the method developed in [@CDEG05] for the case of a linear plate interacting with nonlinear Navier-Stokes equations).

Let $\{\psi_i\}_{i\in {{\mathbb N}}}$ be the orthonormal basis in $\widetilde X
=\{v\in X : v\big\vert_{{\Omega}}=0\}$ consisting of the eigenvectors of the Stokes problem: $$\begin{gathered}
-{\Delta}\psi_i +\nabla p_i =\mu_i\psi_i  \quad \mbox{in} \; {{\cal O}},~~~
{\rm div}\psi_i=0, \quad \psi_i|_{{\partial}{{\cal O}}}=0.\end{gathered}$$ Here $0<\mu_1\le \mu_2\le \cdots$ are the corresponding eigenvalues. Denote by $\{\xi_i\}_{i\in{{\mathbb N}}}$ the basis in ${\widehat H^2_{0}({{\Omega}})}$ which consists of the eigenfunctions of the following problem $$({\Delta}\xi_i,{\Delta}w)_{{\Omega}}=\hat{\kappa}_i (M_\alpha\xi_i, w)_{{\Omega}},~~~\forall\, w\in {\widehat{H}}^2_0({{\Omega}}),$$ with the eigenvalues $0<\hat{\kappa}_1\le\hat{\kappa}_2\le\ldots$ and such that $(M_\alpha\xi_i,\xi_j)_{{{\Omega}}}=\delta_{ij}$. Further, let $\{\eta_i\}_{i\in{{\mathbb N}}}$ be the basis in $H^1_0({{\Omega}}) \times H^1_0({{\Omega}})$ which consists of eigenfunctions of the problem $$a(\eta_i,w)= \tilde{\kappa}_i (\eta_i, w)_{{\Omega}}, \qquad
\forall\, w\in H^1_0({{\Omega}})\times H^1_0({{\Omega}}),$$ with the eigenvalues $0<\tilde{\kappa}_1\le\tilde{\kappa}_2\le\ldots$ and $||\eta_i||_{{{\Omega}}}=1$. The form $a(\eta,w)$ is given by .

Let ${\hat{\phi}}_i=N_0(0;0;\xi_i)$ and ${\tilde{\phi}}_i=N_0(\eta_i;0)$, where the operator $N_0$ is defined by (\[fl.n0\]). One can see that $\eta_i \in [H^{3/2-\delta}({{\Omega}})]^2$. Therefore by Proposition \[pr:stokes\] we have that ${\hat{\phi}}_i, {\tilde{\phi}}_i \in [H^{2-\delta}({{\cal O}})]^3\cap V$ for every $\delta>0$.

In what follows we suppose $\phi_i={\tilde{\phi}}_i$, $\zeta_i=(\eta_i;0)$, $\kappa_i= \tilde{\kappa}_i$, and ${{\gamma}}_i=0$; $\phi_{n+i}={\hat{\phi}}_i$, $\zeta_{n+i}=(0,0,\xi_i)$, $\kappa_{n+i}=\hat{\kappa}_i$, and ${{\gamma}}_{n+i}={{\gamma}}$ for $i=1,\ldots,n$. We also define the parameter ${{\varrho}}_i$ by the relations: ${{\varrho}}_i={{\varrho}}$ for $i=1,\ldots,n$ and 