2017/12/29 by George Avalos, Avalos, George, Pelin G. Geredeli +1 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1801.00051
openalex publication_date 2017/12/29 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
In this study we consider a coupled system of partial differential equations\n(PDE's) which describes a certain structural acoustics interaction. One\ncomponent of this PDE system is a wave equation, which serves to model the\ninterior acoustic wave medium within a given three dimensional chamber %\n\Ω . This acoustic wave equation is coupled on a boundary interface (%\n\Γ 0) to a two dimensional system of thermoelasticity: this\nthermoelastic PDE comprises a structural beam or plate equation, which governs\nthe vibrations of flexible wall portion \Γ 0 of the chamber \Ω ;\nthe elastic dynamics is coupled to a heat equation which also evolves on\n\Γ 0, and which imparts a thermal damping onto the entire structural\nacoustic system. As we said, the interaction between the wave and thermoelastic\nPDE components takes place on the boundary interface % \Γ 0, and\ninvolves coupling boundary terms which are above the level of finite energy. We\nanalyze the stability properties of this coupled structural acoustics PDE\nmodel, in the absence of any additive feedback dissipation on the hard walls\n\Γ 1 of the boundary \∂ \Ω . Under a certain geometric\nassumption on \Γ 1, an assumption which has appeared in the literature\nin conection with structural acoustic flow, and which allows for the invocation\nof a recently derived microlocal boundary trace estimate, we show that\nclassical solutions of this thermally damped structural acoustics PDE decay\nuniformly to zero, with a rational rate of decay.\n