Exponential stability of a nondissipative, compressible flow–structure PDE model

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 the associated state equation for the 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 nonzero ambient flow profile $$\mathbf {U}$$ and will generally be nondissipative. By way of obtaining the aforesaid exponential stability, a “frequency domain” approach is adopted here, an approach which is predicated on obtaining a uniform estimate on the resolvent of the associated flow–structure semigroup generator.