term is well defined and belong to $L^2(0, T; H_\#^1(\Omega_0))$ for $T \le T(R, M)$. By using Holder's inequality, we get that there exists $C > 0$ such that

$$ \|h^a\|_{L^2(0,T;H_\#^1(\Omega_0))} \le C \left( \|A_{\zeta,\eta} - A_{\zeta_0,\eta_0}\|_{L^\infty(0,T;L_\#^\infty(\Omega_0))} \|z\|_{L^2(0,T;H_\#^2(\Omega_0))} + \|(\nabla(A_{\zeta,\eta} - A_{\zeta_0,\eta_0})\nabla)z\|_{L^2(0,T;L_\#^2(\Omega_0))} \right), $$

Since $\|(z, r)\|_{X_{f,T}} \le M$ and by using Lemma 2.1, we obtain, for $s > 0$,

$$ \|h^a\|_{L^2(0,T;H_\#^1(\Omega_0))} \le CM \left( \|A_{\zeta,\eta} - A_{\zeta_0,\eta_0}\|_{L^\infty(0,T;L_\#^\infty(\Omega_0))} + \|A_{\zeta,\eta} - A_{\zeta_0,\eta_0}\|_{L^\infty(0,T;H_\#^{1+s}(\Omega_0))} \right), $$

Thus by using the Sobolev injection of $H_\#^{1+s}(\Omega_0) \subset L_\#^\infty(\Omega_0)$ and Lemma 4.1, we get, for $0 < s < 1/2$

$$ \|h^a\|_{L^2(0,T;H_\#^1(\Omega_0))} \le CM^2 T^{\frac{1}{2}-s}. \quad (4.13) $$

The two other terms can be estimated in the same way. By combining (4.6) with the above lemma, we deduce the following corollary.

**Corollary 4.1** For $T$ small enough, $\mathcal{L}_T^0$ maps $K_T(M)$ into itself for $M \ge M_0$.

We turn now to show that for $T$ small enough and $M \ge M_0$, the mapping $\mathcal{L}_T^0: K_T(M) \to K_T(M)$ is a contraction. Let $(z^{(1)}, r^{(1)}, \bar{\zeta}^{(1)}, \bar{\eta}^{(1)})$ and $(z^{(2)}, r^{(2)}, \bar{\zeta}^{(2)}, \bar{\eta}^{(2)})$ in $K_T(M)$. For $i = 1, 2$, denote by

$$ (v^{(i)}, q^{(i)}, \bar{\chi}^{(i)}, \bar{\mathcal{Y}}^{(i)}) = \mathcal{L}_T^0(z^{(i)}, r^{(i)}, \bar{\zeta}^{(i)}, \bar{\eta}^{(i)}). $$

We recall that $(\bar{\zeta}^{(i)}, \bar{\eta}^{(i)}, \chi^{(i)}, \mathcal{Y}^{(i)}) = (\zeta^{(i)} - \zeta_0, \eta^{(i)} - \eta_0, \chi^{(i)} - \zeta_0, \mathcal{Y}^{(i)} - \eta_0)$ with $(v^{(i)}, q^{(i)}, \chi^{(i)}, \mathcal{Y}^{(i)})$ is a solution of Problem (3.1)-(3.6) with $(a, b) = (\zeta_0, \eta_0)$ and source term $(f^{(i)}, h^{(i)}g^{(i)}, 0, 0)$ defined as follows:

$$ f^{(i)} = f[z^{(i)}, \zeta^{(i)}, \eta^{(i)}], \quad h^{(i)} = h[z^{(i)}, r^{(i)}, \zeta^{(i)}, \eta^{(i)}], \quad g^{(i)} = g[z^{(i)}, \zeta^{(i)}, \eta^{(i)}]. \quad (4.14) $$

Then

$$ \mathcal{L}_T^0(z^{(1)}, r^{(1)}, \bar{\zeta}^{(1)}, \bar{\eta}^{(1)}) - \mathcal{L}_T^0(z^{(2)}, r^{(2)}, \bar{\zeta}^{(2)}, \bar{\eta}^{(2)}) = (v^{(1)}, q^{(1)}, \zeta^{(1)}, \eta^{(1)}) - (v^{(2)}, q^{(2)}, \zeta^{(2)}, \eta^{(2)}) \quad (4.15) $$

Denote by $(v, q, \zeta, \eta) = (v^{(1)}, q^{(1)}, \zeta^{(1)}, \eta^{(1)}) - (v^{(2)}, q^{(2)}, \zeta^{(2)}, \eta^{(2)})$. Then $(v, q, \zeta, \eta)$ is a solution of Problem (3.1) with $(a, b) = (\zeta_0, \eta_0)$, zero initial data and source term $(f, h, g, 0, 0) = (f^{(1)} - f^{(2)}, h^{(1)} - h^{(2)}, g^{(1)} - g^{(2)}, 0, 0)$. Hence if the source term $(f, h, g, 0, 0) \in S_T$, then by using Proposition 3.2 for $(a, b) = (\zeta_0, \eta_0)$ we get

$$ \|(v, q)\|_{X_{f,T}} + \|(\zeta, \eta)\|_{X_{s,T}} \le C(R) \|(f, h, g, 0, 0)\|_{S_T}, $$

and thus in view of (4.15) we get

$$ \|\mathcal{L}_T^0(z^{(1)}, r^{(1)}, \bar{\zeta}^{(1)}, \bar{\eta}^{(1)}) - \mathcal{L}_T^0(z^{(2)}, r^{(2)}, \bar{\zeta}^{(2)}, \bar{\eta}^{(2)})\|_{X_{f,T} \times X_{s,T}} \le C(R) \|(f^{(1)} - f^{(2)}, h^{(1)} - h^{(2)}, g^{(1)} - g^{(2)}, 0, 0)\|_{S_T}. \quad (4.16) $$

To proceed we have to bound the source term $(f^{(1)} - f^{(2)}, h^{(1)} - h^{(2)}, g^{(1)} - g^{(2)}, 0, 0)$ defined by (4.14) respectively in $S_T$ in the right hand side of (4.16). In order to establish such bounds, we gather in the following lemma some properties of the matrix $B_{\zeta,\eta}$ and $A_{\zeta,\eta}$ defined respectively by (2.11), (2.12), that can be derived in the very same way as in Lemma 4.1.
**Lemma 4.3** Consider two couples $(\zeta^{(1)}, \eta^{(1)})$ and $(\zeta^{(2)}, \eta^{(2)})$ in $L^{\infty}(0, T; H_{\#}^2(0, L)) \cap W^{1, \infty}(0, T; H_{\#}^1(0, L))$ such that
$(\zeta^{(1)}(0), \eta^{(1)}(0)) = (\zeta^{(2)}(0), \eta^{(2)}(0)) = (\zeta_0, \eta_0)$. Then, for $0 < s < \frac{1}{2}$,

$$
\begin{aligned}
\|B_{\zeta^{(1)}, \eta^{(1)}} - B_{\zeta^{(2)}, \eta^{(2)}}\|_{L^{\infty}(0, T; H_{\#}^{1+s}(\Omega_0))} &\le CT^{\frac{1}{2}-s} \|(\bar{\zeta}, \bar{\eta})\|_{L^{\infty}(0, T; H_{\#}^2(0, L))}^{\frac{1}{2}+s} \|(\bar{\zeta}, \bar{\eta})\|_{W^{1, \infty}(0, T; H_{\#}^1(0, L))}^{\frac{1}{2}-s} \\
\|\partial_t B_{\zeta^{(1)}, \eta^{(1)}} - \partial_t B_{\zeta^{(2)}, \eta^{(2)}}\|_{L^{\infty}(0, T; L_{\#}^4(\Omega_0))} &\le C \|(\bar{\zeta}, \bar{\eta})\|_{L^{\infty}(0, T; H_{\#}^2(0, L)) \cap W^{1, \infty}(0, T; H_{\#}^1(0, L))},
\end{aligned}
$$

Furthermore, if we assume that

$$
\| (\zeta_0, \eta_0) \|_{H_{\#}^2(0, L)} + \|(\det \nabla \chi_{\zeta_0, \eta_0})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R,
$$

$$
\| (\zeta^{(i)}, \eta^{(i)}) \|_{L^{\infty}(0, T; H_{\#}^2(0, L)) \cap W^{1, \infty}(0, T; H_{\#}^1(0, L))} \le M,
$$

there exists a time $T(R, M)$ depending on $R$ and $M$ such that for $T \le T(R, M)$, $\det(\nabla\chi_{\zeta^{(i)},\eta^{(i)}})^{-1}$ and $A_{\zeta^{(i)},\eta^{(i)}}$ are well defined in $C([0,T]; H_{\#}^{1+s}(\Omega_0))$, for $0 < s < \frac{1}{2}$, and

$$
\|(\det \nabla \chi_{\zeta^{(1)}, \eta^{(1)}})^{-1} - (\det \nabla \chi_{\zeta^{(2)}, \eta^{(2)}})^{-1}\|_{L^{\infty}(0, T; H_{\#}^{1+s}(\Omega_0))} \le C(R, M)T^{\frac{1}{2}-s},
$$

$$
\| A_{\zeta^{(1)}, \eta^{(1)}} - A_{\zeta^{(2)}, \eta^{(2)}} \|_{L^{\infty}(0, T; H_{\#}^{1+s}(\Omega_0))} \le C(R, M) T^{\frac{1}{2}-s},
$$

We move now to bound the source terms.

