Rational rates of uniform decay for strong solutions to a fluid-structure PDE system

Abstract: In this work we investigate the uniform stability properties of solutions to a well-established partial differential equation (PDE) model for a fluid-structure interaction. The PDE system under consideration comprises a Stokes flow which evolves within a three-dimensional cavity; moreover, a Kirchhoff plate equation is invoked to describe the displacements along a (fixed) portion -- say, Omega -- of the cavity wall. Contact between the respective fluid and structure dynamics occurs on the boundary interface Omega. The main result in the paper is as follows: the solutions to the composite PDE system, corresponding to smooth initial data, decay at the rate of O(1/t). Our method of proof hinges upon the appropriate invocation of a relatively recent resolvent criterion for polynomial decays of strongly continuous semigroups. While the characterization provided by said criterion originates in the context of operator theory and functional analysis, the work entailed here is wholly within the realm of PDE.