Analysis of Properties of Effective Hamiltonians with Applications

许多物理系统（如复合材料和湍流燃烧）在实际应用中涉及小结构。重要的是通过合适的数学模型（称为“均质化”）对这些小尺度进行“平均”，并研究与原始系统近似的宏观结构（“有效量”）。一个特别有趣和与物理相关的问题是了解这些有效量如何依赖于原始物理系统。有效哈密顿量来源于均匀化哈密顿-雅可比方程，它是偏微分方程（PDE），在控制理论和晶体生长、火焰传播等建模中起着重要作用。例如，通过流行的g方程模型，有效哈密顿量是模拟湍流燃烧（如强风吹起的野火蔓延、发动机内部燃烧）中最重要的量之一湍流火焰速度的重要方法。该项目的一个目标是了解湍流火焰速度如何取决于气流强度（如风速）和其他重要物理量（如曲率效应），这可以为实验中观察到的现象提供理论依据。曲率效应是捕捉沿火焰锋面燃烧温度变化的一种现象学方法。本项目为研究生提供研究训练机会。项目的第一部分旨在确定哈密顿-雅可比方程周期均匀化的最优收敛速率。正如首席研究员及其合作者在之前的工作中所揭示的，最优收敛速率与相应的有效哈密顿量的形状密切相关。项目的第二和第三部分侧重于从数学兴趣和湍流燃烧实际应用的角度研究有效哈密顿量的重要性质。尽管自lions - papanicolau - varadhan（1987）的基础工作以来，在许多情况下已经建立了有效哈密顿量的存在性，但由于PDEs中标准方法的局限性，理解有效哈密顿量的更精细性质仍然是均匀化理论中的一个主要开放问题。有效哈密顿量和动力系统的性质之间有很深的联系。特别是，首席研究者需要从遍历理论、奥布里-马瑟理论/弱鉴理论中运用大量的工具/方法。此外，与二人零和微分博弈相关的动力学在处理非凸哈密顿量和曲率项（曲率效应）的存在方面也很重要。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many physical systems (e.g. composite materials and turbulent combustion) in practical applications involve small structures. It is important to “average” those small scales through suitable mathematical models (called “homogenization”) and investigate resulting macroscopic structures (“effective quantities”) that are approximations of original systems. A particularly interesting and physically relevant question is to understand how those effective quantities depend on original physical systems. Effective Hamiltonian arises from homogenizing Hamilton-Jacobi equations, which are partial differential equations (PDE) that play fundamental roles in control theory and the modeling of things like crystal growth, flame propagations, etc. For example, effective Hamiltonian serves as an important approach to model turbulent flame speed, one of the most important quantities in turbulent combustion (e.g., the spread of wild fire fanned by strong wind, the burning inside an engine), via the popular G-equation model. One goal of this project is to understand how the turbulent flame speed depends on the strength of the flow (e.g., wind velocity) and other important physical quantities like curvature effects, which could provide theoretical justifications for phenomena observed in experiments. The curvature effect is a phenomenological way to capture variance of burning temperature along the flame front. The project provides research training opportunities for graduate students.The first part of the project aims to identify optimal convergence rate in periodic homogenization of Hamilton-Jacobi equations. As the principal investigator and his collaborators have revealed in previous works, the optimal convergence rate is closely related to the shape of corresponding effective Hamiltonian. The second and third part of the project focus on studying important properties of effective Hamiltonian from perspectives of both the mathematical interest and practical applications in turbulent combustion. Although the existence of effect Hamiltonian has been established in many situations since the fundamental work of Lions-Papanicolaou-Varadhan (1987), understanding finer properties of effective Hamiltonian remains a major open problem in homogenization theory due to limitations of standard methods in PDEs. There are deep connections between properties of effective Hamiltonian and dynamical systems. In particular, the principal investigator needs to employ substantial tools/methods from ergodic theory, Aubry-Mather theory/weak KAM theory. In addition, dynamics associated with two-person zero-sum differential games is also important in dealing with non-convex Hamiltonian and the presence of curvature term (curvature effect).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.