of (1.13)). Next, we state our main result. We end this section by proving an elliptic result for a steady Stokes-like problem set in the reference configuration. In Section 3, we tackle the linear problem and we prove existence and uniqueness of strong solutions. We first study the linear problem with homogeneous divergence constraint and then we study the non-homogeneous case by lifting the divergence term using the elliptic result of the steady Stokes-like problem established in the previous section. Note that, here, in order to lift the divergence constraint one needs to both change the fluid unknowns and the structure unknowns, contrary to the case of only transversal motions. In the last section, we introduce a mapping for which a solution of the nonlinear coupled problem can be defined as a fixed point of this mapping. By applying
Picard fixed point Theorem, we obtain the existence of a unique fixed point. Compared to the only transversal motion case additional terms appears. We emphasize that the main novelty is to take into account the longitudinal displacement of the elastic beam. Moreover we do not add any damping to the equations but take into account the inertia of rotation effect.

# 2 Notations, Change of Variables, and Main Result

## 2.1 Notations

Let us fix the notations and introduce some functional spaces. Let $\Omega \subset \mathbb{R}^2$ be an open and connected set denoting the reference cell that can be reproduced by periodicity in $x$ to generate the domain $\mathcal{O}$ such that $\bar{\mathcal{O}} = \cup_{k \in \mathbb{Z}}(\bar{\Omega} + kLe_1)$. We denote by $C_{\#}^{\infty}(\Omega)$ the restriction to $\Omega$ of functions of $C_{\#}^{\infty}(\mathcal{O})$ of infinitely differentiable functions which are $L$-periodic with respect to the first space variable in $\mathcal{O}$. We denote by $H_{\#}^s(\Omega)$ with $s > 0$ and $L_{\#}^p(\Omega)$ with $0 \le p \le \infty$ the closure of $C_{\#}^{\infty}(\Omega)$ in the usual Sobolev space $H^s(\Omega)$ and the Lebesgue space $L^p(\Omega)$ respectively. In a similar way, we define $H_{\#}^s(0, L)$ with $s \ge 0$, $L_{\#}^p(0, L)$ with $0 \le p \le \infty$, $C_{\#}(0, L)$. We note that $H_{\#}^0(0, L) = L_{\#}^2(0, L)$. Note that any function of $H_{\#}^s(\Omega)$ can be extended by periodicity as a function of $H_{\#}^s(\mathcal{O})$. The same notations shall be used for the restriction or the extension. We emphasize that contrary to the usual convention, we consider that the time variable is the last variable of a function so that we may use unified definition for the periodic functions whether they have only one space variable or two space variables. Moreover in all what follows we will not differentiate spaces of vector valued functions from scalar valued functions.

## 2.2 Construction of a Change of Variables

One of the classical difficulties in fluid-structure interaction problems is that we are dealing with a free boundary problems. To overcome this difficulty, we introduce in this subsection a change of variables which maps the reference configuration $\bar{\mathcal{F}}_0$ into a deformed one and we state some useful properties of this change of variables.

Let us introduce a general change of variables and consider a given structure displacement $d = (a, b)^T$, where $(a, b) \in H_{\#}^{\frac{3}{2}+s}(0, L)$ with $s > 0$ and such that

$$ \min_{x \in (0, L)} (1 + b(x)) > 0; \qquad (2.1) $$

$$ \text{the mapping } S_{a,b}: x \mapsto (x + a(x), 1 + b(x)) \text{ is injective as a function in } \mathbb{R} \qquad (2.2) $$

Thanks to the periodicity of $a$ and $b$, one has $S_{a,b}(x + kL) = S_{a,b}(x) + kLe_1$. Let $\Gamma_{a,b} = \cup_{k \in \mathbb{Z}}(S_{a,b}([0, L]) + kLe_1)$. Let $\mathcal{F}_{a,b}$ the subset of $\mathbb{R}_+^2$ that has a rigid lower boundary $\mathbb{R} \times \{0\}$ and its upper boundary is $\Gamma_{a,b}$. We are going to build a change of variables that maps $\mathcal{F}_0$ onto $\mathcal{F}_{a,b}$. We consider $\chi_{a,b}$ a $L$-periodic in $x$ mapping, such that $\chi_{a,b}(x, 1) = (x + a(x), 1 + b(x))$ and $\chi_{a,b}(x, 0) = (x, 0)$ for any $x \in [0, L]$. One can define $\chi_{a,b}$ as follows

$$ \chi_{a,b} = I_2 + (Ra, Rb), $$

where $\mathcal{R}$ is a continuous linear lifting operator from $H_{\#}^{\theta}(0, L) \to H_{\#}^{\theta+\frac{1}{2}}(\Omega_0)$, $\theta \ge \frac{1}{2}$, where $H_{\#}^{\theta+\frac{1}{2}}(\Omega_0)$ denotes the subspace of $H_{\#}^{\theta+\frac{1}{2}}(\Omega_0)$ of functions that have a zero trace on $\Gamma_0$. With a slight abuse of notation we set
Picard fixed point Theorem, we obtain the existence of a unique fixed point. Compared to the only transversal motion case additional terms appears. We emphasize that the main novelty is to take into account the longitudinal displacement of the elastic beam. Moreover we do not add any damping to the equations but take into account the inertia of rotation effect.

# 2 Notations, Change of Variables, and Main Result

## 2.1 Notations

Let us fix the notations and introduce some functional spaces. Let $\Omega \subset \mathbb{R}^2$ be an open and connected set denoting the reference cell that can be reproduced by periodicity in $x$ to generate the domain $\mathcal{O}$ such that $\bar{\mathcal{O}} = \cup_{k \in \mathbb{Z}}(\bar{\Omega} + kLe_1)$. We denote by $C_{\#}^{\infty}(\Omega)$ the restriction to $\Omega$ of functions of $C_{\#}^{\infty}(\mathcal{O})$ of infinitely differentiable functions which are $L$-periodic with respect to the first space variable in $\mathcal{O}$. We denote by $H_{\#}^s(\Omega)$ with $s > 0$ and $L_{\#}^p(\Omega)$ with $0 \le p \le \infty$ the closure of $C_{\#}^{\infty}(\Omega)$ in the usual Sobolev space $H^s(\Omega)$ and the Lebesgue space $L^p(\Omega)$ respectively. In a similar way, we define $H_{\#}^s(0, L)$ with $s \ge 0$, $L_{\#}^p(0, L)$ with $0 \le p \le \infty$, $C_{\#}(0, L)$. We note that $H_{\#}^0(0, L) = L_{\#}^2(0, L)$. Note that any function of $H_{\#}^s(\Omega)$ can be extended by periodicity as a function of $H_{\#}^s(\mathcal{O})$. The same notations shall be used for the restriction or the extension. We emphasize that contrary to the usual convention, we consider that the time variable is the last variable of a function so that we may use unified definition for the periodic functions whether they have only one space variable or two space variables. Moreover in all what follows we will not differentiate spaces of vector valued functions from scalar valued functions.

## 2.2 Construction of a Change of Variables

One of the classical difficulties in fluid-structure interaction problems is that we are dealing with a free boundary problems. To overcome this difficulty, we introduce in this subsection a change of variables which maps the reference configuration $\bar{\mathcal{F}}_0$ into a deformed one and we state some useful properties of this change of variables.

Let us introduce a general change of variables and consider a given structure displacement $d = (a, b)^T$, where $(a, b) \in H_{\#}^{\frac{3}{2}+s}(0, L)$ with $s > 0$ and such that

$$ \min_{x \in (0, L)} (1 + b(x)) > 0; \qquad (2.1) $$

$$ \text{the mapping } S_{a,b}: x \mapsto (x + a(x), 1 + b(x)) \text{ is injective as a function in } \mathbb{R} \qquad (2.2) $$

Thanks to the periodicity of $a$ and $b$, one has $S_{a,b}(x + kL) = S_{a,b}(x) + kLe_1$. Let $\Gamma_{a,b} = \cup_{k \in \mathbb{Z}}(S_{a,b}([0, L]) + kLe_1)$. Let $\mathcal{F}_{a,b}$ the subset of $\mathbb{R}_+^2$ that has a rigid lower boundary $\mathbb{R} \times \{0\}$ and its upper boundary is $\Gamma_{a,b}$. We are going to build a change of variables that maps $\mathcal{F}_0$ onto $\mathcal{F}_{a,b}$. We consider $\chi_{a,b}$ a $L$-periodic in $x$ mapping, such that $\chi_{a,b}(x, 1) = (x + a(x), 1 + b(x))$ and $\chi_{a,b}(x, 0) = (x, 0)$ for any $x \in [0, L]$. One can define $\chi_{a,b}$ as follows

$$ \chi_{a,b} = I_2 + (Ra, Rb), $$

where $\mathcal{R}$ is a continuous linear lifting operator from $H_{\#}^{\theta}(0, L) \to H_{\#}^{\theta+\frac{1}{2}}(\Omega_0)$, $\theta \ge \frac{1}{2}$, where $H_{\#}^{\theta+\frac{1}{2}}(\Omega_0)$ denotes the subspace of $H_{\#}^{\theta+\frac{1}{2}}(\Omega_0)$ of functions that have a zero trace on $\Gamma_0$. With a slight abuse of notation we set
$\mathcal{R}(a,b) = (\mathcal{R}a, \mathcal{R}b)$. Given $(a,b) \in H_{\#}^{3/2+s}(0,L)$ with $s > 0$, we have $\mathcal{R}(a,b) \in H_{\#}^{2+s}(\Omega_0)$ with

$$ \|\mathcal{R}(a,b)\|_{H_{\#}^{2+s}(\Omega_0)} \le C \|(a,b)\|_{H_{\#}^{3/2+s}(0,L)}, $$

where $C > 0$ depending on $\Omega_0$.

Setting $\Omega_{a,b} = \chi_{a,b}(\Omega_0)$, if $\chi_{a,b}$ is injective on $\overline{\mathcal{F}}_0$, we have $\overline{\mathcal{F}}_{a,b} = \chi_{a,b}(\overline{\mathcal{F}}_0) = \cup_{k \in \mathbb{Z}} (\overline{\Omega}_{a,b} + kLe_1)$ (see [15]).

Consequently we have built a mapping from the reference configuration onto a given deformed one.

One can show easily the following proposition

**Proposition 2.1** Let $(a,b) \in H_{\#}^{3/2+s}(0,L)$ with $s > 0$. There exists a positive constant $M$ such that, for all $0 < R \le M$, if $\|(a,b)\|_{H_{\#}^{3/2+s}(0,L)} + \|(\det\nabla\chi_{a,b})^{-1}\|_{C_{\#}(\overline{\Omega}_0)} \le R$, then

* the mapping $\chi_{a,b}$ is a $C^1$-diffeomorphism from $\overline{\mathcal{F}}_0$ onto $\overline{\mathcal{F}}_{a,b}$;

* for any $m \le 2$, the mapping $f \mapsto f \circ \chi_{a,b}$ realizes a linear homeomorphism from $H_{\#}^m(\Omega_{a,b})$ into $H_{\#}^m(\Omega_0)$;

* for any $m \le 2$, there exists a non-decreasing constant $K_m : [0, \infty) \to (0, \infty)$ such that

$$ \|f \circ \chi_{a,b}\|_{H_{\#}^m(\Omega_0)} \le K_m(R) \|f\|_{H_{\#}^m(\Omega_{a,b})}, \quad \|f\|_{H_{\#}^m(\Omega_{a,b})} \le K_m(R) \|f \circ \chi_{a,b}\|_{H_{\#}^m(\Omega_0)}; $$

The first point can be obtained adapting results in [15] and the second and third point can be proved as in [27]. Thus we have sufficient conditions ensuring that $\chi_{a,b}$ is a change of variables. This type of change of variables will enable us to write system (1.1)-(1.12) in the reference configuration and show the existence of a unique strong solution as long as we are able to build such an injective mapping whose jacobian remains strictly positive.

**Remark 2.1** Proposition 2.1 gives sufficient conditions to obtain a $C^1$-diffeomorphism from $\overline{\mathcal{F}}_0$ onto $\overline{\mathcal{F}}_{a,b} = \chi_{a,b}(\overline{\mathcal{F}}_0)$. One can build such a change of variables as long as $\det\nabla\chi_{a,b} > 0$ on $\overline{\Omega}_0$ and $\chi_{a,b}$ is injective on $\overline{\mathcal{F}}_0$.

## 2.3 Main Result

Before stating our main result, we introduce the notion of strong solutions to the periodic problem (1.1)-(1.11).

**Definition 2.1** Let $T > 0$. Consider $(u_0, \zeta_0, \zeta_1, \eta_0, \eta_1) \in H_{\#}^1(\mathcal{F}_{\zeta_0, \eta_0}) \times H_{\#}^2(0, L) \times H_{\#}^1(0, L) \times H_{\#}^3(0, L) \times H_{\#}^1(0, L)$ satisfying the following compatibility conditions

$$ \left\{ \begin{array}{ll} u_0(x + \zeta_0(x), 1 + \eta_0(x)) = (\zeta_1(x), \eta_1(x)), & x \in (0, L), \\ u_0(x, 0) = 0, & x \in (0, L), \\ \text{div } u_0 = 0, & \text{in } \mathcal{F}_{\zeta_0, \eta_0}. \end{array} \right. \qquad (2.3) $$

Assume that the initial conditions are such that there exists a mapping $\chi_{\zeta_0, \eta_0}$ satisfying $\det\nabla\chi_{\zeta_0, \eta_0} > 0$ on $\overline{\Omega}_0$ and $\chi_{\zeta_0, \eta_0}$ is injective on $\overline{\mathcal{F}}_0$.

A strong solution to Problem (1.1)-(1.12) on $(0, T)$ associated with the initial conditions $(u_0, \zeta_0, \zeta_1, \eta_0, \eta_1)$ as above is a quadruplet $(u, p, \zeta, \eta)$ such that
$\mathcal{R}(a, b) = (\mathcal{R}a, \mathcal{R}b)$. Given $(a, b) \in H_{\#}^{3/2+s}(0, L)$ with $s > 0$, we have $\mathcal{R}(a, b) \in H_{\#}^{2+s}(\Omega_0)$ with

$$ \|\mathcal{R}(a, b)\|_{H_{\#}^{2+s}(\Omega_0)} \le C \|(a, b)\|_{H_{\#}^{3/2+s}(0, L)}, $$

where $C > 0$ depending on $\Omega_0$.

Setting $\Omega_{a,b} = \chi_{a,b}(\Omega_0)$, if $\chi_{a,b}$ is injective on $\bar{\mathcal{F}}_0$, we have $\bar{\mathcal{F}}_{a,b} = \chi_{a,b}(\bar{\mathcal{F}}_0) = \cup_{k \in \mathbb{Z}} (\bar{\Omega}_{a,b} + kLe_1)$ (see [15]).

Consequently we have built a mapping from the reference configuration onto a given deformed one.

One can show easily the following proposition

**Proposition 2.1** Let $(a, b) \in H_{\#}^{3/2+s}(0, L)$ with $s > 0$. There exists a positive constant $M$ such that, for all $0 < R \le M$, if $\|(a, b)\|_{H_{\#}^{3/2+s}(0, L)} + \|(\det\nabla\chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R$, then

* the mapping $\chi_{a,b}$ is a $C^1$-diffeomorphism from $\bar{\mathcal{F}}_0$ onto $\bar{\mathcal{F}}_{a,b}$;

* for any $m \le 2$, the mapping $f \mapsto f \circ \chi_{a,b}$ realizes a linear homeomorphism from $H_{\#}^m(\Omega_{a,b})$ into $H_{\#}^m(\Omega_0)$;

* for any $m \le 2$, there exists a non-decreasing constant $K_m : [0, \infty) \to (0, \infty)$ such that

$$ \|f \circ \chi_{a,b}\|_{H_{\#}^m(\Omega_0)} \le K_m(R) \|f\|_{H_{\#}^m(\Omega_{a,b})}, \quad \|f\|_{H_{\#}^m(\Omega_{a,b})} \le K_m(R) \|f \circ \chi_{a,b}\|_{H_{\#}^m(\Omega_0)}; $$

The first point can be obtained adapting results in [15] and the second and third point can be proved as in [27]. Thus we have sufficient conditions ensuring that $\chi_{a,b}$ is a change of variables. This type of change of variables will enable us to write system (1.1)-(1.12) in the reference configuration and show the existence of a unique strong solution as long as we are able to build such an injective mapping whose jacobian remains strictly positive.

**Remark 2.1** Proposition 2.1 gives sufficient conditions to obtain a $C^1$-diffeomorphism from $\bar{\mathcal{F}}_0$ onto $\bar{\mathcal{F}}_{a,b} = \chi_{a,b}(\bar{\mathcal{F}}_0)$. One can build such a change of variables as long as $\det\nabla\chi_{a,b} > 0$ on $\bar{\Omega}_0$ and $\chi_{a,b}$ is injective on $\bar{\mathcal{F}}_0$.

## 2.3 Main Result

Before stating our main result, we introduce the notion of strong solutions to the periodic problem (1.1)-(1.11).

**Definition 2.1** Let $T > 0$. Consider $(u_0, \zeta_0, \zeta_1, \eta_0, \eta_1) \in H_{\#}^1(\mathcal{F}_{\zeta_0, \eta_0}) \times H_{\#}^2(0, L) \times H_{\#}^1(0, L) \times H_{\#}^3(0, L) \times H_{\#}^1(0, L)$ satisfying the following compatibility conditions

$$ \left\{ \begin{array}{ll} u_0(x + \zeta_0(x), 1 + \eta_0(x)) = (\zeta_1(x), \eta_1(x)), & x \in (0, L), \\ u_0(x, 0) = 0, & x \in (0, L), \\ \operatorname{div} u_0 = 0, & \text{in } \mathcal{F}_{\zeta_0, \eta_0}. \end{array} \right. \qquad (2.3) $$

Assume that the initial conditions are such that there exists a mapping $\chi_{\zeta_0, \eta_0}$ satisfying $\det\nabla\chi_{\zeta_0, \eta_0} > 0$ on $\bar{\Omega}_0$ and $\chi_{\zeta_0, \eta_0}$ is injective on $\bar{\mathcal{F}}_0$.

A strong solution to Problem (1.1)-(1.12) on $(0, T)$ associated with the initial conditions $(u_0, \zeta_0, \zeta_1, \eta_0, \eta_1)$ as above is a quadruplet $(u, p, \zeta, \eta)$ such that
* it satisfies the following regularities

$$ 
\left\{ 
\begin{array}{l}
\partial_t u \in L_{\#}^2(F_T), \nabla^2 u \in L_{\#}^2(F_T), p \in L_{\#}^2(F_T), \nabla p \in L_{\#}^2(F_T), \\
\zeta \in H^2(0, T; L_{\#}^2(0, L)) \cap W^{1,\infty}(0, T; H_{\#}^1(0, L)) \cap L^{\infty}(0, T; H_{\#}^2(0, L)) \cap H^1(0, T; H_{\#}^{\frac{3}{2}}(0, L)), \\
\eta \in H^2(0, L; H_{\#}^1(0, L)) \cap W^{1,\infty}(0, T; H_{\#}^2(0, L)) \cap L^{\infty}(0, T; H_{\#}^3(0, L));
\end{array}
\right. 
$$

* equations (1.1)-(1.2) are satisfied a.e. in $\mathcal{F}_T$;

* equation (1.3) is satisfied a.e in $(0, L) \times (0, T)$;

* equation (1.4) is satisfied in $L^2(0, T; H_{\#}^{-1}(0, L))$;

* equations (1.6)-(1.7) are satisfied a.e in $(0, T) \times (0, L)$;

* equation (1.12) is satisfied a.e in $\mathcal{F}_{\zeta_0, \eta_0}$ and equations (1.8)-(1.11) are satisfied a.e in $(0, L)$.

We underline that the compatibility conditions on the initial data (2.3) (the equality of the velocities at the interface together with the incompressibility constraint) imply

$$ \int_0^L (-\zeta_1(x)\partial_x\eta_0(x) + \eta_1(x)(1 + \partial_x\zeta_0(x))) dx = 0. \quad (2.4) $$

This is a non linear constraint that has to be verified by the initial conditions. Our main result now reads as follows:

**Theorem 2.1** Consider initial data $(u_0, \zeta_0, \zeta_1, \eta_0, \eta_1) \in H_{\#}^1(\mathcal{F}_{\zeta_0, \eta_0}) \times H_{\#}^2(0, L) \times H_{\#}^1(0, L) \times H_{\#}^3(0, L) \times H_{\#}^1(0, L)$ such that (2.3) is satisfied and such that there exists a mapping $\chi_{\zeta_0, \eta_0}$ satisfying $\det\nabla\chi_{\zeta_0, \eta_0} > 0$ on $\bar{\Omega}_0$ and $\chi_{\zeta_0, \eta_0}$ is injective on $\bar{\mathcal{F}}_0$. Then there exists a time $T_0$ depending decreasingly on

$$ \|u_0\|_{H_{\#}^1(\mathcal{F}_{\zeta_0, \eta_0})} + \|\zeta_0\|_{H_{\#}^2(0, L)} + \|\zeta_1\|_{H_{\#}^1(0, L)} + \|\eta_0\|_{H_{\#}^3(0, L)} + \|\eta_1\|_{H_{\#}^1(0, L)} + \|(\det \nabla \chi_{\zeta_0, \eta_0})^{-1}\|_{C_{\#}^2(\bar{\Omega}_0)} $$

such that Problem (1.1)-(1.11) admits a unique strong solution on $(0, T_0)$.

Moreover, the solution exits as long as

i. One can build a mapping $\chi_{\zeta, \eta} \in C^0([0, T]; H_{\#}^{2+s}(\Omega_0))$, for $s > 0$ and such that for $t \le T$

$$ \chi_{\zeta(t), \eta(t)}(x, 1) = (x + \zeta(x, t), 1 + \eta(x, t))) \quad \text{and} \quad \chi_{\zeta(t), \eta(t)}(x, 0) = (x, 0), \quad \forall x \in [0, L], \quad (2.5) $$

$$ \det \nabla \chi_{\zeta(t), \eta(t)} > 0, \quad \text{on } \bar{\Omega}_0, \quad (2.6) $$

$$ \chi_{\zeta(t), \eta(t)} \text{ is injective on } \bar{\mathcal{F}}_0. \quad (2.7) $$

$$ \text{ii. } \|u(t)\|_{H^1(\mathcal{F}(t))} + \|\zeta(t)\|_{H_{\#}^2(0, L)} + \|\partial_t \zeta(t)\|_{H_{\#}^1(0, L)} + \|\eta(t)\|_{H_{\#}^3(0, L)} + \|\partial_t \eta(t)\|_{H_{\#}^1(0, L)} < +\infty. $$

**Remark 2.2** Note that one can not extend the solution if we have self-contact of the elastic beam or contact with the bottom of the fluid cavity in particular if for some $T^*$ we have either $1 + \eta(T^*) = 0$ or the mapping $x \mapsto (x + \zeta(x, T_*), 1 + \eta(x, T_*))$ is not injective on $\mathbb{R}$.
* it satisfies the following regularities

$$ 
\left\{ 
\begin{array}{l}
\partial_t u \in L_{\#}^2(F_T), \nabla^2 u \in L_{\#}^2(F_T), p \in L_{\#}^2(F_T), \nabla p \in L_{\#}^2(F_T), \\
\zeta \in H^2(0, T; L_{\#}^2(0, L)) \cap W^{1,\infty}(0, T; H_{\#}^1(0, L)) \cap L^{\infty}(0, T; H_{\#}^2(0, L)) \cap H^1(0, T; H_{\#}^{\frac{3}{2}}(0, L)), \\
\eta \in H^2(0, L; H_{\#}^1(0, L)) \cap W^{1,\infty}(0, T; H_{\#}^2(0, L)) \cap L^{\infty}(0, T; H_{\#}^3(0, L));
\end{array}
\right. 
$$

* equations (1.1)-(1.2) are satisfied a.e. in $\mathcal{F}_T$;

* equation (1.3) is satisfied a.e in $(0, L) \times (0, T)$;

* equation (1.4) is satisfied in $L^2(0, T; H_{\#}^{-1}(0, L))$;

* equations (1.6)-(1.7) are satisfied a.e in $(0, T) \times (0, L)$;

* equation (1.12) is satisfied a.e in $\mathcal{F}_{\zeta_0, \eta_0}$ and equations (1.8)-(1.11) are satisfied a.e in $(0, L)$.

We underline that the compatibility conditions on the initial data (2.3) (the equality of the velocities at the interface together with the incompressibility constraint) imply

$$ \int_0^L (-\zeta_1(x)\partial_x\eta_0(x) + \eta_1(x)(1 + \partial_x\zeta_0(x))) dx = 0. \quad (2.4) $$

This is a non linear constraint that has to be verified by the initial conditions. Our main result now reads as follows:

**Theorem 2.1** Consider initial data $(u_0, \zeta_0, \zeta_1, \eta_0, \eta_1) \in H_{\#}^1(\mathcal{F}_{\zeta_0, \eta_0}) \times H_{\#}^2(0, L) \times H_{\#}^1(0, L) \times H_{\#}^3(0, L) \times H_{\#}^1(0, L)$ such that (2.3) is satisfied and such that there exists a mapping $\chi_{\zeta_0, \eta_0}$ satisfying $\det\nabla\chi_{\zeta_0, \eta_0} > 0$ on $\bar{\Omega}_0$ and $\chi_{\zeta_0, \eta_0}$ is injective on $\bar{\mathcal{F}}_0$. Then there exists a time $T_0$ depending decreasingly on

$$ \|u_0\|_{H_{\#}^1(\mathcal{F}_{\zeta_0, \eta_0})} + \|\zeta_0\|_{H_{\#}^2(0, L)} + \|\zeta_1\|_{H_{\#}^1(0, L)} + \|\eta_0\|_{H_{\#}^3(0, L)} + \|\eta_1\|_{H_{\#}^1(0, L)} + \|(\det \nabla \chi_{\zeta_0, \eta_0})^{-1}\|_{C_{\#}^2(\bar{\Omega}_0)} $$

such that Problem (1.1)-(1.11) admits a unique strong solution on $(0, T_0)$.

