2019/04/24 by Justin T. Webster, Webster, Justin T.
Engineering · Mathematics · #35B41 #35M33 #35Q74 #37L25 #74F10 #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical methods for differential equations #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.1904.11016
openalex publication_date 2019/04/24 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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 H02(Ω) (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.