Quasi-stability and Exponential Attractors for A Non-Gradient System---Applications to Piston-Theoretic Plates with Internal Damping

摘要

We consider a nonlinear (Berger or Von Karman) clamped plate model with a{\em piston-theoretic} right hand side---which include non-dissipative,non-conservative lower order terms. The model arises in aeroelasticity when apanel is immersed in a high velocity linear potential flow; in this case theeffect of the flow can be captured by a dynamic pressure term written in termsof the material derivative of the plate's displacement. The effect offully-supported internal damping is studied for both Berger and von Karmandynamics. The non-dissipative nature of the dynamics preclude the use of strongtools such as backward-in-time smallness of velocities and finiteness of thedissipation integral. Modern quasi-stability techniques are utilized to showthe existence of compact global attractors and generalized fractal exponentialattractors. Specific results depending on the size of the damping parameter andthe nonlinearity in force. For the Berger plate, in the presence of largedamping, the existence of a proper global attractor (whose fractal dimension isfinite in the state space) is shown via a decomposition of the nonlineardynamics. This leads to the construction of a compact set upon whichquasi-stability theory can be implemented. Numerical investigations forappropriate 1-D models are presented which explore and support the abstractresults presented herein.