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观察到的众所周知的物理现象。很容易看到烟雾或牛奶中烟雾径的混乱行为，很容易看到咖啡。的确，在半个千年前，莱昂纳多·达·芬奇（Leonardo da Vinci）勾勒出动荡的流程。尽管它具有显着性，但湍流仍然是理论和数学物理学中最大的谜团之一。在为该理论开发更好的数学基础方面取得的任何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：在流体动力的湍流中，这对应于涡流的形成，这些涡流形成形成较小的涡流，因此过程重复。该研究项目的目的是研究这种级联中存在的数学结构。该项目以两个基本目标为指导：更好地了解动力学中动能的异常耗散，并为波湍流理论开发更强的数学基础。具体而言，与第一个目标有关，研究人员打算致力于解决著名的Lars Onsager的猜想，该猜想与具有动能不能保存的Euler方程的薄弱解决方案有关。这项工作将建立在先前开发的凸集成方案的基础上，并将在迄今为止的任何凸集成中介绍看不见的新想法。关于第二个目标，研究人员打算开发新的数学方法，以便更好地了解物理文献中制作的启发式衍生物如何变得严格。这项研究的一个关键方面将是使用分析数理论的方法来表征共振和准谐振相互作用，这些相互作用被怀疑在调查的波浪湍流方案的非线性动力学中起着基本作用。