# Attractors and Determining Functionals for A Flutter Model: Finite Dimensionality Out of Thin Air

## TL;DR

This paper proves the finite dimensionality of the dynamics for a flow-plate interaction PDE model in aeroelasticity, constructing attractors and determining functionals without relying on structural damping.

## Contribution

It establishes the existence of a smooth global attractor, an exponential attractor, and finite determining functionals for a nonlinear flow-plate PDE model without structural damping.

## Key findings

- Constructed a smooth, finite-dimensional global attractor.
- Fattened the attractor to obtain an exponential attractor.
- Proved the existence of finite determining functionals using the completeness defect.

## Abstract

We establish the effective {\em finite dimensionality} of the dynamics corresponding to a flow-plate interaction PDE model arising in aeroelasticity: a nonlinear panel, in the absence of rotational inertia, immersed in an inviscid potential flow. An intrinsic component of the analysis is the study of a plate equation with a {\it delay} term---a fundamentally non-gradient dynamics. First, we construct a compact global attractor and observe that the attractor is smooth, with finite fractal dimension in the state space. Secondly, by fattening the attractor, we obtain an exponential attractor, though with finite dimension only in an extended space. Lastly, we show that a finite set of {\em determining functionals} exists by considering the {\em completeness defect} for some practical functionals on $H_0^2(\Omega)$ (e.g., nodes, modes, and averages). The primary tool here is the recent quasi-stability theory of Chueshov and Lasiecka. All of the main results require {\em no imposed structural damping}, as dissipative effects are contributed by the flow through the coupling. In the final section, we discuss additional results and conjectures when imposed structural damping is present.