This work is motivated by a longstanding interest in the long time behavior of flow‐structure interaction (FSI) PDE dynamics. We consider a linearized compressible flow structure interaction (FSI) PDE model with a view of analyzing the stability properties of both the compressible flow and plate solution components. In our earlier work, we gave an answer in the affirmative to question of uniform stability for finite energy solutions of said compressible flow‐structure system, by means of a “frequency domain” approach. However, the frequency domain method of proof in that work is not “robust” (insofar as we can see), when one wishes to study longtime behavior of solutions of compressible flow‐structure PDE models, which track the appearance of the ambient state onto the boundary interface. Nor is a frequency domain approach in this earlier work availing when one wishes to consider the dynamics, in long time, of solutions to physically relevant nonlinear versions of the compressible flow‐structure PDE system under present consideration (e.g., the Navier–Stokes nonlinearity in the PDE flow component or a nonlinearity of Berger/Von Karman type in the plate equation). Accordingly, in the present work, we operate in the time domain by way of obtaining the necessary energy estimates, which culminate in an alternative proof for the uniform stability of finite energy compressible flow‐structure solutions. Since there is a need to avoid steady states in our stability analysis, as a prerequisite result, we also show here that zero is an eigenvalue for the generators of flow‐structure systems, whether the material derivative term be absent or present. Moreover, we provide a clean characterization of the (one dimensional) zero eigenspace, with or without material derivative, under an appropriate assumption on the underlying ambient vector field.

中文翻译：

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线性可压缩流结构PDE系统指数稳定性的时域方法

长期以来，人们对流动-结构相互作用（FSI）PDE动力学的长期行为产生了浓厚的兴趣。我们考虑分析可压缩流和板溶液成分的稳定性，考虑了线性可压缩流结构相互作用（FSI）PDE模型。在我们较早的工作中，我们通过“频域”方法肯定地回答了所述可压缩流结构系统有限能量解的均匀稳定性问题。但是，当人们希望研究可压缩流结构PDE模型的解的长期行为时，频域证明工作的“稳健性”（正如我们所看到的那样不多），可以跟踪环境状态的出现。边界接口。当人们希望长时间考虑当前考虑的可压缩流结构PDE系统的物理相关非线性版本的解决方案的动力学问题时（例如，Navier–Stokes非线性），这种早期工作中的频域方法也无济于事。平板方程中的PDE流量分量或Berger / Von Karman类型的非线性）。因此，在当前工作中，我们通过获取必要的能量估计值在时域中进行操作，最终得出了有限的能量可压缩流结构解的均匀稳定性的替代证明。由于需要在稳定性分析中避免稳态，因此，作为前提条件，我们在这里还表明，零是流结构系统生成器的特征值，物质导数项不存在还是不存在。此外，在适当的基础环境矢量场假设下，我们提供了（一维）零本征空间的清晰表征，无论有无材料导数。