Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Flow-plate interactions: Well-posedness and long-time behavior

Pages: 925 - 965, Volume 7, Issue 5, October 2014      doi:10.3934/dcdss.2014.7.925

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Igor Chueshov - Kharkov University, Department of Mathematics and Mechanics, 4 Svobody sq, 61077 Kharkov, Ukraine (email)
Irena Lasiecka - University of Memphis, Department of Mathematical Sciences, 373 Dunn Hall, Memphis, TN 38152, United States (email)
Justin Webster - Oregon State University, Department of Mathematics, 368 Kidder Hall, Corvallis, OR 97330, United States (email)

Abstract: We consider flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are treated with Neumann type flow conditions, and a novel treatment of the so called Kutta-Joukowsky flow conditions are given in the subsonic case. The goal of the paper is threefold: (i) to provide an accurate review of recent results on existence, uniqueness, and stability of weak solutions, (ii) to present a construction of finite dimensional, attracting sets corresponding to the structural dynamics and discuss convergence of trajectories, and (iii) to state several open questions associated with the topic. This second task is based on a decoupling technique which reduces the analysis of the full flow-structure system to a PDE system with delay.

Keywords:  Flow-structure interaction, nonlinear plate, nonlinear semigroups, well-posedness, global attractors, PDE with delay.
Mathematics Subject Classification:  Primary: 35M33, 74F10; Secondary: 35B41, 35Q74.

Received: March 2013;      Revised: June 2013;      Available Online: May 2014.