Analytic Methods in Hydrodynamic and Wave Turbulence

湍流是日常生活中可以观察到的一种众所周知的物理现象。香烟或牛奶倒进咖啡时产生的烟雾痕迹看起来很混乱，这很容易看到；事实上，50多年前，列奥纳多·达·芬奇就画出了著名的湍流。尽管它引人注目，但湍流仍然是理论和数学物理中最大的谜团之一。在为该理论建立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，这与漩涡的形成相对应，漩涡破裂后形成更小的漩涡，然后重复这个过程。本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，并为波浪湍流理论发展更强大的数学基础。具体地说，关于第一个目标，研究者打算致力于解决Lars Onsager关于欧拉方程弱解存在性的著名猜想，其动能不守恒。这项工作将建立在以前开发的凸集成方案上，并将引入迄今为止在任何凸集成中未见过的新思想。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将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动湍流的非线性动力学中起着基本作用。

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.