**Lemma 4.4** Given $(z^{(1)}, r^{(1)}, \bar{\zeta}^{(1)}, \bar{\eta}^{(1)})$ and $(z^{(2)}, r^{(2)}, \bar{\zeta}^{(2)}, \bar{\eta}^{(2)})$ in $K_T(M)$ and $(\zeta_0, \eta_0) \in H_{\#}^2(0, L) \times H_{\#}^3(0, L)$. Let $f^{(i)}$, $h^{(i)}$, $g^{(i)}$ be defined as (4.14) with $(\zeta^{(i)}, \eta^{(i)}) = (\bar{\zeta}^{(i)} + \zeta_0, \bar{\eta}^{(i)} + \eta_0)$ for $i = 1, 2$, then there exists a positive constant $C$ and a time $T(M, R)$ such that for $T \le T(M, R)$

$$
\|f^{(1)} - f^{(2)}\|_{L^2(0,T; L_{\#}^2(\Omega_0))} \le C(R)MT^{\frac{1}{4}} \|(z^{(1)} - z^{(2)}, 0, \bar{\zeta}^{(1)} - \bar{\zeta}^{(2)}, \bar{\eta}^{(1)} - \bar{\eta}^{(2)})\|_{X_{f,T} \times X_{s,T}}, \quad (4.17)
$$

$$
\|h^{(1)} - h^{(2)}\|_{L^2(0,T; H_{\#}^1(\Omega_0))} \le CT^{\frac{1}{4}}M \|(z^{(1)} - z^{(2)}, r^{(1)} - r^{(2)}, \bar{\zeta}^{(1)} - \bar{\zeta}^{(2)}, \bar{\eta}^{(1)} - \bar{\eta}^{(2)})\|_{X_{f,T} \times X_{s,T}}, \quad (4.18)
$$

$$
\|g^{(1)} - g^{(2)}\|_{L^2(0,T; H_{\#}^1(\Omega_0))} \le CMT^{\frac{1}{4}} \|(z^{(1)} - z^{(2)}, 0, \bar{\zeta}^{(1)} - \bar{\zeta}^{(2)}, \bar{\eta}^{(1)} - \bar{\eta}^{(2)})\|_{X_{f,T} \times X_{s,T}}, \quad (4.19)
$$

$$
\|g^{(1)} - g^{(2)}\|_{H_{00}^1(0, T; (H_{\#}^1(\Omega_0)))'} \le CMT^{\frac{1}{4}} \|(z^{(1)} - z^{(2)}, 0, \bar{\zeta}^{(1)} - \bar{\zeta}^{(2)}, \bar{\eta}^{(1)} - \bar{\eta}^{(2)})\|_{X_{f,T} \times X_{s,T}}, \quad (4.20)
$$

Proof. We skip the proof of the first three estimates as they can be derived thanks to Lemma 2.1, Lemma 4.3 and by following similar arguments as in [28]. We focus on the last estimate as it makes appear terms that vanish in the only transversal case and require a special attention. The estimate will once again be done by duality. Let $w \in H_{\#}^1(\Omega_0)$ and $\varphi \in C_c^{\infty}(0, T)$. We have

$$
\int_0^T \int_{\Omega_0} (g^{(1)} - g^{(2)}) w \partial_t \varphi = \int_0^T \int_{\Omega_0} \operatorname{div} \left( (B_{\zeta_0, \eta_0}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T)(z^{(1)} - z^{(2)}) \right) w \partial_t \varphi \\
\qquad - \int_0^T \int_{\Omega_0} \operatorname{div} \left( (B_{\zeta^{(1)}, \eta^{(1)}}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) z^{(1)} \right) w \partial_t \varphi \quad (4.21)
$$

By performing integration by parts with respect to space we obtain

$$
\begin{align*}
\int_0^T \int_{\Omega_0} (g^{(1)} - g^{(2)}) w \partial_t \varphi &= - \int_0^T \int_{\Omega_0} \left( (B_{\zeta_0, \eta_0}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T)(z^{(1)} - z^{(2)}) \right) \cdot \nabla w \partial_t \varphi \\
&\quad + \int_0^T \int_{\Omega_0} \left( (B_{\zeta^{(1)}, \eta^{(1)}}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) z^{(1)} \right) \cdot \nabla w \partial_t \varphi \\
&\quad + \int_0^T \int_0^L \left[ \partial_x \bar{\eta}^{(2)} (\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}) - \partial_x \bar{\zeta}^{(2)} (\partial_t \eta^{(1)} - \partial_t \eta^{(2)}) \right] w|_{y=1} \partial_t \varphi \\
&\quad - \int_0^T \int_0^L \left( (\partial_x \eta^{(1)} - \partial_x \eta^{(2)}) \partial_t \zeta^{(1)} - (\partial_x \zeta^{(1)} - \partial_x \zeta^{(2)}) \partial_t \eta^{(1)} \right) w|_{y=1} \partial_t \varphi.
\end{align*}
$$
**Lemma 4.3** Consider two couples $(\zeta^{(1)}, \eta^{(1)})$ and $(\zeta^{(2)}, \eta^{(2)})$ in $L^{\infty}(0, T; H_{\#}^2(0, L)) \cap W^{1, \infty}(0, T; H_{\#}^1(0, L))$ such that
$(\zeta^{(1)}(0), \eta^{(1)}(0)) = (\zeta^{(2)}(0), \eta^{(2)}(0)) = (\zeta_0, \eta_0)$. Then, for $0 < s < \frac{1}{2}$,

$$
\begin{align*}
\|B_{\zeta^{(1)}, \eta^{(1)}} - B_{\zeta^{(2)}, \eta^{(2)}}\|_{L^{\infty}(0, T; H_{\#}^{1+s}(\Omega_0))} &\le CT^{\frac{1}{2}-s} \|(\bar{\zeta}, \bar{\eta})\|_{L^{\infty}(0, T; H_{\#}^2(0, L))}^{\frac{1}{2}+s} \|(\bar{\zeta}, \bar{\eta})\|_{W^{1, \infty}(0, T; H_{\#}^1(0, L))}^{\frac{1}{2}-s} \\
\|\partial_t B_{\zeta^{(1)}, \eta^{(1)}} - \partial_t B_{\zeta^{(2)}, \eta^{(2)}}\|_{L^{\infty}(0, T; L_{\#}^4(\Omega_0))} &\le C \|(\bar{\zeta}, \bar{\eta})\|_{L^{\infty}(0, T; H_{\#}^2(0, L)) \cap W^{1, \infty}(0, T; H_{\#}^1(0, L))},
\end{align*}
$$

Furthermore, if we assume that

$$
\| (\zeta_0, \eta_0) \|_{H_{\#}^2(0, L)} + \|(\det \nabla \chi_{\zeta_0, \eta_0})^{-1}\|_{C_{\#}(\bar{\Omega}_0)} \le R,
$$

$$
\| (\zeta^{(i)}, \eta^{(i)}) \|_{L^{\infty}(0, T; H_{\#}^2(0, L)) \cap W^{1, \infty}(0, T; H_{\#}^1(0, L))} \le M,
$$

there exists a time $T(R, M)$ depending on $R$ and $M$ such that for $T \le T(R, M)$, $\det(\nabla\chi_{\zeta^{(i)},\eta^{(i)}})^{-1}$ and $A_{\zeta^{(i)},\eta^{(i)}}$
are well defined in $C([0,T]; H_{\#}^{1+s}(\Omega_0))$, for $0 < s < \frac{1}{2}$, and

$$
\| (\det \nabla \chi_{\zeta^{(1)}, \eta^{(1)}})^{-1} - (\det \nabla \chi_{\zeta^{(2)}, \eta^{(2)}})^{-1} \|_{L^{\infty}(0, T; H_{\#}^{1+s}(\Omega_0))} \le C(R, M) T^{\frac{1}{2}-s},
$$

$$
\| A_{\zeta^{(1)}, \eta^{(1)}} - A_{\zeta^{(2)}, \eta^{(2)}} \|_{L^{\infty}(0, T; H_{\#}^{1+s}(\Omega_0))} \le C(R, M) T^{\frac{1}{2}-s},
$$

We move now to bound the source terms.

