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Fluid–Structure Interaction with Kelvin–Voigt Damping: Analyticity, Spectral Analysis, Exponential Decay

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Abstract

We consider a fluid–structure interaction model defined on a doughnut-like domain. It consists of the dynamic Stokes equations evolving on the exterior sub-domain, coupled with an elastic structure occupying the interior sub-domain. A key factor—a novelty over past literature—is that the structure equation includes a strong (viscoelastic) damping term of Kelvin–Voigt type at the interior. This affects the boundary conditions at the interface between the two media and accounts for a highly unbounded “perturbation”. Results include: (i) analyticity of s.c semigroup of contractions defining the overall coupled system, (ii) its (uniform) exponential decay, along with (iii) sharp spectral properties of its generator. Some results are geometry-dependant.

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Acknowledgements

The authors wish to thank two referees for their much appreciated comments. Research partially supported by the National Science Foundation under Grant DMS-1713506. This paper was written while R.T was a MSRI Research Professor during the 2021 Spring Semester, “Mathematical problems in fluid dynamics”, held at the Mathematical Sciences Research Institute, University of California, Berkeley.

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  1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Rasika Mahawattege & Roberto Triggiani

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  1. Rasika Mahawattege
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  2. Roberto Triggiani
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Correspondence to Rasika Mahawattege.

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Mahawattege, R., Triggiani, R. Fluid–Structure Interaction with Kelvin–Voigt Damping: Analyticity, Spectral Analysis, Exponential Decay. Appl Math Optim 84 (Suppl 2), 1821–1863 (2021). https://doi.org/10.1007/s00245-021-09812-5

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