A Time Domain Approach for the Exponential Stability of a Nondissipative Linearized Compressible Flow-Structure PDE System

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.