In this work, a result of exponential stability is obtained for solutions of a compressible flow–structure partial differential equation (PDE) model which has recently appeared in the literature. In particular, a compressible flow PDE and the associated state equation for the pressure variable, each evolving within a three-dimensional domain \(\mathcal {O}\), are coupled to a fourth-order plate equation which holds on a flat portion \(\Omega \) of the boundary \(\partial \mathcal {O}\). Moreover, since this coupled PDE model is the result of a linearization of the compressible Navier–Stokes equations about an arbitrary state, the flow PDE component contains a nonzero ambient flow profile \(\mathbf {U}\) and will generally be nondissipative. By way of obtaining the aforesaid exponential stability, a “frequency domain” approach is adopted here, an approach which is predicated on obtaining a uniform estimate on the resolvent of the associated flow–structure semigroup generator.

中文翻译：

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非耗散可压缩流结构PDE模型的指数稳定性

在这项工作中，最近出现在文献中的可压缩流动结构偏微分方程（PDE）模型的解获得了指数稳定性的结果。特别地，分别在三维域\（\ mathcal {O} \）内演化的可压缩流量PDE和压力变量的相关状态方程式与保持在平坦部分上的四阶板方程式耦合。边界\（\ partial \ mathcal {O} \）的\（\ Omega \）。此外，由于此耦合的PDE模型是可压缩的Navier–Stokes方程关于任意状态的线性化的结果，因此流量PDE分量包含非零的环境流量分布\（\ mathbf {U} \），并且通常为非耗散的。通过获得上述指数稳定性，此处采用“频域”方法，该方法基于对相关流结构半群发生器的分解器进行统一估计。