Moreover, the solution exits as long as

i. One can build a mapping $\chi_{\zeta, \eta} \in C^0([0, T]; H_{\#}^{2+s}(\Omega_0))$, for $s > 0$ and such that for $t \le T$

$$ \chi_{\zeta(t), \eta(t)}(x, 1) = (x + \zeta(x, t), 1 + \eta(x, t))) \quad \text{and} \quad \chi_{\zeta(t), \eta(t)}(x, 0) = (x, 0), \quad \forall x \in [0, L], \quad (2.5) $$

$$ \det \nabla \chi_{\zeta(t), \eta(t)} > 0, \quad \text{on } \bar{\Omega}_0, \quad (2.6) $$

$$ \chi_{\zeta(t), \eta(t)} \text{ is injective on } \bar{\mathcal{F}}_0. \quad (2.7) $$

$$ \text{ii. } \|u(t)\|_{H^1(\mathcal{F}(t))} + \|\zeta(t)\|_{H_{\#}^2(0, L)} + \|\partial_t \zeta(t)\|_{H_{\#}^1(0, L)} + \|\eta(t)\|_{H_{\#}^3(0, L)} + \|\partial_t \eta(t)\|_{H_{\#}^1(0, L)} < +\infty. $$

**Remark 2.2** Note that one can not extend the solution if we have self-contact of the elastic beam or contact with the bottom of the fluid cavity in particular if for some $T^*$ we have either $1 + \eta(T^*) = 0$ or the mapping $x \mapsto (x + \zeta(x, T_*), 1 + \eta(x, T_*))$ is not injective on $\mathbb{R}$.
**Remark 2.3** The constraint (1.13) is satisfied by any solution and is indeed a volume preserving constraint. Indeed thanks to the identity (see [15])

$$ \frac{d}{dt} \int_{\Omega_0} \det \nabla \chi_{\zeta(t), \eta(t)} dxdy = \int_0^L \partial_t \chi_{\zeta(t), \eta(t)} \cof \nabla \chi_{\zeta(t), \eta(t)} \cdot e_2 dx = \int_0^L (-\partial_t \zeta(x,t) \partial_x \eta(x,t) + \partial_t \eta(x,t)(1 + \partial_x \zeta(x,t))) dx = 0, $$
we get

$$ \int_{\Omega_0} \det \nabla \chi_{\zeta(t), \eta(t)} dxdy = \int_{\Omega_0} \det \nabla \chi_{\zeta_0, \eta_0} dxdy, $$

and implying that the volume of the initial reference cell is preserved by $\chi_{\zeta(t), \eta(t)}$ as it is injective, namely we have $|\Omega_{\zeta(t), \eta(t)}| = |\Omega_{\zeta_0, \eta_0}|$.

We are going to prove the well-posedness of Problem (1.1)-(1.12) by following quite standard steps: write the system in the reference configuration, study the associated non homogeneous linear problem, prove existence and uniqueness of solution by perturbation for small time. One main difficulty comes from the coupling between both longitudinal and transversal components of the elastic displacement that are coupled through the flow and the non linear volume preserving constraint (1.13).

## 2.4 Equivalent system in a reference configuration

The first step to prove Theorem 2.1 is standard and consists in rewriting the free boundary problem (1.1)-(1.12) in a reference configuration, here $\mathcal{F}_0$ (or equivalently in the reference cell $\Omega_0$). We do this by using a change of variable $\chi_{\zeta(t), \eta(t)}$ which maps the $\mathcal{F}_0$ onto $\mathcal{F}_{\zeta(t), \eta(t)}$ by mapping the reference cell $\Omega_0$ onto $\Omega_{\zeta(t), \eta(t)} = \chi_{\zeta(t), \eta(t)}(\Omega_0)$. To this end, we introduce the new unknowns

$$ v(x, y, t) = u(\chi_{\zeta(t), \eta(t)}(x, y), t) \text{ and } q(x, y, t) = p(\chi_{\zeta(t), \eta(t)}(x, y), t), \quad (x, y, t) \in \Omega_0 \times (0, T). $$

Then the Navier-Stokes equations on the reference cell configuration $\Omega_0 \times (0, T)$ write as follows:

$$ \rho_{f, \zeta, \eta} \partial_t v + \rho_f ((v - \partial_t \chi_{\zeta, \eta}) \cdot (B_{\zeta, \eta} \nabla)) v - \mu \operatorname{div}((A_{\zeta, \eta} \nabla)v) + (B_{\zeta, \eta} \nabla)q = 0, \quad (2.8) $$

$$ \operatorname{div}(B_{\zeta, \eta}^T v) = 0, \quad (2.9) $$

where

$$ \rho_{f, \zeta, \eta} = \rho_f \det \nabla \chi_{\zeta, \eta}, \quad (2.10) $$

$$ B_{\zeta, \eta} = \cof \nabla \chi_{\zeta, \eta}, \quad (2.11) $$

$$ A_{\zeta, \eta} = \frac{1}{\det \nabla \chi_{\zeta, \eta}} B_{\zeta, \eta}^T B_{\zeta, \eta}. \quad (2.12) $$

With these new unknowns, the structure equations on $(0, L)$ write

$$ \rho_s \partial_{tt} \zeta - \beta \partial_{xx} \zeta = - \left( \mu (A_{\zeta, \eta} \nabla) v + \frac{\mu}{\det \nabla \chi_{\zeta, \eta}} B_{\zeta, \eta} \nabla v^T B_{\zeta, \eta} - q B_{\zeta, \eta} \right)_{|_{y=1}} e_2 \cdot e_1, \quad (2.13) $$

$$ \rho_s \partial_{tt} \eta - \gamma \partial_{xxt} \eta + \alpha \partial_{xxxx} \eta = - \left( \mu (A_{\zeta, \eta} \nabla) v + \frac{\mu}{\det \nabla \chi_{\zeta, \eta}} B_{\zeta, \eta} \nabla v^T B_{\zeta, \eta} - q B_{\zeta, \eta} \right)_{|_{y=1}} e_2 \cdot e_2. \quad (2.14) $$
**Remark 2.3** The constraint (1.13) is satisfied by any solution and is indeed a volume preserving constraint. Indeed thanks to the identity (see [15])

$$ \frac{d}{dt} \int_{\Omega_0} \det \nabla \chi_{\zeta(t), \eta(t)} dxdy = \int_0^L \partial_t \chi_{\zeta(t), \eta(t)} \cof \nabla \chi_{\zeta(t), \eta(t)} \cdot e_2 dx = \int_0^L (-\partial_t \zeta(x,t) \partial_x \eta(x,t) + \partial_t \eta(x,t)(1 + \partial_x \zeta(x,t))) dx = 0, $$
we get

$$ \int_{\Omega_0} \det \nabla \chi_{\zeta(t), \eta(t)} dxdy = \int_{\Omega_0} \det \nabla \chi_{\zeta_0, \eta_0} dxdy, $$

and implying that the volume of the initial reference cell is preserved by $\chi_{\zeta(t), \eta(t)}$ as it is injective, namely we have $|\Omega_{\zeta(t), \eta(t)}| = |\Omega_{\zeta_0, \eta_0}|$.

We are going to prove the well-posedness of Problem (1.1)-(1.12) by following quite standard steps: write the system in the reference configuration, study the associated non homogeneous linear problem, prove existence and uniqueness of solution by perturbation for small time. One main difficulty comes from the coupling between both longitudinal and transversal components of the elastic displacement that are coupled through the flow and the non linear volume preserving constraint (1.13).

## 2.4 Equivalent system in a reference configuration

The first step to prove Theorem 2.1 is standard and consists in rewriting the free boundary problem (1.1)-(1.12) in a reference configuration, here $\mathcal{F}_0$ (or equivalently in the reference cell $\Omega_0$). We do this by using a change of variable $\chi_{\zeta(t), \eta(t)}$ which maps the $\mathcal{F}_0$ onto $\mathcal{F}_{\zeta(t), \eta(t)}$ by mapping the reference cell $\Omega_0$ onto $\Omega_{\zeta(t), \eta(t)} = \chi_{\zeta(t), \eta(t)}(\Omega_0)$. To this end, we introduce the new unknowns

$$ v(x, y, t) = u(\chi_{\zeta(t), \eta(t)}(x, y), t) \text{ and } q(x, y, t) = p(\chi_{\zeta(t), \eta(t)}(x, y), t), \quad (x, y, t) \in \Omega_0 \times (0, T). $$

Then the Navier-Stokes equations on the reference cell configuration $\Omega_0 \times (0, T)$ write as follows:

$$ \rho_{f, \zeta, \eta} \partial_t v + \rho_f ((v - \partial_t \chi_{\zeta, \eta}) \cdot (B_{\zeta, \eta} \nabla)) v - \mu \operatorname{div}((A_{\zeta, \eta} \nabla)v) + (B_{\zeta, \eta} \nabla)q = 0, \quad (2.8) $$

$$ \operatorname{div}(B_{\zeta, \eta}^T v) = 0, \quad (2.9) $$

where

$$ \rho_{f, \zeta, \eta} = \rho_f \det \nabla \chi_{\zeta, \eta}, \quad (2.10) $$

$$ B_{\zeta, \eta} = \cof \nabla \chi_{\zeta, \eta}, \quad (2.11) $$

$$ A_{\zeta, \eta} = \frac{1}{\det \nabla \chi_{\zeta, \eta}} B_{\zeta, \eta}^T B_{\zeta, \eta}. \quad (2.12) $$

With these new unknowns, the structure equations on $(0, L)$ write

$$ \rho_s \partial_{tt} \zeta - \beta \partial_{xx} \zeta = - \left( \mu (A_{\zeta, \eta} \nabla) v + \frac{\mu}{\det \nabla \chi_{\zeta, \eta}} B_{\zeta, \eta} \nabla v^T B_{\zeta, \eta} - q B_{\zeta, \eta} \right)_{|_{y=1}} e_2 \cdot e_1, \quad (2.13) $$

$$ \rho_s \partial_{tt} \eta - \gamma \partial_{xxt} \eta + \alpha \partial_{xxxx} \eta = - \left( \mu (A_{\zeta, \eta} \nabla) v + \frac{\mu}{\det \nabla \chi_{\zeta, \eta}} B_{\zeta, \eta} \nabla v^T B_{\zeta, \eta} - q B_{\zeta, \eta} \right)_{|_{y=1}} e_2 \cdot e_2. \quad (2.14) $$
**Remark 2.4** An important remark here is that since $\text{div}(\nabla u^T) = 0$ (the divergence operator in this equality is the one associated to the variables in the deformed domain $\mathcal{F}_{\zeta(t),\eta(t)}$), then we have

$$ \text{div}\left(\frac{1}{\det\nabla\chi_{\zeta,\eta}} B_{\zeta,\eta} \nabla v^T B_{\zeta,\eta}\right) = 0, $$

where the divergence operator in the above equality is the one which is associated to the variables in the reference configuration $\mathcal{F}_0$. Note however that contrary to the case where the longitudinal displacement is neglected and the beam moves in the transversal direction only, this term gives a non zero contribution in the right hand side of the structure equations (2.13), (2.14).

Moreover, the boundary conditions satisfied by $(v, \zeta, \eta)$ are

* $v|_{y=1} = \partial_t \zeta e_1 + \partial_t \eta e_2$;

* L-periodicity w.r.t $x$ for the velocity field $v$, the pressure field $q$ and structure displacement $(\zeta, \eta)$;

* no-slip boundary conditions on the bottom of the fluid container: $v|_{y=0} = 0$.

Concerning the initial conditions we define

$$ v_0(x, y) := v(x, y, 0) = u_0(\chi_{\zeta_0, \eta_0}(x, y)), \qquad (2.15) $$

and, in view of (2.3),

$$ \left\{ \begin{array}{ll} \text{div}(B_{\zeta_0, \eta_0}^T v_0) = 0, & \text{in } \Omega_0, \\ v_{0|_{y=1}} = \zeta_1 e_1 + \eta_1 e_2, & \\ v_{0|_{y=0}} = 0, & \end{array} \right. \qquad (2.16) $$

whereas the initial conditions for the structure (1.8)-(1.11) are still valid.

For $T > 0$ such that there exists a mapping $\chi_{\zeta,\eta} \in C^0([0, T]; H_{\#}^{2+s}(\Omega_0))$, with $s > 0$ verifying

$$ \begin{array}{l} \chi_{\zeta(t),\eta(t)}(x, 1) = (x + \zeta(x, t), 1 + \eta(x, t))) \text{ and } \chi_{\zeta(t),\eta(t)}(x, 0) = (x, 0), \forall (x, t) \in [0, L] \times [0, T], \\ \det \nabla \chi_{\zeta(t),\eta(t)} > 0, \text{ on } \bar{\Omega}_0, \forall t \in [0, T], \\ \chi_{\zeta(t),\eta(t)} \text{ is injective on } \bar{\mathcal{F}}_0, \forall t \in [0, T], \\ \partial_t \chi_{\zeta,\eta} \in L^\infty(0, T; H^1(\Omega_0)), \end{array} $$

the existence of a unique strong solution $(u, p, \zeta, \eta)$ to Problem (1.1)-(1.12) with initial conditions $(u_0, \zeta_0, \zeta_1, \eta_0, \eta_1)$ satisfying (2.3) is equivalent to the existence and uniqueness of strong solution $(v, q, \zeta, \eta)$ of Problem (2.8)-(2.14) on $\Omega_0 \times (0, T)$ together with the above boundary conditions and initial conditions $(v_0, \zeta_0, \zeta_1, \eta_0, \eta_1)$ defined through (2.15), (1.8)-(1.12) and satisfying (2.16). For the sake of simplicity the problem in the reference configuration $\Omega_0 \times (0, T)$ is called $(FS)_{\text{ref}}$ problem.

Next, we apply a standard perturbation method to linearize the whole problem $(FS)_{\text{ref}}$ around the initial configuration. We note that the constraint (1.13) can be rewritten as follows:

$$ \int_0^L \left( -\partial_t \zeta(x, t) \partial_x \eta_0(x) + \partial_t \eta(x, t) (1 + \partial_x \zeta_0(x)) \right) dx = V(t), \qquad (2.17) $$
**Remark 2.4** An important remark here is that since $\text{div}(\nabla u^T) = 0$ (the divergence operator in this equality is the one associated to the variables in the deformed domain $\mathcal{F}_{\zeta(t),\eta(t)}$), then we have

$$ \text{div}\left(\frac{1}{\det\nabla\chi_{\zeta,\eta}} B_{\zeta,\eta} \nabla v^T B_{\zeta,\eta}\right) = 0, $$

where the divergence operator in the above equality is the one which is associated to the variables in the reference configuration $\mathcal{F}_0$. Note however that contrary to the case where the longitudinal displacement is neglected and the beam moves in the transversal direction only, this term gives a non zero contribution in the right hand side of the structure equations (2.13), (2.14).

Moreover, the boundary conditions satisfied by $(v, \zeta, \eta)$ are

* $v|_{y=1} = \partial_t \zeta e_1 + \partial_t \eta e_2$;

* L-periodicity w.r.t $x$ for the velocity field $v$, the pressure field $q$ and structure displacement $(\zeta, \eta)$;

* no-slip boundary conditions on the bottom of the fluid container: $v|_{y=0} = 0$.

Concerning the initial conditions we define

$$ v_0(x, y) := v(x, y, 0) = u_0(\chi_{\zeta_0, \eta_0}(x, y)), \quad (2.15) $$

and, in view of (2.3),

$$ \left\{ \begin{array}{ll} \text{div}(B_{\zeta_0, \eta_0}^T v_0) = 0, & \text{in } \Omega_0, \\ v_{0|_{y=1}} = \zeta_1 e_1 + \eta_1 e_2, & \\ v_{0|_{y=0}} = 0, & \end{array} \right. \quad (2.16) $$

whereas the initial conditions for the structure (1.8)-(1.11) are still valid.

For $T > 0$ such that there exists a mapping $\chi_{\zeta,\eta} \in C^0([0, T]; H_{\#}^{2+s}(\Omega_0))$, with $s > 0$ verifying

$$ \begin{array}{l} \chi_{\zeta(t),\eta(t)}(x, 1) = (x + \zeta(x, t), 1 + \eta(x, t))) \text{ and } \chi_{\zeta(t),\eta(t)}(x, 0) = (x, 0), \forall (x, t) \in [0, L] \times [0, T], \\ \det\nabla\chi_{\zeta(t),\eta(t)} > 0, \text{ on } \bar{\Omega}_0, \forall t \in [0, T], \\ \chi_{\zeta(t),\eta(t)} \text{ is injective on } \bar{\mathcal{F}}_0, \forall t \in [0, T], \\ \partial_t \chi_{\zeta,\eta} \in L^\infty(0, T; H^1(\Omega_0)), \end{array} $$

the existence of a unique strong solution $(u, p, \zeta, \eta)$ to Problem (1.1)-(1.12) with initial conditions $(u_0, \zeta_0, \zeta_1, \eta_0, \eta_1)$ satisfying (2.3) is equivalent to the existence and uniqueness of strong solution $(v, q, \zeta, \eta)$ of Problem (2.8)-(2.14) on $\Omega_0 \times (0, T)$ together with the above boundary conditions and initial conditions $(v_0, \zeta_0, \zeta_1, \eta_0, \eta_1)$ defined through (2.15), (1.8)-(1.12) and satisfying (2.16). For the sake of simplicity the problem in the reference configuration $\Omega_0 \times (0, T)$ is called $(FS)_{\text{ref}}$ problem.

Next, we apply a standard perturbation method to linearize the whole problem $(FS)_{\text{ref}}$ around the initial configuration. We note that the constraint (1.13) can be rewritten as follows:

$$ \int_0^L \left( -\partial_t \zeta(x, t) \partial_x \eta_0(x) + \partial_t \eta(x, t) (1 + \partial_x \zeta_0(x)) \right) dx = V(t), \quad (2.17) $$
with

$$V(t) = \int_{0}^{L} \left( \partial_t \zeta(x,t) \partial_x (\eta(x,t) - \eta_0(x)) - \partial_t \eta(x,t) \partial_x (\zeta(x,t) - \zeta_0(x)) \right) dx. \quad (2.18)$$

Similarly Problem (FS)$_{ref}$ can be written as follows:

$$\left\{
\begin{array}{ll}
\rho_{f,\zeta_0,\eta_0} \partial_t v - \mu \operatorname{div} \left( (A_{\zeta_0,\eta_0} \nabla) v + \frac{1}{\det \nabla \chi_{\zeta_0,\eta_0}} B_{\zeta_0,\eta_0} \nabla v^T B_{\zeta_0,\eta_0} - q B_{\zeta_0,\eta_0} \right) & \\
\qquad = f_1 + f_2 + \operatorname{div} h[v, q, \zeta, \eta], & \text{in } \Omega_0 \times (0,T), \\
\operatorname{div} (B_{\zeta_0,\eta_0}^T v) = g[v, \zeta, \eta], & \text{in } \Omega_0 \times (0,T), \\
v(x, 1, t) = (\partial_t \zeta(x,t), \partial_t \eta(x,t)), & \text{on } (0, L) \times (0,T), \\
v(x, 0, t) = 0, & \text{on } (0, L) \times (0,T), \\
\rho_s \partial_{tt} \zeta - \beta \partial_{xx} \zeta = -T_{\zeta_0,\eta_0} [v, q] \cdot e_1 - h[v, q, \zeta, \eta] e_2 \cdot e_1, & \text{in } (0, L) \times (0,T), \\
\rho_s \partial_{tt} \eta + \alpha \partial_{xxxx} \eta - \gamma \partial_{txx} \eta = -T_{\zeta_0,\eta_0} [v, q] \cdot e_2 - h[v, q, \zeta, \eta] e_2 \cdot e_2, & \text{in } (0, L) \times (0,T),
\end{array}
\right.
\quad (2.19)$$

with

$$T_{\zeta_0, \eta_0} [v, q] = \left( \mu (A_{\zeta_0, \eta_0} \nabla) v \right) e_2 + \left. \frac{\mu}{\det \nabla \chi_{\zeta_0, \eta_0}} B_{\zeta_0, \eta_0} \nabla v^T B_{\zeta_0, \eta_0} e_2 - q B_{\zeta_0, \eta_0} e_2 \right|_{y=1}, \quad (2.20)$$

$$f_1[v, \zeta, \eta] = \rho_f(\det \nabla \chi_{\zeta_0, \eta_0} - \det \nabla \chi_{\zeta, \eta})\partial_t v, \quad (2.21)$$

$$f_2[v, \zeta, \eta] = -\rho_f \left( (v - \partial_t \chi_{\zeta, \eta}) \cdot (B_{\zeta, \eta} \nabla) v \right), \quad (2.22)$$

$$g[v, \zeta, \eta] = \operatorname{div} \left( (B_{\zeta_0, \eta_0}^T - B_{\zeta, \eta}^T) v \right), \quad (2.23)$$

$$h[v, q, \zeta, \eta] = -\mu \left[ (A_{\zeta_0, \eta_0} - A_{\zeta, \eta}) \nabla \right] v - \mu \left( \frac{1}{\det \nabla \chi_{\zeta_0, \eta_0}} B_{\zeta_0, \eta_0} \nabla v^T B_{\zeta_0, \eta_0} - \frac{1}{\det \nabla \chi_{\zeta, \eta}} B_{\zeta, \eta} \nabla v^T B_{\zeta, \eta} \right) + q(B_{\zeta_0, \eta_0} - B_{\zeta, \eta}). \quad (2.24)$$

Consequently, we shall consider at first $f_1, f_2, h, g$ as a source terms of the above linear problem with initial data ($v_0, \zeta_0, \zeta_1, \eta_0, \eta_1$) and unknowns ($v, q, \zeta, \eta$). We underline that the right hand side of the structure equations contains terms that are the counter part of div $h$ in the fluid equation. Moreover, we remark that by integrating the divergence constraint over $\Omega_0$ and integrating by parts, we obtain

$$\begin{aligned}
\int_{\Omega_0} g \,dx dy &= \int_{\Omega_0} \operatorname{div} (B_{\zeta_0, \eta_0}^T v) \,dx dy \\
&= \int_{\partial \Omega_0} B_{\zeta_0, \eta_0}^T v \cdot n \,d\gamma \\
&= \int_0^L B_{\zeta_0, \eta_0|_{y=1}}^T (\partial_t \zeta, \partial_t \eta) \cdot e_2 \,dx.
\end{aligned}$$

Therefore, by (2.17)

$$\int_{\Omega_0} g \,dx dy = V(t). \quad (2.25)$$

In particular $g$ does not have a zero mean as it is the case where the longitudinal displacement $\zeta$ is neglected. As a consequence one can not lift this divergence constraint with a velocity satisfying homogeneous boundary conditions on the upper and lower boundaries. Moreover, the decomposition of the pressure in (1.15) yields to

$$q(x, y, t) = p_0(\chi_{\zeta(t),\eta(t)}(x, y), t) + c(t) := q_0(x, y, t) + c(t).$$
with

$$V(t) = \int_{0}^{L} \left( \partial_t \zeta(x,t) \partial_x (\eta(x,t) - \eta_0(x)) - \partial_t \eta(x,t) \partial_x (\zeta(x,t) - \zeta_0(x)) \right) dx. \quad (2.18)$$

Similarly Problem (FS)$_{ref}$ can be written as follows:

$$\left\{
\begin{aligned}
& \rho_{f,\zeta_0,\eta_0} \partial_t v - \mu \operatorname{div} \left( (A_{\zeta_0,\eta_0} \nabla) v + \frac{1}{\det \nabla \chi_{\zeta_0,\eta_0}} B_{\zeta_0,\eta_0} \nabla v^T B_{\zeta_0,\eta_0} - q B_{\zeta_0,\eta_0} \right) \\
& = f_1 + f_2 + \operatorname{div} h[v, q, \zeta, \eta], \quad \text{in } \Omega_0 \times (0,T), \\
& \operatorname{div} (B_{\zeta_0,\eta_0}^T v) = g[v, \zeta, \eta], \quad \text{in } \Omega_0 \times (0,T), \\
& v(x, 1, t) = (\partial_t \zeta(x,t), \partial_t \eta(x,t)), \quad \text{on } (0,L) \times (0,T), \\
& v(x, 0, t) = 0, \quad \text{on } (0,L) \times (0,T), \\
& \rho_s \partial_{tt} \zeta - \beta \partial_{xx} \zeta = -T_{\zeta_0,\eta_0} [v, q] \cdot e_1 - h[v, q, \zeta, \eta] e_2 \cdot e_1, \quad \text{in } (0,L) \times (0,T), \\
& \rho_s \partial_{tt} \eta + \alpha \partial_{xxxx} \eta - \gamma \partial_{txx} \eta = -T_{\zeta_0,\eta_0} [v, q] \cdot e_2 - h[v, q, \zeta, \eta] e_2 \cdot e_2, \quad \text{in } (0,L) \times (0,T),
\end{aligned}
\right.
\quad (2.19)$$

with

$$T_{\zeta_0, \eta_0} [v, q] = \left( \mu (A_{\zeta_0, \eta_0} \nabla) v \right) e_2 + \left. \frac{\mu}{\det \nabla \chi_{\zeta_0, \eta_0}} B_{\zeta_0, \eta_0} \nabla v^T B_{\zeta_0, \eta_0} e_2 - q B_{\zeta_0, \eta_0} e_2 \right|_{y=1}, \quad (2.20)$$

$$f_1[v, \zeta, \eta] = \rho_f(\det \nabla \chi_{\zeta_0, \eta_0} - \det \nabla \chi_{\zeta, \eta}) \partial_t v, \quad (2.21)$$

$$f_2[v, \zeta, \eta] = -\rho_f \left( (v - \partial_t \chi_{\zeta, \eta}) \cdot (B_{\zeta, \eta} \nabla) v \right), \quad (2.22)$$

$$g[v, \zeta, \eta] = \operatorname{div} \left( (B_{\zeta_0, \eta_0}^T - B_{\zeta, \eta}^T) v \right), \quad (2.23)$$

$$h[v, q, \zeta, \eta] = -\mu \left[ (A_{\zeta_0, \eta_0} - A_{\zeta, \eta}) \nabla \right] v - \mu \left( \frac{1}{\det \nabla \chi_{\zeta_0, \eta_0}} B_{\zeta_0, \eta_0} \nabla v^T B_{\zeta_0, \eta_0} - \frac{1}{\det \nabla \chi_{\zeta, \eta}} B_{\zeta, \eta} \nabla v^T B_{\zeta, \eta} \right) + q(B_{\zeta_0, \eta_0} - B_{\zeta, \eta}). \quad (2.24)$$

Consequently, we shall consider at first $f_1, f_2, h, g$ as a source terms of the above linear problem with initial data ($v_0, \zeta_0, \zeta_1, \eta_0, \eta_1$) and unknowns ($v, q, \zeta, \eta$). We underline that the right hand side of the structure equations contains terms that are the counter part of div $h$ in the fluid equation. Moreover, we remark that by integrating the divergence constraint over $\Omega_0$ and integrating by parts, we obtain

$$\begin{aligned}
\int_{\Omega_0} g \,dx\,dy &= \int_{\Omega_0} \operatorname{div} (B_{\zeta_0, \eta_0}^T v) \,dx\,dy \\
&= \int_{\partial \Omega_0} B_{\zeta_0, \eta_0}^T v \cdot n \,d\gamma \\
&= \int_0^L B_{\zeta_0, \eta_0|_{y=1}}^T (\partial_t \zeta, \partial_t \eta) \cdot e_2 \,dx.
\end{aligned}$$

Therefore, by (2.17)

$$\int_{\Omega_0} g \,dx\,dy = V(t). \quad (2.25)$$

In particular $g$ does not have a zero mean as it is the case where the longitudinal displacement $\zeta$ is neglected. As a consequence one can not lift this divergence constraint with a velocity satisfying homogeneous boundary conditions on the upper and lower boundaries. Moreover, the decomposition of the pressure in (1.15) yields to

$$q(x, y, t) = p_0(\chi_{\zeta(t),\eta(t)}(x, y), t) + c(t) := q_0(x, y, t) + c(t).$$
Due to (1.16), $q_0$ shall satisfy the following constraint:

$$ \int_{\Omega_0} q_0 \det \nabla \chi_{\zeta, \eta} dxdy = 0. \qquad (2.26) $$

However, the constraint in (2.26) is not easy to deal with as it is a non linear combination of unknowns $q_0$, $\zeta$ and $\eta$. To this end, we split again the pressure as follows:

$$ q(x, y, t) = r_0(x, y, t) + d(t), \qquad (2.27) $$

with

$$ \int_{\Omega_0} r_0 dxdy = 0, \qquad (2.28) $$

and

$$ d(t) = \frac{1}{\|n_0\|_{L^2(0,L)}^2} \left( \rho_s \frac{d}{dt} \int_0^L \left( -\partial_t \zeta \partial_x \eta_0 + \partial_t \eta (1 + \partial_x \zeta_0) \right) dx - \beta \int_0^L \zeta \partial_{xxx} \eta_0 dx - \alpha \int_0^L \partial_{xxx} \eta \partial_{xx} \zeta_0 dx \right. \\ \left. + \gamma \int_0^L \partial_{xtt} \eta \partial_{xx} \zeta_0 dx + \int_0^L T[v, r_0] \cdot n_0 dx + \int_0^L h_{|_{y=1}} e_2 \cdot n_0 dx \right), \quad (2.29) $$

where $n_0 = -\partial_x \eta_0 e_1 + (1 + \partial_x \zeta_0) e_2$ is a non unitary normal vector to $\Gamma_{\zeta_0, \eta_0}$.