**Lemma 4.4** Given $(z^{(1)}, r^{(1)}, \bar{\zeta}^{(1)}, \bar{\eta}^{(1)})$ and $(z^{(2)}, r^{(2)}, \bar{\zeta}^{(2)}, \bar{\eta}^{(2)})$ in $K_T(M)$ and $(\zeta_0, \eta_0) \in H_{\#}^2(0, L) \times H_{\#}^3(0, L)$. Let $f^{(i)}$, $h^{(i)}$, $g^{(i)}$ be defined as (4.14) with $(\zeta^{(i)}, \eta^{(i)}) = (\bar{\zeta}^{(i)} + \zeta_0, \bar{\eta}^{(i)} + \eta_0)$ for $i = 1, 2$, then there exists a positive constant $C$ and a time $T(M, R)$ such that for $T \le T(M, R)$

$$
\| f^{(1)} - f^{(2)} \|_{L^2(0,T; L_{\#}^2(\Omega_0))} \le C(R)MT^{\frac{1}{4}} \|(z^{(1)} - z^{(2)}, 0, \bar{\zeta}^{(1)} - \bar{\zeta}^{(2)}, \bar{\eta}^{(1)} - \bar{\eta}^{(2)}) \|_{X_{f,T} \times X_{s,T}}, \quad (4.17)
$$

$$
\|h^{(1)} - h^{(2)}\|_{L^2(0, T; H_{\#}^1(\Omega_0))} \le CT^{\frac{1}{4}} M \|(z^{(1)} - z^{(2)}, r^{(1)} - r^{(2)}, \bar{\zeta}^{(1)} - \bar{\zeta}^{(2)}, \bar{\eta}^{(1)} - \bar{\eta}^{(2)})\|_{X_{f,T} \times X_{s,T}}, \quad (4.18)
$$

$$
\| g^{(1)} - g^{(2)} \|_{L^2(0, T; H_{\#}^1(\Omega_0))} \le CMT^{\frac{1}{4}} \|(z^{(1)} - z^{(2)}, 0, \bar{\zeta}^{(1)} - \bar{\zeta}^{(2)}, \bar{\eta}^{(1)} - \bar{\eta}^{(2)}) \|_{X_{f,T} \times X_{s,T}}, \quad (4.19)
$$

$$
\| g^{(1)} - g^{(2)} \|_{H_{00}^1(0, T; (H_{\#}^1(\Omega_0)))'} \le CMT^{\frac{1}{4}} \|(z^{(1)} - z^{(2)}, 0, \bar{\zeta}^{(1)} - \bar{\zeta}^{(2)}, \bar{\eta}^{(1)} - \bar{\eta}^{(2)}) \|_{X_{f,T} \times X_{s,T}}, \quad (4.20)
$$

Proof. We skip the proof of the first three estimates as they can be derived thanks to Lemma 2.1, Lemma 4.3 and by following similar arguments as in [28]. We focus on the last estimate as it makes appear terms that vanish in the only transversal case and require a special attention. The estimate will once again be done by duality. Let $w \in H_{\#}^1(\Omega_0)$ and $\varphi \in C_c^{\infty}(0, T)$. We have

$$
\int_0^T \int_{\Omega_0} (g^{(1)} - g^{(2)}) w \partial_t \varphi = \int_0^T \int_{\Omega_0} \operatorname{div} \left( (B_{\zeta_0, \eta_0}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T)(z^{(1)} - z^{(2)}) \right) w \partial_t \varphi \\
\qquad - \int_0^T \int_{\Omega_0} \operatorname{div} \left( (B_{\zeta^{(1)}, \eta^{(1)}}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) z^{(1)} \right) w \partial_t \varphi \quad (4.21)
$$

By performing integration by parts with respect to space we obtain

$$
\begin{align*}
\int_0^T \int_{\Omega_0} (g^{(1)} - g^{(2)}) w \partial_t \varphi &= - \int_0^T \int_{\Omega_0} \left( (B_{\zeta_0, \eta_0}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T)(z^{(1)} - z^{(2)}) \right) \cdot \nabla w \partial_t \varphi \\
&\quad + \int_0^T \int_{\Omega_0} \left( (B_{\zeta^{(1)}, \eta^{(1)}}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) z^{(1)} \right) \cdot \nabla w \partial_t \varphi \\
&\quad + \int_0^T \int_0^L \left[ \partial_x \bar{\eta}^{(2)} (\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}) - \partial_x \bar{\zeta}^{(2)} (\partial_t \eta^{(1)} - \partial_t \eta^{(2)}) \right] w|_{y=1} \partial_t \varphi \\
&\quad - \int_0^T \int_0^L \left( (\partial_x \eta^{(1)} - \partial_x \eta^{(2)}) \partial_t \zeta^{(1)} - (\partial_x \zeta^{(1)} - \partial_x \zeta^{(2)}) \partial_t \eta^{(1)} \right) w|_{y=1} \partial_t \varphi.
\end{align*}
$$
Now, by performing integration by parts with respect to time we get for any $\varphi \in C_c^\infty(0, T)$

$$
\begin{aligned}
\int_0^T \int_{\Omega_0} (g^{(1)} - g^{(2)}) w \partial_t \varphi &= \int_0^T \int_{\Omega_0} (-\partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T (z^{(1)} - z^{(2)}) + (B_{\zeta_0, \eta_0}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) (\partial_t z^{(1)} - \partial_t z^{(2)})) \cdot \nabla w \varphi \\
&\quad - \int_0^T \int_{\Omega_0} ((\partial_t B_{\zeta^{(1)}, \eta^{(1)}}^T - \partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T) z^{(1)} + (B_{\zeta^{(1)}, \eta^{(1)}}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) \partial_t z^{(1)}) \cdot \nabla w \varphi \\
&\quad - \int_0^T \int_0^L \partial_t (\partial_x \bar{\eta}^{(2)} (\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}) - \partial_x \bar{\zeta}^{(2)} (\partial_t \eta^{(1)} - \partial_t \eta^{(2)})) w_{|_{y=1}} \varphi \\
&\quad + \int_0^T \int_0^L \partial_t ((\partial_x \eta^{(1)} - \partial_x \eta^{(2)}) \partial_t \zeta^{(1)} - (\partial_x \zeta^{(1)} - \partial_x \zeta^{(2)}) \partial_t \eta^{(1)}) w_{|_{y=1}} \varphi.
\end{aligned}
$$

Consequently, we have

$$
\begin{aligned}
\langle \partial_t (g^{(1)} - g^{(2)}), w \rangle_{(H_\sharp^1(\Omega_0))', H_\sharp^1(\Omega_0)} &= \int_{\Omega_0} (-\partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T (z^{(1)} - z^{(2)}) + (B_{\zeta_0, \eta_0}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) (\partial_t z^{(1)} - \partial_t z^{(2)})) \cdot \nabla w \\
&\quad - \int_{\Omega_0} ((\partial_t B_{\zeta^{(1)}, \eta^{(1)}}^T - \partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T) z^{(1)} + (B_{\zeta^{(1)}, \eta^{(1)}}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) \partial_t z^{(1)}) \cdot \nabla w \\
&\quad - \int_0^L (\partial_{xt} \bar{\eta}^{(2)} (\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}) - \partial_{xt} \bar{\zeta}^{(2)} (\partial_t \eta^{(1)} - \partial_t \eta^{(2)}) - \partial_x \bar{\eta}^{(2)} (\partial_{tt} \zeta^{(1)} - \partial_{tt} \zeta^{(2)}) + \partial_x \bar{\zeta}^{(2)} (\partial_{tt} \eta^{(1)} - \partial_{tt} \eta^{(2)})) w_{|_{y=1}} \\
&\quad + \int_0^L ((\partial_{xt} \eta^{(1)} - \partial_{xt} \eta^{(2)}) \partial_t \zeta^{(1)} - (\partial_{xt} \zeta^{(1)} - \partial_{xt} \zeta^{(2)}) \partial_t \eta^{(1)} + (\partial_x \eta^{(1)} - \partial_x \eta^{(2)}) \partial_{tt} \zeta^{(1)} - (\partial_x \zeta^{(1)} - \partial_x \zeta^{(2)}) \partial_{tt} \eta^{(1)}) w_{|_{y=1}}.
\end{aligned}
$$

