A time domain approach for the exponential stability of a linearized compressible flow‐structure PDE system

Abstract

This work is motivated by a longstanding interest in the long time behavior of flow‐structure interaction (FSI) PDE dynamics. 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. 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. Since there is a need to avoid steady states in our stability analysis, as a prerequisite result, we also show here that zero is an eigenvalue for the generators of flow‐structure systems, whether the material derivative term be absent or present. Moreover, we provide a clean characterization of the (one dimensional) zero eigenspace, with or without material derivative, under an appropriate assumption on the underlying ambient vector field.