Analytic Methods in Hydrodynamic and Wave Turbulence

湍流是在日常生活中可观察到的众所周知的物理现象。香烟或牛奶倒入咖啡时，烟迹的看似混乱的行为很容易看到;事实上，湍流是列奥纳多达芬奇在500多年前著名的素描。尽管它引人注目，湍流仍然是理论物理和数学物理中最大的谜团之一。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：在流体动力学湍流中，这对应于涡旋的形成，涡旋分解形成较小的涡旋，然后该过程重复。本研究项目的目的是调查在这样的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，并为波动湍流理论建立更坚实的数学基础。具体而言，关于第一个目标，研究人员打算努力解决一个著名的猜想Lars Onsager有关的存在弱解的欧拉方程的动能不守恒。这项工作将建立在以前开发的凸积分计划，并将介绍新的想法看不见的任何凸积分日期。关于第二个目标，研究者打算开发新的数学方法，以便更好地理解物理学文献中的启发式推导如何变得严格。这项调查的一个关键方面将是使用解析数论的方法，以表征共振和准共振相互作用，这被怀疑在调查中的波动湍流的制度的非线性动力学中发挥了根本性的作用。

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.