Strong stability and the Schiffer Conjecture for the fluid-elastic semigroup

In a series of papers, Avalos and Triggiani established the fluid-elastic semigroup for the coupled Stokes-Lamé system modelling the coupled dynamics of a linearly elastic structure immersed in a viscous Newtonian fluid. They analyzed the spectrum of its generator and proved that the semigroup is strongly stable, if the domain of the structure satisfies a geometric condition, i.e. it is not a bad domain. We extend these results in two directions: first, for bad domains, we prove a decomposition of the dynamics into a strongly stable part and a pressure wave, a special solution of the Dirichlet-Lamé system, that can be determined from the initial values. This fully characterizes the long-time behaviour of the semigroup. Secondly, we show that the characterization of bad domains is equivalent to the Schiffer problem. This strengthens the conjecture that balls are the only bad domains and establishes a direct connection to geometric analysis. We also discuss implications for associated nonlinear systems.