Thanks to Holder's inequality, Sobolev injections and trace theorem we have

$$
\begin{aligned}
& |\langle \partial_t (g^{(1)} - g^{(2)}), w \rangle_{(H_\sharp^1(\Omega_0))', H_\sharp^1(\Omega_0)}| \\
& \leq (C \|\partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T\|_{L_\sharp^4(\Omega_0)} \|z^{(1)} - z^{(2)}\|_{H_\sharp^1(\Omega_0)} + \|B_{\zeta_0, \eta_0} - B_{\zeta^{(2)}, \eta^{(2)}}\|_{L_\sharp^\infty(\Omega_0)} \|\partial_t z^{(1)} - \partial_t z^{(2)}\|_{L_\sharp^2(\Omega_0)} \\
& \quad + C \|\partial_t B_{\zeta^{(1)}, \eta^{(1)}}^T - \partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T\|_{L_\sharp^4(\Omega_0)} \|z^{(1)}\|_{H_\sharp^1(\Omega_0)} + \|B_{\zeta^{(1)}, \eta^{(1)}} - B_{\zeta^{(2)}, \eta^{(2)}}\|_{L_\sharp^\infty(\Omega_0)} \|\partial_t z^{(1)}\|_{L_\sharp^2(\Omega_0)}) \|w\|_{H_\sharp^1(\Omega_0)} \\
& \quad + C (\|\partial_t \bar{\eta}^{(2)}\|_{L_\sharp^2(0,L)} \|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{L_\sharp^\infty(0,L)} + \|\partial_t \bar{\zeta}^{(2)}\|_{L_\sharp^2(0,L)} \|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L_\sharp^\infty(0,L)} \\
& \quad + \|\partial_x \bar{\eta}^{(2)}\|_{L_\sharp^\infty(0,L)} \|\partial_{tt} \zeta^{(1)} - \partial_{tt} \zeta^{(2)}\|_{L_\sharp^2(0,L)} + \|\partial_x \bar{\zeta}^{(2)}\|_{L_\sharp^\infty(0,L)} \|\partial_{tt} \eta^{(1)} - \partial_{tt} \eta^{(2)}\|_{L_\sharp^2(0,L)} \\
& \quad + \|\partial_{xt} \eta^{(1)} - \partial_{xt} \eta^{(2)}\|_{L_\sharp^2(0,L)} \|\partial_t \zeta^{(1)}\|_{L_\sharp^\infty(0,L)} + \|\partial_{xt} \zeta^{(1)} - \partial_{xt} \zeta^{(2)}\|_{L_\sharp^2(0,L)} \|\partial_t \eta^{(1)}\|_{L_\sharp^\infty(0,L)} \\
& \quad + \|\partial_x \eta^{(1)} - \partial_x \eta^{(2)}\|_{L_\sharp^\infty(0,L)} \|\partial_{tt} \zeta^{(1)}\|_{L_\sharp^2(0,L)} + \|\partial_x \zeta^{(1)} - \partial_x \zeta^{(2)}\|_{L_\sharp^\infty(0,L)} \|\partial_{tt} \eta^{(1)}\|_{L_\sharp^2(0,L)}) \|w\|_{H_\sharp^1(\Omega_0)}.
\end{aligned}
$$

Using the Sobolev injection of $H^{\frac{1}{2}+s}(0, L) \subset C_{\#}^1(0, L)$ for any $s > 0$ it follows that

$$
\begin{aligned}
\|\partial_t (g^{(1)} - g^{(2)})\|_{(H_\sharp^1(\Omega_0))'} &\le C \|\partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T\|_{L_\sharp^4(\Omega_0)} \|z^{(1)} - z^{(2)}\|_{H_\sharp^1(\Omega_0)} + \|B_{\zeta_0, \eta_0} - B_{\zeta^{(2)}, \eta^{(2)}}\|_{L_\sharp^\infty(\Omega_0)} \|\partial_t z^{(1)} - \partial_t z^{(2)}\|_{L_\sharp^2(\Omega_0)} \\
&\quad + C \|\partial_t B_{\zeta^{(1)}, \eta^{(1)}}^T - \partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T\|_{L_\sharp^4(\Omega_0)} \|z^{(1)}\|_{H_\sharp^1(\Omega_0)} + \|B_{\zeta^{(1)}, \eta^{(1)}} - B_{\zeta^{(2)}, \eta^{(2)}}\|_{L_\sharp^\infty(\Omega_0)} \|\partial_t z^{(1)}\|_{L_\sharp^2(\Omega_0)} \\
&\quad + C (\|\partial_t \bar{\eta}^{(2)}\|_{H_\sharp^1(0,L)} \|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{H_\sharp^1(0,L)} + \|\partial_t \bar{\zeta}^{(2)}\|_{H_\sharp^1(0,L)} \|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{H_\sharp^1(0,L)} \\
&\quad + \|\bar{\eta}^{(2)}\|_{H_\sharp^{\frac{3}{2}+s}(0,L)} \|\partial_{tt} \zeta^{(1)} - \partial_{tt} \zeta^{(2)}\|_{L_\sharp^2(0,L)} + \|\bar{\zeta}^{(2)}\|_{H_\sharp^{\frac{3}{2}+s}(0,L)} \|\partial_{tt} \eta^{(1)} - \partial_{tt} \eta^{(2)}\|_{L_\sharp^2(0,L)} \\
&\quad + \|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{H_\sharp^1(0,L)} \|\partial_t \zeta^{(1)}\|_{H_\sharp^1(0,L)} + \|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{H_\sharp^1(0,L)} \|\partial_t \eta^{(1)}\|_{H_\sharp^1(0,L)} \\
&\quad + \|\eta^{(1)} - \eta^{(2)}\|_{H_\sharp^{\frac{3}{2}+s}(0,L)} \|\partial_{tt} \zeta^{(1)}\|_{L_\sharp^2(0,L)} + \|\zeta^{(1)} - \zeta^{(2)}\|_{H_\sharp^{\frac{3}{2}+s}(0,L)} \|\partial_{tt} \eta^{(1)}\|_{L_\sharp^2(0,L)}).
\end{aligned}
$$
Now, by performing integration by parts with respect to time we get for any $\varphi \in C_c^\infty(0, T)$

$$
\begin{aligned}
\int_0^T \int_{\Omega_0} (g^{(1)} - g^{(2)}) w \partial_t \varphi &= \int_0^T \int_{\Omega_0} (-\partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T (z^{(1)} - z^{(2)}) + (B_{\zeta_0, \eta_0}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) (\partial_t z^{(1)} - \partial_t z^{(2)})) \cdot \nabla w \varphi \\
&\quad - \int_0^T \int_{\Omega_0} ((\partial_t B_{\zeta^{(1)}, \eta^{(1)}}^T - \partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T) z^{(1)} + (B_{\zeta^{(1)}, \eta^{(1)}}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) \partial_t z^{(1)}) \cdot \nabla w \varphi \\
&\quad - \int_0^T \int_0^L \partial_t (\partial_x \bar{\eta}^{(2)} (\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}) - \partial_x \bar{\zeta}^{(2)} (\partial_t \eta^{(1)} - \partial_t \eta^{(2)})) w|_{y=1} \varphi \\
&\quad + \int_0^T \int_0^L \partial_t ((\partial_x \eta^{(1)} - \partial_x \eta^{(2)}) \partial_t \zeta^{(1)} - (\partial_x \zeta^{(1)} - \partial_x \zeta^{(2)}) \partial_t \eta^{(1)}) w|_{y=1} \varphi.
\end{aligned}
$$

