Strong Stabilization of a 3D Potential Flow via a Weakly Damped von Karman Plate

TL;DR

This paper proves that weak damping in a 2D elastic plate embedded in a 3D potential flow can stabilize the flow-plate system, overcoming mathematical challenges related to boundary regularity and hyperbolic boundary conditions.

Contribution

It demonstrates strong stabilization of a 3D flow by a 2D elastic structure without regularization, using microlocal analysis to handle boundary regularity issues.

Findings

Weak plate damping achieves system stability.

Microlocal analysis compensates for boundary regularity issues.

Stability results extend to wave and plate subsystems.

Abstract

The elimination of aeroelastic instability (resulting in sustained oscillations of bridges, buildings, airfoils) is a central engineering and design issue. Mathematically, this translates to strong asymptotic stabilization of a 3D flow by a 2D elastic structure. The stabilization (convergence to the stationary set) of a aerodynamic wave-plate model is established here. A 3D potential flow on the half-space has a spatially-bounded von Karman plate embedded in the boundary. The physical model, then, is a Neumann wave equation with low regularity of coupling conditions. Motivated on empirical observations, we examine if intrinsic panel damping can stabilize the subsonic flow-plate system to a stationary point. Several partial results have been established through partial regularization of the model. Without doing so, classical approaches attempting to treat the given wave boundary data have fallen short, owing to the failure of the Lopatinski condition (in the sense of Kreiss, Sakamoto) and the associated regularity defect of the hyperbolic Neumann mapping. Here, we operate on the panel model as in the engineering literature with no regularization or modifications; we completely resolve the question of stability by demonstrating that weak plate damping strongly stabilizes system trajectories. This is accomplished by microlocalizing the wave data (given by the plate) and observing an "anisotropic" a microlocal compensation by the plate dynamics precisely where the regularity of the 3D wave is compromsed (in the characteristic sector). Several additional stability results for both wave and plate subsystems are established to "push" strong stability of the plate onto the flow.