AI 中文总结
本文结合数学分析、科学计算与物理学框架，研究流体流动的极端行为，推导严格上界，通过变分优化求解尖锐边界，揭示极端行为的物理机制，还列出开放问题并探讨方法改进。
AI 中文摘要
极端流动是指通过合适的初始条件或施加的外力，实现关注量在瞬时或有限时间内的最大增长，这类关注量通常用于衡量小尺度特性，因此能反映流动的正则性。极端行为是流体力学中多个开放问题的核心，包括湍流中的耗散异常以及各类流体流动模型中奇点的形成。本文阐述了一种结合数学分析、科学计算与物理学的框架，用于系统研究此类极端行为。第一步是推导给定模型解中关注量增长的严格上界，这些不等式表达了所有可容许解中可能出现的最极端行为的基本限制。但由于推导方式，这些边界可能是保守的，会高估系统中实际可实现的增长。为探究这种可能性，下一步是建立变分优化问题，在合适的约束下最大化关注量的增长，这类问题的求解依赖现代数值优化方法。当由此得到的最大化子的特性与边界匹配时，该边界被认定为尖锐，无法从根本上改进。最后，饱和边界的解的特性揭示了实现极端行为的物理机制。本文综述了该研究计划已产生尖锐边界及饱和这些边界的极端流动的问题，随后列出一系列开放问题，并在文末讨论了可能的方法学改进方向。
英文摘要
Extreme flows realize the largest possible growth, either instantaneously or in finite time, of certain quantities of interest which is achieved by a suitable choice of the initial condition or the applied forcing. The quantities of interest usually measure some small-scale properties and therefore provide information about the regularity of the flow. Extreme behavior is at the heart of several open problems in fluid mechanics including the dissipation anomaly in turbulence and formation of singularities in various models of fluid flow. In this essay we describe a framework making it possible to study such extreme behavior systematically by combining mathematical analysis, scientific computation and physics. As a first step, one aims to deduce rigorous upper bounds on the growth of the quantities of interest in the solutions of a given model. These inequalities express fundamental limitations on the most extreme behavior possible among {\em all} admissible solutions. However, given how they are obtained, these bounds may be conservative and overestimate the growth actually realizable in the system. In order to probe this possibility, as the next step, we set up variational optimization problems where the growth of the quantity of interest is maximized under suitable constraints. Solution of such problems is enabled by modern methods of numerical optimization. When properties of the thus obtained maximizers match the bounds, the bounds are declared sharp and therefore cannot be fundamentally improved. Finally, properties of the solutions saturating the bounds reveal insights about the physical mechanisms realizing the extreme behavior. We survey problems where this research program has produced sharp bounds together with extreme flows saturating these bounds. A collection of open problems is then presented and we close the essay with a discussion of possible methodological improvements.
Comments77 pages, 16 figures