Consequently, we have

$$
\begin{aligned}
\langle \partial_t (g^{(1)} - g^{(2)}), w \rangle_{(H_\sharp^1(\Omega_0))', H_\sharp^1(\Omega_0)} &= \int_{\Omega_0} (-\partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T (z^{(1)} - z^{(2)}) + (B_{\zeta_0, \eta_0}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) (\partial_t z^{(1)} - \partial_t z^{(2)})) \cdot \nabla w \\
&\quad - \int_{\Omega_0} ((\partial_t B_{\zeta^{(1)}, \eta^{(1)}}^T - \partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T) z^{(1)} + (B_{\zeta^{(1)}, \eta^{(1)}}^T - B_{\zeta^{(2)}, \eta^{(2)}}^T) \partial_t z^{(1)}) \cdot \nabla w \\
&\quad - \int_0^L (\partial_{xt} \bar{\eta}^{(2)} (\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}) - \partial_{xt} \bar{\zeta}^{(2)} (\partial_t \eta^{(1)} - \partial_t \eta^{(2)}) - \partial_x \bar{\eta}^{(2)} (\partial_{tt} \zeta^{(1)} - \partial_{tt} \zeta^{(2)}) + \partial_x \bar{\zeta}^{(2)} (\partial_{tt} \eta^{(1)} - \partial_{tt} \eta^{(2)})) w|_{y=1} \\
&\quad + \int_0^L ((\partial_{xt} \eta^{(1)} - \partial_{xt} \eta^{(2)}) \partial_t \zeta^{(1)} - (\partial_{xt} \zeta^{(1)} - \partial_{xt} \zeta^{(2)}) \partial_t \eta^{(1)} + (\partial_x \eta^{(1)} - \partial_x \eta^{(2)}) \partial_{tt} \zeta^{(1)} - (\partial_x \zeta^{(1)} - \partial_x \zeta^{(2)}) \partial_{tt} \eta^{(1)}) w|_{y=1}.
\end{aligned}
$$

Thanks to Holder's inequality, Sobolev injections and trace theorem we have

$$
\begin{aligned}
& |\langle \partial_t (g^{(1)} - g^{(2)}), w \rangle_{(H_\sharp^1(\Omega_0))', H_\sharp^1(\Omega_0)}| \\
& \leq (C \|\partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T\|_{L_\sharp^4(\Omega_0)} \|z^{(1)} - z^{(2)}\|_{H_\sharp^1(\Omega_0)} + \|B_{\zeta_0, \eta_0} - B_{\zeta^{(2)}, \eta^{(2)}}\|_{L_\sharp^\infty(\Omega_0)} \|\partial_t z^{(1)} - \partial_t z^{(2)}\|_{L_\sharp^2(\Omega_0)} \\
& \quad + C \|\partial_t B_{\zeta^{(1)}, \eta^{(1)}}^T - \partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T\|_{L_\sharp^4(\Omega_0)} \|z^{(1)}\|_{H_\sharp^1(\Omega_0)} + \|B_{\zeta^{(1)}, \eta^{(1)}} - B_{\zeta^{(2)}, \eta^{(2)}}\|_{L_\sharp^\infty(\Omega_0)} \|\partial_t z^{(1)}\|_{L_\sharp^2(\Omega_0)}) \|w\|_{H_\sharp^1(\Omega_0)} \\
& \quad + C (\|\partial_t \bar{\eta}^{(2)}\|_{L_\sharp^2(0,L)} \|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{L_\sharp^\infty(0,L)} + \|\partial_t \bar{\zeta}^{(2)}\|_{L_\sharp^2(0,L)} \|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L_\sharp^\infty(0,L)} \\
& \quad + \|\partial_x \bar{\eta}^{(2)}\|_{L_\sharp^\infty(0,L)} \|\partial_{tt} \zeta^{(1)} - \partial_{tt} \zeta^{(2)}\|_{L_\sharp^2(0,L)} + \|\partial_x \bar{\zeta}^{(2)}\|_{L_\sharp^\infty(0,L)} \|\partial_{tt} \eta^{(1)} - \partial_{tt} \eta^{(2)}\|_{L_\sharp^2(0,L)} \\
& \quad + \|\partial_x \eta^{(1)} - \partial_x \eta^{(2)}\|_{L_\sharp^\infty(0,L)} \|\partial_t \zeta^{(1)}\|_{L_\sharp^\infty(0,L)} + \|\partial_x \zeta^{(1)} - \partial_x \zeta^{(2)}\|_{L_\sharp^2(0,L)} \|\partial_t \eta^{(1)}\|_{L_\sharp^\infty(0,L)} \\
& \quad + \|\partial_x \eta^{(1)} - \partial_x \eta^{(2)}\|_{L_\sharp^\infty(0,L)} \|\partial_{tt} \zeta^{(1)}\|_{L_\sharp^2(0,L)} + \|\partial_x \zeta^{(1)} - \partial_x \zeta^{(2)}\|_{L_\sharp^\infty(0,L)} \|\partial_{tt} \eta^{(1)}\|_{L_\sharp^2(0,L)}) \|w\|_{H_\sharp^1(\Omega_0)}.
\end{aligned}
$$

Using the Sobolev injection of $H^{\frac{1}{2}+s}(0, L) \subset C_{\#}^1(0, L)$ for any $s > 0$ it follows that

$$
\begin{aligned}
\|\partial_t (g^{(1)} - g^{(2)})\|_{(H_\sharp^1(\Omega_0))'} &\le C \|\partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T\|_{L_\sharp^4(\Omega_0)} \|z^{(1)} - z^{(2)}\|_{H_\sharp^1(\Omega_0)} + \|B_{\zeta_0, \eta_0} - B_{\zeta^{(2)}, \eta^{(2)}}\|_{L_\sharp^\infty(\Omega_0)} \|\partial_t z^{(1)} - \partial_t z^{(2)}\|_{L_\sharp^2(\Omega_0)} \\
&\quad + C \|\partial_t B_{\zeta^{(1)}, \eta^{(1)}}^T - \partial_t B_{\zeta^{(2)}, \eta^{(2)}}^T\|_{L_\sharp^4(\Omega_0)} \|z^{(1)}\|_{H_\sharp^1(\Omega_0)} + \|B_{\zeta^{(1)}, \eta^{(1)}} - B_{\zeta^{(2)}, \eta^{(2)}}\|_{L_\sharp^\infty(\Omega_0)} \|\partial_t z^{(1)}\|_{L_\sharp^2(\Omega_0)} \\
&\quad + C (\|\partial_t \bar{\eta}^{(2)}\|_{H_\sharp^1(0,L)} \|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{H_\sharp^1(0,L)} + \|\partial_t \bar{\zeta}^{(2)}\|_{H_\sharp^1(0,L)} \|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{H_\sharp^1(0,L)} \\
&\quad + \|\bar{\eta}^{(2)}\|_{H_\sharp^{\frac{3}{2}+s}(0,L)} \|\partial_{tt} \zeta^{(1)} - \partial_{tt} \zeta^{(2)}\|_{L_\sharp^2(0,L)} + \|\bar{\zeta}^{(2)}\|_{H_\sharp^{\frac{3}{2}+s}(0,L)} \|\partial_{tt} \eta^{(1)} - \partial_{tt} \eta^{(2)}\|_{L_\sharp^2(0,L)} \\
&\quad + \|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{H_\sharp^1(0,L)} \|\partial_t \zeta^{(1)}\|_{H_\sharp^1(0,L)} + \|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{H_\sharp^1(0,L)} \|\partial_t \eta^{(1)}\|_{H_\sharp^1(0,L)} \\
&\quad + \|\eta^{(1)} - \eta^{(2)}\|_{H_\sharp^{\frac{3}{2}+s}(0,L)} \|\partial_{tt} \zeta^{(1)}\|_{L_\sharp^2(0,L)} + \|\zeta^{(1)} - \zeta^{(2)}\|_{H_\sharp^{\frac{3}{2}+s}(0,L)} \|\partial_{tt} \eta^{(1)}\|_{L_\sharp^2(0,L)}).
\end{aligned}
$$
Thus,