Thus, in view of (2.17), $d(t)$ can be rewritten as

$$ d(t) = \frac{1}{\|n_0\|_{L^2(0,L)}^2} \left( \rho_s \dot{V}(t) - \beta \int_0^L \zeta \partial_{xxx} \eta_0 dx - \alpha \int_0^L \partial_{xx} \eta \partial_{xx} \zeta_0 dx + \gamma \int_0^L \partial_{xtt} \eta \partial_{xx} \zeta_0 dx \right. \\ \left. + \int_0^L T[v, r_0] \cdot n_0 dx + \int_0^L h_{|_{y=1}} e_2 \cdot n_0 dx \right), \quad (2.30) $$

where $\dot{V}$ denotes the time derivative of $V$. We underline that from the knowledge of $(r_0, d)$, we can recover $(q_0, c)$.

In fact, we may rewrite

$$ q = (r_0 + d(t) - c(t)) + c(t) = q_0 + c(t). \qquad (2.31) $$

In view of (2.26), we obtain

$$ \int_{\Omega_0} (r_0 + d(t) - c(t)) \det \nabla \chi_{\zeta(t), \eta(t)} dxdy = 0. $$

It follows that

$$ \int_{\Omega_0} r_0 \det \nabla \chi_{\zeta(t), \eta(t)} dxdy + (d(t) - c(t)) |\Omega_{\zeta(t), \eta(t)}| = 0. $$

Therefore,

$$ c(t) = d(t) + \frac{1}{|\Omega_{\zeta(t), \eta(t)}|} \int_{\Omega_0} r_0 \det \nabla \chi_{\zeta(t), \eta(t)} dxdy. \qquad (2.32) $$

By substituting $c(t)$ in (2.31), we get

$$ q_0 = r_0 - \frac{1}{|\Omega_{\zeta(t), \eta(t)}|} \int_{\Omega_0} r_0 \det \nabla \chi_{\zeta(t), \eta(t)} dxdy. \qquad (2.33) $$
Due to (1.16), $q_0$ shall satisfy the following constraint:

$$ \int_{\Omega_0} q_0 \det \nabla \chi_{\zeta, \eta} dxdy = 0. \qquad (2.26) $$

However, the constraint in (2.26) is not easy to deal with as it is a non linear combination of unknowns $q_0$, $\zeta$ and $\eta$. To this end, we split again the pressure as follows:

$$ q(x, y, t) = r_0(x, y, t) + d(t), \qquad (2.27) $$

with

$$ \int_{\Omega_0} r_0 dxdy = 0, \qquad (2.28) $$

and

$$ d(t) = \frac{1}{\|n_0\|_{L^2(0,L)}^2} \left( \rho_s \frac{d}{dt} \int_0^L \left( -\partial_t \zeta \partial_x \eta_0 + \partial_t \eta (1 + \partial_x \zeta_0) \right) dx - \beta \int_0^L \zeta \partial_{xxx} \eta_0 dx - \alpha \int_0^L \partial_{xxx} \eta \partial_{xx} \zeta_0 dx \right. \\ \left. + \gamma \int_0^L \partial_{xtt} \eta \partial_{xx} \zeta_0 dx + \int_0^L T[v, r_0] \cdot n_0 dx + \int_0^L h_{|_{y=1}} e_2 \cdot n_0 dx \right), \quad (2.29) $$

where $n_0 = -\partial_x \eta_0 e_1 + (1 + \partial_x \zeta_0) e_2$ is a non unitary normal vector to $\Gamma_{\zeta_0, \eta_0}$.

Thus, in view of (2.17), $d(t)$ can be rewritten as

$$ d(t) = \frac{1}{\|n_0\|_{L^2(0,L)}^2} \left( \rho_s \dot{V}(t) - \beta \int_0^L \zeta \partial_{xxx} \eta_0 dx - \alpha \int_0^L \partial_{xx} \eta \partial_{xx} \zeta_0 dx + \gamma \int_0^L \partial_{xtt} \eta \partial_{xx} \zeta_0 dx \right. \\ \left. + \int_0^L T[v, r_0] \cdot n_0 dx + \int_0^L h_{|_{y=1}} e_2 \cdot n_0 dx \right), \quad (2.30) $$

where $\dot{V}$ denotes the time derivative of $V$. We underline that from the knowledge of $(r_0, d)$, we can recover $(q_0, c)$.

In fact, we may rewrite

$$ q = (r_0 + d(t) - c(t)) + c(t) = q_0 + c(t). \qquad (2.31) $$

In view of (2.26), we obtain

$$ \int_{\Omega_0} (r_0 + d(t) - c(t)) \det \nabla \chi_{\zeta(t), \eta(t)} dxdy = 0. $$

It follows that

$$ \int_{\Omega_0} r_0 \det \nabla \chi_{\zeta(t), \eta(t)} dxdy + (d(t) - c(t)) |\Omega_{\zeta(t), \eta(t)}| = 0. $$

Therefore,

$$ c(t) = d(t) + \frac{1}{|\Omega_{\zeta(t), \eta(t)}|} \int_{\Omega_0} r_0 \det \nabla \chi_{\zeta(t), \eta(t)} dxdy. \qquad (2.32) $$

By substituting $c(t)$ in (2.31), we get

$$ q_0 = r_0 - \frac{1}{|\Omega_{\zeta(t), \eta(t)}|} \int_{\Omega_0} r_0 \det \nabla \chi_{\zeta(t), \eta(t)} dxdy. \qquad (2.33) $$
## 2.5 Useful propositions and Elliptic Result

We first state a useful uniform coercivity property and a uniform Korn inequality.

**Proposition 2.2** Let $s > 0$. For any $(a,b) \in H_{\#}^{3+s}(0,L)$ such that $\det \nabla \chi_{a,b} > 0$ on $\bar{\Omega}_0$, then the symmetric matrix $A_{a,b} \in H_{\#}^{1+s}(\Omega_0) \subset C_{\#}(\bar{\Omega}_0)$. Moreover, there exists two non negative functions $\alpha_1, \alpha_2$, resp. non increasing and non decreasing with respect to R such that for any $(a,b) \in H_{\#}^{3+s}(0,L)$ satisfying $\|(a,b)\|_{H_{\#}^{3+s}(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R$

$$ \alpha_1(R)I_2 \le A_{a,b}(x,y) \le \alpha_2(R)I_2, \quad \text{for any } (x,y) \in \bar{\Omega}_0. \qquad (2.34) $$

In the same way, we have a uniform Korn like inequality: there exists a constant $\lambda(R) > 0$, non increasing with respect to R, such that, for any $(a,b) \in H_{\#}^{3+s}(0,L)$ such that $\chi_{a,b}$ is injective on $\bar{\Omega}_0$ and $\|(a,b)\|_{H_{\#}^{3+s}(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R,$

$$ \int_{\Omega_0} \frac{1}{\det \nabla \chi_{a,b}} |((B_{a,b} \nabla) v + ((B_{a,b} \nabla) v)^T)|^2 dxdy \ge \lambda(R) \int_{\Omega_0} |\nabla v|^2 dxdy, \quad \forall v \in H_{\#}^{1}(\Omega_0), v|_{y=0} = 0. \quad (2.35) $$

**Proof.** Since $H_{\#}^{1+s}(\Omega_0)$ is an algebra, if $\det \nabla \chi_{a,b} > 0$ on $\bar{\Omega}_0$, then $A_{a,b} \in H_{\#}^{1+s}(\Omega_0)$. The proof of (2.34) relies on the fact that $H_{\#}^{1+s}(\Omega_0)$ compactly embed in $C_{\#}(\bar{\Omega}_0)$. For the Korn inequality, since $\chi_{a,b}$ is injective on $\bar{\Omega}_0$ and $\det \nabla \chi_{a,b} > 0$ on $\bar{\Omega}_0$, it is a $C^1$-diffeomorphism from $\Omega_0$ onto $\Omega_{a,b} = \chi_{a,b}(\Omega_0)$. Applying this change of variables, we observe

$$ \int_{\Omega_0} \frac{1}{\det \nabla \chi_{a,b}} |((B_{a,b} \nabla) v + ((B_{a,b} \nabla) v)^T)|^2 dxdy = 2 \int_{\Omega_0} \left( (A_{a,b} \nabla) v + \frac{1}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v^T B_{a,b} \right) : \nabla v dxdy \\ = \int_{\Omega_{a,b}} |\nabla \tilde{v} + (\nabla \tilde{v})^T|^2 dx'dy'. $$

where $\tilde{v}(x', y') = v(\chi_{a,b}^{-1}(x', y'))$ for any vector field $v$.

Hence, by applying Korn inequality in $\Omega_{a,b}$ and there exists $\tilde{\lambda}_{a,b} > 0$ such that, $\forall v \in H_{\#}^{1}(\Omega_0)$ with $v|_{y=0} = 0$,

$$ \int_{\Omega_0} \frac{1}{\det \nabla \chi_{a,b}} |((B_{a,b} \nabla) v + ((B_{a,b} \nabla) v)^T)|^2 dxdy \ge \tilde{\lambda}_{a,b} \int_{\Omega_{a,b}} |\nabla \tilde{v}|^2 dx'dy'. $$

Changing of variables we deduce that there exists $\lambda_{a,b} > 0$ such that, $\forall v \in H_{\#}^{1}(\Omega_0), v|_{y=0} = 0$,

$$ \int_{\Omega_0} \frac{1}{\det \nabla \chi_{a,b}} |((B_{a,b} \nabla) v + ((B_{a,b} \nabla) v)^T)|^2 dxdy \ge \lambda_{a,b} \int_{\Omega_0} |\nabla v|^2 dxdy. $$

The next step is to prove that we can choose a uniform constant for any $(a,b) \in H_{\#}^{3+s}(0,L)$ with $\chi_{a,b}$ is injective on $\bar{\Omega}_0$ and $\|(a,b)\|_{H_{\#}^{3+s}(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R$, depending only on $R$. By choosing $\lambda_{a,b}$ as the best constant and defining $\lambda(R)$ as the infimum over the set of $(a,b) \in H_{\#}^{3+s}(0,L)$ with $\chi_{a,b}$ is injective on $\bar{\Omega}_0$ and $\|(a,b)\|_{H_{\#}^{3+s}(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R$, it is easy to show by contradiction that $\lambda(R)$ is not equal to zero following the same steps as in [35]. It is clear that the mapping $R \mapsto \lambda(R)$ is a non increasing function.

**Remark 2.5** Note that in the only transversal case this uniform Korn inequality is not needed for the study of the full coupled problem as, in this case, the contribution of $((B_{a,b} \nabla) v)^T$ disappears (see [14], and Remark 2.4).
## 2.5 Useful propositions and Elliptic Result

We first state a useful uniform coercivity property and a uniform Korn inequality.

**Proposition 2.2** Let $s > 0$. For any $(a,b) \in H_{\#}^{3+s}(0,L)$ such that $\det \nabla \chi_{a,b} > 0$ on $\bar{\Omega}_0$, then the symmetric matrix $A_{a,b} \in H_{\#}^{1+s}(\Omega_0) \subset C_{\#}(\bar{\Omega}_0)$. Moreover, there exists two non negative functions $\alpha_1, \alpha_2$, resp. non increasing and non decreasing with respect to R such that for any $(a,b) \in H_{\#}^{3+s}(0,L)$ satisfying $\|(a,b)\|_{H_{\#}^{3+s}(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R$

$$ \alpha_1(R)I_2 \le A_{a,b}(x,y) \le \alpha_2(R)I_2, \quad \text{for any } (x,y) \in \bar{\Omega}_0. \qquad (2.34) $$

In the same way, we have a uniform Korn like inequality: there exists a constant $\lambda(R) > 0$, non increasing with respect to R, such that, for any $(a,b) \in H_{\#}^{3+s}(0,L)$ such that $\chi_{a,b}$ is injective on $\bar{\Omega}_0$ and $\|(a,b)\|_{H_{\#}^{3+s}(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R,$

$$ \int_{\Omega_0} \frac{1}{\det \nabla \chi_{a,b}} |((B_{a,b} \nabla) v + ((B_{a,b} \nabla) v)^T)|^2 dxdy \ge \lambda(R) \int_{\Omega_0} |\nabla v|^2 dxdy, \quad \forall v \in H_{\#}^{1}(\Omega_0), v|_{y=0} = 0. \qquad (2.35) $$

**Proof.** Since $H_{\#}^{1+s}(\Omega_0)$ is an algebra, if $\det \nabla \chi_{a,b} > 0$ on $\bar{\Omega}_0$, then $A_{a,b} \in H_{\#}^{1+s}(\Omega_0)$. The proof of (2.34) relies on the fact that $H_{\#}^{1+s}(\Omega_0)$ compactly embed in $C_{\#}(\bar{\Omega}_0)$. For the Korn inequality, since $\chi_{a,b}$ is injective on $\bar{\Omega}_0$ and $\det \nabla \chi_{a,b} > 0$ on $\bar{\Omega}_0$, it is a $C^1$-diffeomorphism from $\Omega_0$ onto $\Omega_{a,b} = \chi_{a,b}(\Omega_0)$. Applying this change of variables, we observe

$$ \int_{\Omega_0} \frac{1}{\det \nabla \chi_{a,b}} |((B_{a,b} \nabla) v + ((B_{a,b} \nabla) v)^T)|^2 dxdy = 2 \int_{\Omega_0} \left( (A_{a,b} \nabla) v + \frac{1}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v^T B_{a,b} \right) : \nabla v dxdy \\ = \int_{\Omega_{a,b}} |\nabla \tilde{v} + (\nabla \tilde{v})^T|^2 dx'dy'. $$

where $\tilde{v}(x', y') = v(\chi_{a,b}^{-1}(x', y'))$ for any vector field $v$.

Hence, by applying Korn inequality in $\Omega_{a,b}$ and there exists $\tilde{\lambda}_{a,b} > 0$ such that, $\forall v \in H_{\#}^{1}(\Omega_0)$ with $v|_{y=0} = 0$,

$$ \int_{\Omega_0} \frac{1}{\det \nabla \chi_{a,b}} |((B_{a,b} \nabla) v + ((B_{a,b} \nabla) v)^T)|^2 dxdy \ge \tilde{\lambda}_{a,b} \int_{\Omega_{a,b}} |\nabla \tilde{v}|^2 dx'dy'. $$

Changing of variables we deduce that there exists $\lambda_{a,b} > 0$ such that, $\forall v \in H_{\#}^{1}(\Omega_0), v|_{y=0} = 0$,

$$ \int_{\Omega_0} \frac{1}{\det \nabla \chi_{a,b}} |((B_{a,b} \nabla) v + ((B_{a,b} \nabla) v)^T)|^2 dxdy \ge \lambda_{a,b} \int_{\Omega_0} |\nabla v|^2 dxdy. $$

The next step is to prove that we can choose a uniform constant for any $(a,b) \in H_{\#}^{3+s}(0,L)$ with $\chi_{a,b}$ is injective on $\bar{\Omega}_0$ and $\|(a,b)\|_{H_{\#}^{3+s}(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R$, depending only on R. By choosing $\lambda_{a,b}$ as the best constant and defining $\lambda(R)$ as the infimum over the set of $(a,b) \in H_{\#}^{3+s}(0,L)$ with $\chi_{a,b}$ is injective on $\bar{\Omega}_0$ and $\|(a,b)\|_{H_{\#}^{3+s}(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R$, it is easy to show by contradiction that $\lambda(R)$ is not equal to zero following the same steps as in [35]. It is clear that the mapping $R \mapsto \lambda(R)$ is a non increasing function.

**Remark 2.5** Note that in the only transversal case this uniform Korn inequality is not needed for the study of the full coupled problem as, in this case, the contribution of $((B_{a,b} \nabla) v)^T$ disappears (see [14], and Remark 2.4).
Next to achieve the desired regularity of the fluid velocity and pressure solution to Problem (2.19), we need an elliptic regularity result of the following periodic steady state Stokes-like problem, for $(a, b) \in H_{\#}^{3+s}(0, L)$ with $s > 0$

$$ -\mu \operatorname{div}((A_{a,b}\nabla)\hat{u} + \frac{1}{\det \nabla\chi_{a,b}} B_{a,b}\nabla\hat{u}^T B_{a,b}) + (B_{a,b}\nabla)\hat{p} = \hat{f}, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.36) $$

$$ \operatorname{div}(B_{a,b}^T \hat{u}) = \hat{g}, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.37) $$

completed with the following boundary conditions

$$ \hat{u}_{|y=1} = \psi, \qquad (2.38) $$

$$ \hat{u}_{|y=0} = 0, \qquad (2.39) $$

where $\hat{f} \in L_{\#}^2(\Omega_0)$, $\hat{g} \in H_{\#}^1(\Omega_0)$ and $\psi = (\psi_1, \psi_2) \in H_{\#}^{3/2}(0, L)$. From Remark 2.4, equation (2.36) writes

$$ -\mu \operatorname{div}((A_{a,b}\nabla)\hat{u}) + (B_{a,b}\nabla)\hat{p} = \hat{f}, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.40) $$

and even if $\frac{1}{\det \nabla\chi_{a,b}} B_{a,b}\nabla\hat{u}^T B_{a,b} \cdot n$ is not equal to zero on the fluid-structure interface, since in this subsection we are interested in the elliptic regularity of a Dirichlet boundary problem, we can work here with (2.40). Here we underline that the couple $(a, b)$ does not depend on time and that, moreover, due to (2.37) and to the boundary conditions (2.38)-(2.39), the data should satisfy the following compatibility condition

$$ \int_{\Omega_0} \hat{g} \, dxdy = \int_0^L (-\psi_1(x)\partial_x b(x) + \psi_2(x)(1 + \partial_x a(x))) dx. \qquad (2.41) $$

We remark that the pressure $\hat{p}$ in (2.40) is defined up to a constant. Consequently, we impose, for instance, the additional condition

$$ \int_{\Omega_0} \hat{p} \, dxdy = 0, $$

to ensure the uniqueness of pressure. To this end, we introduce the following space

$$ L_{\#}^2(\Omega_0) = \{p \in L_{\#}^2(\Omega_0) : \int_{\Omega_0} p \, dxdy = 0\}. $$

We underline that Problem (2.40)-(2.39) is equivalent to the stationary Stokes problem written in the domain $\Omega_{a,b}$ whose boundary is only $H^{3+s}$ regular, which does not embed into $C^{1,1}$ or $W^{2,\infty}$ as in [9]. Thus, standard regularity results of the stationary Stokes problem does not apply here because it requires that the boundary of the domain to be smooth enough. Moreover a key issue is to control the different constants uniformly with respect to $(a, b)$. In [27] an elliptic result has been established for Problem (2.40)-(2.39) in the case $a = 0$ and $b \in H_{\#}^2(0, L)$. In [9] the regularity of the Stokes problem is derived for domains with $W^{2,\infty}$ boundary and in [10] the regularity of the solution of a divergence free Stokes system in non smooth domains is derived.

A key argument in all these $H^2 \times H^1$ regularity results is that the matrices $A_{a,b}$, and $B_{a,b}$ belongs to a multiplier space of $H^1(\Omega_0)$, and, in our case, we moreover rely on the following lemma

**Lemma 2.1** Given a function $k \in H_{\#}^s(\Omega_0)$ with $0 < s < 1$, a function $\varphi \in H_{\#}^1(\Omega_0)$. Then there exists a positive constant $C$ which depends on the domain $\Omega_0$ and for each $\delta > 0$ there exists a constant $C_\delta > 0$ such that

$$ \|k\varphi\|_{L_{\#}^2(\Omega_0)} \le C \|k\|_{H_{\#}^s(\Omega_0)} \|\varphi\|_{H_{\#}^{1-s}(\Omega_0)} \le C \|k\|_{H_{\#}^s(\Omega_0)} (C_\delta \|\varphi\|_{L_{\#}^2(\Omega_0)} + \delta \|\nabla\varphi\|_{L_{\#}^2(\Omega_0)}). $$
Next to achieve the desired regularity of the fluid velocity and pressure solution to Problem (2.19), we need an elliptic regularity result of the following periodic steady state Stokes-like problem, for $(a, b) \in H_{\#}^{3+s}(0, L)$ with $s > 0$

$$ -\mu \operatorname{div}((A_{a,b}\nabla)\hat{u} + \frac{1}{\det \nabla\chi_{a,b}} B_{a,b} \nabla \hat{u}^T B_{a,b}) + (B_{a,b}\nabla)\hat{p} = \hat{f}, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.36) $$

$$ \operatorname{div}(B_{a,b}^T \hat{u}) = \hat{g}, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.37) $$

completed with the following boundary conditions

$$ \hat{u}_{|y=1} = \psi, \qquad (2.38) $$

$$ \hat{u}_{|y=0} = 0, \qquad (2.39) $$

where $\hat{f} \in L_{\#}^2(\Omega_0)$, $\hat{g} \in H_{\#}^1(\Omega_0)$ and $\psi = (\psi_1, \psi_2) \in H_{\#}^{3/2}(0, L)$. From Remark 2.4, equation (2.36) writes

$$ -\mu \operatorname{div}((A_{a,b}\nabla)\hat{u}) + (B_{a,b}\nabla)\hat{p} = \hat{f}, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.40) $$

and even if $\frac{1}{\det \nabla\chi_{a,b}} B_{a,b} \nabla \hat{u}^T B_{a,b} \cdot n$ is not equal to zero on the fluid-structure interface, since in this subsection we are interested in the elliptic regularity of a Dirichlet boundary problem, we can work here with (2.40). Here we underline that the couple $(a, b)$ does not depend on time and that, moreover, due to (2.37) and to the boundary conditions (2.38)-(2.39), the data should satisfy the following compatibility condition

$$ \int_{\Omega_0} \hat{g} \, dxdy = \int_0^L \left( -\psi_1(x)\partial_x b(x) + \psi_2(x)(1 + \partial_x a(x)) \right) dx. \qquad (2.41) $$

We remark that the pressure $\hat{p}$ in (2.40) is defined up to a constant. Consequently, we impose, for instance, the additional condition

$$ \int_{\Omega_0} \hat{p} \, dxdy = 0, $$

to ensure the uniqueness of pressure. To this end, we introduce the following space

$$ L_{\#}^2(\Omega_0) = \{p \in L_{\#}^2(\Omega_0) : \int_{\Omega_0} p \, dxdy = 0\}. $$

We underline that Problem (2.40)-(2.39) is equivalent to the stationary Stokes problem written in the domain $\Omega_{a,b}$ whose boundary is only $H^{3+s}$ regular, which does not embed into $C^{1,1}$ or $W^{2,\infty}$ as in [9]. Thus, standard regularity results of the stationary Stokes problem does not apply here because it requires that the boundary of the domain to be smooth enough. Moreover a key issue is to control the different constants uniformly with respect to $(a, b)$. In [27] an elliptic result has been established for Problem (2.40)-(2.39) in the case $a = 0$ and $b \in H_{\#}^2(0, L)$. In [9] the regularity of the Stokes problem is derived for domains with $W^{2,\infty}$ boundary and in [10] the regularity of the solution of a divergence free Stokes system in non smooth domains is derived.

A key argument in all these $H^2 \times H^1$ regularity results is that the matrices $A_{a,b}$, and $B_{a,b}$ belongs to a multiplier space of $H^1(\Omega_0)$, and, in our case, we moreover rely on the following lemma

**Lemma 2.1** Given a function $k \in H_{\#}^s(\Omega_0)$ with $0 < s < 1$, a function $\varphi \in H_{\#}^1(\Omega_0)$. Then there exists a positive constant $C$ which depends on the domain $\Omega_0$ and for each $\delta > 0$ there exists a constant $C_{\delta} > 0$ such that

$$ \|k\varphi\|_{L_{\#}^2(\Omega_0)} \le C \|k\|_{H_{\#}^s(\Omega_0)} \|\varphi\|_{H_{\#}^{1-s}(\Omega_0)} \le C \|k\|_{H_{\#}^s(\Omega_0)} (C_{\delta} \|\varphi\|_{L_{\#}^2(\Omega_0)} + \delta \|\nabla\varphi\|_{L_{\#}^2(\Omega_0)}). $$
Proof. Let $k \in H_{\#}^s(\Omega_0)$ with $0 < s < 1$ and consider a function $\varphi \in H_{\#}^1(\Omega_0)$, we have

$$
\begin{aligned}
\|k\varphi\|_{L_{\#}^2(\Omega_0)} &\le \|k\|_{L_{\#}^{\frac{2}{1-s}}(\Omega_0)} \|\varphi\|_{L_{\#}^s(\Omega_0)}^{\frac{2}{1-s}} \\
&\le C \|k\|_{H_{\#}^s(\Omega_0)} \|\varphi\|_{H_{\#}^{1-s}(\Omega_0)} \\
&\le C \|k\|_{H_{\#}^s(\Omega_0)} \|\varphi\|_{L_{\#}^2(\Omega_0)}^s \|\varphi\|_{H_{\#}^1(\Omega_0)}^{1-s}
\end{aligned}
$$

where $C > 0$ depends on $\Omega_0$.

By Young's inequality, we get

$$
\|k\varphi\|_{L_{\#}^2(\Omega_0)} \le C \|k\|_{H_{\#}^s(\Omega_0)} (C_{\delta}\|\varphi\|_{L_{\#}^2(\Omega_0)} + \delta\|\nabla\varphi\|_{L_{\#}^2(\Omega_0)}). \quad (2.42)
$$

We note that $C_\delta$ is a decreasing function of $\delta$.

We now state an elliptic result of the Stokes-like problem (2.40)-(2.39).

