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On the interaction of an elasticwall with a poiseuille-type flow

On the interaction of an elasticwall with a poiseuille-type flow

We study the dynamics of a coupled system formed by the 3D Navier–Stokes equations linearized near a certain Poiseuille-type flow in an (unbounded) domain and a classical (possibly nonlinear) equation for transverse displacements of an elastic plate in a flexible flat part of the boundary. We first...

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Bibliographic Details
Main Authors: Chueshov, I., Ryzhkova, I.
Format: Article
Language:English
Published: Інститут математики НАН України 2013
Series:Український математичний журнал
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Online Access:http://dspace.nbuv.gov.ua/handle/123456789/164930
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Summary:We study the dynamics of a coupled system formed by the 3D Navier–Stokes equations linearized near a certain Poiseuille-type flow in an (unbounded) domain and a classical (possibly nonlinear) equation for transverse displacements of an elastic plate in a flexible flat part of the boundary. We first show that this problem generates an evolution semigroup St in an appropriate phase space. Then, under some conditions imposed on the underlying (Poiseuille-type) flow, we prove the existence of a compact finite-dimensional global attractor for this semigroup and also show that St is an exponentially stable C₀ -semigroup of linear operators in the completely linear case. Since we do not assume any kind of mechanical damping in the plate component, this means that the dissipation of energy in the flow of fluid caused by viscosity is sufficient to stabilize the system.

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