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.Comment: 31 page
