AI 中文总结
本文针对部分耗散系统引入共振分类，通过值域失效层级与预解式增长刻画无界增长，并在热-波系统中证明存在周期导致无限导数损失及光滑强迫共振。
AI 中文摘要
我们为具有弱耗散或部分耗散的系统引入了若干共振概念：$\dot u=Au+f(t)$，其中$A$是希尔伯特空间$H$上强稳定半群$S(t)$的生成元。我们将共振称为这样一种现象：时间周期函数$f$可能产生无界的$u$，经典上这发生在$A$具有纯虚特征值时。然而在无穷维空间中，无界增长可能依赖于拓扑选择，并且可能发生几种类型的“共振”。经典上，在$T$-周期$f$下存在$T$-周期$u$等价于一个值域条件：$\mathcal{R}(I-S(T))=H$。因此，该条件的失效允许无界解的增长，我们通过$S(t)$的性质阐明了这一联系。我们描述了值域失效的层级结构，并提供了相应的共振分类法。$||(A-\lambda I)^{-1}||_H$在$i\R$上的增长率决定了$u$与$f$之间的正则性间隙，并且对于$\lambda$的快速增长，该间隙可以是无限的。在上述情形下，我们展示了其对周期可解性的影响，以及构造光滑共振力的机制。这里发展的理论受到一个双曲-抛物系统的启发（并针对该系统进行了演示），其中后一个分量提供了系统唯一的耗散。这种动力学是流体-结构现象的简化，该现象表现出强稳定性但不支持无条件的周期适定性。尽管点谱共振被排除，我们注意到$\mathcal R(I-S(T))\neq H$。我们的主要结果表明，对于精心构造的几何结构，一个经典的热-波系统具有周期$T$，这些周期产生无限的导数损失，并且通过辅助结果，产生具有光滑强迫的共振。
英文摘要
We introduce several notions of resonance for systems exhibiting weak or partial dissipation: $\dot u=Au+f(t)$, with $A$ the generator of a strongly stable semigroup $S(t)$ on a Hilbert space $H$. We refer to resonance as the phenomenon where a time-periodic $f$ may yield an unbounded $u$, classically occurring when $A$ has imaginary eigenvalues. In infinite dimensions, however, unbounded growth may depend on topological choices, and several sorts of``resonance" may occur. Classically, existence of $T$-periodic $u$ under $T$-periodic $f$ is equivalent to a range condition: $\mathcal{R}(I-S(T))=H$. Consequently, its failure permits unbounded solution growth and we elucidate that connection through properties of $S(t)$. We describe a hierarchy of range failures, and provide an associated resonance taxonomy. The growth rate of ~$||(A-λI)^{-1}||_H$ on ~$i\R$ dictates a regularity gap between $u$ and $f$, and, for rapid growth in $λ$, that gap can be infinite. In said case, we demonstrate implications for periodic solvability, and a mechanism for constructing smooth resonant forces. The theory developed here is motivated by (and demonstrated for) a hyperbolic-parabolic system, where the latter component provides the only system dissipation. Such dynamics are a simplification of fluid-structure phenomena, which demonstrate strong stability but do not support unconditional periodic well-posedness. Though point-spectral resonance is ruled out, we note that $\mathcal R(I-S(T))\neq H$. Our main result here shows that, for a carefully constructed geometry, a classical heat-wave system possesses periods $T$ which yield infinite derivative loss and, via supporting results, yield resonance with smooth forcing.
Comments6 figures, 46 pages
