Eliminating flutter for clamped von Karman plates immersed in subsonic flows
We address the long-time behavior of a non-rotational von Karman plate in an inviscid potential ow. The model arises in aeroelasticity and models the interaction between a thin, nonlinear panel and a ow 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 ow-plate system, full trajectories-both plate and ow-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 ows 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 ow-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 utter (a non-static end behavior) does not occur in subsonic dynamics. While such results were known for rotational (compact/regular) plate dynamics  (and references therein), the result presented herein is the first such result obtained for non-regularized-the most physically relevant-models.
Communications on Pure and Applied Analysis
Lasiecka, I., & Webster, J. (2014). Eliminating flutter for clamped von Karman plates immersed in subsonic flows. Communications on Pure and Applied Analysis, 13 (5), 1935-1969. https://doi.org/10.3934/cpaa.2014.13.1935