Ukrainian Mathematical Journal

, Volume 65, Issue 1, pp 158–177 | Cite as

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

  • I. Chueshov
  • I. Ryzhkova

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 S t 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 S t is an exponentially stable C 0 -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.


Weak Solution Global Attractor Elastic Plate Extension Operator Plate Model 
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© Springer Science+Business Media New York 2013

Authors and Affiliations

  • I. Chueshov
    • 1
  • I. Ryzhkova
    • 1
  1. 1.Kharkov National UniversityKharkovUkraine

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