has dimension one. ]{}

Stokes problem
--------------

In further considerations we need some regularity properties of the terms responsible for fluid–plate interaction. To this end we consider the following Stokes problem $$\begin{aligned}
  -\nu\Delta v+\nabla p= g, \quad
   {{\rm div\, }}v=0 \quad {\rm in}\quad {{\cal O}}; \nonumber
\\
 v=0 ~~ {\rm on}~S;
\quad
v=\psi=(\psi^1;\psi^2;\psi^3) ~~{\rm on} ~ {{\Omega}},\label{stokes}\end{aligned}$$ where $g\in [L^2({{\cal O}})]^3$ and $\psi\in [L^2({{\Omega}})]^2 \times {\widehat L_2({{\Omega}})}$ are given. This type of boundary value problems for the Stokes equation was studied by many authors (see, e.g., [@lad-NSbook] and [@temam-NS] and references therein). We collect some properties of solutions to in the following assertion.

\[pr:stokes\] The following statements hold.

-   Let $g\in [H^{-1+\sigma}({{\cal O}})]^3$, and $\psi\in [H^{1/2+\sigma}_*({{\Omega}})]^3$ with $\int_{{\Omega}}\psi^3(x')dx'=0$. Then for every $0\le \sigma\le  1$ problem has a unique solution $\{v;p\}$ in $[H^{1+\sigma}({{\cal O}})]^3\times[ H^{\sigma}({{\cal O}})/{{\mathbb R}}]$ such that $$\label{stokes-bnd1}
    \|v\|_{[H^{1+\sigma}({{\cal O}})]^3}+\|p\|_{H^{\sigma}({{\cal O}})/{{\mathbb R}}}
    \le c_0\left\{\|g\|_{[H^{-1+\sigma}({{\cal O}})]^3}+\|\psi\|_{[H_*^{\sigma+1/2}({{\Omega}})]^3} \right\}.$$

-   If $g=0$, $\psi\in [H_*^{-1/2+{{\sigma}}}({{\Omega}})]^3$, $0\le {{\sigma}}\le 1$, $\int_{{\Omega}}\psi^3(x')dx'=0$, then $$\label{stokes-bnd2}
    \|v\|_{[H^{{{\sigma}}}({{\cal O}})]^3}+\|p\|_{H^{-1+{{\sigma}}}({{\cal O}})/{{\mathbb R}}}
    \le c_0\|\psi\|_{[H_*^{-1/2+{{\sigma}}}({{\Omega}})]^3}.$$ In particular, we can define a linear operator $N_0 : [L^2({{\Omega}})]^2 \times {\widehat L_2({{\Omega}})}\mapsto [H^{1/2}({{\cal O}})]^3$ by the formula $$\label{fl.n0}
    N_0\psi=w ~~\mbox{iff}~~\left\{
    \begin{array}{l}
     -\nu\Delta w+\nabla p=0, \quad
       {{\rm div\, }}w=0 \quad {\rm in}\quad {{\cal O}};
    \\
     