$$
\begin{aligned}
\|\partial_t(g^{(1)} - g^{(2)})\|_{L^2(0,T;H_\frac{1}{2}^1(\Omega_0))'} &\le CT^{\frac{1}{2}}\|\partial_t B_{\zeta^{(2)},\eta^{(2)}}^T\|_{L^\infty(0,T;L_\frac{1}{2}^4(\Omega_0))} \|z^{(1)} - z^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(\Omega_0))} \\
&\quad + \|B_{\zeta_0,\eta_0} - B_{\zeta^{(2)},\eta^{(2)}}\|_{L^\infty(0,T;L_\frac{1}{2}^\infty(\Omega_0))} \|\partial_t z^{(1)} - \partial_t z^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(\Omega_0))} \\
&\quad + CT^{\frac{1}{2}}\|\partial_t B_{\zeta^{(1)},\eta^{(1)}}^T - \partial_t B_{\zeta^{(2)},\eta^{(2)}}^T\|_{L^\infty(0,T;L_\frac{1}{2}^4(\Omega_0))} \|z^{(1)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(\Omega_0))} \\
&\quad + \|B_{\zeta^{(1)},\eta^{(1)}} - B_{\zeta^{(2)},\eta^{(2)}}\|_{L^\infty(0,T;L_\frac{1}{2}^\infty(\Omega_0))} \|\partial_t z^{(1)}\|_{L^2(0,T;L_\frac{1}{2}^2(\Omega_0))} \\
&\quad + C(T^{\frac{1}{2}}\|\partial_t \bar{\eta}^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \bar{\zeta}^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + \|\bar{\eta}^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{\frac{3}{2}+s}(0,L))} \|\partial_t t\zeta^{(1)} - \partial_t t\zeta^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))} \\
&\quad + \|\bar{\zeta}^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{\frac{3}{2}+s}(0,L))} \|\partial_t t\eta^{(1)} - \partial_t t\eta^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \|\partial_t \zeta^{(1)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \|\partial_t \zeta^{(1)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + \|\eta^{(1)} - \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{\frac{3}{2}+s}(0,L))} \|\partial_t t\zeta^{(1)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))} \\
&\quad + \|\zeta^{(1)} - \zeta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{\frac{3}{2}+s}(0,L))} \|\partial_t t\eta^{(1)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))}
\end{aligned}
$$

By using inequality (4.7), Lemma 4.1 and noting that $(z^{(i)}, 0, \zeta^{(i)}, \eta^{(i)}) \in \mathcal{K}_{T,M}$ for $i=1,2$, we obtain

$$
\begin{aligned}
\|\partial_t(g^{(1)} - g^{(2)})\|_{L^2(0,T;H_\frac{1}{2}^1(\Omega_0))'} &\le CM(T^{\frac{1}{2}}\|z^{(1)} - z^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(\Omega_0))} + T^s\|\partial_t z^{(1)} - \partial_t z^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(\Omega_0))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t B_{\zeta^{(1)},\eta^{(1)}}^T - \partial_t B_{\zeta^{(2)},\eta^{(2)}}^T\|_{L^\infty(0,T;L_\frac{1}{2}^4(\Omega_0))} \\
&\quad + \|B_{\zeta^{(1)},\eta^{(1)}} - B_{\zeta^{(2)},\eta^{(2)}}\|_{L^\infty(0,T;L_\frac{1}{2}^\infty(\Omega_0))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + T^{\frac{1}{2}-s}\|\partial_t t\zeta^{(1)} - \partial_t t\zeta^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))} \\
&\quad + T^{\frac{1}{2}-s}\|\partial_t t\eta^{(1)} - \partial_t t\eta^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + \|\eta^{(1)} - \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{\frac{3}{2}+s}(0,L))} \\
&\quad + \|\zeta^{(1)} - \zeta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{\frac{3}{2}+s}(0,L))}
\end{aligned}
$$

Finally, by using Lemma 4.3 and (4.7), and taking $s = 1/4$ we obtain we establish the estimate (4.20).

Combining the result of the above Lemma with (4.16), we deduce the following:

**Corollary 4.2** For $T$ small enough and $M \ge M_0$, the mapping $\mathcal{L}_T^0: \mathcal{K}_T(M) \to \mathcal{K}_T(M)$ is a contraction.

Consequently we have proved existence and uniqueness of strong solution of a periodic two dimensional fluid-beam interaction problem for which no additional viscosity has been added to the structure equations and without neglecting the longitudinal displacement. There no gap between the initial regularity and the one obtain at any positive time. Note that additional regularity may surely be derived for the transversal displacement by using properties of the equation with inertia of rotation, yet this is out of the scope of the present study. As for the fluid-wave case the dissipation of the fluid enables to control the velocity of the structure and it is therefore likely to prove that the semi-group associated to the linear problem is analytic paving the way to the study of the long time behavior of this coupled system. Note that considering other boundary conditions at the inlet and outlet induce additional difficulties and in particular on may loose the elliptic result.

## References

[1] M. BADRA AND T. TAKAHASHI, *Feedback boundary stabilization of 2d fluid-structure interaction systems*, Discrete and Continuous Dynamical Systems - Series A, 37 (5), pp.2315-2373, 2017.
Thus,

$$
\begin{aligned}
\|\partial_t(g^{(1)} - g^{(2)})\|_{L^2(0,T;H_\frac{1}{2}^1(\Omega_0))'} &\le CT^{\frac{1}{2}}\|\partial_t B_{\zeta^{(2)},\eta^{(2)}}^T\|_{L^\infty(0,T;L_\frac{1}{2}^4(\Omega_0))} \|z^{(1)} - z^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(\Omega_0))} \\
&\quad + \|B_{\zeta_0,\eta_0} - B_{\zeta^{(2)},\eta^{(2)}}\|_{L^\infty(0,T;L_\frac{1}{2}^\infty(\Omega_0))} \|\partial_t z^{(1)} - \partial_t z^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(\Omega_0))} \\
&\quad + CT^{\frac{1}{2}}\|\partial_t B_{\zeta^{(1)},\eta^{(1)}}^T - \partial_t B_{\zeta^{(2)},\eta^{(2)}}^T\|_{L^\infty(0,T;L_\frac{1}{2}^4(\Omega_0))} \|z^{(1)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(\Omega_0))} \\
&\quad + \|B_{\zeta^{(1)},\eta^{(1)}} - B_{\zeta^{(2)},\eta^{(2)}}\|_{L^\infty(0,T;L_\frac{1}{2}^\infty(\Omega_0))} \|\partial_t z^{(1)}\|_{L^2(0,T;L_\frac{1}{2}^2(\Omega_0))} \\
&\quad + C(T^{\frac{1}{2}}\|\partial_t \bar{\eta}^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \bar{\zeta}^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + \|\bar{\eta}^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{3+s}(0,L))} \|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))} + \|\bar{\zeta}^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{3+s}(0,L))} \|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \|\partial_t \zeta^{(1)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \|\partial_t \zeta^{(1)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + \|\eta^{(1)} - \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{3+s}(0,L))} \|\partial_t \zeta^{(1)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))} + \|\zeta^{(1)} - \zeta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{3+s}(0,L))} \|\partial_t \eta^{(1)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))} \).
\end{aligned}
$$

By using inequality (4.7), Lemma 4.1 and noting that $(z^{(i)}, 0, \zeta^{(i)}, \eta^{(i)}) \in \mathcal{K}_{T,M}$ for $i=1,2$, we obtain

$$
\begin{aligned}
\|\partial_t(g^{(1)} - g^{(2)})\|_{L^2(0,T;H_\frac{1}{2}^1(\Omega_0))'} &\le CM(T^{\frac{1}{2}}\|z^{(1)} - z^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(\Omega_0))} + T^s\|\partial_t z^{(1)} - \partial_t z^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(\Omega_0))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t B_{\zeta^{(1)},\eta^{(1)}}^T - \partial_t B_{\zeta^{(2)},\eta^{(2)}}^T\|_{L^\infty(0,T;L_\frac{1}{2}^4(\Omega_0))} + \|B_{\zeta^{(1)},\eta^{(1)}} - B_{\zeta^{(2)},\eta^{(2)}}\|_{L^\infty(0,T;L_\frac{1}{2}^\infty(\Omega_0))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} + T^{\frac{1}{2}}\|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + T^{\frac{1}{2}-s}\|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))} + T^{\frac{1}{2}-s}\|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^2(0,T;L_\frac{1}{2}^2(0,L))} \\
&\quad + T^{\frac{1}{2}}\|\partial_t \eta^{(1)} - \partial_t \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} + T^{\frac{1}{2}}\|\partial_t \zeta^{(1)} - \partial_t \zeta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^1(0,L))} \\
&\quad + \|\eta^{(1)} - \eta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{3+s}(0,L))} + \|\zeta^{(1)} - \zeta^{(2)}\|_{L^\infty(0,T;H_\frac{1}{2}^{3+s}(0,L))} \).
\end{aligned}
$$

Finally, by using Lemma 4.3 and (4.7), and taking $s = 1/4$ we obtain we establish the estimate (4.20).

