论文标题
非隔离性线性压缩流程PDE系统的指数稳定性的时域方法
A Time Domain Approach for the Exponential Stability of a Nondissipative Linearized Compressible Flow-Structure PDE System
论文作者
论文摘要
我们考虑一个线性化的可压缩流结构相互作用(FSI)PDE模型,它可以分析可压缩流和板溶液组件的稳定性。在我们较早的工作中,我们通过``频域''的方法给出了一个肯定的答案,以解决上述可压缩流程系统的有限能量解决方案的统一稳定性问题。但是,当人们希望研究可压缩流程结构PDE模型的解决方案的长期行为时,频率域的证明方法不是``稳健''(在我们所看到的,这是``稳健''(在我们所看到的,这些方法都不是``强大''(即,在流量PDE组件(2)中,可以跟踪环境状态的外观(即,$κ= 1 $)。当人们希望在当前考虑的情况下(例如,Navier-Stokes-Stokes非线性在PDE流量组件中,PDE流量成分或非线性的berger/von Karman类型的非线性),当人们希望在很长一段时间内采用动态解决方案的动态,或者在较早的情况下进行频率域的方法。因此,在目前的工作中,我们通过获得必要的能量估计值在时间域中运行,这些能量估计值在替代证明中,以证明有限能量可压缩流量结构解决方案的均匀稳定性。这种新颖的时域证明将在我们即将发表的论文中使用,该论文介绍了相应的非线性耦合系统的紧凑全球吸引子,其中将考虑到相互作用表面的材料衍生物(结合上述环境状态)。
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 (i.e., $κ=1$ in flow PDE component (2)). 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. This novel time domain proof will be used in our forthcoming paper which addresses the existence of compact global attractors for the corresponding nonlinear coupled system in which the material derivative -- which incorporates the aforesaid ambient state -- of the interaction surface will be taken into account.