CAREER: Order and Disorder in Two-dimensional Fluid Motion

许多地球物理和天体物理系统，如海流、大尺度天气模式和行星大气，都是用二维流体方程近似地描述的，因为这些系统的垂直范围通常比水平范围小得多。因此，了解二维流体的长期动力学是天气预报、气候科学和天体物理学的基础。该项目旨在从几何和动力系统的角度推进我们对二维流体的理解，并通过培训初级研究人员和组织小型课程和会议来弥合这些学科之间的差距。这项工作的重点将是研究所有这些系统的原型，强迫二维不可压缩的Navier-Stokes方程和它们的非粘性，非强制的对应物，欧拉方程。我们将在不同的极限数量级上分析长时间和弱耗散/强迫的物理相关状态。这个体系的一个相当令人惊讶和神秘的特征是，在一个体系（欧拉）中，它以大飓风的形式（如漩涡）捕捉到了秩序的诞生和持久，而在另一个体系（纳维-斯托克斯）中，它描述了一种无序的、看似随机的、湍流的漩涡汤。也就是说，理想的欧拉流体通过涡旋合并和混合的过程，随着时间的推移趋于有序，而具有非常轻微粘度和作用力的纳维-斯托克斯流体则通过一系列不稳定性趋于无序，这与Kolmogorov的长期猜想是一致的。该项目旨在通过偏微分方程、几何和动力系统的镜头，通过研究接近和远离平衡的解的行为，促进我们对流体运动这些基本方面的理解。计算机实验将用于严谨的数学分析和框架问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many geophysical and astrophysical systems – such as oceanic currents, large-scale weather patterns and planetary atmospheres – are described, to good approximation, by two-dimensional fluid equations, since the vertical extent of these systems is typically much smaller than the horizontal. Understanding the long-term dynamics of two-dimensional fluids is thus fundamental to weather prediction, climate science, and astrophysics. This project aims to advance our understanding of two-dimensional fluids from the perspectives of both geometry and dynamical systems, as well as to bridge the gap between these disciplines through the training of junior researchers and the organizing of mini-courses and conferences.The focus of this work will be the study of the prototype of all such systems, the forced two-dimensional incompressible Navier-Stokes equations and their inviscid, unforced counterpart, the Euler equations. We will analyze the physically relevant regimes of long time and weak dissipation/forcing, in various orders of limits. A rather surprising and mysterious feature of this system is that, in one regime (Euler), it captures the birth and permanence of order in the form of large hurricane – like whirls – while in another regime (Navier-Stokes) it describes a disorderly, seemingly random, turbulent soup of eddies. That is, an ideal Euler fluid tends towards order over time through a process of vortex mergers and mixing, whereas a Navier-Stokes fluid with very slight viscosity and forcing tends towards disorder through a cascade of instabilities in accord with longstanding conjectures of Kolmogorov. The project aims to advance our understanding of these fundamental aspects of fluid motion by studying the behavior of solutions both near and far from equilibrium through the lens of partial differential equations, geometry, and dynamical systems. Computer experiments will be used to inform rigorous mathematical analysis and frame questions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.