  We consider a linearized compressible flow structure interaction (FSI) PDE
model with a view of analyzing the stability properties of both the
compressible flow and plate solution components. In our earlier work, we gave
an answer in the affirmative to question of uniform stability for finite energy
solutions of said compressible flow-structure system, by means of a ``frequency
domain'' approach. However, the frequency domain method of proof in that work
is not ``robust'' (insofar as we can see), when one wishes to study longtime
behavior of solutions of compressible flow-structure PDE models which track the
appearance of the ambient state onto the boundary interface (i.e., $\kappa =1$
in flow PDE component (2)). Nor is a frequency domain approach in this earlier
work availing when one wishes to consider the dynamics, in long time, of
solutions to physically relevant nonlinear versions of the compressible
flow-structure PDE system under present consideration (e.g., the Navier-Stokes
nonlinearity in the PDE flow component, or a nonlinearity of Berger/Von Karman
type in the plate equation). Accordingly, in the present work, we operate in
the time domain by way of obtaining the necessary energy estimates which
culminate in an alternative proof for the uniform stability of finite energy
compressible flow-structure solutions. This novel time domain proof will be
used in our forthcoming paper which addresses the existence of compact global
attractors for the corresponding nonlinear coupled system in which the material
derivative -- which incorporates the aforesaid ambient state -- of the
interaction surface will be taken into account.
