标题： 一种严格的弗洛凯方法用于抛物型偏微分方程，应用于行走液滴统计学 显示英文标题

标题： A Rigorous Floquet Approach to Parabolic PDEs, with Applications to Walking Droplet Statistics

摘要： 弗洛凯理论为研究时间周期性动力学提供了一个强大的工具，使得任何周期性线性常微分方程的稳定性及长时间行为都可以重新表述为线性时不变系统。 然而，除了在处理希尔伯特空间或某些良好行为的操作符类时，弗洛凯理论无法在不进行维数约简的情况下严格应用于偏微分方程。 我们描述了弗洛凯理论的一个简单且严格的扩展，适用于巴拿赫空间上的广泛一类抛物型偏微分方程，使经典理论可以直接应用，只需最小的修改。 为了展示扩展的弗洛凯形式主义的实用性，我们研究了“行走液滴”：能够在振动油浴表面自我推进的毫米级油滴。 行走液滴表现出显著的、涌现的统计模式，这些模式尚未完全理解。 例如，已经观察到液滴的统计特性与引导波的时间平均之间存在广泛对应关系，但仅在无限域均匀深度中的单个液滴情况下得到了严格证明（Durey, Milewski 和 Bush, 2018）。 我们应用弗洛凯结果来推导液滴-波统计对应关系在一个高度一般的情况下，允许任意几何形状、多个液滴相互作用以及非共振液滴弹跳。 除了行走液滴之外，我们的框架为更广泛地分析周期性偏微分方程提供了一个灵活的工具，具有在日周期下的潮汐流动和营养盐-浮游植物-浮游动物（NPZ）模型中的潜在应用。 显示英文摘要

摘要： Floquet theory provides a powerful tool for studying time-periodic dynamics, allowing the stability and long-time behavior of any periodic, linear ODE to be recast in terms of a linear time-invariant system. However, except when dealing with Hilbert spaces or certain well-behaved classes of operators, Floquet theory cannot be rigorously applied to PDEs without dimensional reduction. We describe a simple, rigorous extension of Floquet theory to a wide class of parabolic PDEs on Banach spaces, enabling direct application with minimal changes from the classical theory. To demonstrate the utility of the extended Floquet formalism, we investigate the statistics of 'walking droplets': millimetric oil droplets that self-propel over the surface of a vibrating oil bath. Walking droplets exhibit striking, emergent statistical patterns that are not yet fully understood. For instance, a widespread correspondence has been observed between the statistics of the droplet and the time-average of its guiding wave, but it has only been proved rigorously for a single droplet in an unbounded domain of uniform depth (Durey, Milewski, and Bush, 2018). We apply our Floquet results to derive the droplet-wave statistical correspondence in a highly general case, allowing for arbitrary geometries, multiple droplet interactions, and non-resonant droplet bouncing. Beyond walking droplets, our framework provides a versatile tool for analyzing periodic PDEs more broadly, with promising applications to tidal flows and nutrient-phytoplankton-zooplankton (NPZ) models under daily cycles.