Analytic Methods in Hydrodynamic and Wave Turbulence

流体动力学和波浪湍流的分析方法

• 批准号: 1820764
• 负责人: Tristan Buckmaster
• 金额: $ 4.89万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2019-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1820764&HistoricalAwards=false
• 关键词: Analytic; Methods; Hydrodynamic; Wave; Turbulence

Turbulence is a well-known physical phenomenon observable in everyday life. The seemingly chaotic behavior of smoke trails from a cigarette or milk as it is poured into coffee is easily seen; indeed, turbulent flow was famously sketched by Leonardo da Vinci over half a millennium ago. Despite its conspicuousness, turbulence remains one of greatest enigmas of both theoretical and mathematical physics. Any small progress made in developing better mathematical underpinnings to the theory could lead to profound developments in the applied sciences. Characteristic to turbulence is a cascade to small scales: in hydrodynamic turbulence this corresponds to the formations of eddies that break up to form smaller eddies, whereupon the process repeats. The aim of this research project is to investigate the mathematical structures present in such a cascade. The project is guided by two underlying goals: to better understand anomalous dissipation of kinetic energy in turbulence, and to develop stronger mathematical foundations for the theory of wave turbulence. Specifically, relating to the first goal, the investigator intends to work towards resolving a famous conjecture of Lars Onsager relating to the existence of weak solutions to the Euler equations whose kinetic energy is not conserved. The work will build on previously-developed convex integration schemes and will introduce new ideas unseen in any convex integration to date. With regard to the second goal, the investigator intends to develop new mathematical methods in order to better understand how heuristic derivations made in the physics literature can be made rigorous. A key aspect of this investigation will be the use of methods from analytic number theory in order to characterize resonant and quasi-resonant interactions, which are suspected to play a fundamental role in the non-linear dynamics of the regimes of wave turbulence under investigation.

湍流是日常生活中可以观察到的众所周知的物理现象。香烟或牛奶倒入咖啡时产生的烟迹看似混乱，这一现象很容易被观察到。事实上，五百年前，列奥纳多·达·芬奇就对湍流进行了著名的描绘。尽管湍流很引人注目，但它仍然是理论物理学和数学物理学中最大的谜团之一。在为该理论发展更好的数学基础方面取得的任何微小进展都可能导致应用科学的深刻发展。湍流的特征是小尺度的级联：在流体动力湍流中，这对应于涡流的形成，涡流分裂形成更小的涡流，从而重复该过程。该研究项目的目的是研究这种级联中存在的数学结构。该项目有两个基本目标：更好地理解湍流中动能的异常耗散，并为波浪湍流理论奠定更坚实的数学基础。具体来说，与第一个目标相关，研究人员打算致力于解决 Lars Onsager 的一个著名猜想，该猜想涉及动能不守恒的欧拉方程的弱解的存在。这项工作将建立在先前开发的凸积分方案的基础上，并将引入迄今为止任何凸积分中未曾见过的新想法。关于第二个目标，研究者打算开发新的数学方法，以便更好地理解如何使物理文献中的启发式推导变得严格。这项研究的一个关键方面是使用解析数论的方法来表征共振和准共振相互作用，这些相互作用被怀疑在所研究的波湍流体系的非线性动力学中发挥着重要作用。