Combining the result of the above Lemma with (4.16), we deduce the following:

**Corollary 4.2** For $T$ small enough and $M \ge M_0$, the mapping $\mathcal{L}_T^0: \mathcal{K}_T(M) \to \mathcal{K}_T(M)$ is a contraction.

Consequently we have proved existence and uniqueness of strong solution of a periodic two dimensional fluid-beam interaction problem for which no additional viscosity has been added to the structure equations and without neglecting the longitudinal displacement. There no gap between the initial regularity and the one obtain at any positive time. Note that additional regularity may surely be derived for the transversal displacement by using properties of the equation with inertia of rotation, yet this is out of the scope of the present study. As for the fluid-wave case the dissipation of the fluid enables to control the velocity of the structure and it is therefore likely to prove that the semi-group associated to the linear problem is analytic paving the way to the study of the long time behavior of this coupled system. Note that considering other boundary conditions at the inlet and outlet induce additional difficulties and in particular on may loose the elliptic result.

## References

[1] M. BADRA AND T. TAKAHASHI, *Feedback boundary stabilization of 2d fluid-structure interaction systems*, Discrete and Continuous Dynamical Systems - Series A, 37 (5), pp.2315-2373, 2017.
[2] M. BADRA AND T. TAKAHASHI, *Gevrey regularity for a system coupling the Navier-Stokes system with a beam equation*, SIAM Journal on Mathematical Analysis 51 (6), 4776-4814, 2019.

[3] M. BADRA AND T. TAKAHASHI, *Gevrey regularity for a system coupling the Navier-Stokes system with a beam: the non-flat case*, Funkcialaj Ekvacioj, 65(1), 63-109, 2022.

[4] M. BADRA AND T. TAKAHASHI, *Analyticity of the semigroup associated with a Stokes-wave interaction system and application to the system of interaction between a viscous incompressible fluid and an elastic structure*, Journal of Evolution Equations, 2022.

[5] H. BEIRÃO DA VEIGA, *On the existence of strong solutions to a coupled fluid-structure evolution problem*, J. Math. Fluid Mech., 6(1):21-52, 2004.

[6] M. BOULAKIA, *Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid*, Journal of Mathematical Fluid Mechanics, 9(2):262-294, 2007.

[7] M. BOULAKIA AND S. GUERRERO, *Regular solutions of a problem coupling a compressible fluid and an elastic structure*, Journal de Mathématiques Pures et Appliquées, 94(4):341-365, 2010.

[8] M. BOULAKIA, E. L. SCHWINDT, AND T. TAKAHASHI, *Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid*, Interfaces Free Bound, 14(3):273-306, 2012.

[9] J. A BELLO, *$L^r$ regularity for the Stokes and Navier-Stokes problems*. Ann. Mat. Pura Appl. 4, 170, 187-206 (1996).

[10] D. BREIT, *Regularity results in 2D fluid-structure interaction*, Mathematische Annalen, 1-44, 2022.

[11] M. BUKAČ, S. ČANIĆ, R. GLOWINSKI, J. TAMBAČA, A. QUAINI, *Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement*, Journal of Computational Physics, 235: 515-541, 2013.

[12] J.-J. CASANOVA, *Fluid structure system with boundary conditions involving the pressure*, Journal of Evolution Equations, 2020.

[13] J.-J. CASANOVA, C. GRANDMONT, M. HILLAIRET. *On an existence theory for a fluid-beam problem encom-passing possible contacts*. Journal de l'École polytechnique, 2021.

[14] A. CHAMBOLLE, B. DESJARDINS, M. J. ESTEBAN, AND C. GRANDMONT, *Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate*, J. Math. Fluid Mech., 7(3):368-404, 2005.

[15] P. G. CIARLET, *Mathematical elasticity. Vol I. Three-dimensional elasticity*, North-Holland Publishing Co., Amsterdam, 1988.

[16] M. DEBAYAN, J.P. RAYMOND, AND A. ROY, it Maximal-in-time existence and uniqueness of strong solution of a 3D fluid-structure interaction model, SIAM Journal on Mathematical Analysis 52.6, 6338-6378, 2020.

[17] B. DESJARDINS, M. J. ESTEBAN, C. GRANDMONT, AND P. LE TALLEC, *Weak solutions for a fluid-elastic structure interaction model*, Rev. Mat. Complut., 14(2):523-538, 2001.
[2] M. BADRA AND T. TAKAHASHI, *Gevrey regularity for a system coupling the Navier-Stokes system with a beam equation*, SIAM Journal on Mathematical Analysis 51 (6), 4776-4814, 2019.

[3] M. BADRA AND T. TAKAHASHI, *Gevrey regularity for a system coupling the Navier-Stokes system with a beam: the non-flat case*, Funkcialaj Ekvacioj, 65(1), 63-109, 2022.

[4] M. BADRA AND T. TAKAHASHI, *Analyticity of the semigroup associated with a Stokes-wave interaction system and application to the system of interaction between a viscous incompressible fluid and an elastic structure*, Journal of Evolution Equations, 2022.

[5] H. BEIRÃO DA VEIGA, *On the existence of strong solutions to a coupled fluid-structure evolution problem*, J. Math. Fluid Mech., 6(1):21-52, 2004.

[6] M. BOULAKIA, *Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid*, Journal of Mathematical Fluid Mechanics, 9(2):262-294, 2007.

[7] M. BOULAKIA AND S. GUERRERO, *Regular solutions of a problem coupling a compressible fluid and an elastic structure*, Journal de Mathématiques Pures et Appliquées, 94(4):341-365, 2010.

[8] M. BOULAKIA, E. L. SCHWINDT, AND T. TAKAHASHI, *Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid*, Interfaces Free Bound, 14(3):273-306, 2012.

[9] J. A BELLO, *$L^r$ regularity for the Stokes and Navier-Stokes problems*. Ann. Mat. Pura Appl. 4, 170, 187-206 (1996).

[10] D. BREIT, *Regularity results in 2D fluid-structure interaction*, Mathematische Annalen, 1-44, 2022.

[11] M. BUKAČ, S. ČANIĆ, R. GLOWINSKI, J. TAMBAČA, A. QUAINI, *Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement*, Journal of Computational Physics, 235: 515-541, 2013.

[12] J.-J. CASANOVA, *Fluid structure system with boundary conditions involving the pressure*, Journal of Evolution Equations, 2020.

[13] J.-J. CASANOVA, C. GRANDMONT, M. HILLAIRET. *On an existence theory for a fluid-beam problem encom-passing possible contacts*. Journal de l'École polytechnique, 2021.

[14] A. CHAMBOLLE, B. DESJARDINS, M. J. ESTEBAN, AND C. GRANDMONT, *Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate*, J. Math. Fluid Mech., 7(3):368-404, 2005.

[15] P. G. CIARLET, *Mathematical elasticity. Vol I. Three-dimensional elasticity*, North-Holland Publishing Co., Amsterdam, 1988.

[16] M. DEBAYAN, J.P. RAYMOND, AND A. ROY, it Maximal-in-time existence and uniqueness of strong solution of a 3D fluid-structure interaction model, SIAM Journal on Mathematical Analysis 52.6, 6338-6378, 2020.

[17] B. DESJARDINS, M. J. ESTEBAN, C. GRANDMONT, AND P. LE TALLEC, *Weak solutions for a fluid-elastic structure interaction model*, Rev. Mat. Complut., 14(2):523-538, 2001.
[18] D. LENGELER, *Weak Solutions for An Incompressible, Generalized Newtonian Fluid Interacting with a Linearly Elastic Koiter Type Shell*, 46(4):2614-2649, 2014.

