Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction

Abstract

This paper is focused on an established, genuinely physical fluid-structure interaction model, herein the structure is immersed in a fluid with coupling taking place at the boundary interface between the two media. Mathematically, the model is a coupled parabolic-hyperbolic system of two partial differential equations in 3-d with nonstandard coupling at the boundary interface. Fluid and structure are mathematically expressed by the (dynamic) Stokes system (parabolic) and the Lame system (hyperbolic), respectively. The main claim presented is a contraction semigroup well-posedness result on the natural space of finite energy. There are two main features in the analysis: (i) a nonstandard elimination of the pressure term, as the boundary coupling between fluid and structure rules out application of the classical Leray/Helmoltz projection; (ii) a nonstandard usage of the Babuska-Brezzi inf-sup theory to assert maximal dissipativity of the candidate generator. A unified treatment includes both undamped and (perhaps, partially) damped boundary conditions at the interface. With the generator explicitly at hand, an analysis of its point spectrum on the imaginary axis is also included. In the undamped case, it depends on the geometry of the structure. In the case of full boundary damping, it implies as a by-product a strong stability result for the solutions by soft methods.