Stable structures and chaotic dynamics in fluid flows

从流体力学稳定性和湍流理论的角度出发，着重从理论上理解流体力学中的长时间动力学问题。通过将几个经典问题，如相干结构的元稳定性和流体流动的遍历/统计性质，放在严格的数学基础上，我的目标是加深对Navier-Stokes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、谐波/随机分析和动力系统方面开发新的数学技术，这将允许超越该领域目前的限制。这些技术将在两个基本问题的背景下设计和成熟。-流体的连贯结构和稳定性：由于其稳定和无限维的性质，流体混合导致熵向小尺度的转移，这种转移在有限时间内是可逆和保守的，但在无限时间内导致信息的不可逆损失。由于没有粘性，它产生无粘性阻尼，而它与扩散的相互作用产生耗散时间尺度，负责亚稳定行为。这些稳定性问题的严谨结果对解决大尺度大气动力学中从层流到湍流状态的转变以及相干涡状结构的形成等经典问题具有里程碑式的意义。-随机流体力学中的级联：从湍流的现象学理论中严格推导出尺度定律是流体力学中的一个核心问题。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，在高度创新的背景下，依赖于对李雅普诺夫指数等经典对象的相关参数的量化。

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.