Stable structures and chaotic dynamics in fluid flows

流体流动中的稳定结构和混沌动力学

• 批准号: EP/X020886/1
• 负责人: Michele Coti Zelati
• 金额: $ 162.63万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX020886%2F1
• 关键词: Stable; structures; chaotic; dynamics; fluid

The proposal focuses on the theoretical understanding of long-time dynamics questions in fluid mechanics, from the point of hydrodynamic stability and turbulence theory. By putting on rigorous mathematical grounds several classical questions, such as the meta-stability of coherent structures and the ergodic/statistical properties of fluid flows, I aim to deepen the understanding of the long- time behavior of solutions to the Navier-Stokes and the Euler equations.The main objective is to establish a research group at Imperial College to develop novel mathematical techniques in the theory of PDEs, harmonic/stochastic analysis, and dynamical systems, that would allow to move beyond the current limits of the field. These techniques will be devised and mature in the context of two fundamental problems.- Coherent structures and stability of fluids: due to its stabilizing and infinite-dimensional nature, fluid mixing causes a transfer of enstrophy to small scales, in a manner which is reversible and conservative for finite times, but results in an irreversible loss of information at infinite times. With no viscosity, it generates inviscid damping, while its interaction with diffusion creates dissipation time-scales responsible for meta-stable behavior. Rigorous results on these stability problems constitute a milestone towards the resolution of classical long-standing questions related to the transition from laminar to turbulent states and the formation of coherent vortex-like structures in large-scale atmospheric dynamics.- Cascades in stochastic fluid mechanics: the rigorous derivation of scaling laws stemming from the phenomenological theory of turbulence is a central problem in fluid mechanics. I aim to understand how randomness affects foundational aspects of the theory, such as ergodicity and anomalous dissipation, in a highly innovative context that relies on the quantification with respect to relevant parameters of classical objects such as Lyapunov exponents.

该提案侧重于从流体动力学稳定性和湍流理论的角度对流体力学中长期动力学问题的理论理解。通过对几个经典问题，如相干结构的亚稳定性和流体流动的遍历/统计性质，提出严格的数学基础，我的目标是加深对Navier-Stokes方程和Euler方程解的长期行为的理解。主要目标是在帝国理工学院建立一个研究小组，在偏微分方程理论中发展新的数学技术，谐波/随机分析和动态系统，这将允许超越目前的限制领域。这些技术将在两个基本问题的背景下设计和成熟。流体的相干结构和稳定性：由于其稳定性和无限维性质，流体混合导致拟能向小尺度的转移，其方式在有限时间内是可逆和保守的，但在无限时间内导致不可逆的信息损失。由于没有粘性，它会产生无粘阻尼，而它与扩散的相互作用会产生耗散时间尺度，从而导致亚稳态行为。这些稳定性问题的严格结果构成了解决经典长期问题的里程碑，这些问题与从层流到湍流状态的过渡以及大尺度大气动力学中相干涡旋状结构的形成有关。随机流体力学中的级联：严格推导源自湍流唯象理论的标度律是流体力学中的一个中心问题。我的目标是了解随机性如何影响理论的基础方面，如遍历性和异常耗散，在一个高度创新的背景下，依赖于量化相对于相关参数的经典对象，如李雅普诺夫指数。