Exponential Stability and Supporting Spectral Analysis of a Linearized Compressible Flow-Structure PDE Model

TL;DR

This paper proves exponential stability for a linearized compressible flow-structure PDE model with spectral analysis, involving coupled PDEs in a 3D domain and a boundary plate, using a frequency domain approach.

Contribution

It introduces a spectral analysis method to establish exponential stability for a linearized compressible flow-structure PDE system with nonzero ambient flow.

Findings

Exponential stability of the coupled PDE system is established.

A uniform resolvent estimate is obtained for the flow-structure semigroup.

The approach handles nonzero ambient flow profiles in the stability analysis.

Abstract

In this work, a result of exponential stability is obtained for solutions of a compressible flow-structure partial differential equation (PDE) model which has recently appeared in the literature. In particular, a compressible flow PDE and its associated state equation for the associated pressure variable, each evolving within a three dimensional domain $\mathcal{O}$, are coupled to a fourth order plate equation which holds on a flat portion $% \Omega $ of the boundary $\partial \mathcal{O}$. Moreover, since this coupled PDE model is the result of a linearization of the compressible Navier-Stokes equations about an arbitrary state, the flow PDE component contains a generally nonzero ambient flow profile $\mathbf{U}$. By way of obtaining the aforesaid exponential stability, a \textquotedblleft frequency domain\textquotedblright\ approach is adopted here, an approach which is predicated on obtaining a uniform estimate on the resolvent of the associated flow-structure semigroup generator.