Evolution Semigroups in Supersonic Flow-Plate Interactions

TL;DR

This paper establishes the well-posedness of a flow-structure interaction model involving a flexible plate in supersonic flow, using semigroup theory and novel analytical techniques to handle both subsonic and supersonic regimes.

Contribution

It introduces a new flow potential variable and employs semigroup analysis to prove well-posedness for the nonlinear model in supersonic flow, addressing an open problem.

Findings

Linearized model is well-posed and generates a strongly continuous semigroup.

Global-in-time well-posedness for the nonlinear model is established.

Novel analytical methods enable handling of supersonic flow regimes.

Abstract

We consider the well-posedness of a model for a flow-structure interaction. This model describes the dynamics of an elastic flexible plate with clamped boundary conditions immersed in a supersonic flow. A perturbed wave equation describes the flow potential. The plate's out-of-plane displacement can be modeled by various nonlinear plate equations (including von Karman and Berger). We show that the linearized model is well-posed on the state space (as given by finite energy considerations) and generates a strongly continuous semigroup. We make use of these results to conclude global-in-time well-posedness for the fully nonlinear model. The proof of generation has two novel features, namely: (1) we introduce a new flow potential velocity-type variable which makes it possible to cover both subsonic and supersonic cases, and to split the dynamics generating operator into a skew-adjoint component and a perturbation acting outside of the state space. Performing semigroup analysis also requires a nontrivial approximation of the domain of the generator. And (2) we make critical use of hidden regularity for the flow component of the model (in the abstract setup for the semigroup problem) which allows us run a fixed point argument and eventually conclude well-posedness. This well-posedness result for supersonic flows (in the absence of rotational inertia) has been hereto open. The use of semigroup methods to obtain well-posedness opens this model to long-time behavior considerations.