The Strong Stability and Instability of a Fluid-Structure Semigroup

Journal Article · Thu Mar 15 04:00:00 EDT 2007 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S00245-006-0884-Z· OSTI ID:21064195

Avalos, George ^[1]

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1. Department of Mathematics, University of Nebraska-Lincoln (United States)

The strong stability problem for a fluid-structure interactive partial differential equation (PDE) is considered. The PDE comprises a coupling of the linearized Stokes equations to the classical system of elasticity, with the coupling occurring on the boundary interface between the fluid and solid media. Because of the nature of the unbounded coupling between fluid and structure, the resolvent of the associated semigroup generator will not be a compact operator. In consequence, the classical solution to the stability problem, by means of the Nagy-Foias decomposition, will not avail here. Moreover, it is not practicable to write down explicitly the resolvent of the fluid-structure generator; this situation thus makes it problematic to use the well-known semigroup stability result of Arendt-Batty and Lyubich-Phong. When a locally supported boundary dissipative mechanism is in place, we derive here a result of strong decay for this fluid-structure PDE. In the absence of said dissipative mechanism, we show the lack of asymptotic decay for solutions corresponding to arbitrary initial data of finite energy.

• OSTI ID: 21064195
• Journal Information: Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 2 Vol. 55; ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ASYMPTOTIC SOLUTIONS
ELASTICITY
FLUIDS
GROUP THEORY
INSTABILITY
MATHEMATICAL OPERATORS
PARTIAL DIFFERENTIAL EQUATIONS
STABILITY