Communications on Pure and Applied Analysis (CPAA)

Eliminating flutter for clamped von Karman plates immersed in subsonic flows

Pages: 1935 - 1969, Volume 13, Issue 5, September 2014      doi:10.3934/cpaa.2014.13.1935

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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 address the long-time behavior of a non-rotational von Karman plate in an inviscid potential flow. The model arises in aeroelasticity and models the interaction between a thin, nonlinear panel and a flow of gas in which it is immersed [6, 21, 23]. Recent results in [16, 18] show that the plate component of the dynamics (in the presence of a physical plate nonlinearity) converge to a global compact attracting set of finite dimension; these results were obtained in the absence of mechanical damping of any type. Here we show that, by incorporating mechanical damping the full flow-plate system, full trajectories---both plate and flow---converge strongly to (the set of) stationary states. Weak convergence results require ``minimal" interior damping, and strong convergence of the dynamics are shown with sufficiently large damping. We require the existence of a ``good" energy balance equation, which is only available when the flows are subsonic. Our proof is based on first showing the convergence properties for regular solutions, which in turn requires propagation of initial regularity on the infinite horizon. Then, we utilize the exponential decay of the difference of two plate trajectories to show that full flow-plate trajectories are uniform-in-time Hadamard continuous. This allows us to pass convergence properties of smooth initial data to finite energy type initial data. Physically, our results imply that flutter (a non-static end behavior) does not occur in subsonic dynamics. While such results were known for rotational (compact/regular) plate dynamics [14] (and references therein), the result presented herein is the first such result obtained for non-regularized---the most physically relevant---models.

Keywords:  Strong stability, nonlinear plates, mathematical aeroelasticity, flutter, nonlinear semigroups.
Mathematics Subject Classification:  74F10, 74K20, 76G25, 35B40, 35G25, 37L15.

Received: January 2014;      Revised: April 2014;      Available Online: June 2014.