@DGHL03] in the case of elastic bodies with the fixed interface; see also the literature cited in these papers.

We also note that the global (asymptotic) dynamics in nonlinear plate-fluid models were studied before in [@Chu_2010; @ChuRyz2011]. The article [@Chu_2010] deals with a class of fluid-plate interaction problems, when the plate, occupying ${{\Omega}}$, oscillates in [*longitudinal*]{} directions only. This kind of models arises in the study of blood flows in large arteries (see, e.g., [@Ggob-jmfm08] and the references therein). A fluid-plate interaction model, accounting for [*only transversal*]{} displacement of the plate, was studied in [@ChuRyz2011]. In contrast our mathematical model formulated below takes into account [*both*]{} transversal and in-plane displacements.

Let ${{\cal O}}\subset {{\mathbb R}}^3$ be a bounded domain with a sufficiently smooth boundary $\partial{{\cal O}}$. We assume that $\partial{{\cal O}}=\Omega\cup S$, where $${{\Omega}}\subset\{ x=(x_1;x_2;0)\, :\,x'\equiv
(x_1;x_2)\in{{\mathbb R}}^2\}$$ with a smooth contour $\Gamma=\partial{{\Omega}}$ and $S$ is a surface which lies in the subspace ${{\mathbb R}}^3_- =\{ x_3\le 0\}$. The exterior normal on $\partial{{\cal O}}$ is denoted by $n$. We have that $n=(0;0;1)$ on ${{\Omega}}$.

We consider the following [*linear*]{} Navier–Stokes equations in ${{\cal O}}$ for the fluid velocity field $v=v(x,t)=(v^1(x,t);v^2(x,t);v^3(x,t))$ and for the pressure $p(x,t)$: $$\label{fl.1}
   v_t-\nu\Delta v+\nabla p=G_f(t)\quad {\rm 