A global attractor for a fluid--plate  interaction model

I. D. Chueshov; Iryna Ryzhkova

doi:10.3934/cpaa.2013.12.1635

Article Contents

2013, Volume 12, Issue 4: 1635-1656. Doi: 10.3934/cpaa.2013.12.1635

This issue Previous Article On Dirichlet, Poncelet and Abel problems Next Article Fracture models as $\Gamma$-limits of damage models

A global attractor for a fluid--plate interaction model

I. D. Chueshov^1, and
Iryna Ryzhkova^2,

1.
Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody sq., 61077, Kharkov
2.
Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody Sq., Kharkov, 61022

Received: January 31, 2011

Revised: May 31, 2012

Published: June 2013

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Abstract

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and a classical (nonlinear) elastic plate equation for transversal displacement on a flexible flat part of the boundary. We show that this problem generates a semiflow on appropriate phase space. Our main result states the existence of a compact finite-dimensional global attractor for this semiflow. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system. To achieve the result we first study the corresponding linearized model and show that this linear model generates strongly continuous exponentially stable semigroup.

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Mathematics Subject Classification: Primary: 35B41, 35Q30; Secondary: 74F10, 74K20.

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