Evolution semigroups in supersonic flow-plate interactions

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). Supersonic regimes corresponding to the flow provide for new mathematical challenge that is related to the loss of ellipticity in a stationary dynamics. This difficulty is present also in the linear model. 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 along with sharp regularity of Airy's stress function (obtained by compensated compactness method) 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. The latter is due to the loss of ellipticity. And (2) we make critical use of hidden trace regularity for the flow component of the model (in the abstract setup for the semigroup problem) which allows us to develop a fixed point argument and eventually conclude well-posedness. This well-posedness result for supersonic flows (in the absence of regularizing rotational inertia) has been hereto open. The use of semigroup methods to obtain well-posedness opens this model to long-time behavior considerations.