多层热-波-波系统的适定性、谱分析和渐近稳定性
Wellposedness, Spectral Analysis and Asymptotic Stability of a Multilayered Heat-Wave-Wave System
摘要：
在这项工作中，我们考虑了一个多层热波系统，其中一个3-D热方程与一个3-D波动方程通过一个2-D界面耦合，其动力学由一个2-D波动方程描述。该系统可视为某种流体-结构相互作用（FSI）PDE模型的简化，其中结构为复合型;即它由\textquotedblleft薄\textquotedblright \层和\textquotedblleft厚\textquotedblright \层组成。我们将系统的适定性与一个强连续半群联系起来，并建立了它的渐近衰减。我们的第一个结果是（FSI）PDE动力系统的半群适定性。利用这里的Lumer-Phillips方法，我们表明，流体结构系统产生一个$C_0$ -半群上选择的有限能量空间的数据。作为第二个结果，我们证明了由$C_0$ -半群生成的（FSI）动力学的解对所有初始数据渐近趋于零状态。也就是说，（FSI）系统的半群是强稳定的。对于这个稳定性的工作，我们分析了频谱的发生器$\mathbf A$，并表明，频谱的$\mathbf A$不相交的虚轴。\vskip流固耦合，热波系统，适定性，半群，强稳定性。
英文摘要：
n this work we consider a multilayered heat-wave system where a 3-D heat equation is coupled with a 3-D wave equation via a 2-D interface whose dynamics is described by a 2-D wave equation. This system can be viewed as a simplification of a certain fluid-structure interaction (FSI) PDE model where the structure is of composite-type; namely it consists of a \textquotedblleft thin\textquotedblright\ layer and a \textquotedblleft thick\textquotedblright\ layer. We associate the wellposedness of the system with a strongly continuous semigroup and establish its asymptotic decay. Our first result is semigroup well-posedness for the (FSI) PDE dynamics. Utilizing here a Lumer-Phillips approach, we show that the fluid-structure system generates a $C_0$-semigroup on a chosen finite energy space of data. As our second result, we prove that the solution to the (FSI) dynamics generated by the $C_0$-semigroup tends asymptotically to the zero state for all initial data. That is, the semigroup of the (FSI) system is strongly stable. For this stability work, we analyze the spectrum of the generator $\mathbf A$ and show that the spectrum of $\mathbf A$ does not intersect the imaginary axis. \vskip.3cm \noindent \textbf{Key terms:} Fluid-structure interaction, heat-wave system, well-posedness, semigroup, strong stability.
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