**Proposition 2.3** Let $s > 0$ and suppose that $(a,b) \in H_{\#}^{3+s}(0,L)$ such that $\chi_{a,b}$ is injective on $\bar{\Omega}_0$. Then for any source terms and boundary condition

$$
(\hat{f}, \hat{g}) \in H_{\#}^{-1}(\Omega_0) \times L_{\#}^2(\Omega_0), \quad \psi \in H_{\#}^{\frac{1}{2}}(0, L),
$$

such that

$$
\int_{\Omega_0} \hat{g} \, dxdy = \int_0^L \left( -\psi_1(x) \partial_x b(x) + \psi_2(x) (1 + \partial_x a(x)) \right) dx, \quad (2.43)
$$

there exists a unique solution $(\hat{u}, \hat{p}) \in H_{\#}^1(\Omega_0) \times L_{\#}^2(\Omega_0)$ to the Stokes-like problem (2.40)-(2.39) such that, if we have $\|(a,b)\|_{H_{\#}^{3+s}(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R$, then

$$
\|\hat{u}\|_{H_{\#}^1(\Omega_0)} + \|\hat{p}\|_{L_{\#}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{H_{\#}^{-1}(\Omega_0)} + \|\hat{g}\|_{L_{\#}^2(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{1}{2}}(0,L)} \right), \quad (2.44)
$$

where $K: [0, \infty) \mapsto (0, \infty)$ is a non decreasing function.

Moreover, if

$$
(\hat{f}, \hat{g}) \in L_{\#}^2(\Omega_0) \times H_{\#}^1(\Omega_0), \quad \psi \in H_{\#}^{\frac{3}{2}}(0, L),
$$

then $(\hat{u}, \hat{p}) \in H_{\#}^2(\Omega_0) \times H_{\#}^1(\Omega_0)$ and satisfies

$$
\|\hat{u}\|_{H_{\#}^2(\Omega_0)} + \|\hat{p}\|_{H_{\#}^1(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right). \quad (2.45)
$$

**Proof.** We split the proof into two steps.

**Step 1.** Let $\hat{f} \in H_{\#}^{-1}(\Omega_0)$, $\hat{g} \in L_{\#}^2(\Omega_0)$ and $\psi \in H_{\#}^{\frac{1}{2}}(0, L)$. Then there exists $u_{\psi} \in H_{\#}^1(\Omega_0)$ such that

$$
u_{\psi|_{y=1}} = \psi_1 e_1 + \psi_2 e_2 \text{ and } u_{\psi|_{y=0}} = 0,
$$

and satisfying

$$
\|u_{\psi}\|_{H_{\#}^{1}(\Omega_{0})} \leq C \|\psi\|_{H_{\#}^{\frac{1}{2}}(0,L)}, \quad (2.46)
$$

where $C$ is a positive constant depending on the domain $\Omega_0$.

Let $\bar{u} = \hat{u} - u_{\psi}$. Then $(\bar{u}, \hat{p})$ satisfies

$$
-\mu \operatorname{div}((A_{a,b} \nabla)\bar{u}) + (B_{a,b} \nabla)\hat{p} = \bar{f}, \quad \text{in } \Omega_0 \times (0,T), \qquad (2.47)
$$

$$
\operatorname{div}(B_{a,b}^T \bar{u}) = \bar{g}, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.48)
$$
Proof. Let $k \in H_{\#}^s(\Omega_0)$ with $0 < s < 1$ and consider a function $\varphi \in H_{\#}^1(\Omega_0)$, we have

$$
\begin{aligned}
\|k\varphi\|_{L_{\#}^2(\Omega_0)} &\le \|k\|_{L_{\#}^{\frac{2}{1-s}}(\Omega_0)} \|\varphi\|_{L_{\#}^s(\Omega_0)}^{\frac{2}{1-s}} \\
&\le C \|k\|_{H_{\#}^s(\Omega_0)} \|\varphi\|_{H_{\#}^{1-s}(\Omega_0)} \\
&\le C \|k\|_{H_{\#}^s(\Omega_0)} \|\varphi\|_{L_{\#}^2(\Omega_0)}^s \|\varphi\|_{H_{\#}^1(\Omega_0)}^{1-s}
\end{aligned}
$$

where $C > 0$ depends on $\Omega_0$.

By Young's inequality, we get

$$
\|k\varphi\|_{L_{\#}^2(\Omega_0)} \le C \|k\|_{H_{\#}^s(\Omega_0)} (C_{\delta}\|\varphi\|_{L_{\#}^2(\Omega_0)} + \delta\|\nabla\varphi\|_{L_{\#}^2(\Omega_0)}). \quad (2.42)
$$

We note that $C_{\delta}$ is a decreasing function of $\delta$.

We now state an elliptic result of the Stokes-like problem (2.40)-(2.39).

**Proposition 2.3** Let $s > 0$ and suppose that $(a,b) \in H_{\#}^{3+s}(0,L)$ such that $\chi_{a,b}$ is injective on $\bar{\Omega}_0$. Then for any source terms and boundary condition

$$
(\hat{f}, \hat{g}) \in H_{\#}^{-1}(\Omega_0) \times L_{\#}^2(\Omega_0), \quad \psi \in H_{\#}^{\frac{1}{2}}(0, L),
$$

such that

$$
\int_{\Omega_0} \hat{g} \, dxdy = \int_0^L \left( -\psi_1(x) \partial_x b(x) + \psi_2(x) (1 + \partial_x a(x)) \right) dx, \quad (2.43)
$$

there exists a unique solution $(\hat{u}, \hat{p}) \in H_{\#}^1(\Omega_0) \times L_{\#}^2(\Omega_0)$ to the Stokes-like problem (2.40)-(2.39) such that, if we have $\|(a,b)\|_{H_{\#}^{3+s}(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R$, then

$$
\|\hat{u}\|_{H_{\#}^1(\Omega_0)} + \|\hat{p}\|_{L_{\#}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{H_{\#}^{-1}(\Omega_0)} + \|\hat{g}\|_{L_{\#}^2(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{1}{2}}(0,L)} \right), \quad (2.44)
$$

where $K: [0, \infty) \mapsto (0, \infty)$ is a non decreasing function.

Moreover, if

$$
(\hat{f}, \hat{g}) \in L_{\#}^2(\Omega_0) \times H_{\#}^1(\Omega_0), \quad \psi \in H_{\#}^{\frac{3}{2}}(0, L),
$$

then $(\hat{u}, \hat{p}) \in H_{\#}^2(\Omega_0) \times H_{\#}^1(\Omega_0)$ and satisfies

$$
\|\hat{u}\|_{H_{\#}^2(\Omega_0)} + \|\hat{p}\|_{H_{\#}^1(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right). \quad (2.45)
$$

**Proof.** We split the proof into two steps.

**Step 1.** Let $\hat{f} \in H_{\#}^{-1}(\Omega_0)$, $\hat{g} \in L_{\#}^2(\Omega_0)$ and $\psi \in H_{\#}^{\frac{1}{2}}(0, L)$. Then there exists $u_{\psi} \in H_{\#}^1(\Omega_0)$ such that

$$
u_{\psi|_{y=1}} = \psi_1 e_1 + \psi_2 e_2 \quad \text{and} \quad u_{\psi|_{y=0}} = 0,
$$

and satisfying

$$
\|u_{\psi}\|_{H_{\#}^{1}(\Omega_{0})} \leq C \|\psi\|_{H_{\#}^{\frac{1}{2}}(0,L)}, \quad (2.46)
$$

where $C$ is a positive constant depending on the domain $\Omega_0$.

Let $\bar{u} = \hat{u} - u_{\psi}$. Then $(\bar{u}, \hat{p})$ satisfies

$$
-\mu \operatorname{div}((A_{a,b} \nabla)\bar{u}) + (B_{a,b} \nabla)\hat{p} = \bar{f}, \quad \text{in } \Omega_0 \times (0, T), \quad (2.47)
$$

$$
\operatorname{div}(B_{a,b}^T \bar{u}) = \bar{g}, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.48)
$$
with

$$ \bar{f} := \hat{f} + \mu \operatorname{div}((A_{a,b}\nabla)u_{\psi}), \quad \bar{g} := \hat{g} - \operatorname{div}(B_{a,b}^{T}u_{\psi}), \qquad (2.49) $$

completed with the following boundary conditions

$$ \bar{u}_{|_{y=1}} = \bar{u}_{|_{y=0}} = 0. \qquad (2.50) $$

Since $\hat{f} \in H_{\#}^{-1}(\Omega_0)$, $A_{a,b} \in H_{\#}^{1+s}(\Omega_0)$, $B_{a,b} \in H_{\#}^{1+s}(\Omega_0)$, and $u_{\psi} \in H_{\#}^{1}(\Omega_0)$ and thanks to the Sobolev injection of $H_{\#}^{1+s}(\Omega_0)$ into $L_{\#}^{\infty}(\Omega_0)$ we get that $\bar{f} \in H_{\#}^{-1}(\Omega_0)$. Similarly since $\hat{g} \in L_{\#}^{2}(\Omega_0)$ and thanks to the fact that $H_{\#}^{1+s}(\Omega_0)$ is a multiplier space of $H_{\#}^{1}(\Omega_0)$, we have $\bar{g} \in L_{\#}^{2}(\Omega_0)$.

Moreover, equation (2.43) implies

$$ \int_{\Omega_0} \bar{g} \, dxdy = 0. $$

Hence, there exists unique $v \in H_{\#}^{1}(\Omega_0)$ such that $\operatorname{div} v = \bar{g}$, in $\Omega_0$, $v_{|_{y=1}} = 0$, $v_{|_{y=0}} = 0$, and satisfying

$$ \|v\|_{H_{\#}^{1}(\Omega_0)} \leq C \|\bar{g}\|_{L_{\#}^{2}(\Omega_0)}, $$

where $C > 0$ depends on $\Omega_0$.

Let us take $w = B_{a,b}^{-T} v$, then $w \in H_{\#}^{1}(\Omega_0)$ and satisfies $\operatorname{div}(B_{a,b}^{T}w) = \bar{g}$, on $\Omega_0$, $w_{|_{y=1}} = 0$, $w_{|_{y=0}} = 0$. Moreover, as $B_{a,b}$ is invertible in $H_{\#}^{1+s}(\Omega_0)$, we obtain

$$ \|w\|_{H_{\#}^{1}(\Omega_0)} \leq K(R) \|\bar{g}\|_{L_{\#}^{2}(\Omega_0)}, $$

where $K$ is a non decreasing positive function of $R$.

Set $U = \bar{u} - w$. Then $U$ satisfies

$$ -\mu \operatorname{div}((A_{a,b}\nabla)U) + (B_{a,b}\nabla)\hat{p} = F, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.51) $$

$$ \operatorname{div}(B_{a,b}^{T}U) = 0, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.52) $$

with

$$ F = \bar{f} + \mu \operatorname{div}((A_{a,b}\nabla)w), $$

completed with the following boundary conditions

$$ U_{|_{y=1}} = 0, \qquad (2.53) $$

$$ U_{|_{y=0}} = 0. \qquad (2.54) $$

We underline that due to (2.52), equation (2.51) is equivalent to

$$ - \mu \operatorname{div}((A_{a,b}\nabla)U) + (B_{a,b}\nabla)\hat{p} = F, \quad \text{in } \Omega_0 \times (0, T). \qquad (2.55) $$

Next, by assumption, we can apply Proposition 2.2 to $A_{a,b}$ which is consequently uniformly coercive. Moreover, since $B_{a,b}$ is invertible in $H_{\#}^{1+s}(\Omega_0)$, we find, by standard arguments (see for instance [23,25,27] for similar arguments in the context of transversal displacement only) a weak solution $(U, \hat{p}) \in H_{\#}^{1}(\Omega_0) \times L_{\#}^{2}(\Omega_0)$ to Problem (2.51)-(2.54) depending continuously on $F$ and satisfying

$$ \|U\|_{H_{\#}^{1}(\Omega_0)} + \|\hat{p}\|_{L_{\#}^{2}(\Omega_0)} \leq K(R) \|F\|_{H_{\#}^{-1}(\Omega_0)}. $$
with

$$ \bar{f} := \hat{f} + \mu \operatorname{div}((A_{a,b}\nabla)u_{\psi}), \quad \bar{g} := \hat{g} - \operatorname{div}(B_{a,b}^{T}u_{\psi}), \qquad (2.49) $$

completed with the following boundary conditions

$$ \bar{u}_{|y=1} = \bar{u}_{|y=0} = 0. \qquad (2.50) $$

Since $\hat{f} \in H_{\#}^{-1}(\Omega_0)$, $A_{a,b} \in H_{\#}^{1+s}(\Omega_0)$, $B_{a,b} \in H_{\#}^{1+s}(\Omega_0)$, and $u_{\psi} \in H_{\#}^{1}(\Omega_0)$ and thanks to the Sobolev injection of $H_{\#}^{1+s}(\Omega_0)$ into $L_{\#}^{\infty}(\Omega_0)$ we get that $\bar{f} \in H_{\#}^{-1}(\Omega_0)$. Similarly since $\hat{g} \in L_{\#}^{2}(\Omega_0)$ and thanks to the fact that $H_{\#}^{1+s}(\Omega_0)$ is a multiplier space of $H_{\#}^{1}(\Omega_0)$, we have $\bar{g} \in L_{\#}^{2}(\Omega_0)$.

Moreover, equation (2.43) implies

$$ \int_{\Omega_0} \bar{g} \, dxdy = 0. $$

Hence, there exists unique $v \in H_{\#}^{1}(\Omega_0)$ such that $\operatorname{div} v = \bar{g}$, in $\Omega_0$, $v|_{y=1} = 0$, $v|_{y=0} = 0$, and satisfying

$$ \|v\|_{H_{\#}^{1}(\Omega_0)} \leq C \|\bar{g}\|_{L_{\#}^{2}(\Omega_0)}, $$

where $C > 0$ depends on $\Omega_0$.

Let us take $w = B_{a,b}^{-T} v$, then $w \in H_{\#}^{1}(\Omega_0)$ and satisfies $\operatorname{div}(B_{a,b}^{T}w) = \bar{g}$, on $\Omega_0$, $w|_{y=1} = 0$, $w|_{y=0} = 0$. Moreover, as $B_{a,b}$ is invertible in $H_{\#}^{1+s}(\Omega_0)$, we obtain

$$ \|w\|_{H_{\#}^{1}(\Omega_0)} \leq K(R) \|\bar{g}\|_{L_{\#}^{2}(\Omega_0)}, $$

where $K$ is a non decreasing positive function of $R$.

Set $U = \bar{u} - w$. Then $U$ satisfies

$$ -\mu \operatorname{div}((A_{a,b}\nabla)U) + (B_{a,b}\nabla)\hat{p} = F, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.51) $$

$$ \operatorname{div}(B_{a,b}^{T}U) = 0, \quad \text{in } \Omega_0 \times (0, T), \qquad (2.52) $$

with

$$ F = \bar{f} + \mu \operatorname{div}((A_{a,b}\nabla)w), $$

completed with the following boundary conditions

$$ U|_{y=1} = 0, \qquad (2.53) $$

$$ U|_{y=0} = 0. \qquad (2.54) $$

We underline that due to (2.52), equation (2.51) is equivalent to

$$ -\mu \operatorname{div}((A_{a,b}\nabla)U) + (B_{a,b}\nabla)\hat{p} = F, \quad \text{in } \Omega_0 \times (0, T). \qquad (2.55) $$

Next, by assumption, we can apply Proposition 2.2 to $A_{a,b}$ which is consequently uniformly coercive. Moreover, since $B_{a,b}$ is invertible in $H_{\#}^{1+s}(\Omega_0)$, we find, by standard arguments (see for instance [23,25,27] for similar arguments in the context of transversal displacement only) a weak solution $(U, \hat{p}) \in H_{\#}^{1}(\Omega_0) \times L_{\#}^{2}(\Omega_0)$ to Problem (2.51)-(2.54) depending continuously on $F$ and satisfying

$$ \|U\|_{H_{\#}^{1}(\Omega_0)} + \|\hat{p}\|_{L_{\#}^{2}(\Omega_0)} \leq K(R) \|F\|_{H_{\#}^{-1}(\Omega_0)}. $$
It follows that

$$ \|\bar{u}\|_{H_{\#}^1(\Omega_0)} + \|\hat{p}\|_{L_{\#0}^2(\Omega_0)} \le K(R) \left( \|\bar{f}\|_{H_{\#}^{-1}(\Omega_0)} + \|\bar{g}\|_{L_{\#0}^2(\Omega_0)} \right). \quad (2.56) $$

By combining the above inequality with (2.46), we get

$$ \|\hat{u}\|_{H_{\#}^1(\Omega_0)} + \|\hat{p}\|_{L_{\#0}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{H_{\#}^{-1}(\Omega_0)} + \|\hat{g}\|_{L_{\#}^2(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{1}{2}}(0,L)} \right). \quad (2.57) $$

**Step 2.** Let $\hat{f} \in L_{\#}^2(\Omega_0)$, $\hat{g} \in H_{\#}^1(\Omega_0)$ and $\psi \in H_{\#}^{\frac{3}{2}}(0, L)$. At this step we prove that the solution $(\hat{u}, \hat{p})$ of System (2.40)-(2.39) belongs to $H_{\#}^2(\Omega_0) \times (H_{\#}^1(\Omega_0) \cap L_{\#0,0}^2(\Omega_0))$ and satisfies the estimate (2.45). We start by lifting the boundary conditions but this time the lifting vector $u_{\psi} \in H_{\#}^2(\Omega_0)$ as $\psi \in H_{\#}^{\frac{3}{2}}(0, L)$ and satisfies

$$ \|u_{\psi}\|_{H_{\#}^2(\Omega_0)} \le C \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)}, \quad (2.58) $$

where $C$ depends only on $\Omega_0$. We already proved that there exists $(\bar{u}, \hat{p}) \in H_{\#}^1(\Omega_0) \times L_{\#0,0}^2(\Omega_0)$, where $\bar{u} = \hat{u} - u_{\psi}$ unique solution of (2.47)-(2.50) and satisfying (2.56). Hence, to prove the estimate (2.45), it remains to bound the second order derivatives of $\bar{u}$ and first order derivatives of $\hat{p}$ in $L_{\#}^2(\Omega_0)$.

Due to the regularity of $(a, b)$, $A_{a,b} \in H_{\#}^{1+s}(\Omega_0)$, $B_{a,b} \in H_{\#}^{1+s}(\Omega_0)$ which is a multiplier space of $H_{\#}^1(\Omega_0)$. Remembering Piola identity (see [15]) that states $\text{div}(B_{a,b}) = 0$, it implies that $\bar{g} = \hat{g} - B_{a,b} : \nabla u_{\psi}$ and thus the new sources terms $(\bar{f}, \bar{g})$ belong to $L_{\#}^2(\Omega_0) \times H_{\#}^1(\Omega_0)$. We now differentiate equations (2.47)-(2.48) satisfied by $(\bar{u}, \hat{p})$ with respect to $x$. Formally, we obtain

$$ -\mu \operatorname{div}((A_{a,b}\nabla)\partial_x \bar{u}) + (B_{a,b}\nabla)\partial_x \hat{p} = \bar{f}, \quad \text{in } \Omega_0 \times (0, T), \quad (2.59) $$

$$ \operatorname{div}(B_{a,b}^T \partial_x \bar{u}) = \partial_x \bar{g} - \partial_x B_{a,b}^T : \nabla \bar{u}, \quad \text{in } \Omega_0 \times (0, T), \quad (2.60) $$

with

$$ \bar{f} := \mu \operatorname{div}((\partial_x A_{a,b} \nabla) \bar{u}) - (\partial_x B_{a,b} \nabla) \hat{p}) + \partial_x \bar{f}. \quad (2.61) $$

We note that $\partial_x \bar{u}$ is $L$-periodic with respect to $x$ and it vanishes on $y = 1$ and $y = 0$. Consequently, thanks to boundary conditions and Piola identity, since $\partial_x B_{a,b} = \text{cot}(\partial_x \nabla \chi_{a,b})$, the right hand side of the divergence constraint satisfies

$$ \int_{\Omega_0} (\partial_x \bar{g} - \partial_x B_{a,b}^T : \nabla \bar{u}) \, dxdy = \int_{\Omega_0} \operatorname{div}(\partial_x B_{a,b}^T \nabla \bar{u}) \, dxdy = 0. $$

Therefore Step 1 implies

$$ \|\partial_x \bar{u}\|_{H_{\#}^1(\Omega_0)} + \|\partial_x \hat{p}\|_{L_{\#0}^2(\Omega_0)} \le K(R) \left( \|\bar{f}\|_{H_{\#}^{-1}(\Omega_0)} + \|\partial_x \bar{g} - \partial_x B_{a,b}^T : \nabla \bar{u}\|_{L_{\#}^2(\Omega_0)} \right). \quad (2.62) $$

Hence to bound the left hand side of the above equation, we have to bound $\tilde{f}$ in $H_{\#}^{-1}(\Omega_0)$ and the term $\partial_x \bar{g} - \partial_x B_{a,b}^T : \nabla \bar{u}$ in $L_{\#}^2(\Omega_0)$. By performing integration by parts, for any $w \in H_{\#}^1(\Omega_0)$ such that $w|_{y=0} = w|_{y=1} = 0$, we have

$$ \int_{\Omega_0} \operatorname{div}((\partial_x A_{a,b} \nabla) \bar{u}) \cdot w \, dxdy = - \int_{\Omega_0} ((\partial_x A_{a,b} \nabla) \bar{u}) : \nabla w \, dxdy, $$

and hence

$$ \|\operatorname{div}((\partial_x A_{a,b} \nabla) \bar{u})\|_{H_{\#}^{-1}(\Omega_0)} \le \|(\partial_x A_{a,b} \nabla) \bar{u}\|_{L_{\#}^2(\Omega_0)}. \quad (2.63) $$
It follows that

$$ \|\bar{u}\|_{H_{\#}^1(\Omega_0)} + \|\hat{p}\|_{L_{\#0,0}^2(\Omega_0)} \le K(R) \left( \|\bar{f}\|_{H_{\#}^{-1}(\Omega_0)} + \|\bar{g}\|_{L_{\#0,0}^2(\Omega_0)} \right). \quad (2.56) $$

By combining the above inequality with (2.46), we get

$$ \|\hat{u}\|_{H_{\#}^1(\Omega_0)} + \|\hat{p}\|_{L_{\#0,0}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{H_{\#}^{-1}(\Omega_0)} + \|\hat{g}\|_{L_{\#}^2(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{1}{2}}(0,L)} \right). \quad (2.57) $$

**Step 2.** Let $\hat{f} \in L_{\#}^2(\Omega_0)$, $\hat{g} \in H_{\#}^1(\Omega_0)$ and $\psi \in H_{\#}^{\frac{3}{2}}(0, L)$. At this step we prove that the solution $(\hat{u}, \hat{p})$ of System (2.40)-(2.39) belongs to $H_{\#}^2(\Omega_0) \times (H_{\#}^1(\Omega_0) \cap L_{\#0,0}^2(\Omega_0))$ and satisfies the estimate (2.45). We start by lifting the boundary conditions but this time the lifting vector $u_{\psi} \in H_{\#}^2(\Omega_0)$ as $\psi \in H_{\#}^{\frac{3}{2}}(0, L)$ and satisfies

$$ \|u_{\psi}\|_{H_{\#}^2(\Omega_0)} \le C \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)}, \quad (2.58) $$

where $C$ depends only on $\Omega_0$. We already proved that there exists $(\bar{u}, \hat{p}) \in H_{\#}^1(\Omega_0) \times L_{\#0,0}^2(\Omega_0)$, where $\bar{u} = \hat{u} - u_{\psi}$ unique solution of (2.47)-(2.50) and satisfying (2.56). Hence, to prove the estimate (2.45), it remains to bound the second order derivatives of $\bar{u}$ and first order derivatives of $\hat{p}$ in $L_{\#}^2(\Omega_0)$.

