  We study dynamics of a coupled system consisting of the 3D Navier--Stokes
equations which is linearized near a certain Poiseuille type flow in an
(unbounded) domain and a classical (possibly nonlinear) elastic plate equation
for transversal displacement on a flexible flat part of the boundary. We first
show that this problem generates an evolution semigroup $S_t$ on an appropriate
phase space. Then under some conditions concerning the underlying (Poiseuille
type) flow we prove the existence of a compact finite-dimensional global
attractor for this semigroup and also show that $S_t $ is an exponentially
stable $C_0$-semigroup of linear operators in the fully linear case. Since we
do not assume any kind of mechanical damping in the plate component, this means
that dissipation of the energy in the fluid flow due to viscosity is sufficient
to stabilize the system.
