A resolvent criterion approach to strong decay of a multilayered Lamé-heat system

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 [23]. 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 $\{e^{\mathcal{A}t}{\}}_{t\ge 0}$ is indeed c.n.u., in part by invoking relatively recent results of global uniqueness for overdetermined Lamé systems on non-smooth domains. Although the entire proof also requires higher regularity results for some trace terms, this ‘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 [Avalos G, Geredeli PG, Muha B. Wellposedness, spectral analysis and asymptotic stability of a multilayered heat-wavewave system. J Differ Equ. 2020;269:7129–7156].

Keywords:

• Fluid-structure interaction
• Lamé-heat system
• semigroup
• strong stability

2010 Mathematics Subject Classifications:

• 34A12
• 74F10
• 35Q35
• 76N10

Acknowledgements

The authors would like to thank the referee for giving them thoughtful feedback which improved the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In [24], the geometry was assumed to be $C^1$. However the details of proof apply readily to piecewise $C^1-$domains. Indeed, Holmgren's uniqueness will hold for Lamé systems on Lipschitz domains; see [40].

Additional information

Funding

The authors G. Avalos and Pelin G. Geredeli would like to thank the National Science Foundation, and acknowledge their partial funding from NSF Grant Division of Mathematical Sciences [DMS-1907823].