Due to the regularity of $(a, b)$, $A_{a,b} \in H_{\#}^{1+s}(\Omega_0)$, $B_{a,b} \in H_{\#}^{1+s}(\Omega_0)$ which is a multiplier space of $H_{\#}^1(\Omega_0)$. Remembering Piola identity (see [15]) that states $\text{div}(B_{a,b}) = 0$, it implies that $\bar{g} = \hat{g} - B_{a,b} : \nabla u_{\psi}$ and thus the new sources terms $(\bar{f}, \bar{g})$ belong to $L_{\#}^2(\Omega_0) \times H_{\#}^1(\Omega_0)$. We now differentiate equations (2.47)-(2.48) satisfied by $(\bar{u}, \hat{p})$ with respect to $x$. Formally, we obtain

$$ -\mu \operatorname{div}((A_{a,b}\nabla)\partial_x \bar{u}) + (B_{a,b}\nabla)\partial_x \hat{p} = \bar{f}, \quad \text{in } \Omega_0 \times (0, T), \quad (2.59) $$

$$ \operatorname{div}(B_{a,b}^T \partial_x \bar{u}) = \partial_x \bar{g} - \partial_x B_{a,b}^T : \nabla \bar{u}, \quad \text{in } \Omega_0 \times (0, T), \quad (2.60) $$

with

$$ \bar{f} := \mu \operatorname{div}((\partial_x A_{a,b} \nabla) \bar{u}) - (\partial_x B_{a,b} \nabla) \hat{p}) + \partial_x \bar{f}. \quad (2.61) $$

We note that $\partial_x \bar{u}$ is $L$-periodic with respect to $x$ and it vanishes on $y = 1$ and $y = 0$. Consequently, thanks to boundary conditions and Piola identity, since $\partial_x B_{a,b} = \operatorname{cot}(\partial_x \nabla \chi_{a,b})$, the right hand side of the divergence constraint satisfies

$$ \int_{\Omega_0} (\partial_x \bar{g} - \partial_x B_{a,b}^T : \nabla \bar{u}) \, dxdy = \int_{\Omega_0} \operatorname{div}(\partial_x B_{a,b}^T \nabla \bar{u}) \, dxdy = 0. $$

Therefore Step 1 implies

$$ \|\partial_x \bar{u}\|_{H_{\#}^1(\Omega_0)} + \|\partial_x \hat{p}\|_{L_{\#0,0}^2(\Omega_0)} \le K(R) \left( \|\bar{f}\|_{H_{\#}^{-1}(\Omega_0)} + \|\partial_x \bar{g} - \partial_x B_{a,b}^T : \nabla \bar{u}\|_{L_{\#}^2(\Omega_0)} \right). \quad (2.62) $$

Hence to bound the left hand side of the above equation, we have to bound $\tilde{f}$ in $H_{\#}^{-1}(\Omega_0)$ and the term $\partial_x \bar{g} - \partial_x B_{a,b}^T : \nabla \bar{u}$ in $L_{\#}^2(\Omega_0)$. By performing integration by parts, for any $w \in H_{\#}^1(\Omega_0)$ such that $w|_{y=0} = w|_{y=1} = 0$, we have

$$ \int_{\Omega_0} \operatorname{div}((\partial_x A_{a,b} \nabla) \bar{u}) \cdot w \, dxdy = - \int_{\Omega_0} ((\partial_x A_{a,b} \nabla) \bar{u}) : \nabla w \, dxdy, $$

and hence

$$ \|\operatorname{div}((\partial_x A_{a,b} \nabla) \bar{u})\|_{H_{\#}^{-1}(\Omega_0)} \le \|(\partial_x A_{a,b} \nabla) \bar{u}\|_{L_{\#}^2(\Omega_0)}. \quad (2.63) $$
By using Lemma 2.1 for $\partial_x A_{a,b} \in H_{\#}^s(\Omega_0)$ and $\nabla \bar{u} \in H_{\#}^1(\Omega_0)$, we get, for any $\delta > 0$,

$$ \| (\partial_x A_{a,b} \nabla) \bar{u} \|_{L_{\#}^2(\Omega_0)} \le K(R) \left( C_{\delta} \| \nabla \bar{u} \|_{L_{\#}^2(\Omega_0)} + \delta \| \nabla^2 \bar{u} \|_{L_{\#}^2(\Omega_0)} \right). \quad (2.64) $$

Combining the above inequality with (2.63) and (2.57), we obtain, for any $\delta > 0$,

$$ \| \mathrm{div}((\partial_x A_{a,b} \nabla) \bar{u}) \|_{H_{\#}^{-1}(\Omega_0)} \le K(R) \left( C_{\delta} \left( \| \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + \| \hat{g} \|_{L_{\#}^2(\Omega_0)} + \| \psi \|_{H_{\#}^{\frac{1}{2}}(0,L)} \right) + \delta \| \nabla^2 \bar{u} \|_{L_{\#}^2(\Omega_0)} \right). \quad (2.65) $$

By performing again integration by parts and using Piola identity, we get that for any $w \in H_{\#}^1(\Omega_0)$ such that $w|_{y=0} = w|_{y=1} = 0$

$$ \int_{\Omega_0} (\partial_x B_{a,b} \nabla) \hat{p} \cdot w \, dxdy = - \int_{\Omega_0} \partial_x B_{a,b} \hat{p} : \nabla w \, dxdy, $$

and thus

$$ \| (\partial_x B_{a,b} \nabla) \hat{p} \|_{H_{\#}^{-1}(\Omega_0)} \le \| \partial_x B_{a,b} \hat{p} \|_{L_{\#}^2(\Omega_0)}. \quad (2.66) $$

Hence applying again Lemma 2.1 (for $\partial_x B_{a,b} \in H_{\#}^s(\Omega_0)$ and $\hat{p} \in H_{\#}^1(\Omega_0)$) leads to

$$ \| \partial_x B_{a,b} \hat{p} \|_{L_{\#}^2(\Omega_0)} \le K(R) \left( C_{\delta} \| \hat{p} \|_{L_{\#}^2(\Omega_0)} + \delta \| \nabla \hat{p} \|_{L_{\#}^2(\Omega_0)} \right), \quad (2.67) $$

for any $\delta > 0$. Consequently, gathering the above estimate (2.67) with (2.57) and (2.66), we obtain, for any $\delta > 0$

$$ \| (\partial_x B_{a,b} \nabla) \hat{p} \|_{H_{\#}^{-1}(\Omega_0)} \le K(R) \left( C_{\delta} \left( \| \hat{f} \|_{L_{\#}^2(\Omega_0)} + \| \hat{g} \|_{L_{\#}^2(\Omega_0)} + \| \psi \|_{H_{\#}^{\frac{1}{2}}(0,L)} \right) + \delta \| \nabla \hat{p} \|_{L_{\#}^2(\Omega_0)} \right). \quad (2.68) $$

Hence, it remains to bound $\partial_x \bar{f}$ in $H_{\#}^{-1}(\Omega_0)$ and $\partial_x \bar{g}$ in $L_{\#}^2(\Omega_0)$. From the definition of $\bar{f}$ and $\bar{g}$ in (2.49), we have

$$ \| \partial_x \bar{f} \|_{H_{\#}^{-1}(\Omega_0)} \le \| \partial_x \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + \|(\partial_x A_{a,b} \nabla) u_{\psi} \|_{L^2(\Omega_0)} + \|(A_{a,b} \nabla) \partial_x u_{\psi} \|_{L^2(\Omega_0)}, \quad (2.69) $$

and

$$ \| \partial_x \bar{g} \|_{L_{\#}^2(\Omega_0)} \le \| \partial_x \hat{g} \|_{L_{\#}^2(\Omega_0)} + \| \partial_x B_{a,b}^T : \nabla u_{\psi} \|_{L_{\#}^2(\Omega_0)} + \| B_{a,b}^T : \partial_x \nabla u_{\psi} \|_{L_{\#}^2(\Omega_0)}. \quad (2.70) $$

Repeating the same arguments we easily get

$$ \| \partial_x \bar{f} \|_{H_{\#}^{-1}(\Omega_0)} \le \| \partial_x \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + K(R) \| u_{\psi} \|_{H_{\#}^2(\Omega_0)}, $$

$$ \| \partial_x \bar{g} \|_{L_{\#}^2(\Omega_0)} \le \| \partial_x \hat{g} \|_{L_{\#}^2(\Omega_0)} + K(R) \| u_{\psi} \|_{H_{\#}^2(\Omega_0)}. $$

Since $u_{\psi} \in H_{\#}^2(\Omega_0)$ and satisfies (2.58), we finally obtain

$$ \| \partial_x \bar{f} \|_{H_{\#}^{-1}(\Omega_0)} \le \| \partial_x \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + K(R) \| \psi \|_{H_{\#}^{\frac{3}{2}}(0,L)}, \quad (2.71) $$

$$ \| \partial_x \bar{g} \|_{L^2(\Omega_0)} \le \| \partial_x \hat{g} \|_{L^2(\Omega_0)} + K(R) \| \psi \|_{H_{\#}^{\frac{3}{2}}(0,L)}. \quad (2.72) $$

Combining the above estimates (2.71) and (2.72) with (2.65) and (2.68), we deduce

$$ \| \tilde{f} \|_{H_{\#}^{-1}(\Omega_0)} \le \| \partial_x \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + K(R) \| \psi \|_{H_{\#}^{\frac{3}{2}}(0,L)} \\ + K(R) \left( C_{\delta} \left( \| \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + \| \hat{g} \|_{L_{\#}^2(\Omega_0)} + \| \psi \|_{H_{\#}^{\frac{1}{2}}(0,L)} \right) + \delta \| \nabla^2 \bar{u} \|_{L_{\#}^2(\Omega_0)} + \delta \| \nabla \hat{p} \|_{L_{\#}^2(\Omega_0)} \right), \quad (2.73) $$
By using Lemma 2.1 for $\partial_x A_{a,b} \in H_{\#}^s(\Omega_0)$ and $\nabla \bar{u} \in H_{\#}^1(\Omega_0)$, we get, for any $\delta > 0$,

$$ \| (\partial_x A_{a,b} \nabla) \bar{u} \|_{L_{\#}^2(\Omega_0)} \le K(R) \left( C_{\delta} \| \nabla \bar{u} \|_{L_{\#}^2(\Omega_0)} + \delta \| \nabla^2 \bar{u} \|_{L_{\#}^2(\Omega_0)} \right). \quad (2.64) $$

Combining the above inequality with (2.63) and (2.57), we obtain, for any $\delta > 0$,

$$ \| \operatorname{div}((\partial_x A_{a,b} \nabla) \bar{u}) \|_{H_{\#}^{-1}(\Omega_0)} \le K(R) \left( C_{\delta} \left( \| \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + \| \hat{g} \|_{L_{\#}^2(\Omega_0)} + \| \psi \|_{H_{\#}^{\frac{1}{2}}(0,L)} \right) + \delta \| \nabla^2 \bar{u} \|_{L_{\#}^2(\Omega_0)} \right). \quad (2.65) $$

By performing again integration by parts and using Piola identity, we get that for any $w \in H_{\#}^1(\Omega_0)$ such that $w|_{y=0} = w|_{y=1} = 0$

$$ \int_{\Omega_0} (\partial_x B_{a,b} \nabla) \hat{p} \cdot w \, dxdy = - \int_{\Omega_0} \partial_x B_{a,b} \hat{p} : \nabla w \, dxdy, $$

and thus

$$ \| (\partial_x B_{a,b} \nabla) \hat{p} \|_{H_{\#}^{-1}(\Omega_0)} \le \| \partial_x B_{a,b} \hat{p} \|_{L_{\#}^2(\Omega_0)}. \quad (2.66) $$

Hence applying again Lemma 2.1 (for $\partial_x B_{a,b} \in H_{\#}^s(\Omega_0)$ and $\hat{p} \in H_{\#}^1(\Omega_0)$) leads to

$$ \| \partial_x B_{a,b} \hat{p} \|_{L_{\#}^2(\Omega_0)} \le K(R) \left( C_{\delta} \| \hat{p} \|_{L_{\#}^2(\Omega_0)} + \delta \| \nabla \hat{p} \|_{L_{\#}^2(\Omega_0)} \right), \quad (2.67) $$

for any $\delta > 0$. Consequently, gathering the above estimate (2.67) with (2.57) and (2.66), we obtain, for any $\delta > 0$

$$ \| (\partial_x B_{a,b} \nabla) \hat{p} \|_{H_{\#}^{-1}(\Omega_0)} \le K(R) \left( C_{\delta} \left( \| \hat{f} \|_{L_{\#}^2(\Omega_0)} + \| \hat{g} \|_{L_{\#}^2(\Omega_0)} + \| \psi \|_{H_{\#}^{\frac{1}{2}}(0,L)} \right) + \delta \| \nabla \hat{p} \|_{L_{\#}^2(\Omega_0)} \right). \quad (2.68) $$

Hence, it remains to bound $\partial_x \bar{f}$ in $H_{\#}^{-1}(\Omega_0)$ and $\partial_x \bar{g}$ in $L_{\#}^2(\Omega_0)$. From the definition of $\bar{f}$ and $\bar{g}$ in (2.49), we have

$$ \| \partial_x \bar{f} \|_{H_{\#}^{-1}(\Omega_0)} \le \| \partial_x \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + \|(\partial_x A_{a,b} \nabla) u_{\psi} \|_{L^2(\Omega_0)} + \|(A_{a,b} \nabla) \partial_x u_{\psi} \|_{L^2(\Omega_0)}, \quad (2.69) $$

and

$$ \| \partial_x \bar{g} \|_{L_{\#}^2(\Omega_0)} \le \| \partial_x \hat{g} \|_{L_{\#}^2(\Omega_0)} + \| \partial_x B_{a,b}^T : \nabla u_{\psi} \|_{L_{\#}^2(\Omega_0)} + \| B_{a,b}^T : \partial_x \nabla u_{\psi} \|_{L_{\#}^2(\Omega_0)}. \quad (2.70) $$

Repeating the same arguments we easily get

$$ \| \partial_x \bar{f} \|_{H_{\#}^{-1}(\Omega_0)} \le \| \partial_x \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + K(R) \| u_{\psi} \|_{H_{\#}^2(\Omega_0)}, $$

$$ \| \partial_x \bar{g} \|_{L_{\#}^2(\Omega_0)} \le \| \partial_x \hat{g} \|_{L_{\#}^2(\Omega_0)} + K(R) \| u_{\psi} \|_{H_{\#}^2(\Omega_0)}. $$

Since $u_{\psi} \in H_{\#}^2(\Omega_0)$ and satisfies (2.58), we finally obtain

$$ \| \partial_x \bar{f} \|_{H_{\#}^{-1}(\Omega_0)} \le \| \partial_x \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + K(R) \| \psi \|_{H_{\#}^{\frac{3}{2}}(0,L)}, \quad (2.71) $$

$$ \| \partial_x \bar{g} \|_{L^2(\Omega_0)} \le \| \partial_x \hat{g} \|_{L^2(\Omega_0)} + K(R) \| \psi \|_{H_{\#}^{\frac{3}{2}}(0,L)}. \quad (2.72) $$

Combining the above estimates (2.71) and (2.72) with (2.65) and (2.68), we deduce

$$ \| \tilde{f} \|_{H_{\#}^{-1}(\Omega_0)} \le \| \partial_x \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + K(R) \| \psi \|_{H_{\#}^{\frac{3}{2}}(0,L)} \\ + K(R) \left( C_{\delta} \left( \| \hat{f} \|_{H_{\#}^{-1}(\Omega_0)} + \| \hat{g} \|_{L_{\#}^2(\Omega_0)} + \| \psi \|_{H_{\#}^{\frac{1}{2}}(0,L)} \right) + \delta \| \nabla^2 \bar{u} \|_{L_{\#}^2(\Omega_0)} + \delta \| \nabla \hat{p} \|_{L_{\#}^2(\Omega_0)} \right), \quad (2.73) $$
and

$$ \|\partial_x \bar{g} - \partial_x B_{a,b}^T : \nabla \bar{u}\|_{H_{\bar{\Omega}_0}^{-1}(\Omega_0)} \le \|\partial_x \hat{g}\|_{L^2(\Omega_0)} + K(R)\|\psi\|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)} \\ + K(R)\left(C_{\delta}\left(\|\hat{f}\|_{H_{\bar{\Omega}_0}^{-1}(\Omega_0)} + \|\hat{g}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \|\psi\|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)}\right) + \delta\|\nabla^2 \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)}\right). \quad (2.74) $$

Therefore by combining (2.73) and (2.74) with (2.62), we get

$$ \|\partial_x \bar{u}\|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + \|\partial_x \hat{p}\|_{L_{\bar{\Omega}_0,0}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \|\hat{g}\|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + \|\psi\|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)} \right) + K(R) \left( \delta \|\nabla^2 \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \delta \|\nabla \hat{p}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \right). $$

For $\delta$ small enough, we obtain

$$ \|\partial_x \bar{u}\|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + \|\partial_x \hat{p}\|_{L_{\bar{\Omega}_0,0}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \|\hat{g}\|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + \|\psi\|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)} \right) + K(R) \left( \delta \|\partial_{yy} \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \delta \|\partial_y \hat{p}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \right). \quad (2.75) $$

It remains to estimate $\|\partial_{yy} \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)}$ and $\|\partial_y \hat{p}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)}$. To obtain a similar estimate on the full second gradient of $\bar{u}$ and the full gradient of $\hat{p}$, we differentiate the divergence equation (2.48) with respect to $y$. This yields

$$ \sum_{j=1}^{2} (B_{a,b})_{j2} \partial_{yy} \bar{u}_j = \partial_y \bar{g} - \sum_{j=1}^{2} \left( \partial_y \left( \partial_x (B_{a,b})_{j1} + \partial_y (B_{a,b})_{j2} \right) \bar{u}_j - 2 \partial_y (B_{a,b})_{j2} \partial_y \bar{u}_j - \partial_y (B_{a,b})_{j1} \partial_x \bar{u}_j \right. \\ \left. - \partial_x (B_{a,b})_{j1} \partial_y \bar{u}_j + (B_{a,b})_{j1} \partial_{xy} \bar{u}_j \right) \quad (2.76) $$

However, by Piola's identity we have $\mathrm{div}(B_{a,b}) = 0$ and hence we get

$$ \partial_x (B_{a,b})_{j1} + \partial_y (B_{a,b})_{j2} = (\mathrm{div}(B_{a,b}))_j = 0. $$

It then follows

$$ \sum_{j=1}^{2} (B_{a,b})_{j2} \partial_{yy} \bar{u}_j = \partial_y \bar{g} - \sum_{j=1}^{2} \left( 2 \partial_y (B_{a,b})_{j2} \partial_y \bar{u}_j - \partial_y (B_{a,b})_{j1} \partial_x \bar{u}_j - \partial_x (B_{a,b})_{j1} \partial_y \bar{u}_j + (B_{a,b})_{j1} \partial_{xy} \bar{u}_j \right) \quad (2.77) $$

By applying Lemma 2.1 with $\varphi = \partial_x \bar{u}$ or $\varphi = \partial_y \bar{u}$ and by noting that the estimate (2.72) remains true for partial derivatives with respect to $y$, we obtain

$$ \| \sum_{j=1}^{2} (B_{a,b})_{j2} \partial_{yy} \bar{u}_j \|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \le \| \hat{g} \|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + K(R) \left( \| \psi \|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)} + C_{\delta} \| \nabla \bar{u} \|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \delta \| \partial_x \nabla \bar{u} \|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \right. \\ \left. + \delta \| \partial_{yy} \bar{u} \|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \| \partial_{xy} \bar{u} \|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \right). \quad (2.78) $$

Combining the above inequality with (2.56) and (2.75) implies

$$ \| \sum_{j=1}^{2} (B_{a,b})_{j2} \partial_{yy} \bar{u}_j \|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \|\hat{g}\|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + \|\psi\|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)} \right) + K(R) \left( \delta \|\partial_{yy} \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \delta \|\partial_y \hat{p}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \right). \quad (2.79) $$

We note that j-th row (2.47) reads as follows

$$ - \mu(A_{a,b})_{22} \partial_{yy} \bar{u}_j + (B_{a,b})_{j2} \partial_y \hat{p} = \bar{f}_j + \mu \partial_x ((A_{a,b} \nabla) \bar{u})_{j1} + \mu ((\partial_y A_{a,b} \nabla) \bar{u})_{j2} \\ + \mu(A_{a,b})_{12} \partial_{xy} \bar{u}_j - (B_{a,b})_{j1} \partial_x \hat{p}. \quad (2.80) $$
and

$$ \|\partial_x \bar{g} - \partial_x B_{a,b}^T : \nabla \bar{u}\|_{H_{\bar{\Omega}_0}^{-1}(\Omega_0)} \le \|\partial_x \hat{g}\|_{L^2(\Omega_0)} + K(R)\|\psi\|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)} \\ + K(R)\left(C_{\delta}\left(\|\hat{f}\|_{H_{\bar{\Omega}_0}^{-1}(\Omega_0)} + \|\hat{g}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \|\psi\|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)}\right) + \delta\|\nabla^2 \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)}\right). \quad (2.74) $$

Therefore by combining (2.73) and (2.74) with (2.62), we get

$$ \|\partial_x \bar{u}\|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + \|\partial_x \hat{p}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \|\hat{g}\|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + \|\psi\|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)} \right) + K(R) \left( \delta \|\nabla^2 \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \delta \|\nabla \hat{p}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \right). $$

For $\delta$ small enough, we obtain

$$ \|\partial_x \bar{u}\|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + \|\partial_x \hat{p}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \|\hat{g}\|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + \|\psi\|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)} \right) + K(R) \left( \delta \|\partial_{yy} \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \delta \|\partial_y \hat{p}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \right). \quad (2.75) $$

It remains to estimate $\|\partial_{yy} \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)}$ and $\|\partial_y \hat{p}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)}$. To obtain a similar estimate on the full second gradient of $\bar{u}$ and the full gradient of $\hat{p}$, we differentiate the divergence equation (2.48) with respect to $y$. This yields

$$ \sum_{j=1}^{2} (B_{a,b})_{j2} \partial_{yy} \bar{u}_j = \partial_y \bar{g} - \sum_{j=1}^{2} \left( \partial_y \left( \partial_x (B_{a,b})_{j1} + \partial_y (B_{a,b})_{j2} \right) \bar{u}_j - 2 \partial_y (B_{a,b})_{j2} \partial_y \bar{u}_j - \partial_y (B_{a,b})_{j1} \partial_x \bar{u}_j \right. \\ \left. - \partial_x (B_{a,b})_{j1} \partial_y \bar{u}_j + (B_{a,b})_{j1} \partial_{xy} \bar{u}_j \right) \quad (2.76) $$

However, by Piola's identity we have $\mathrm{div}(B_{a,b}) = 0$ and hence we get

$$ \partial_x (B_{a,b})_{j1} + \partial_y (B_{a,b})_{j2} = (\mathrm{div}(B_{a,b}))_j = 0. $$

It then follows

$$ \sum_{j=1}^{2} (B_{a,b})_{j2} \partial_{yy} \bar{u}_j = \partial_y \bar{g} - \sum_{j=1}^{2} \left( 2 \partial_y (B_{a,b})_{j2} \partial_y \bar{u}_j - \partial_y (B_{a,b})_{j1} \partial_x \bar{u}_j - \partial_x (B_{a,b})_{j1} \partial_y \bar{u}_j + (B_{a,b})_{j1} \partial_{xy} \bar{u}_j \right) \quad (2.77) $$

By applying Lemma 2.1 with $\varphi = \partial_x \bar{u}$ or $\varphi = \partial_y \bar{u}$ and by noting that the estimate (2.72) remains true for partial derivatives with respect to $y$, we obtain

$$ \| \sum_{j=1}^{2} (B_{a,b})_{j2} \partial_{yy} \bar{u}_j \|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \le \|\hat{g}\|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + K(R) \left( \|\psi\|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)} + C_{\delta} \|\nabla \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \delta \|\partial_x \nabla \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \right. \\ \left. + \delta \|\partial_{yy} \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \|\partial_{xy} \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \right). \quad (2.78) $$

Combining the above inequality with (2.56) and (2.75) implies

$$ \| \sum_{j=1}^{2} (B_{a,b})_{j2} \partial_{yy} \bar{u}_j \|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \|\hat{g}\|_{H_{\bar{\Omega}_0}^1(\Omega_0)} + \|\psi\|_{H_{\bar{\Omega}_0}^{\frac{3}{2}}(0,L)} \right) + K(R) \left( \delta \|\partial_{yy} \bar{u}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} + \delta \|\partial_y \hat{p}\|_{L_{\bar{\Omega}_0}^2(\Omega_0)} \right). \quad (2.79) $$

We note that j-th row (2.47) reads as follows

$$ - \mu(A_{a,b})_{22} \partial_{yy} \bar{u}_j + (B_{a,b})_{j2} \partial_y \hat{p} = \bar{f}_j + \mu \partial_x ((A_{a,b} \nabla) \bar{u})_{j1} + \mu ((\partial_y A_{a,b} \nabla) \bar{u})_{j2} \\ + \mu(A_{a,b})_{12} \partial_{xy} \bar{u}_j - (B_{a,b})_{j1} \partial_x \hat{p}. \quad (2.80) $$
Next, we multiply the above equation by $(B_{a,b})_{j2} = (B_{a,b})_{2j}^T$, then we sum the resulting equations over $j = 1$ and $j = 2$. Noting that

$$ \sum_{j=1}^{2} (B_{a,b})_{2j}^{T} (B_{a,b})_{j2} = (\det \nabla \chi_{a,b}) (A_{a,b})_{22}, $$

we get

$$ (\det \nabla \chi_{a,b}) (A_{a,b})_{22} \partial_y \hat{p} = \sum_{j=1}^{2} (B_{a,b})_{j2} f_j^* + 2\mu (A_{a,b})_{22} \sum_{j=1}^{2} (B_{a,b})_{j1} \partial_{yy} \bar{u}_j, $$

where $f_j^*$ is the right hand side of (2.80).

By performing similar arguments as before to bound $f_j^*$ in $L_{\#}^2(\Omega_0)$, we obtain

$$ \|\partial_y \hat{p}\|_{L_{\#}^2(\Omega_0)} \le K(R) \left( \|\bar{f}\|_{L_{\#}^2(\Omega_0)} + C_{\delta} \|\nabla \bar{u}\|_{L_{\#}^2(\Omega_0)} + \delta \|\nabla^2 \bar{u}\|_{L_{\#}^2(\Omega_0)} + \|\nabla \partial_x \bar{u}\|_{L_{\#}^2(\Omega_0)} \right. \\ \left. + \|\partial_x \hat{p}\|_{L_{\#}^2(\Omega_0)} + \sum_{j=1}^{2} (B_{a,b})_{j1} \partial_{yy} \bar{u}_j \|_{L_{\#}^2(\Omega_0)} \right). $$

From (2.46), (2.57), (2.75) and (2.79), it follows that

$$ \|\partial_y \hat{p}\|_{L_{\#}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right) + K(R) \left( \delta \|\partial_{yy} \bar{u}\|_{L_{\#}^2(\Omega_0)} + \delta \|\partial_y \hat{p}\|_{L_{\#}^2(\Omega_0)} \right). $$

Hence, for $\delta$ small enough, we obtain

$$ \|\partial_y \hat{p}\|_{L_{\#}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right) + \delta K(R) \|\partial_{yy} \bar{u}\|_{L_{\#}^2(\Omega_0)}. \quad (2.81) $$

Going back to (2.80) and using (2.75), we get

$$ \|\partial_{yy} \bar{u}\|_{L_{\#}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right) + \delta K(R) \|\partial_{yy} \bar{u}\|_{L_{\#}^2(\Omega_0)}. $$

Hence, for $\delta$ small enough, we have

$$ \|\partial_{yy} \bar{u}\|_{L_{\#}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right). \quad (2.82) $$

and thus in view with (2.81), we get

$$ \|\partial_y \hat{p}\|_{L_{\#}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right). \quad (2.83) $$

Combining (2.83) and (2.82) with (2.75), we obtain

$$ \|\nabla \bar{u}\|_{H_{\#}^1(\Omega_0)} + \|\nabla \hat{p}\|_{L_{\#}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right). \quad (2.84) $$

By gathering (2.57) with (2.84), we finally deduce

$$ \|\hat{u}\|_{H_{\#}^2(\Omega_0)} + \|\hat{p}\|_{H_{\#}^1(\Omega_0) \cap L_{\#}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right). $$

This ends the proof of the elliptic regularity result. Note that the same could be done in the case where the whole boundary of the fluid domain is elastic by localisation strategy and quotient method. The result relies mainly on the fact that the matrices $A_{a,b}$, $B_{a,b}$ belong to multiplier spaces of $H^1$, on the Piola identity and on an interpolation result similar to Lemma 2.1 (see [10, 23, 24] for a perturbative method and [9, 25, 27, 39] for a similar method as the one used here).
Next, we multiply the above equation by $(B_{a,b})_{j2} = (B_{a,b})_{2j}^T$, then we sum the resulting equations over $j=1$ and $j=2$. Noting that

$$\sum_{j=1}^{2} (B_{a,b})_{2j}^{T} (B_{a,b})_{j2} = (\det \nabla \chi_{a,b}) (A_{a,b})_{22},$$

we get

$$(\det \nabla \chi_{a,b}) (A_{a,b})_{22} \partial_y \hat{p} = \sum_{j=1}^{2} (B_{a,b})_{j2} f_j^* + 2\mu (A_{a,b})_{22} \sum_{j=1}^{2} (B_{a,b})_{j1} \partial_{yy} \bar{u}_j,$$

where $f_j^*$ is the right hand side of (2.80).

