A Resolvent Criterion Approach to Strong Decay of a Multilayered Lam\'e-Heat System

TL;DR

This paper proves the strong decay (stability) of a complex multilayered PDE system coupling thermal and elastic dynamics using a resolvent criterion approach in the frequency domain.

Contribution

It introduces a resolvent criterion method to establish strong decay for a multilayered Lamé-Heat system, avoiding complex PDE multipliers.

Findings

Solution asymptotically stabilizes to zero.

Semigroup associated with the system is completely non-unitary.

Established regularity results for trace terms.

Abstract

We consider a multilayer hyperbolic-parabolic PDE system which constitutes a coupling of 3D thermal - 2D elastic - 3D elastic dynamics, in which the boundary interface coupling between 3D fluid and 3D structure is realized via a 2D elastic equation. Our main result here is one of strong decay for the given multilayered - heat system. That is, the solution to this composite PDE system is stabilized asymptotically to the zero state. Our proof of strong stability takes place in the "frequency domain" and ultimately appeals to the pointwise resolvent condition introduced by Tomilov [45]. This very useful result, however, requires that the semigroup associated with our multilayered FSI system be completely non-unitary (c.n.u). Accordingly, we firstly establish that the semigroup $\{e^{\mathcal {A}t}\}_{t\geq 0}$ is indeed c.n.u., in part by invoking relatively recent results of global uniqueness for overdetermined Lam\'e systems on nonsmooth domains. Although the entire proof also requires higher regularity results for some trace terms, this \textit{"resolvent criterion approach"} allows us to establish a "classially soft" proof of strong decay. In particular, it avoids the sort of technical PDE multipliers invoked in [9].