Fluid-structure interaction  with and without internal dissipation  of the structure: A contrast study in stability

George Avalos; Roberto Triggiani

doi:10.3934/eect.2013.2.563

Article Contents

2013, Volume 2, Issue 4: 563-598. Doi: 10.3934/eect.2013.2.563

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Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability

George Avalos^1, and
Roberto Triggiani^2,

1.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588
2.
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240

Received: May 2013

Revised: September 2013

Published: December 2013

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Abstract

We consider a coupled parabolic--hyperbolic PDE system arising in fluid--structure interaction, where the coupling is exercised at the interface between the two media. This paper is a study in contrast on stability properties of the overall coupled system under two scenarios: the case with interior dissipation of the structure, and the case without. In the first case, uniform stabilization is achieved (by a $\lambda$-domain analysis) without geometrical conditions on the structure, but only on an explicitly identified space $Ĥ$ of codimension one with respect to the original energy state space $H$ where semigroup well-posedness holds. In the second case, only rational (a fortiori strong) stability is possible, again only on the space $Ĥ$, however, under geometrical conditions of the structure, which e.g., exclude a sphere. Many classes of good geometries are identified. Recent papers [6,9] show uniform stabilization on all of $H$, and without geometrical conditions; however, with dissipation at the boundary interface.

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Mathematics Subject Classification: Primary: 35M13, 93D20.

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