By performing similar arguments as before to bound $f_j^*$ in $L_{\#}^2(\Omega_0)$, we obtain

$$\|\partial_y \hat{p}\|_{L_{\#}^2(\Omega_0)} \le K(R) \left( \|\bar{f}\|_{L_{\#}^2(\Omega_0)} + C_{\delta} \|\nabla \bar{u}\|_{L_{\#}^2(\Omega_0)} + \delta \|\nabla^2 \bar{u}\|_{L_{\#}^2(\Omega_0)} + \|\nabla \partial_x \bar{u}\|_{L_{\#}^2(\Omega_0)} \right. \\ \left. + \|\partial_x \hat{p}\|_{L_{\#}^2(\Omega_0)} + \sum_{j=1}^{2} (B_{a,b})_{j1} \partial_{yy} \bar{u}_j \|_{L^2(\Omega_0)} \right).$$

From (2.46), (2.57), (2.75) and (2.79), it follows that

$$\|\partial_y \hat{p}\|_{L_{\#}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right) + K(R) \left( \delta \|\partial_{yy} \bar{u}\|_{L_{\#}^2(\Omega_0)} + \delta \|\partial_y \hat{p}\|_{L_{\#}^2(\Omega_0)} \right).$$

Hence, for $\delta$ small enough, we obtain

$$\|\partial_y \hat{p}\|_{L_{\#}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right) + \delta K(R) \|\partial_{yy} \bar{u}\|_{L_{\#}^2(\Omega_0)}. \quad (2.81)$$

Going back to (2.80) and using (2.75), we get

$$\|\partial_{yy} \bar{u}\|_{L_{\#}^2(\Omega_0)} \le K_{\delta}(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right) + \delta K(R) \|\partial_{yy} \bar{u}\|_{L_{\#}^2(\Omega_0)}.$$

Hence, for $\delta$ small enough, we have

$$\|\partial_{yy} \bar{u}\|_{L_{\#}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right). \quad (2.82)$$

and thus in view with (2.81), we get

$$\|\partial_y \hat{p}\|_{L_{\#}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right). \quad (2.83)$$

Combining (2.83) and (2.82) with (2.75), we obtain

$$\|\nabla \bar{u}\|_{H_{\#}^1(\Omega_0)} + \|\nabla \hat{p}\|_{L_{\#}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right). \quad (2.84)$$

By gathering (2.57) with (2.84), we finally deduce

$$\|\hat{u}\|_{H_{\#}^2(\Omega_0)} + \|\hat{p}\|_{H_{\#}^1(\Omega_0) \cap L_{\#}^2(\Omega_0)} \le K(R) \left( \|\hat{f}\|_{L_{\#}^2(\Omega_0)} + \|\hat{g}\|_{H_{\#}^1(\Omega_0)} + \|\psi\|_{H_{\#}^{\frac{3}{2}}(0,L)} \right).$$

This ends the proof of the elliptic regularity result. Note that the same could be done in the case where the whole boundary of the fluid domain is elastic by localisation strategy and quotient method. The result relies mainly on the fact that the matrices $A_{a,b}$, $B_{a,b}$ belong to multiplier spaces of $H^1$, on the Piola identity and on an interpolation result similar to Lemma 2.1 (see [10, 23, 24] for a perturbative method and [9, 25, 27, 39] for a similar method as the one used here).
### 3 Study of a linear system

Throughout this section, we fix a couple $(a, b) \in H_{\#}^2(0, L)$ such that $\chi_{a,b}$ is injective on $\bar{\mathcal{F}}_0$. We recall that this implies that $\det \nabla \chi_{a,b} \in H_{\#}^3(\Omega_0)$ and hence $\det \nabla \chi_{a,b} \in C_{\#}(\bar{\Omega}_0)$. We moreover assume that $\|(a, b)\|_{H_{\#}^2(0, L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R$. We consider the following linear problem:

$$ 
\left\{
\begin{array}{ll}
\rho_{f,a,b} \partial_t v - \mu \operatorname{div} \left( (A_{a,b} \nabla) v + \frac{1}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v^T B_{a,b} - q B_{a,b} \right) & \\
\qquad = f + \operatorname{div} h, & \text{in } \Omega_0 \times (0, T), \\
\operatorname{div} (B_{a,b}^T v) = g, & \text{in } \Omega_0 \times (0, T), \\
v(x, 1, t) = (\partial_t \zeta(x, t), \partial_t \eta(x, t)), & \text{on } (0, L) \times (0, T), \\
v(x, 0, t) = 0, & \text{on } (0, L) \times (0, T); \\
\rho_s \partial_{tt} \zeta - \beta \partial_{xx} \zeta = -T_{a,b}[v, q] \cdot e_1 - h_{|_{y=1}} e_2 \cdot e_1 + \ell, & \text{in } (0, L), \\
\rho_s \partial_{tt} \eta - \gamma \partial_{xxt} \eta + \alpha \partial_{xxxx} \eta = -T_{a,b}[v, q] \cdot e_2 - h_{|_{y=1}} e_2 \cdot e_2 + m, & \text{in } (0, L),
\end{array}
\right.
\quad (3.1) $$

where

$$ T_{a,b}[v, q] = (\mu(A_{a,b} \nabla) v \, e_2 + \frac{\mu}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v^T B_{a,b} e_2 - q B_{a,b} e_2)_{|_{y=1}}. $$

$L$-periodic boundary conditions w.r.t. $x$ for the fluid velocity and the structure displacement are considered together with the following initial conditions at $t = 0$:

$$ v(\cdot, 0) = v_0(\cdot), \quad \text{in } \Omega_0, \qquad (3.2) $$

$$ \zeta(\cdot, 0) = \zeta_0(\cdot), \quad \text{on } (0, L), \qquad (3.3) $$

$$ \partial_t \zeta(\cdot, 0) = \zeta_1(\cdot), \quad \text{on } (0, L), \qquad (3.4) $$

$$ \eta(\cdot, 0) = \eta_0(\cdot), \quad \text{on } (0, L), \qquad (3.5) $$

$$ \partial_t \eta(\cdot, 0) = \eta_1(\cdot), \quad \text{on } (0, L), \qquad (3.6) $$

satisfying the following compatibility conditions

$$ 
\left\{
\begin{array}{ll}
\operatorname{div} (B_{a,b}^T v_0) = 0, & \text{in } \Omega_0, \\
v_{0|_{y=1}} = (\zeta_1, \eta_1), & \\
v_{0|_{y=0}} = 0.
\end{array}
\right.
\qquad (3.7) $$

The equality of the initial velocities and the divergence free constraint implies that the initial conditions have to satisfy

$$ \int_0^L \left( -\zeta_1(x) \partial_x b(x) + \eta_1(x) (1 + \partial_x a(x)) \right) dx = 0. \qquad (3.8) $$

Moreover we emphasize that any solution satisfies

$$ \delta_{a,b}(\partial_t \zeta, \partial_t \eta)(t) = \int_{\Omega_0} g \, dx dy, \qquad (3.9) $$

with $\delta_{a,b}(\xi, \varphi)$ defined by

$$ \delta_{a,b}(\xi, \varphi) = \int_{0}^{L} \left( -\xi \partial_x b + \varphi (1 + \partial_x a) \right) dx. \qquad (3.10) $$
### 3 Study of a linear system

Throughout this section, we fix a couple $(a, b) \in H_{\#}^2(0, L)$ such that $\chi_{a,b}$ is injective on $\bar{\mathcal{F}}_0$. We recall that this implies that $\det \nabla\chi_{a,b} \in H_{\#}^3(\Omega_0)$ and hence $\det \nabla\chi_{a,b} \in C_{\#}(\bar{\Omega}_0)$. We moreover assume that $\|(a, b)\|_{H_{\#}^2(0, L)} + \|(\det \nabla\chi_{a,b})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R$. We consider the following linear problem:

$$ 
\left\{
\begin{array}{ll}
\rho_{f,a,b} \partial_t v - \mu \operatorname{div} \left( (A_{a,b} \nabla) v + \frac{1}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v^T B_{a,b} - q B_{a,b} \right) & \\
\qquad = f + \operatorname{div} h, & \text{in } \Omega_0 \times (0, T), \\
\operatorname{div} (B_{a,b}^T v) = g, & \text{in } \Omega_0 \times (0, T), \\
v(x, 1, t) = (\partial_t \zeta(x, t), \partial_t \eta(x, t)), & \text{on } (0, L) \times (0, T), \\
v(x, 0, t) = 0, & \text{on } (0, L) \times (0, T); \\
\rho_s \partial_{tt} \zeta - \beta \partial_{xx} \zeta = -T_{a,b}[v, q] \cdot e_1 - h_{|_{y=1}} e_2 \cdot e_1 + \ell, & \text{in } (0, L), \\
\rho_s \partial_{tt} \eta - \gamma \partial_{xxt} \eta + \alpha \partial_{xxxx} \eta = -T_{a,b}[v, q] \cdot e_2 - h_{|_{y=1}} e_2 \cdot e_2 + m, & \text{in } (0, L),
\end{array}
\right.
\quad (3.1) $$

where

$$ T_{a,b}[v, q] = (\mu(A_{a,b}\nabla)v \, e_2 + \frac{\mu}{\det \nabla\chi_{a,b}} B_{a,b} \nabla v^T B_{a,b} e_2 - q B_{a,b} e_2)_{|_{y=1}}. $$

$L$-periodic boundary conditions w.r.t. $x$ for the fluid velocity and the structure displacement are considered together with the following initial conditions at $t = 0$:

$$ v(\cdot, 0) = v_0(\cdot), \quad \text{in } \Omega_0, \quad (3.2) $$

$$ \zeta(\cdot, 0) = \zeta_0(\cdot), \quad \text{on } (0, L), \quad (3.3) $$

$$ \partial_t \zeta(\cdot, 0) = \zeta_1(\cdot), \quad \text{on } (0, L), \quad (3.4) $$

$$ \eta(\cdot, 0) = \eta_0(\cdot), \quad \text{on } (0, L), \quad (3.5) $$

$$ \partial_t \eta(\cdot, 0) = \eta_1(\cdot), \quad \text{on } (0, L), \quad (3.6) $$

satisfying the following compatibility conditions

$$ 
\left\{
\begin{array}{ll}
\operatorname{div} (B_{a,b}^T v_0) = 0, & \text{in } \Omega_0, \\
v_{0|_{y=1}} = (\zeta_1, \eta_1), & \\
v_{0|_{y=0}} = 0.
\end{array}
\right.
\quad (3.7) $$

The equality of the initial velocities and the divergence free constraint implies that the initial conditions have to satisfy

$$ \int_0^L \left( -\zeta_1(x) \partial_x b(x) + \eta_1(x) (1 + \partial_x a(x)) \right) dx = 0. \quad (3.8) $$

Moreover we emphasize that any solution satisfies

$$ \delta_{a,b}(\partial_t \zeta, \partial_t \eta)(t) = \int_{\Omega_0} g \, dx dy, \quad (3.9) $$

with $\delta_{a,b}(\xi, \varphi)$ defined by

$$ \delta_{a,b}(\xi, \varphi) = \int_{0}^{L} \left( -\xi \partial_{x} b + \varphi (1 + \partial_{x} a) \right) dx. \quad (3.10) $$
We note that, at initial time, necessarily $g(0) = 0$ and $\delta_{a,b}(\partial_t\zeta, \partial_t\eta)(0) = \delta_{a,b}(\zeta_1, \eta_1) = 0$.

As mentioned in the previous section, the pressure of this system is uniquely defined and can be decomposed as follows

$$q(x, y, t) = s_0(x, y, t) + k(t), \qquad (3.11)$$

with

$$\int_{\Omega_0} s_0 \, dxdy = 0, \qquad (3.12)$$

and

$$k(t) := \frac{1}{\|(-\partial_x b, 1 + \partial_x a)\|_{L^2(0,L)}^2} (\rho_s \frac{d}{dt}(\delta_{a,b}(\partial_t \zeta, \partial_t \eta))(t) + \beta \int_0^L \partial_{xx}\zeta \partial_x b \, dx - \alpha \int_0^L \partial_{xxx}\eta \partial_{xx} a \, dx + \gamma \int_0^L \partial_{xtt}\eta \partial_{xx} a \, dx \\ + \int_0^L (T_{a,b}[v, s_0] + h)(x, 1)e_2 \cdot (-\partial_x b, 1 + \partial_x a) \, dx - \int_0^L (\ell, m) \cdot (-\partial_x b, 1 + \partial_x a) \, dx). \quad (3.13)$$

In order to prove existence of a unique strong solution for the linear system (3.1) we first consider the case of a divergence free constraint ($g = 0$), then the case of a non homogeneous divergence constraint. Note that the average of the right hand side $g$ is not equal to zero preventing to lift this non homogeneous divergence with a velocity satisfying homogeneous Dirichlet boundary conditions on $\Gamma_0$ and on $\Gamma_e$ and thus preventing to modify only the fluid unknowns in the lifting process. This one of the major difference compared to the case where only the transversal displacement is considered.

## 3.1 Functional spaces

Let us consider the following functional spaces respectively for the initial data, solution and source terms in which we solve the linear problem associated to (2.19). Concerning initial data, we set

$$X^0 := \{ (v_0, \zeta_0, \zeta_1, \eta_0, \eta_1) \in X_f^0 \times X_s^0 : v_{0|_{y=1}} = (\zeta_1, \eta_1) \},$$

where

$$X_f^0 := \{ v_0 \in H_{\#}^1(\Omega_0) : \operatorname{div}(B_{a,b}^T v_0) = 0 \text{ in } \Omega_0 \text{ and } v_{0|_{y=0}} = 0 \},$$

$$X_s^0 := H_{\#}^2(0, L) \times H_{\#}^1(0, L) \times H_{\#}^3(0, L) \times H_{\#}^2(0, L).$$

The space $X^0$ is endowed with the product-norm:

$$\|(v_0, \zeta_0, \zeta_1, \eta_0, \eta_1)\|_{X^0} = \|v_0\|_{H_{\#}^1(\Omega_0)} + \|\zeta_0\|_{H_{\#}^2(\Omega_0)} + \|\zeta_1\|_{H_{\#}^1(\Omega_0)} + \|\eta_0\|_{H_{\#}^3(\Omega_0)} + \|\eta_1\|_{H_{\#}^2(\Omega_0)}.$$

For any $(v_0, \zeta_0, \zeta_1, \eta_0, \eta_1) \in X^0$, the compatibility condition (3.8) is satisfied. Given $T > 0$, the source term $(f, h, g, \ell, m)$ of the problem under consideration belongs to the following space:

$$S_T = \left\{ (f, h, g, \ell, m) : \begin{array}{l} f \in L^2(0, T; L_{\#}^2(\Omega_0)), h \in L^2(0, T; H_{\#}^1(\Omega_0)), \\ (\ell, m) \in (L^2(0, T; H_{\#}^{\frac{1}{2}}(0, L))^2, g \in L^2(0, T; H_{\#}^1(\Omega_0)) \cap H_{0,0}^1(0, T; (H_{\#}^1(\Omega_0))'), \end{array} \right\},$$
We note that, at initial time, necessarily $g(0) = 0$ and $\delta_{a,b}(\partial_t\zeta, \partial_t\eta)(0) = \delta_{a,b}(\zeta_1, \eta_1) = 0$.

As mentioned in the previous section, the pressure of this system is uniquely defined and can be decomposed as follows

$$q(x, y, t) = s_0(x, y, t) + k(t), \quad (3.11)$$

with

$$\int_{\Omega_0} s_0 \, dxdy = 0, \quad (3.12)$$

and

$$k(t) := \frac{1}{\|(-\partial_x b, 1 + \partial_x a)\|_{L^2(0,L)}^2} \left( \rho_s \frac{d}{dt} (\delta_{a,b}(\partial_t \zeta, \partial_t \eta))(t) + \beta \int_0^L \partial_{xx}\zeta \partial_x b \, dx - \alpha \int_0^L \partial_{xxx}\eta \partial_{xx} a \, dx + \gamma \int_0^L \partial_{xtt}\eta \partial_{xx} a \, dx \right. \\ \left. + \int_0^L (T_{a,b}[v, s_0] + h)(x, 1) e_2 \cdot (-\partial_x b, 1 + \partial_x a) \, dx - \int_0^L (\ell, m) \cdot (-\partial_x b, 1 + \partial_x a) \, dx \right). \quad (3.13)$$

In order to prove existence of a unique strong solution for the linear system (3.1) we first consider the case of a divergence free constraint ($g = 0$), then the case of a non homogeneous divergence constraint. Note that the average of the right hand side $g$ is not equal to zero preventing to lift this non homogeneous divergence with a velocity satisfying homogeneous Dirichlet boundary conditions on $\Gamma_0$ and on $\Gamma_e$ and thus preventing to modify only the fluid unknowns in the lifting process. This one of the major difference compared to the case where only the transversal displacement is considered.

## 3.1 Functional spaces

Let us consider the following functional spaces respectively for the initial data, solution and source terms in which we solve the linear problem associated to (2.19). Concerning initial data, we set

$$X^0 := \{ (v_0, \zeta_0, \zeta_1, \eta_0, \eta_1) \in X_f^0 \times X_s^0 : v_{0|_{y=1}} = (\zeta_1, \eta_1) \},$$

where

$$X_f^0 := \{ v_0 \in H_{\#}^1(\Omega_0) : \operatorname{div}(B_{a,b}^T v_0) = 0 \text{ in } \Omega_0 \text{ and } v_{0|_{y=0}} = 0 \},$$

$$X_s^0 := H_{\#}^2(0, L) \times H_{\#}^1(0, L) \times H_{\#}^3(0, L) \times H_{\#}^2(0, L).$$

The space $X^0$ is endowed with the product-norm:

$$\|(v_0, \zeta_0, \zeta_1, \eta_0, \eta_1)\|_{X^0} = \|v_0\|_{H_{\#}^1(\Omega_0)} + \|\zeta_0\|_{H_{\#}^2(\Omega_0)} + \|\zeta_1\|_{H_{\#}^1(\Omega_0)} + \|\eta_0\|_{H_{\#}^3(\Omega_0)} + \|\eta_1\|_{H_{\#}^2(\Omega_0)}.$$

For any $(v_0, \zeta_0, \zeta_1, \eta_0, \eta_1) \in X^0$, the compatibility condition (3.8) is satisfied. Given $T > 0$, the source term $(f, h, g, \ell, m)$ of the problem under consideration belongs to the following space:

$$S_T = \left\{ (f, h, g, \ell, m) : \begin{array}{l} f \in L^2(0, T; L_{\#}^2(\Omega_0)), h \in L^2(0, T; H_{\#}^1(\Omega_0)), \\ (\ell, m) \in (L^2(0, T; H_{\#}^{1/2}(0, L))^2, g \in L^2(0, T; H_{\#}^1(\Omega_0)) \cap H_{0,0}^1(0, T; (H_{\#}^1(\Omega_0))'), \end{array} \right\},$$
where

$$H_{0,0}^{1}(0, T; (H_{\#}^{1}(\Omega_{0})))' = \{g \in H^{1}(0, T; (H_{\#}^{1}(\Omega_{0})))' : g(\cdot, 0) = 0\}.$$

We underline that we enforce the vanishing condition at initial time so that (3.9) is satisfied at the initial time since $\delta_{a,b}(0) = 0$ because of (3.8). The space $S_T$ is endowed by the product norm.

Now, we introduce the spaces in which, respectively, the fluid and structure unknowns belong:

$$X_{f,T} := (H^1(0, T; L_{\#}^2(\Omega_0)) \cap L^\infty(0, T; H_{\#}^1(\Omega_0)) \cap L^2(0, T; H_{\#}^2(\Omega_0))) \times L^2(0, T; H_{\#}^1(\Omega_0)),$$ $$X_{s,T} := \left\{ (\zeta, \eta) : \begin{array}{ll} \zeta \in H^2(0, T; L_{\#}^2(0, L)) \cap W^{1,\infty}(0, T; H_{\#}^1(0, L)) \cap L^\infty(0, T; H_{\#}^2(0, L)), \\ \eta \in H^2(0, L; H_{\#}^1(0, L)) \cap W^{1,\infty}(0, T; H_{\#}^2(0, L)) \cap L^\infty(0, T; H_{\#}^3(0, L)) \end{array} \right\}$$

Again, these spaces are endowed with the obvious product-intersection norms denoted $\|\cdot\|_{X_{f,T}}$ and $\|\cdot\|_{X_{s,T}}$.

## 3.2 Homogeneous divergence constraint

We first focuss on the case $g = 0$. This subsection is devoted to the proof of the following proposition:

**Proposition 3.1** Consider $(a,b) \in H_{\#}^2(0,L)$ such that $\chi_{a,b}$ is a injective on $\overline{\mathcal{F}}_0$. Given $0 < T \le T^*$, initial data $(v_0, \zeta_0, \zeta_1, \eta_0, \eta_1) \in X^0$ and a source term $(f, h, 0, \ell, m) \in S_T$. Then there exists a unique solution $(v, q, \zeta, \eta) \in X_{f,T} \times X_{s,T}$ to Problem (3.1)-(3.7) and for any $T \le T_0$ a non decreasing function $K_{T^*} : [0, \infty) \to [0, \infty)$ such that assuming $\|(a,b)\|_{H_{\#}^2(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}^2(\Omega_0)} \le R$, the solution $(v, q, \zeta, \eta)$ satisfies

$$\|(v, q)\|_{X_{f,T}} + \|(\zeta, \eta)\|_{X_{s,T}} \le K_{T_0}(R)(\|v_0, \zeta_0, \zeta_1, \eta_0, \eta_1)\|_{X^0} + \|(f, h, 0, \ell, m)\|_{S_T}, \forall T \le T^*. \quad (3.14)$$

The proof of the above proposition is divided into three steps similar to the ones followed in [28]. In the first step, we prove the existence of strong solution for a regularized Problem of (3.1)-(3.7) by adding a dissipative term $\varepsilon\partial_{xxxxt}\zeta$ to the wave equation and $\varepsilon\partial_{xxxxt}\eta$ to the beam equation with $\varepsilon > 0$. Next, we derive, for the solution of the regularized problem, estimates independent of the regularizing parameter $\varepsilon$. These uniform estimates enable us to pass into the limit in the final step. Since some steps are similar to [28] we emphasize at which are the main differences. Note that we assume that $(a,b) \in H_{\#}^2(0,L)$ which is more than $(a,b) \in H_{\#}^{3+s}(0,L)$ that was sufficient for the elliptic regularity result. Requiring this extra regularity seems to be crucial to estimate pressure boundary terms appearing in the regularity estimate. These terms comes from the fact that, contrary to the Stokes steady state case studied at Section 2.5, we do not deal only with Dirichlet boundary conditions but with a coupled fluid-structure system. Note that these extra terms do not appears in the case where the longitudinal displacement is neglected ($a = \zeta = 0$).

### 3.2.1 Regularized problem

The regularized problem of Problem (3.1) reads as follows: for $\varepsilon > 0$, $T > 0$ and a source term $(f_1, f_2, h, 0, \ell, m) \in S_T$, find $(v_\varepsilon, q_\varepsilon, \zeta_\varepsilon, \eta_\varepsilon) \in X_{f,T} \times X_{s,T}$ such that

$$\rho_{f,a,b} \partial_t v_\varepsilon - \mu \text{div} \left( (A_{a,b} \nabla) v_\varepsilon + \frac{1}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v_\varepsilon^T B_{a,b} - q_\varepsilon B_{a,b} \right) = f + \text{div} h, \quad \text{in } \Omega_0 \times (0, T), \quad (3.15)$$
where

$$H_{0,0}^{1}(0, T; (H_{\#}^{1}(\Omega_{0})))' = \{g \in H^{1}(0, T; (H_{\#}^{1}(\Omega_{0})))' : g(\cdot, 0) = 0\}.$$

We underline that we enforce the vanishing condition at initial time so that (3.9) is satisfied at the initial time since $\delta_{a,b}(0) = 0$ because of (3.8). The space $S_T$ is endowed by the product norm.

Now, we introduce the spaces in which, respectively, the fluid and structure unknowns belong:

$$X_{f,T} := (H^1(0, T; L_{\#}^2(\Omega_0)) \cap L^\infty(0, T; H_{\#}^1(\Omega_0)) \cap L^2(0, T; H_{\#}^2(\Omega_0))) \times L^2(0, T; H_{\#}^1(\Omega_0)),$$ $$X_{s,T} := \left\{ \begin{array}{ll} (\zeta, \eta) : & \zeta \in H^2(0, T; L_{\#}^2(0, L)) \cap W^{1,\infty}(0, T; H_{\#}^1(0, L)) \cap L^\infty(0, T; H_{\#}^2(0, L)), \\ & \eta \in H^2(0, L; H_{\#}^1(0, L)) \cap W^{1,\infty}(0, T; H_{\#}^2(0, L)) \cap L^\infty(0, T; H_{\#}^3(0, L)) \end{array} \right\}$$

Again, these spaces are endowed with the obvious product-intersection norms denoted $\|\cdot\|_{X_{f,T}}$ and $\|\cdot\|_{X_{s,T}}$.

## 3.2 Homogeneous divergence constraint

We first focuss on the case $g = 0$. This subsection is devoted to the proof of the following proposition:

**Proposition 3.1** Consider $(a,b) \in H_{\#}^2(0, L)$ such that $\chi_{a,b}$ is a injective on $\overline{\mathcal{F}}_0$. Given $0 < T \le T^*$, initial data $(v_0, \zeta_0, \zeta_1, \eta_0, \eta_1) \in X^0$ and a source term $(f, h, 0, \ell, m) \in S_T$. Then there exists a unique solution $(v, q, \zeta, \eta) \in X_{f,T} \times X_{s,T}$ to Problem (3.1)-(3.7) and for any $T \le T_0$ a non decreasing function $K_{T^*} : [0, \infty) \to [0, \infty)$ such that assuming $\|(a,b)\|_{H_{\#}^2(0,L)} + \|(\det \nabla \chi_{a,b})^{-1}\|_{C_{\#}^2(\Omega_0)} \le R$, the solution $(v, q, \zeta, \eta)$ satisfies

$$\|(v, q)\|_{X_{f,T}} + \|(\zeta, \eta)\|_{X_{s,T}} \le K_{T_0}(R)(\|v_0, \zeta_0, \zeta_1, \eta_0, \eta_1)\|_{X^0} + \|(f, h, 0, \ell, m)\|_{S_T}, \forall T \le T^*. \quad (3.14)$$

The proof of the above proposition is divided into three steps similar to the ones followed in [28]. In the first step, we prove the existence of strong solution for a regularized Problem of (3.1)-(3.7) by adding a dissipative term $\varepsilon\partial_{xxxxt}\zeta$ to the wave equation and $\varepsilon\partial_{xxxxt}\eta$ to the beam equation with $\varepsilon > 0$. Next, we derive, for the solution of the regularized problem, estimates independent of the regularizing parameter $\varepsilon$. These uniform estimates enable us to pass into the limit in the final step. Since some steps are similar to [28] we emphasize at which are the main differences. Note that we assume that $(a,b) \in H_{\#}^2(0, L)$ which is more than $(a,b) \in H_{\#}^{3+s}(0, L)$ that was sufficient for the elliptic regularity result. Requiring this extra regularity seems to be crucial to estimate pressure boundary terms appearing in the regularity estimate. These terms comes from the fact that, contrary to the Stokes steady state case studied at Section 2.5, we do not deal only with Dirichlet boundary conditions but with a coupled fluid-structure system. Note that these extra terms do not appears in the case where the longitudinal displacement is neglected ($a = \zeta = 0$).

