  We study asymptotic dynamics of a coupled system consisting of linearized 3D
Navier--Stokes equations in a bounded domain and the classical (nonlinear)
elastic plate equation for in-plane motions on a flexible flat part of the
boundary. The main peculiarity of the model is the assumption that the
transversal displacements of the plate are negligible relative to in-plane
displacements. This kind of models arises in the study of blood flows in large
arteries. Our main result states the existence of a compact global attractor of
finite dimension. We also show that the corresponding linearized system
generates exponentially stable $C_0$-semigroup. We do not assume any kind of
mechanical damping in the plate component. Thus our results means that
dissipation of the energy in the fluid due to viscosity is sufficient to
stabilize the system.
