The Strong Stability and Instability of a Fluid-Structure Semigroup

The strong stability problem for a fluid-structure interactive partial differential equation (PDE) is considered. The PDE comprises a coupling of the linearized Stokes equations to the classical system of elasticity, with the coupling occurring on the boundary interface between the fluid and solid media. Because of the nature of the unbounded coupling between fluid and structure, the resolvent of the associated semigroup generator will not be a compact operator. In consequence, the classical solution to the stability problem, by means of the Nagy-Foias decomposition, will not avail here. Moreover, it is not practicable to write down explicitly the resolvent of the fluid-structure generator; this situation thus makes it problematic to use the well-known semigroup stability result of Arendt-Batty and Lyubich-Phong. When a locally supported boundary dissipative mechanism is in place, we derive here a result of strong decay for this fluid-structure PDE. In the absence of said dissipative mechanism, we show the lack of asymptotic decay for solutions corresponding to arbitary initial data of finite energy.