### 3.2.1 Regularized problem

The regularized problem of Problem (3.1) reads as follows: for $\varepsilon > 0$, $T > 0$ and a source term $(f_1, f_2, h, 0, \ell, m) \in S_T$, find $(v_\varepsilon, q_\varepsilon, \zeta_\varepsilon, \eta_\varepsilon) \in X_{f,T} \times X_{s,T}$ such that

$$\rho_{f,a,b} \partial_t v_\varepsilon - \mu \text{div} \left( (A_{a,b} \nabla) v_\varepsilon + \frac{1}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v_\varepsilon^T B_{a,b} - q_\varepsilon B_{a,b} \right) = f + \text{div} h, \quad \text{in } \Omega_0 \times (0, T), \quad (3.15)$$
$$ \mathrm{div}(B_{a,b}^T v_\epsilon) = 0, \quad \text{in } \Omega_0 \times (0, T), \quad (3.16) $$

$$ v_{\epsilon}(x, 1, t) = (\partial_t \zeta_{\epsilon}(x, t), \partial_t \eta_{\epsilon}(x, t)), \quad \text{on } (0, L) \times (0, T), \quad (3.17) $$

$$ v_{\epsilon}(x, 0, t) = 0, \quad \text{on } (0, L) \times (0, T), \quad (3.18) $$

$$ \rho_s \partial_{tt} \zeta_{\epsilon} - \beta \partial_{xx} \zeta_{\epsilon} + \epsilon \partial_{xxxxt} \zeta_{\epsilon} = -T_{a,b}[v_{\epsilon}, q_{\epsilon}] \cdot e_1 - h_{|y=1} e_2 \cdot e_1 + \ell, \quad \text{in } (0, L), \quad (3.19) $$

$$ \rho_s \partial_{tt} \eta_{\epsilon} - \gamma \partial_{xxt} \eta_{\epsilon} + \alpha \partial_{xxxx} \eta_{\epsilon} + \epsilon \partial_{xxxxt} \eta_{\epsilon} = -T_{a,b}[v_{\epsilon}, q_{\epsilon}] \cdot e_2 - h_{|y=1} e_2 \cdot e_2 + m, \quad \text{in } (0, L). \quad (3.20) $$

As we have regularized the solution of the linear problem (3.1), one has also to regularize the initial data as well. There exists $(\zeta_0^\epsilon, \zeta_1^\epsilon, \eta_0^\epsilon, \bar{\eta}_1^\epsilon) \in (H_\#^4(0, L) \times H_\#^3(0, L))^2$ such that

$$ (\zeta_0^\epsilon, \zeta_1^\epsilon, \eta_0^\epsilon, \bar{\eta}_1^\epsilon) \to (\zeta_0, \zeta_1, \eta_0, \bar{\eta}_1) \quad \text{in } X_s^0 \quad \text{when } \epsilon \to 0, \\ \epsilon \left( \| (\zeta_0^\epsilon, \eta_0^\epsilon) \|_{H_\#^4(0,L)} + \| (\zeta_1^\epsilon, \eta_1^\epsilon) \|_{H_\#^3(0,L)} \right) \text{ is bounded independently of } \epsilon > 0. $$

We observe that $\int_0^L (-\zeta_1^\epsilon(x) \partial_x b(x) + \bar{\eta}_1^\epsilon(x)(1+\partial_x a(x))) dx = I_\epsilon$, with $I_\epsilon \to 0$ as $\epsilon \to 0$. Thus we modify the vertical displacement so that the regularized initial beam velocities satisfy the constraint (3.8). Let us define $\eta_1^\epsilon = \bar{\eta}_1^\epsilon - \frac{I_\epsilon}{L}$, we get that $\eta_1^\epsilon \in H_\#^3(0, L)$ and $(\zeta_1^\epsilon, \eta_1^\epsilon)$ satisfies (3.8). Moreover, since $I_\epsilon \to 0$ as $\epsilon \to 0$, we have

$$ (\zeta_0^\epsilon, \zeta_1^\epsilon, \eta_0^\epsilon, \eta_1^\epsilon) \to (\zeta_0, \zeta_1, \eta_0, \eta_1) \quad \text{in } X_s^0 \quad \text{when } \epsilon \to 0, \\ \epsilon \left( \| (\zeta_0^\epsilon, \eta_0^\epsilon) \|_{H_\#^4(0,L)} + \| (\zeta_1^\epsilon, \eta_1^\epsilon) \|_{H_\#^3(0,L)} \right) \text{ is bounded independently of } \epsilon > 0. $$

Due to the compatibility condition (3.7) that should be satisfied by the initial data of the linear problem, one has to approximate the initial velocity $v_0$ by $v_0^\epsilon$ such that

$$ \left\{ \begin{array}{ll} \mathrm{div}(B_{a,b}^T v_0^\epsilon) = 0, & \text{in } \Omega_0, \\ v_0^\epsilon|_{y=1} = (\zeta_1^\epsilon, \eta_1^\epsilon), & \\ v_0^\epsilon|_{y=0} = 0. & \end{array} \right. \quad (3.21) $$

To this end, we define

$$ v_0^\epsilon = v_0 - U(0, 0, \zeta_1^\epsilon - \zeta_1, \eta_1^\epsilon - \eta_1), \quad (3.22) $$

where $U(0, 0, \zeta_1^\epsilon - \zeta_1, \eta_1^\epsilon - \eta_1) \in H_\#^2(\Omega_0)$ is the velocity solution of the periodic Stokes-like Problem (2.40)-(2.39) with $\hat{f} = 0$, $\hat{g} = 0$ and $\psi = (\zeta_1^\epsilon - \zeta_1, \eta_1^\epsilon - \eta_1)$. Thanks to (3.8) that is verified by both $(\zeta_1, \eta_1)$ and $(\zeta_1^\epsilon, \eta_1^\epsilon)$ the compatibility condition

$$ \int_0^L \left( -(\zeta_1^\epsilon(x) - \zeta_1(x)) \partial_x b(x) + (\eta_1^\epsilon(x) - \eta_1(x))(1 + \partial_x a(x)) \right) dx = 0 $$

holds true. With this construction, we have $v_0^\epsilon \in H_\#^1(\Omega_0)$ as $v_0 \in H_\#^1(\Omega_0)$ and $v_0^\epsilon$ satisfies (3.21). Moreover, we have

$$ v_0^\epsilon \to v_0, \quad \text{in } H_\#^1(\Omega_0) \text{ as } \epsilon \to 0. \quad (3.23) $$

**Variational formulation of the problem.** Let us consider the following space

$$ V = \{ (w, \xi, \varphi) \in H_\#^1(\Omega_0) \times H_\#^2(0, L) \times H_\#^2(0, L), \mathrm{div}(B_{a,b}^T w) = 0, w|_{y=1} = (\xi, \varphi), w|_{y=0} = 0 \}. $$

Note that for any $(w, \xi, \varphi) \in V$, then $\delta_{a,b}(\xi, \varphi) = 0$, where $\delta_{a,b}(\xi, \varphi)$ is defined by (3.10). Let $(w, \xi, \varphi) \in V$ and let us suppose that $(v_\epsilon, q_\epsilon, \zeta_\epsilon, \eta_\epsilon)$ is a classical solution of (3.15)-(3.20). By multiplying the fluid equation (3.15) by $w$,
$$ \mathrm{div}(B_{a,b}^T v_\epsilon) = 0, \quad \text{in } \Omega_0 \times (0, T), \qquad (3.16) $$

$$ v_{\epsilon}(x, 1, t) = (\partial_t \zeta_{\epsilon}(x, t), \partial_t \eta_{\epsilon}(x, t)), \quad \text{on } (0, L) \times (0, T), \qquad (3.17) $$

$$ v_{\epsilon}(x, 0, t) = 0, \quad \text{on } (0, L) \times (0, T), \qquad (3.18) $$

$$ \rho_s \partial_{tt} \zeta_{\epsilon} - \beta \partial_{xx} \zeta_{\epsilon} + \epsilon \partial_{xxxxt} \zeta_{\epsilon} = -T_{a,b}[v_{\epsilon}, q_{\epsilon}] \cdot e_1 - h_{|y=1} e_2 \cdot e_1 + \ell, \quad \text{in } (0, L), \qquad (3.19) $$

$$ \rho_s \partial_{tt} \eta_{\epsilon} - \gamma \partial_{xxt} \eta_{\epsilon} + \alpha \partial_{xxxx} \eta_{\epsilon} + \epsilon \partial_{xxxxt} \eta_{\epsilon} = -T_{a,b}[v_{\epsilon}, q_{\epsilon}] \cdot e_2 - h_{|y=1} e_2 \cdot e_2 + m, \quad \text{in } (0, L). \qquad (3.20) $$

As we have regularized the solution of the linear problem (3.1), one has also to regularize the initial data as well. There exists $(\zeta_0^\epsilon, \zeta_1^\epsilon, \eta_0^\epsilon, \bar{\eta}_1^\epsilon) \in (H_\#^4(0, L) \times H_\#^3(0, L))^2$ such that

$$ (\zeta_0^\epsilon, \zeta_1^\epsilon, \eta_0^\epsilon, \bar{\eta}_1^\epsilon) \to (\zeta_0, \zeta_1, \eta_0, \bar{\eta}_1) \quad \text{in } X_s^0 \quad \text{when } \epsilon \to 0, \\ \epsilon \left( \| (\zeta_0^\epsilon, \eta_0^\epsilon) \|_{H_\#^4(0,L)} + \| (\zeta_1^\epsilon, \eta_1^\epsilon) \|_{H_\#^3(0,L)} \right) \text{ is bounded independently of } \epsilon > 0. $$

We observe that $\int_0^L (-\zeta_1^\epsilon(x) \partial_x b(x) + \bar{\eta}_1^\epsilon(x)(1+\partial_x a(x))) dx = I_\epsilon$, with $I_\epsilon \to 0$ as $\epsilon \to 0$. Thus we modify the vertical displacement so that the regularized initial beam velocities satisfy the constraint (3.8). Let us define $\eta_1^\epsilon = \bar{\eta}_1^\epsilon - \frac{I_\epsilon}{L}$, we get that $\eta_1^\epsilon \in H_\#^3(0, L)$ and $(\zeta_1^\epsilon, \eta_1^\epsilon)$ satisfies (3.8). Moreover, since $I_\epsilon \to 0$ as $\epsilon \to 0$, we have

$$ (\zeta_0^\epsilon, \zeta_1^\epsilon, \eta_0^\epsilon, \eta_1^\epsilon) \to (\zeta_0, \zeta_1, \eta_0, \eta_1) \quad \text{in } X_s^0 \quad \text{when } \epsilon \to 0, \\ \epsilon \left( \| (\zeta_0^\epsilon, \eta_0^\epsilon) \|_{H_\#^4(0,L)} + \| (\zeta_1^\epsilon, \bar{\eta}_1^\epsilon) \|_{H_\#^3(0,L)} \right) \text{ is bounded independently of } \epsilon > 0. $$

Due to the compatibility condition (3.7) that should be satisfied by the initial data of the linear problem, one has to approximate the initial velocity $v_0$ by $v_0^\epsilon$ such that

$$ \left\{ \begin{array}{ll} \mathrm{div}(B_{a,b}^T v_0^\epsilon) = 0, & \text{in } \Omega_0, \\ v_0^\epsilon|_{y=1} = (\zeta_1^\epsilon, \eta_1^\epsilon), & \\ v_0^\epsilon|_{y=0} = 0. & \end{array} \right. \qquad (3.21) $$

To this end, we define

$$ v_0^\epsilon = v_0 - U(0, 0, \zeta_1^\epsilon - \zeta_1, \eta_1^\epsilon - \eta_1), \qquad (3.22) $$

where $U(0, 0, \zeta_1^\epsilon - \zeta_1, \eta_1^\epsilon - \eta_1) \in H_\#^2(\Omega_0)$ is the velocity solution of the periodic Stokes-like Problem (2.40)-(2.39) with $\hat{f} = 0$, $\hat{g} = 0$ and $\psi = (\zeta_1^\epsilon - \zeta_1, \eta_1^\epsilon - \eta_1)$. Thanks to (3.8) that is verified by both $(\zeta_1, \eta_1)$ and $(\zeta_1^\epsilon, \eta_1^\epsilon)$ the compatibility condition

$$ \int_0^L \left( -(\zeta_1^\epsilon(x) - \zeta_1(x)) \partial_x b(x) + (\eta_1^\epsilon(x) - \eta_1(x))(1 + \partial_x a(x)) \right) dx = 0 $$

holds true. With this construction, we have $v_0^\epsilon \in H_\#^1(\Omega_0)$ as $v_0 \in H_\#^1(\Omega_0)$ and $v_0^\epsilon$ satisfies (3.21). Moreover, we have

$$ v_0^\epsilon \to v_0, \quad \text{in } H_\#^1(\Omega_0) \text{ as } \epsilon \to 0. \qquad (3.23) $$

**Variational formulation of the problem.** Let us consider the following space

$$ V = \{ (w, \xi, \varphi) \in H_\#^1(\Omega_0) \times H_\#^2(0, L) \times H_\#^2(0, L), \mathrm{div}(B_{a,b}^T w) = 0, w|_{y=1} = (\xi, \varphi), w|_{y=0} = 0 \}. $$

Note that for any $(w, \xi, \varphi) \in V$, then $\delta_{a,b}(\xi, \varphi) = 0$, where $\delta_{a,b}(\xi, \varphi)$ is defined by (3.10). Let $(w, \xi, \varphi) \in V$ and let us suppose that $(v_\epsilon, q_\epsilon, \zeta_\epsilon, \eta_\epsilon)$ is a classical solution of (3.15)-(3.20). By multiplying the fluid equation (3.15) by $w$,
(3.19) by $\xi$ and (3.20) by $\varphi$ and performing integration by parts, we obtain, by using (3.16)-(3.18) and the peridic boundary conditions, that

$$ 
\begin{aligned}
& \int_{\Omega_0} \rho_{f,a,b} \partial_t v_\varepsilon \cdot w \, dxdy + \rho_s \int_0^L \partial_{tt}\zeta_\varepsilon \xi \, dx + \rho_s \int_0^L \partial_{tt}\eta_\varepsilon \varphi \, dx + \gamma \int_0^L \partial_{xxt}\eta_\varepsilon \partial_x \varphi \, dx \\
& \qquad + \beta \int_0^L \partial_x \zeta_\varepsilon \partial_x \xi \, dx + \alpha \int_0^L \partial_{xx}\eta_\varepsilon \partial_{xx}\varphi \, dx + \int_{\Omega_0} \left( \mu(A_{a,b}\nabla)v_\varepsilon + \frac{\mu}{\det\nabla\chi_{a,b}} B_{a,b} \nabla v_\varepsilon^T B_{a,b} \right) : \nabla w \, dxdy \\
& \qquad + \varepsilon \left( \int_0^L \partial_{xxt}\zeta_\varepsilon \partial_{xx}\xi \, dx + \int_0^L \partial_{xxt}\eta_\varepsilon \partial_{xx}\varphi \, dx \right) \\
& = \int_{\Omega_0} f \cdot w \, dxdy - \int_{\Omega_0} h : \nabla w \, dxdy + \int_0^L (\ell, m) \cdot (\xi, \varphi) \, dx. \quad (3.24)
\end{aligned}
$$

This leads to the following weak formulation of the regularized problem (3.15)-(3.20): For given $(f, h, 0, \ell, m) \in S_T$ and $(v_0^\varepsilon, \zeta_0^\varepsilon, \zeta_1^\varepsilon, \eta_0^\varepsilon, \eta_1^\varepsilon)$ as above, find

$$ (v_\varepsilon, \zeta_\varepsilon, \eta_\varepsilon) \in L^2(0, T; H_\#^1(\Omega_0)) \cap L^\infty(0, T; L_\#^2(\Omega_0)) \times W^{1,\infty}(0, T; H_\#^2(0, L)) \times W^{1,\infty}(0, T; H_\#^2(\Omega_0)), $$

such that (3.24) holds for all $(w, \xi, \varphi) \in V$ with

$$ (v_\varepsilon(0), \zeta_\varepsilon(0), \partial_t \zeta_\varepsilon(0), \eta_\varepsilon(0), \partial_t \eta_\varepsilon(0)) = (v_0^\varepsilon, \zeta_0^\varepsilon, \zeta_1^\varepsilon, \eta_0^\varepsilon, \eta_1^\varepsilon). \quad (3.25) $$

In order to obtain the existence of a solution and a priori estimates we make use of the uniform Korn-like inequality derived at Proposition 2.2. Let us mention that in the case where the longitudinal displacement is neglected and the beam only moves in the transversal direction, the Korn inequality is trivially satisfied since the symmetric part of the gradient of the fluid velocity has no contribution in the applied force on the structure. Here, even if $\text{div} \left( \frac{1}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v_\varepsilon^T B_{a,b} \right) = 0$ due to the divergence free constraint, this term has a non zero contribution on the boundary when integrated by parts and thus its normal trace acts on the structure.

We use Faedo-Galerkin method to prove the existence of weak solutions of Problem (3.24)-(3.25). To this end, we approximate the space $V$ by a space $V_m$ by following the same lines in [14] and we prove existence and uniqueness of solutions to the approximated problem. We remark that the estimates that will be derived later on for the solution $(v_\varepsilon, \zeta_\varepsilon, \eta_\varepsilon)$ can be justified by passing over the limit $m \to \infty$ of the estimates satisfied by the approximated solution of the approximated problem as in the approximated problem any time derivative of the solution is a valid test function.

### 3.2.2 Energy estimates

We take $(v_\varepsilon, \partial_t \zeta_\varepsilon, \partial_t \eta_\varepsilon)$ as test function in the weak (3.24). Thus, setting $(w, \xi, \varphi) = (v_\varepsilon, \partial_t \zeta_\varepsilon, \partial_t \eta_\varepsilon)$ in (3.24), we get

$$ 
\begin{aligned}
& \frac{1}{2} \frac{d}{dt} \left[ \int_{\Omega_0} \rho_{f,a,b} |v_\varepsilon|^2 \, dxdy + \rho_s \int_0^L |\partial_t \zeta_\varepsilon(x)|^2 \, dx + \rho_s \int_0^L |\partial_t \eta_\varepsilon(x)|^2 \, dx + \beta \int_0^L |\partial_x \zeta_\varepsilon(x)|^2 \, dx \right. \\
& \qquad + \alpha \int_0^L |\partial_{xx} \eta_\varepsilon(x)|^2 \, dx] + \int_{\Omega_0} \left( \mu(A_{a,b} \nabla) v_\varepsilon + \frac{\mu}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v_\varepsilon^T B_{a,b} \right) : \nabla v_\varepsilon \, dxdy \\
& \qquad + \frac{\gamma}{2} \frac{d}{dt} \left( \int_0^L |\partial_{tx} \eta_\varepsilon(x)|^2 \, dx \right) + \varepsilon \left( \int_0^L |\partial_{xxt} \zeta_\varepsilon(x)|^2 \, dx + \int_0^L |\partial_{xxt} \eta_\varepsilon(x)|^2 \, dx \right) \\
& = \int_{\Omega_0} f \cdot v_\varepsilon \, dxdy - \int_{\Omega_0} h : \nabla v_\varepsilon \, dxdy + \int_0^L (\ell, m) \cdot (\partial_t \zeta_\varepsilon, \partial_t \eta_\varepsilon) \, dx.
\end{aligned}
$$
(3.19) by $\xi$ and (3.20) by $\varphi$ and performing integration by parts, we obtain, by using (3.16)-(3.18) and the peridic boundary conditions, that

$$ 
\begin{aligned}
& \int_{\Omega_0} \rho_{f,a,b} \partial_t v_\varepsilon \cdot w \, dxdy + \rho_s \int_0^L \partial_{tt}\zeta_\varepsilon \xi \, dx + \rho_s \int_0^L \partial_{tt}\eta_\varepsilon \varphi \, dx + \gamma \int_0^L \partial_{xxt}\eta_\varepsilon \partial_x \varphi \, dx \\
& \quad + \beta \int_0^L \partial_x \zeta_\varepsilon \partial_x \xi \, dx + \alpha \int_0^L \partial_{xx}\eta_\varepsilon \partial_{xx}\varphi \, dx + \int_{\Omega_0} \left( \mu(A_{a,b}\nabla)v_\varepsilon + \frac{\mu}{\det\nabla\chi_{a,b}} B_{a,b} \nabla v_\varepsilon^T B_{a,b} \right) : \nabla w \, dxdy \\
& \quad + \varepsilon \left( \int_0^L \partial_{xxt}\zeta_\varepsilon \partial_{xx}\xi \, dx + \int_0^L \partial_{xxt}\eta_\varepsilon \partial_{xx}\varphi \, dx \right) \\
& = \int_{\Omega_0} f \cdot w \, dxdy - \int_{\Omega_0} h : \nabla w \, dxdy + \int_0^L (\ell, m) \cdot (\xi, \varphi) \, dx. \quad (3.24)
\end{aligned}
$$

This leads to the following weak formulation of the regularized problem (3.15)-(3.20): For given $(f, h, 0, \ell, m) \in S_T$ and $(v_0^\varepsilon, \zeta_0^\varepsilon, \zeta_1^\varepsilon, \eta_0^\varepsilon, \eta_1^\varepsilon)$ as above, find

$$ (v_\varepsilon, \zeta_\varepsilon, \eta_\varepsilon) \in L^2(0, T; H_\#^1(\Omega_0)) \cap L^\infty(0, T; L_\#^2(\Omega_0)) \times W^{1,\infty}(0, T; H_\#^2(0, L)) \times W^{1,\infty}(0, T; H_\#^2(\Omega_0)), $$

such that (3.24) holds for all $(w, \xi, \varphi) \in V$ with

$$ (v_\varepsilon(0), \zeta_\varepsilon(0), \partial_t \zeta_\varepsilon(0), \eta_\varepsilon(0), \partial_t \eta_\varepsilon(0)) = (v_0^\varepsilon, \zeta_0^\varepsilon, \zeta_1^\varepsilon, \eta_0^\varepsilon, \eta_1^\varepsilon). \quad (3.25) $$

In order to obtain the existence of a solution and a priori estimates we make use of the uniform Korn-like inequality derived at Proposition 2.2. Let us mention that in the case where the longitudinal displacement is neglected and the beam only moves in the transversal direction, the Korn inequality is trivially satisfied since the symmetric part of the gradient of the fluid velocity has no contribution in the applied force on the structure. Here, even if $\text{div} \left( \frac{1}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v_\varepsilon^T B_{a,b} \right) = 0$ due to the divergence free constraint, this term has a non zero contribution on the boundary when integrated by parts and thus its normal trace acts on the structure.

We use Faedo-Galerkin method to prove the existence of weak solutions of Problem (3.24)-(3.25). To this end, we approximate the space $V$ by a space $V_m$ by following the same lines in [14] and we prove existence and uniqueness of solutions to the approximated problem. We remark that the estimates that will be derived later on for the solution $(v_\varepsilon, \zeta_\varepsilon, \eta_\varepsilon)$ can be justified by passing over the limit $m \to \infty$ of the estimates satisfied by the approximated solution of the approximated problem as in the approximated problem any time derivative of the solution is a valid test function.

### 3.2.2 Energy estimates

We take $(v_\varepsilon, \partial_t \zeta_\varepsilon, \partial_t \eta_\varepsilon)$ as test function in the weak (3.24). Thus, setting $(w, \xi, \varphi) = (v_\varepsilon, \partial_t \zeta_\varepsilon, \partial_t \eta_\varepsilon)$ in (3.24), we get

$$ 
\begin{aligned}
& \frac{1}{2} \frac{d}{dt} \left[ \int_{\Omega_0} \rho_{f,a,b} |v_\varepsilon|^2 \, dxdy + \rho_s \int_0^L |\partial_t \zeta_\varepsilon(x)|^2 \, dx + \rho_s \int_0^L |\partial_t \eta_\varepsilon(x)|^2 \, dx + \beta \int_0^L |\partial_x \zeta_\varepsilon(x)|^2 \, dx \right. \\
& \quad + \alpha \int_0^L |\partial_{xx} \eta_\varepsilon(x)|^2 \, dx] + \int_{\Omega_0} \left( \mu(A_{a,b} \nabla) v_\varepsilon + \frac{\mu}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v_\varepsilon^T B_{a,b} \right) : \nabla v_\varepsilon \, dxdy \\
& \quad + \frac{\gamma}{2} \frac{d}{dt} \left( \int_0^L |\partial_{tx} \eta_\varepsilon(x)|^2 \, dx \right) + \varepsilon \left( \int_0^L |\partial_{xxt} \zeta_\varepsilon(x)|^2 \, dx + \int_0^L |\partial_{xxt} \eta_\varepsilon(x)|^2 \, dx \right) \\
& = \int_{\Omega_0} f \cdot v_\varepsilon \, dxdy - \int_{\Omega_0} h : \nabla v_\varepsilon \, dxdy + \int_0^L (\ell, m) \cdot (\partial_t \zeta_\varepsilon, \partial_t \eta_\varepsilon) \, dx.
\end{aligned}
$$
Since

$$ \int_{\Omega_0} \left( \mu(A_{a,b} \nabla) v_{\varepsilon} + \frac{\mu}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v_{\varepsilon}^T B_{a,b} \right) : \nabla v_{\varepsilon} \, dxdy = \frac{\mu}{2} \int_{\Omega_0} \frac{1}{\det \nabla \chi_{a,b}} \left| \left( B_{a,b} \nabla v + \left( B_{a,b} \nabla v_{\varepsilon} \right)^T \right) \right|^2 dxdy $$

By using (2.35), Cauchy-Schwartz, Poincaré and Young's inequalities, there exists $C > 0$ such that

$$ \begin{aligned} & \frac{1}{2} \frac{d}{dt} \left( \int_{\Omega_0} \rho_{f,a,b} |v_{\varepsilon}|^2 dxdy + \rho_s \int_0^L |\partial_t \zeta_{\varepsilon}(x)|^2 dx + \rho_s \int_0^L |\partial_t \eta_{\varepsilon}(x)|^2 dx + \beta \int_0^L |\partial_x \zeta_{\varepsilon}(x)|^2 dx \right. \\ & \qquad \left. + \alpha \int_0^L |\partial_{xx} \eta_{\varepsilon}(x)|^2 dx \right) + \frac{\mu \lambda(R)}{2} \int_{\Omega_0} |\nabla v_{\varepsilon}|^2 dxdy + \frac{\gamma}{2} \frac{d}{dt} \left( \int_0^L |\partial_{tx} \eta_{\varepsilon}(x)|^2 dx \right) \\ & \qquad + \varepsilon \left( \int_0^L |\partial_{xxt} \zeta_{\varepsilon}(x)|^2 dx + \int_0^L |\partial_{xxt} \eta_{\varepsilon}(x)|^2 dx \right) \le \frac{C}{\lambda(R)} \left( \|f\|_{L_{\#}^2(\Omega_0)}^2 + \|h\|_{L_{\#}^2(\Omega_0)}^2 \right) \\ & \qquad \qquad + \frac{\rho_s}{4} \left( \|\partial_t \zeta_{\varepsilon}\|_{L_{\#}^2(0,L)}^2 + \|\partial_t \eta_{\varepsilon}\|_{L_{\#}^2(0,L)}^2 \right) + \frac{1}{\rho_s} \left( \|\ell\|_{L_{\#}^2(0,L)}^2 + \|\partial_t m\|_{L_{\#}^2(0,L)}^2 \right). \end{aligned} $$

By noting that $\rho_{f,a,b} \ge \frac{\rho_f}{R}$ and integrating with respect to time on $[0, t] \subset [0, T)$, we obtain

$$ \begin{aligned} & \frac{\rho_f}{2R} \|v_{\varepsilon}\|_{L^{\infty}(0,t,L_{\#}^2(\Omega_0))}^2 + \frac{\rho_s}{4} \|\partial_t \zeta_{\varepsilon}\|_{L^{\infty}(0,t,L_{\#}^2(0,L))}^2 + \frac{\rho_s}{4} \|\partial_t \eta_{\varepsilon}\|_{L^{\infty}(0,t,L_{\#}^2(0,L))}^2 + \frac{\beta}{2} \|\partial_x \zeta_{\varepsilon}\|_{L^{\infty}(0,t,L_{\#}^2(0,L))}^2 \\ & \qquad + \frac{\alpha}{2} \|\partial_{xx} \eta_{\varepsilon}\|_{L^{\infty}(0,t,L_{\#}^2(0,L))}^2 + \frac{\mu \lambda(R)}{2} \|\nabla v_{\varepsilon}\|_{L^2(0,t;L^2(\Omega_0))}^2 + \frac{\gamma}{2} \|\partial_{tx} \eta_{\varepsilon}\|_{L^{\infty}(0,t;L_{\#}^2(0,L))}^2 \\ & \qquad + \varepsilon \left( \|\partial_{xxt} \zeta_{\varepsilon}\|_{L^2(0,t;L_{\#}^2(0,L))}^2 + \|\partial_{xxt} \eta_{\varepsilon}\|_{L^2(0,t;L_{\#}^2(0,L))}^2 \right) \le \frac{C}{\lambda(R)} \left( \|f\|_{L^2(0,t;L_{\#}^2(\Omega_0))}^2 + \|h\|_{L^2(0,t;L_{\#}^2(\Omega_0))}^2 \right) \\ & \qquad + \frac{1}{\rho_s} \left( \|\ell\|_{L^2(0,t;L_{\#}^2(0,L))}^2 + \|m\|_{L^2(0,t;L_{\#}^2(0,L))}^2 \right) + C_0. \end{aligned} \quad (3.26) $$

where $C_0 > 0$ depends only on the initial data, and $C > 0$ independent of $T > 0$.

