Uniform stability for solutions of a structural acoustics PDE model with no added dissipative feedback

Abstract

Communicated by P. SacksA rate of rational decay is obtained for smooth solutions of a PDE model, which has been used in the literature to describe structural acoustic flows. This structural acoustics model is composed of two distinct PDE systems: (i) a wave equation, to model the interior acoustic flow within the given cavity and (ii) a structurally damped elastic equation, to describe timeevolving displacements along the flexible portion 0 of the cavity walls. Moreover, the extent of damping in this elastic component is quantified by parameter Á 2 OE0, 1. The coupling between these two distinct dynamics occurs across the boundary interface 0 . Our main result is the derivation of uniform decay rates for classical solutions of this particular structural acoustic PDE, decay rates that are obtained without incorporating any additional boundary dissipative feedback mechanisms. In particular, in the case that full Kelvin-Voight damping is present in fourth-order elastic dynamics, that is, the structural acoustics system as it appears in the literature, solutions that correspond to smooth initial data decay at a rate of O t 1 6 Á. By way of deriving these stability results, necessary a priori inequalities for a certain static structural acoustics PDE model are generated here; these inequalities ultimately allow for an application of a recently derived resolvent criterion for rational decay.

Stability analysis of coupled structural acoustics PDE models under thermal effects and with no additional dissipation

“…This gives, We thus have formally a perturbed fourth order elastic equation in w , with “square root” structural damping. Subsequently, from Theorem 2(a) of , with damping parameter , this formality suggests a decay rate of for smooth solutions of –.…”

“…Remark The decay rate is formally predicted by the results , a work which provides rational decay rates for structural acoustic systems in which the structural PDE component is described by a structurally damped plate. Indeed, taking here parameters and , we can use the heat equation to substitute for in the mechanical equation.…”

“…Remark The geometric conditions (Geometry.1) and (Geometry.2) have been invoked before, in the context of controlling and stabilizing structural acoustic flows; see and ; these assumptions essentially dictate that the “roof” Γ 1 of the acoustic chamber Ω not be “too deep”. Since Γ 0 is only a portion of the boundary, the imposition of geometric conditions on uncontrolled boundary portion Γ 1 is fully expected; see .…”

“…Since Γ 0 is only a portion of the boundary, the imposition of geometric conditions on uncontrolled boundary portion Γ 1 is fully expected; see . The assumptions (Geometry.1) and (Geometry.2) allow for the construction of a special vector field – this vector is generated outright in Appendix II of ; see also – which, in tandem with the classic wave equation identities, give rise to the static wave estimate Theorem below (see ).…”

“…In addition, the paper gives an unspecified, uniform rate of decay for solutions of a wellknown PDE model for structural acoustic flows, for zero wave initial data. More recently, in , rates of rational decay were obtained for classical solutions of the aforesaid well known structural acoustic PDE model: In , the interior wave equation in appears as it does here – the acoustic PDE component does not seem to change from model to model (dating from ) – however, the thermoelastic system is replaced by either a wave or plate equation under some degree of structural damping (from [weak] viscous to [strong] Kelvin–Voight). In , the primary issue is appropriately dealing with the wave solution component for general structural acoustic systems, as well as the boundary traces of both wave displacement and velocity – the main wave estimates of are applied in the present work.…”