Analytic Methods in Hydrodynamic and Wave Turbulence

湍流是一种在日常生活中可以观察到的众所周知的物理现象。当香烟或牛奶被倒进咖啡时，看起来似乎很混乱的烟雾尾迹很容易看到；事实上，湍流是达·芬奇在5000多年前绘制的著名草图。尽管湍流很显眼，但它仍然是理论和数学物理中最大的谜团之一。在为这一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：在流体动力湍流中，这对应于涡流的形成，这些涡流分裂形成较小的涡流，然后重复这一过程。这项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：更好地了解湍流中动能的反常耗散，并为波浪湍流理论发展更坚实的数学基础。具体地说，与第一个目标有关，研究者打算致力于解决Lars Onsager的一个著名猜想，该猜想与动能不守恒的欧拉方程存在弱解有关。这项工作将建立在以前开发的凸积分方案的基础上，并将引入迄今为止在任何凸积分中未曾见过的新想法。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，这两种相互作用被怀疑在所研究的波浪湍流区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.