Coupled parabolic–hyperbolic Stokes–Lamé PDE system: limit behaviour of the resolvent operator on the imaginary axis

Abstract

This article is focused on an established, genuinely physical fluid-structure interaction model, whereby 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 three dimensions with non-standard coupling at the boundary interface: the (dynamic) Stokes system (parabolic, modelling the fluid) and the Lamé system (hyperbolic, modelling the structure). This system generates a contraction semigroup on the natural energy space [G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, Part I: explicit semigroup generator and its spectral properties, Fluids and Waves, Amer. Math. Soc. Contemp. Math. 440 (2007), pp. 15–59] (canonical model) and [G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Contin. Dyn. Sys. Series S, 2(3) (2009), pp. 417–447]. The boundary interface may or may not include a ‘damping’ (or dissipative) term. If damping is active on the entire interface, then uniform (exponential) stabilization is ensured, regardless of the geometry of the structure [G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discrete Contin. Dyn. Syst. 22(4) 2008, pp. 817–835, special issue, invited paper] (canonical model) and [G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system, J. Evol. Eqns 9(2009), pp. 341–370]. This article emphasizes the case of, at most, partial damping. At any rate, the main result is a precise uniform-operator limit behaviour of the resolvent operator of the semigroup generator on the imaginary axis of interest in itself, which holds true with or without damping. It, in turn, then implies a fortiori strong stability results: most notably, on the whole state space, under at least partial damping at the interface; and, in the absence of damping, on the whole state space, after factoring out an explicit one-dimensional null eigenspace, at least for a large class of geometries of the structure: these are characterized by a uniqueness property of a special over-determined elliptic problem.