Analysis of Flow-Plate Interactions: Semigroup Well-Posedness and Long-Time Behavior

(PDEs) analysis of linear and nonlinear flow - plate interactions. Such mathematical models arise in the PDE modeling of flow and fluid structure interactions, typically involving the coupling of two dynamics at an interface. Here, we are primarily interested in the coupling between a nonlinear plate and a perturbed wave equation. This particular model arises in aeroelasticity and, more generally, in the oscillatory behavior of structures immersed in a potential flow. ln this text we examine well - posedness (and related considerations) in a semigroup context for this flow - plate model. Following our well - posedness analysis, we begin an analysis of long - time behavior of corresponding dynamical systems in the presence of feedback control mechanisms. Specifically, we present results on the existence and properties of global attracting sets for the dynamical system generated by solutions to the PDE model. ln our analysis, two key parameters arise - the unperturbed flow velocity and the inertial term for the plate - which create multiple model regimes necessitating varied techniques, most of which are novel in the context of this model. ln all, (1) we present new proofs of some older results, (2) we prove results which essentially end the hereto open question of well - posedness of the system in all parameter regimes, and lastly, (3) we begin the study of long - time behavior, which has been hereto open for most parameter values and dissipation types. This dissertation is principally dedicated to Bill and Betty Webster and Ken and Jackie Archibald. Secondly, the author would like to dedicate this work to the educators and lifelong friends who motivated him to question everything and inspired him to continue in his education: Bill Rash, Diane Hofloss, and Daniel Sheehan.

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