In this talk we discuss a perturbed wave equation (corresponding to a potential flow) which exhibits a bifurcation in the parameter corresponding to the velocity of the flow. We will discuss a classical flow-plate system arising in aeroelasticity which models the dynamics of a fluttering plate and involves the aforementioned flow equation. The analysis of the entire flow-structure system constitues a rather involved endeavor, and in this talk we focus specifically on the flow dynamics, and investigate the bifurcation which occurs in the flow model, as the flow velocity moves through the transonic barrier.
Mathematically, the equation which describes the flow represents a (linear) perturbed wave equation with inhomogenous Neumann data driving the dynamics. For subsonic flows, it may be analyzed with classical hyperbolic semigroup theory. However, for the transonic and supersonic regimes, the spatial component of the flow equation loses strong ellipticity in the direction of the flow. Well-posedness for the full flow-plate model in the presence of supersonic flows has been an open question for some time; this issue was recently resolved by demonstrating that through a change of state variable, the aforementioned "perturbed wave" equation can in fact be handled via semigroup theory. Moreover, the dynamics exhibit additional "hidden" Dirichlet trace regularity (akin to the classical Neumann wave equation) which are critical in the analysis of the full flow-plate system.