Thus, we obtain uniform bounds independent of $\varepsilon$ in the following spaces:

* $v_{\varepsilon}$ in $L^{\infty}(0, T; L_{\#}^2(\Omega_0)) \cap L^2(0, T; H_{\#}^1(\Omega_0))$;

* $\zeta_{\varepsilon}$ in $L^{\infty}(0, T; H_{\#}^1(0, L)) \cap W^{1,\infty}(0, T; L_{\#}^2(0, L))$;

* $\eta_{\varepsilon}$ in $L^{\infty}(0, T; H_{\#}^2(0, L)) \cap W^{1,\infty}(0, T; H_{\#}^1(0, L))$;

* $\sqrt{\varepsilon} \zeta_{\varepsilon}$ and $\sqrt{\varepsilon} \eta_{\varepsilon}$ in $H^1(0, T; H_{\#}^2(0, L))$.

### 3.2.3 Additional estimates for $\varepsilon > 0$

Next one takes $(w, \xi, \varphi) = (\partial_t v_{\varepsilon}, \partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon})$ as a test function in the weak formulation (3.24) and performing integration by parts to the terms involving $h$ that

$$ \begin{aligned} & \int_{\Omega_0} \rho_{f,a,b} |\partial_t v_{\varepsilon}|^2 dxdy + \rho_s \int_0^L |\partial_{tt} \zeta_{\varepsilon}|^2 dx + \rho_s \int_0^L |\partial_{tt} \eta_{\varepsilon}|^2 dx + \gamma \int_0^L |\partial_{xxt} \eta_{\varepsilon}|^2 dx \\ & \qquad + \frac{\varepsilon}{2} \frac{d}{dt} \int_0^L |\partial_{xxt} \zeta_{\varepsilon}|^2 dx + \frac{\varepsilon}{2} \frac{d}{dt} \int_0^L |\partial_{xxt} \eta_{\varepsilon}|^2 dx + \int_{\Omega_0} \left( \mu(A_{a,b} \nabla) v_{\varepsilon} + \frac{\mu}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v_{\varepsilon}^T B_{a,b} \right) : \nabla \partial_t v_{\varepsilon} \, dxdy \\ & \qquad + \beta \int_0^L \partial_x \zeta_{\varepsilon} \partial_{xxt} \zeta_{\varepsilon} \, dx + \alpha \int_0^L \partial_{xx} \eta_{\varepsilon} \partial_{xxt} \eta_{\varepsilon} \, dx = \int_{\Omega_0} f \cdot \partial_t v_{\varepsilon} \, dxdy \\ & \qquad + \int_{\Omega_0} \operatorname{div} h \cdot \partial_t v_{\varepsilon} \, dxdy - \int_0^L (h e_2)_{|_{y=1}} \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx + \int_0^L (\ell, m) \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx. \end{aligned} $$
Since

$$ \int_{\Omega_0} \left( \mu(A_{a,b} \nabla) v_{\varepsilon} + \frac{\mu}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v_{\varepsilon}^T B_{a,b} \right) : \nabla v_{\varepsilon} \, dxdy = \frac{\mu}{2} \int_{\Omega_0} \frac{1}{\det \nabla \chi_{a,b}} \left| \left( B_{a,b} \nabla v + \left( B_{a,b} \nabla v_{\varepsilon} \right)^T \right) \right|^2 dxdy $$

By using (2.35), Cauchy-Schwartz, Poincaré and Young's inequalities, there exists $C > 0$ such that

$$ \begin{aligned} & \frac{1}{2} \frac{d}{dt} \left( \int_{\Omega_0} \rho_{f,a,b} |v_{\varepsilon}|^2 dxdy + \rho_s \int_0^L |\partial_t \zeta_{\varepsilon}(x)|^2 dx + \rho_s \int_0^L |\partial_t \eta_{\varepsilon}(x)|^2 dx + \beta \int_0^L |\partial_x \zeta_{\varepsilon}(x)|^2 dx \right. \\ & \qquad \left. + \alpha \int_0^L |\partial_{xx} \eta_{\varepsilon}(x)|^2 dx \right) + \frac{\mu \lambda(R)}{2} \int_{\Omega_0} |\nabla v_{\varepsilon}|^2 dxdy + \frac{\gamma}{2} \frac{d}{dt} \left( \int_0^L |\partial_{tx} \eta_{\varepsilon}(x)|^2 dx \right) \\ & \qquad + \varepsilon \left( \int_0^L |\partial_{xxt} \zeta_{\varepsilon}(x)|^2 dx + \int_0^L |\partial_{xxt} \eta_{\varepsilon}(x)|^2 dx \right) \le \frac{C}{\lambda(R)} \left( \|f\|_{L_{\#}^2(\Omega_0)}^2 + \|h\|_{L_{\#}^2(\Omega_0)}^2 \right) \\ & \qquad \qquad + \frac{\rho_s}{4} \left( \|\partial_t \zeta_{\varepsilon}\|_{L_{\#}^2(0,L)}^2 + \|\partial_t \eta_{\varepsilon}\|_{L_{\#}^2(0,L)}^2 \right) + \frac{1}{\rho_s} \left( \|\ell\|_{L_{\#}^2(0,L)}^2 + \|\partial_t m\|_{L_{\#}^2(0,L)}^2 \right). \end{aligned} $$

By noting that $\rho_{f,a,b} \ge \frac{\rho_f}{R}$ and integrating with respect to time on $[0, t] \subset [0, T)$, we obtain

$$ \begin{aligned} & \frac{\rho_f}{2R} \|v_{\varepsilon}\|_{L^{\infty}(0,t,L_{\#}^2(\Omega_0))}^2 + \frac{\rho_s}{4} \|\partial_t \zeta_{\varepsilon}\|_{L^{\infty}(0,t,L_{\#}^2(0,L))}^2 + \frac{\rho_s}{4} \|\partial_t \eta_{\varepsilon}\|_{L^{\infty}(0,t,L_{\#}^2(0,L))}^2 + \frac{\beta}{2} \|\partial_x \zeta_{\varepsilon}\|_{L^{\infty}(0,t,L_{\#}^2(0,L))}^2 \\ & \qquad + \frac{\alpha}{2} \|\partial_{xx} \eta_{\varepsilon}\|_{L^{\infty}(0,t,L_{\#}^2(0,L))}^2 + \frac{\mu \lambda(R)}{2} \|\nabla v_{\varepsilon}\|_{L^2(0,t;L^2(\Omega_0))}^2 + \frac{\gamma}{2} \|\partial_{tx} \eta_{\varepsilon}\|_{L^{\infty}(0,t;L_{\#}^2(0,L))}^2 \\ & \qquad + \varepsilon \left( \|\partial_{xxt} \zeta_{\varepsilon}\|_{L^2(0,t;L_{\#}^2(0,L))}^2 + \|\partial_{xxt} \eta_{\varepsilon}\|_{L^2(0,t;L_{\#}^2(0,L))}^2 \right) \le \frac{C}{\lambda(R)} \left( \|f\|_{L^2(0,t;L_{\#}^2(\Omega_0))}^2 + \|h\|_{L^2(0,t;L_{\#}^2(\Omega_0))}^2 \right) \\ & \qquad + \frac{1}{\rho_s} \left( \|\ell\|_{L^2(0,t;L_{\#}^2(0,L))}^2 + \|m\|_{L^2(0,t;L_{\#}^2(0,L))}^2 \right) + C_0. \end{aligned} \quad (3.26) $$

where $C_0 > 0$ depends only on the initial data, and $C > 0$ independent of $T > 0$.

Thus, we obtain uniform bounds independent of $\varepsilon$ in the following spaces:

* $v_{\varepsilon}$ in $L^{\infty}(0, T; L_{\#}^2(\Omega_0)) \cap L^2(0, T; H_{\#}^1(\Omega_0))$;

* $\zeta_{\varepsilon}$ in $L^{\infty}(0, T; H_{\#}^1(0, L)) \cap W^{1,\infty}(0, T; L_{\#}^2(0, L))$;

* $\eta_{\varepsilon}$ in $L^{\infty}(0, T; H_{\#}^2(0, L)) \cap W^{1,\infty}(0, T; H_{\#}^1(0, L))$;

* $\sqrt{\varepsilon} \zeta_{\varepsilon}$ and $\sqrt{\varepsilon} \eta_{\varepsilon}$ in $H^1(0, T; H_{\#}^2(0, L))$.

### 3.2.3 Additional estimates for $\varepsilon > 0$

Next one takes $(w, \xi, \varphi) = (\partial_t v_{\varepsilon}, \partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon})$ as a test function in the weak formulation (3.24) and performing integration by parts to the terms involving $h$ that

$$ \begin{aligned} & \int_{\Omega_0} \rho_{f,a,b} |\partial_t v_{\varepsilon}|^2 dxdy + \rho_s \int_0^L |\partial_{tt} \zeta_{\varepsilon}|^2 dx + \rho_s \int_0^L |\partial_{tt} \eta_{\varepsilon}|^2 dx + \gamma \int_0^L |\partial_{xxt} \eta_{\varepsilon}|^2 dx \\ & \qquad + \frac{\varepsilon}{2} \frac{d}{dt} \int_0^L |\partial_{xxt} \zeta_{\varepsilon}|^2 dx + \frac{\varepsilon}{2} \frac{d}{dt} \int_0^L |\partial_{xxt} \eta_{\varepsilon}|^2 dx + \int_{\Omega_0} \left( \mu(A_{a,b} \nabla) v_{\varepsilon} + \frac{\mu}{\det \nabla \chi_{a,b}} B_{a,b} \nabla v_{\varepsilon}^T B_{a,b} \right) : \nabla \partial_t v_{\varepsilon} \, dxdy \\ & \qquad + \beta \int_0^L \partial_x \zeta_{\varepsilon} \partial_{xxt} \zeta_{\varepsilon} \, dx + \alpha \int_0^L \partial_{xx} \eta_{\varepsilon} \partial_{xxt} \eta_{\varepsilon} \, dx = \int_{\Omega_0} f \cdot \partial_t v_{\varepsilon} \, dxdy \\ & \qquad + \int_{\Omega_0} \operatorname{div} h \cdot \partial_t v_{\varepsilon} \, dxdy - \int_0^L (h e_2)_{|_{y=1}} \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx + \int_0^L (\ell, m) \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx. \end{aligned} $$
Next, integrating the above equation with respect to time on any time interval $[0, t] \subset [0, T)$ and using the following equality

$$ \int_{\Omega_0} \left( \mu(A_{a,b}\nabla)v_{\varepsilon} + \frac{\mu}{\det\nabla\chi_{a,b}} B_{a,b}\nabla v_{\varepsilon}^T B_{a,b} \right) : \nabla\partial_t v_{\varepsilon} \,dx dy = \frac{1}{4} \frac{d}{dt} \int_{\Omega_0} \frac{1}{\det\nabla\chi_{a,b}} \left| ((B_{a,b}\nabla)v_{\varepsilon} + ((B_{a,b}\nabla)v_{\varepsilon})^T) \right|^2 dx dy. $$

lead to

$$ \begin{aligned} & \int_0^t \int_{\Omega_0} \rho_{f,a,b} |\partial_t v_{\varepsilon}|^2 dx dy ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \zeta_{\varepsilon}|^2 dx ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \eta_{\varepsilon}|^2 v dx ds + \gamma \int_0^t \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx ds \\ & + \frac{1}{2} \left( \int_{\Omega_0} \frac{\mu}{2 \det \nabla \chi_{a,b}} \left| ((B_{a,b}\nabla)v_{\varepsilon}(t) + ((B_{a,b}\nabla)v_{\varepsilon}(t))^T \right|^2 dx dy + \varepsilon \int_0^L |\partial_{xxt} \zeta_{\varepsilon}(t)|^2 dx + \varepsilon \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx \right) \\ & \leq -\beta \int_0^t \int_0^L \partial_x \zeta_{\varepsilon} \partial_{xxt} \zeta_{\varepsilon} \, dx ds - \alpha \int_0^t \int_0^L \partial_{xx} \eta_{\varepsilon} \partial_{xxt} \eta_{\varepsilon} \, dx ds + \int_0^t \int_{\Omega_0} f \cdot \partial_t v_{\varepsilon} \, dx dy ds \\ & + \int_0^t \int_{\Omega_0} \operatorname{div} h \cdot \partial_t v_{\varepsilon} \, dx dy ds - \int_0^t \int_0^L (h e_2)_{|y=1} \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds + \int_0^t \int_0^L (\ell, m) \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds \\ & + \frac{1}{2} \left( \int_{\Omega_0} \frac{\mu}{2 \det \nabla \chi_{a,b}} \left| ((B_{a,b}\nabla)v_0^{\varepsilon} + ((B_{a,b}\nabla)v_0^{\varepsilon})^T \right|^2 dx dy + \varepsilon \int_0^L |\partial_{xx} \zeta_1^{\varepsilon}|^2 dx + \varepsilon \int_0^L |\partial_{xx} \eta_1^{\varepsilon}|^2 dx \right). \end{aligned} $$

It follows then by using the uniform Korn's inequality (2.35) and the assumption on $(a, b)$ that there exists $K(R)$ such that

$$ \begin{aligned} & \int_0^t \int_{\Omega_0} \rho_{f,a,b} |\partial_t v_{\varepsilon}|^2 dx dy ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \zeta_{\varepsilon}|^2 dx ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \eta_{\varepsilon}|^2 v dx ds + \gamma \int_0^t \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx ds \\ & + \frac{1}{2} \left( \frac{\mu \lambda(R)}{2} \int_{\Omega_0} |\nabla v_{\varepsilon}(t)|^2 dx dy + \varepsilon \int_0^L |\partial_{xxt} \zeta_{\varepsilon}(t)|^2 dx + \varepsilon \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx \right) \\ & \leq -\beta \int_0^t \int_0^L \partial_x \zeta_{\varepsilon} \partial_{xxt} \zeta_{\varepsilon} \, dx ds - \alpha \int_0^t \int_0^L \partial_{xx} \eta_{\varepsilon} \partial_{xxt} \eta_{\varepsilon} \, dx ds + \int_0^t \int_{\Omega_0} f \cdot \partial_t v_{\varepsilon} \, dx dy ds \\ & + \int_0^t \int_{\Omega_0} \operatorname{div} h \cdot \partial_t v_{\varepsilon} \, dx dy ds - \int_0^t \int_0^L (h e_2)_{|y=1} \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds + \int_0^t \int_0^L (\ell, m) \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds \\ & + \frac{1}{2} \left( \frac{\mu K(R)}{2} \int_{\Omega_0} |\nabla v_0^{\varepsilon}|^2 dx dy + \varepsilon \int_0^L |\partial_{xx} \zeta_1^{\varepsilon}|^2 dx + \varepsilon \int_0^L |\partial_{xx} \eta_1^{\varepsilon}|^2 dx \right). \end{aligned} $$

Since $\rho_{f,a,b} \ge \frac{\rho_f}{R}$, and by performing integration by parts in time in the first two terms on the right hand side of the above inequality, we further obtain

$$ \begin{aligned} & \frac{\rho_f}{R} \int_0^t \int_{\Omega_0} |\partial_t v_{\varepsilon}|^2 dx dy ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \zeta_{\varepsilon}|^2 dx ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \eta_{\varepsilon}|^2 dx ds + \gamma \int_0^t \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx ds \\ & + \frac{1}{2} \left( \frac{\mu \lambda(R)}{2} \int_{\Omega_0} |\nabla v_{\varepsilon}(t)|^2 dx dy + \varepsilon \int_0^L |\partial_{xxt} \zeta_{\varepsilon}(t)|^2 dx + \varepsilon \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx \right) \\ & \leq \beta \int_0^t \int_0^L |\partial_{xt} \zeta_{\varepsilon}|^2 dx ds + \alpha \int_0^t \int_0^L |\partial_{xxt} \eta_{\varepsilon}|^2 dx ds + \int_0^t \int_{\Omega_0} f \cdot \partial_t v_{\varepsilon} \, dx dy ds \\ & + \int_0^t \int_{\Omega_0} \operatorname{div} h \cdot \partial_t v_{\varepsilon} \, dx dy ds - \int_0^t \int_0^L (h e_2)_{|y=1} \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds + \int_0^t \int_0^L (\ell, m) \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds \\ & + \frac{1}{2} \left( \frac{\mu K(R)}{2} \int_{\Omega_0} |\nabla v_0^{\varepsilon}|^2 dx dy + \varepsilon \int_0^L |\partial_{xx} \zeta_1^{\varepsilon}|^2 dx + \varepsilon \int_0^L |\partial_{xx} \eta_1^{\varepsilon}|^2 dx \right) \\ & - \beta \int_0^L \partial_x \zeta_{\varepsilon}(t) \partial_{xt} \zeta_{\varepsilon}(t) dx - \alpha \int_0^L \partial_{xx} \eta_{\varepsilon}(t) \partial_{xxt} \eta_{\varepsilon}(t) dx + \beta \int_0^L \partial_x \zeta_0^{\varepsilon} \partial_x \zeta_1^{\varepsilon} dx + \alpha \int_0^L \partial_{xx} \eta_0^{\varepsilon} \partial_{xx} \eta_1^{\varepsilon} dx. \end{aligned} $$
Next, integrating the above equation with respect to time on any time interval $[0, t] \subset [0, T)$ and using the following equality

$$ \int_{\Omega_0} \left( \mu(A_{a,b}\nabla)v_{\varepsilon} + \frac{\mu}{\det\nabla\chi_{a,b}} B_{a,b}\nabla v_{\varepsilon}^T B_{a,b} \right) : \nabla\partial_t v_{\varepsilon} \,dx dy = \frac{1}{4} \frac{d}{dt} \int_{\Omega_0} \frac{1}{\det\nabla\chi_{a,b}} \left| ((B_{a,b}\nabla)v_{\varepsilon} + ((B_{a,b}\nabla)v_{\varepsilon})^T) \right|^2 dx dy. $$

lead to

$$ \begin{aligned} & \int_0^t \int_{\Omega_0} \rho_{f,a,b} |\partial_t v_{\varepsilon}|^2 dx dy ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \zeta_{\varepsilon}|^2 dx ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \eta_{\varepsilon}|^2 v dx ds + \gamma \int_0^t \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx ds \\ & + \frac{1}{2} \left( \int_{\Omega_0} \frac{\mu}{2 \det \nabla \chi_{a,b}} \left| ((B_{a,b}\nabla)v_{\varepsilon}(t) + ((B_{a,b}\nabla)v_{\varepsilon}(t))^T \right|^2 dx dy + \varepsilon \int_0^L |\partial_{xxt} \zeta_{\varepsilon}(t)|^2 dx + \varepsilon \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx \right) \\ & \leq -\beta \int_0^t \int_0^L \partial_x \zeta_{\varepsilon} \partial_{xxt} \zeta_{\varepsilon} \, dx ds - \alpha \int_0^t \int_0^L \partial_{xx} \eta_{\varepsilon} \partial_{xxt} \eta_{\varepsilon} \, dx ds + \int_0^t \int_{\Omega_0} f \cdot \partial_t v_{\varepsilon} \, dx dy ds \\ & + \int_0^t \int_{\Omega_0} \operatorname{div} h \cdot \partial_t v_{\varepsilon} \, dx dy ds - \int_0^t \int_0^L (h e_2)_{|y=1} \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds + \int_0^t \int_0^L (\ell, m) \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds \\ & + \frac{1}{2} \left( \int_{\Omega_0} \frac{\mu}{2 \det \nabla \chi_{a,b}} \left| ((B_{a,b}\nabla)v_0^{\varepsilon} + ((B_{a,b}\nabla)v_0^{\varepsilon})^T \right|^2 dx dy + \varepsilon \int_0^L |\partial_{xx} \zeta_1^{\varepsilon}|^2 dx + \varepsilon \int_0^L |\partial_{xx} \eta_1^{\varepsilon}|^2 dx \right). \end{aligned} $$

It follows then by using the uniform Korn's inequality (2.35) and the assumption on $(a, b)$ that there exists $K(R)$ such that

$$ \begin{aligned} & \int_0^t \int_{\Omega_0} \rho_{f,a,b} |\partial_t v_{\varepsilon}|^2 dx dy ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \zeta_{\varepsilon}|^2 dx ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \eta_{\varepsilon}|^2 v dx ds + \gamma \int_0^t \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx ds \\ & + \frac{1}{2} \left( \frac{\mu \lambda(R)}{2} \int_{\Omega_0} |\nabla v_{\varepsilon}(t)|^2 dx dy + \varepsilon \int_0^L |\partial_{xxt} \zeta_{\varepsilon}(t)|^2 dx + \varepsilon \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx \right) \\ & \leq -\beta \int_0^t \int_0^L \partial_x \zeta_{\varepsilon} \partial_{xxt} \zeta_{\varepsilon} \, dx ds - \alpha \int_0^t \int_0^L \partial_{xx} \eta_{\varepsilon} \partial_{xxt} \eta_{\varepsilon} \, dx ds + \int_0^t \int_{\Omega_0} f \cdot \partial_t v_{\varepsilon} \, dx dy ds \\ & + \int_0^t \int_{\Omega_0} \operatorname{div} h \cdot \partial_t v_{\varepsilon} \, dx dy ds - \int_0^t \int_0^L (h e_2)_{|y=1} \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds + \int_0^t \int_0^L (\ell, m) \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds \\ & + \frac{1}{2} \left( \frac{\mu K(R)}{2} \int_{\Omega_0} |\nabla v_0^{\varepsilon}|^2 dx dy + \varepsilon \int_0^L |\partial_{xx} \zeta_1^{\varepsilon}|^2 dx + \varepsilon \int_0^L |\partial_{xx} \eta_1^{\varepsilon}|^2 dx \right). \end{aligned} $$

Since $\rho_{f,a,b} \ge \frac{\rho_f}{R}$, and by performing integration by parts in time in the first two terms on the right hand side of the above inequality, we further obtain

$$ \begin{aligned} & \frac{\rho_f}{R} \int_0^t \int_{\Omega_0} |\partial_t v_{\varepsilon}|^2 dx dy ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \zeta_{\varepsilon}|^2 dx ds + \rho_s \int_0^t \int_0^L |\partial_{tt} \eta_{\varepsilon}|^2 dx ds + \gamma \int_0^t \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx ds \\ & + \frac{1}{2} \left( \frac{\mu \lambda(R)}{2} \int_{\Omega_0} |\nabla v_{\varepsilon}(t)|^2 dx dy + \varepsilon \int_0^L |\partial_{xxt} \zeta_{\varepsilon}(t)|^2 dx + \varepsilon \int_0^L |\partial_{xxt} \eta_{\varepsilon}(t)|^2 dx \right) \\ & \leq \beta \int_0^t \int_0^L |\partial_{xt} \zeta_{\varepsilon}|^2 dx ds + \alpha \int_0^t \int_0^L |\partial_{xxt} \eta_{\varepsilon}|^2 dx ds + \int_0^t \int_{\Omega_0} f \cdot \partial_t v_{\varepsilon} \, dx dy ds \\ & + \int_0^t \int_{\Omega_0} \operatorname{div} h \cdot \partial_t v_{\varepsilon} \, dx dy ds - \int_0^t \int_0^L (h e_2)_{|y=1} \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds + \int_0^t \int_0^L (\ell, m) \cdot (\partial_{tt} \zeta_{\varepsilon}, \partial_{tt} \eta_{\varepsilon}) \, dx ds \\ & + \frac{1}{2} \left( \frac{\mu K(R)}{2} \int_{\Omega_0} |\nabla v_0^{\varepsilon}|^2 dx dy + \varepsilon \int_0^L |\partial_{xx} \zeta_1^{\varepsilon}|^2 dx + \varepsilon \int_0^L |\partial_{xx} \eta_1^{\varepsilon}|^2 dx \right) \\ & - \beta \int_0^L \partial_x \zeta_{\varepsilon}(t) \partial_{xt} \zeta_{\varepsilon}(t) dx - \alpha \int_0^L \partial_{xx} \eta_{\varepsilon}(t) \partial_{xxt} \eta_{\varepsilon}(t) dx + \beta \int_0^L \partial_x \zeta_0^{\varepsilon} \partial_x \zeta_1^{\varepsilon} dx + \alpha \int_0^L \partial_{xx} \eta_0^{\varepsilon} \partial_{xx} \eta_1^{\varepsilon} dx. \end{aligned} $$