A Linearized Viscous, Compressible Flow-Plate Interaction with Non-dissipative Coupling

Abstract

We address semigroup well-posedness for a linear, compressible viscous fluid interacting at its boundary with an elastic plate. We derive the model by linearizing the compressible Navier-Stokes equations about an arbitrary flow state, so the fluid PDE includes an ambient flow profile U . In contrast to model in [Avalos, Geredeli, Webster, 2017], we track the effect of this term at the flow-structure interface, yielding a velocity matching condition involving the material derivative of the structure; this destroys the dissipative nature of the coupling of the dynamics. We adopt here a Lumer-Phillips approach, with a view of associating fluid-structure solutions with a C₀ -semigroup {eᴬᵗ}ₜ≥₀ on a chosen finite energy space of data. Given this approach, the challenge becomes establishing the maximal dissipativity of an operator A , yielding the flow-structure dynamics.