多层Lamé-Heat系统强衰变的预解判据方法
A Resolvent Criterion Approach to Strong Decay of a Multilayered Lamé-Heat System
摘要：
本文考虑了一个多层双曲-抛物型偏微分方程系统，该系统由三维热-二维弹性-三维弹性动力学耦合组成，其中三维流体与三维结构之间的边界界面耦合通过二维弹性方程实现。我们的主要结果是对给定的多层热系统的强衰变.也就是说，该复合偏微分方程系统的解渐近稳定到零状态。我们证明的强稳定性发生在“频域“，并最终呼吁逐点预解条件引进Tomilov [45]。然而，这个非常有用的结果要求与我们的多层FSI系统相关联的半群是完全非酉的（c.n.u）。因此，我们首先建立半群$\{e^{\mathcal {A}t}\}_{t\geq 0}$确实是c.n.u.，部分是通过调用非光滑域上超定Lamé系统的全局唯一性的相对较新的结果。虽然整个证明也需要更高的规律性结果的一些跟踪条款，这\textit{"resolvent criterion approach"}允许我们建立一个“经典软”证明强衰减。特别是，它避免了在[9]中调用的技术PDE乘数。
英文摘要：
We consider a multilayer hyperbolic-parabolic PDE system which constitutes a coupling of 3D thermal - 2D elastic - 3D elastic dynamics, in which the boundary interface coupling between 3D fluid and 3D structure is realized via a 2D elastic equation. Our main result here is one of strong decay for the given multilayered - heat system. That is, the solution to this composite PDE system is stabilized asymptotically to the zero state. Our proof of strong stability takes place in the "frequency domain" and ultimately appeals to the pointwise resolvent condition introduced by Tomilov [45]. This very useful result, however, requires that the semigroup associated with our multilayered FSI system be completely non-unitary (c.n.u). Accordingly, we firstly establish that the semigroup $\{e^{\mathcal {A}t}\}_{t\geq 0}$ is indeed c.n.u., in part by invoking relatively recent results of global uniqueness for overdetermined Lamé systems on nonsmooth domains. Although the entire proof also requires higher regularity results for some trace terms, this \textit{"resolvent criterion approach"} allows us to establish a "classially soft" proof of strong decay. In particular, it avoids the sort of technical PDE multipliers invoked in [9].
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