[19] M. DEBAYAN AND T. TAKAHASHI, *$L^p$ Theory for the Interaction Between the Incompressible Navier-Stokes System and a Damped Plate*, Journal of Mathematical Fluid Mechanics 23, no. 4, 103, 2021.

[20] F. FLORI AND P. ORENGA, *On a nonlinear fluid-structure interaction problem defined on a domain depending on time*, Nonlinear Anal. Ser. B: Real World Appl. 38(5), 549-569, 1999.

[21] F. FLORI AND P. ORENGA, *Fluid-structure interaction: analysis of a 3-D compressible model*, Ann. Inst. H. Poincaré Anal. Non Linéaire 17(6), 753-777, 2000.

[22] G. P. GALDI AND M. KYED, *Steady flow of a Navier-Stokes liquid past an elastic body*, Arch. Ration. Mech. Anal., 194(3):849-875, 2009.

[23] C. GRANDMONT, *Existence et unicité de solutions d'un problème de couplage fluide-structure bidimensionnel stationnaire*, C. R. Acad. Sci. Paris Sér. I Math., 326(5):651:656, 1998.

[24] C. GRANDMONT, *Existence for a three-dimensional steady state fluid-structure interaction problem*, J. Math. Fluid Mech., 4(1):76-94, 2002.

[25] C. GRANDMONT, *On an unsteady fluid beam interaction problem*, preprint, 2004.

[26] C. GRANDMONT, *Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate*, SIAM J. Math. Anal., 40(2):716-737, 2008.

[27] C. GRANDMONT AND M. HILLAIRET, *Existence of Global Strong Solutions to a Beam-Fluid Interaction System*, Arch. Ratio. Mech. Anal., 220(3): 205-255, 2016.

[28] C. GRANDMONT, M. HILLAIRET, J. LEQUEUERRE, *Existence of Local Strong Solutions to Fluid-Beam and Fluid-rod Interaction Systems*, Ann. Inst. H. Poincaré Anal. Non Linéaire 36, 2019.

[29] C. GRANDMONT AND Y. MADAY, *Existence for an unsteady fluid-structure interaction problem*, M2AN Math. Model. Numer. Anal. 34(3), 609-636, 2000.

[30] J. LEQUEURRE, *Existence of strong solutions to a fluid-structure system*, SIAM J. Math. Anal., 43(1):389-410, 2011.

[31] J. LEQUEURRE, *Existence of strong solutions for a system coupling the Navier-Stokes equations and a damped wave equation*, J. Math. Fluid Mech., 15(2):249-271, 2013.

[32] B. MUHA AND S. CANIĆ, *Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls*, Arch. Ration. Mech. Anal., 207(3):919-968, 2013.

[33] B. MUHA AND S. SCHWARZACHER *Existence and regularity for weak solutions for a fluid interacting with a non-linear shell in three dimensions*. Annal. l'Inst. Henri Poincaré / Analyse non linéaire, 39: 1369-141, 2022.
[18] D. LENGELER, *Weak Solutions for An Incompressible, Generalized Newtonian Fluid Interacting with a Linearly Elastic Koiter Type Shell*, 46(4):2614-2649, 2014.

[19] M. DEBAYAN AND T. TAKAHASHI, *$L^p$ Theory for the Interaction Between the Incompressible Navier-Stokes System and a Damped Plate*, Journal of Mathematical Fluid Mechanics 23, no. 4, 103, 2021.

[20] F. FLORI AND P. ORENGA, *On a nonlinear fluid-structure interaction problem defined on a domain depending on time*, Nonlinear Anal. Ser. B: Real World Appl. 38(5), 549-569, 1999.

[21] F. FLORI AND P. ORENGA, *Fluid-structure interaction: analysis of a 3-D compressible model*, Ann. Inst. H. Poincaré Anal. Non Linéaire 17(6), 753-777, 2000.

[22] G. P. GALDI AND M. KYED, *Steady flow of a Navier-Stokes liquid past an elastic body*, Arch. Ration. Mech. Anal., 194(3):849-875, 2009.

[23] C. GRANDMONT, *Existence et unicité de solutions d'un problème de couplage fluide-structure bidimensionnel stationnaire*, C. R. Acad. Sci. Paris Sér. I Math., 326(5):651:656, 1998.

[24] C. GRANDMONT, *Existence for a three-dimensional steady state fluid-structure interaction problem*, J. Math. Fluid Mech., 4(1):76-94, 2002.

[25] C. GRANDMONT, *On an unsteady fluid beam interaction problem*, preprint, 2004.

[26] C. GRANDMONT, *Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate*, SIAM J. Math. Anal., 40(2):716-737, 2008.

[27] C. GRANDMONT AND M. HILLAIRET, *Existence of Global Strong Solutions to a Beam-Fluid Interaction System*, Arch. Ratio. Mech. Anal., 220(3): 205-255, 2016.

[28] C. GRANDMONT, M. HILLAIRET, J. LEQUEUERRE, *Existence of Local Strong Solutions to Fluid-Beam and Fluid-rod Interaction Systems*, Ann. Inst. H. Poincaré Anal. Non Linéaire 36, 2019.

[29] C. GRANDMONT AND Y. MADAY, *Existence for an unsteady fluid-structure interaction problem*, M2AN Math. Model. Numer. Anal. 34(3), 609-636, 2000.

[30] J. LEQUEURRE, *Existence of strong solutions to a fluid-structure system*, SIAM J. Math. Anal., 43(1):389-410, 2011.

[31] J. LEQUEURRE, *Existence of strong solutions for a system coupling the Navier-Stokes equations and a damped wave equation*, J. Math. Fluid Mech., 15(2):249-271, 2013.

[32] B. MUHA AND S. CANIĆ, *Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls*, Arch. Ration. Mech. Anal., 207(3):919-968, 2013.

[33] B. MUHA AND S. SCHWARZACHER *Existence and regularity for weak solutions for a fluid interacting with a non-linear shell in three dimensions*. Annal. l'Inst. Henri Poincaré / Analyse non linéaire, 39: 1369-141, 2022.
[34] S. SCHWARZACHER, P. SU *Existence of strong solutions for a perfect elastic beam interacting with Navier-Stokes equations*, preprint.

[35] W. POMPE *Korn's First Inequality with variable coefficients and its generalization* Commentationes Mathematicae Universitatis Carolinae, Vol. 44 (2003), No. 1, 57-70

[36] A. QUARTERONI, M. TUVERI, AND VENEZIANI, *Computational vascular fluid dynamics: problems, models and methods*, Comput Visual Sci 2, 163-197, 2000.

[37] A. RAOULT, *Construction d'un modèle d'évolution de plaques avec terme d'inerte de rotation*. Annali di Matematica Pura ed applicata, 139:361-400, 1985.

[38] S. SCHWARZACHER AND M.SROCZINSKI, *Weak-Strong Uniqueness for an Elastic Plate Interacting with the Navier-Stokes Equation*, SIAM Journal on Mathematical Analysis 54(4), 4104-4138, 2022.

[39] L. TARTAR *An Abstract Regularity Theorem*. Note 87.9 CMU, 1989, 1970

[40] R. TEMAM, *Navier-Stokes equations: theory and numerical analysis*, volume 343., American Mathematical Soc., 33-35, 2001.
[34] S. SCHWARZACHER, P. SU *Existence of strong solutions for a perfect elastic beam interacting with Navier-Stokes equations*, preprint.

[35] W. POMPE *Korn's First Inequality with variable coefficients and its generalization* Commentationes Mathematicae Universitatis Carolinae, Vol. 44 (2003), No. 1, 57-70

[36] A. QUARTERONI, M. TUVERI, AND VENEZIANI, *Computational vascular fluid dynamics: problems, models and methods*, Comput Visual Sci 2, 163-197, 2000.

[37] A. RAOULT, *Construction d'un modèle d'évolution de plaques avec terme d'inerte de rotation*. Annali di Matematica Pura ed applicata, 139:361-400, 1985.

[38] S. SCHWARZACHER AND M.SROCZINSKI, *Weak-Strong Uniqueness for an Elastic Plate Interacting with the Navier-Stokes Equation*, SIAM Journal on Mathematical Analysis 54(4), 4104-4138, 2022.

[39] L. TARTAR *An Abstract Regularity Theorem*. Note 87.9 CMU, 1989, 1970

[40] R. TEMAM, *Navier-Stokes equations: theory and numerical analysis*, volume 343., American Mathematical Soc., 33-35, 2001.