Coherent Structure, Chaos, and Turbulence in Fluid Mechanics

我们每天接触的流体（如水和空气）表现出各种各样的动力学。一个极端是混沌和湍流，其中流体形成复杂的分形图案，由于对初始条件的微小变化非常敏感，其细节不能在实验中精确地重复。湍流经常在流体中观察到，例如在飞机、车辆的尾迹中，甚至在建筑物和桥梁等障碍物中，以及在海洋和大气的大规模动力学中。另一个极端是漩涡细丝的运动，最著名的例子是龙卷风和通常从机翼和直升机叶片脱落的翼尖漩涡。涡旋细丝倾向于以可预测的方式移动，并在较长时间内保持其结构完整性。这个项目的目标是在流体力学中对这些相反的现象有更好的数学理解。涡旋细丝和湍流的准确预测在各种科学和工业应用中至关重要，包括空中、陆地和海上交通工具的设计，以及对气候和天气等复杂流体系统的理解。拥有坚实的数学基础可以帮助其他应用研究人员获得更深入的见解，从而更好地建模。此外，对这些极端的理解有助于为更好地理解相互作用和中间状态铺平道路，在中间状态下，流混合了结构和混乱。最后，克服这些问题的数学挑战将需要创新，这将引起更广泛的数学界的兴趣。这些研究项目还与研究生和年轻科学家在数学和STEM方面的培训相结合。PI将对在高雷诺数下不可压缩流体中观察到的两种行为进行更严格的数学理解：(1)涡旋细丝的相干运动；(2)“一般”、统计稳定强迫下的湍流。激发PI的基本问题是：(A)常用的几何演化模型，如局部感应近似（LIA），对于涡量集中在光滑曲线上的流体中涡丝的运动有多精确？(B)在高雷诺数极限下，我们能否提供实验观察到的Lyapunov正指数和异常耗散（如著名的Kolmogorov 4/5定律）的证明？这需要许多未探索的数学概念，目前还没有明确的方法来解决这些问题。相反，PI已经确定了几个独立有趣的问题，以建立必要的数学基础。对于(A)， PI和他的合作者将研究由Gross-Pitaevskii方程控制的量子流体中的涡旋细丝。由于涡量的量子化和稍微易于线性化的算符，量子情况预计会比经典情况更容易。首先，PI和合作者将研究二维Gross-Pitaevskii涡旋解的稳定性，为理解丝芯提供必要的基础。接下来，PI和合作者将证明LIA准确地描述了近乎直的、可能更普遍的量子涡旋细丝的运动，这些细丝足够长，可以做出有用的预测。对于(B)， PI和他的合作者将：(1)在随机动力系统的启发下，为随机偏微分方程中的李雅普诺夫指数发展新的定性理论；(2)改进了高维系统的定量半椭圆正则性和Lyapunov指数估计工具；(3)研究拉格朗日混沌的定量和非线性方面，即如何将流体中粒子的混沌动力学转化为流体本身的非线性动力学。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。

Fluids that we interact with on a daily basis (such as water and air) display a remarkable variety of dynamics. At one extreme is chaos and turbulence, wherein the fluid forms complicated fractal-like patterns, the details of which cannot be exactly repeated in experiments due to being very sensitive to small changes in the initial conditions. Turbulence is observed often in fluids, such as in the wakes of aircraft, vehicles, and even obstacles such as buildings and bridges and in the large-scale dynamics of the ocean and atmosphere. At the opposite extreme is the motion of vortex filaments, the most notable examples being tornadoes and the wing-tip vortices commonly shed from wings and helicopter blades. Vortex filaments tend to move in a predictable manner and maintain their structural integrity for extended periods of time. The goal of this project is to develop a better mathematical understanding of these opposite phenomena in fluid mechanics. The accurate prediction of vortex filaments and turbulence is crucial in a variety of scientific and industrial applications, including in the design of air, land, and sea vehicles and in the understanding of complex fluid systems such as the climate and weather. Having a firm mathematical foundation could help other applied researchers obtain deeper insights and lead to better modeling. Further, an understanding of these extremes helps pave the way for a better understanding of the interactions and intermediate regimes, where flows have a mix of structure and chaos. Finally, overcoming the mathematical challenges to these questions will require innovations that will be of interest to the wider mathematical community. The research projects are also integrated with the training of graduate students and younger scientists in mathematics and STEM. The PI will develop a more mathematically rigorous understanding of two behaviors observed in incompressible fluids at high Reynolds numbers: (1) the coherent motion of vortex filaments; (2) turbulence under "generic", statistically steady forcing. The fundamental questions motivating the PI are: (A) how accurate are the commonly used geometric evolution models such as the Local Induction Approximation (LIA) for the motion of a vortex filament in a fluid with vorticity concentrated on a smooth curve? (B) can we provide a proof for the experimentally observed positive Lyapunov exponents and anomalous dissipation (e.g., as the celebrated Kolmogorov 4/5 law) from the stochastically forced 3D Navier-Stokes in the high Reynolds number limit? These require a number of unexplored mathematical ideas and currently, there exists no clear way to attack them yet. Instead, the PI has identified several independently interesting problems to build necessary mathematical foundations. For (A), the PI and his collaborators will study vortex filaments in quantum fluids governed by the Gross-Pitaevskii equation. The quantum case is expected to be easier than the classical case due to the quantization of vorticity and slightly more amenable linearized operators. First, the PI and collaborators will study the stability of vortex solutions in 2d Gross-Pitaevskii, providing necessary ground for understanding the filament core. Next, the PI and collaborators will show that the LIA accurately describes the motion of nearly-straight, and potentially more general, quantum vortex filaments long enough to make useful predictions. For (B), the PI and his collaborators will: (1) develop novel qualitative theory for Lyapunov exponents in stochastic PDEs inspired by ideas from random dynamical systems; (2) develop better tools for quantitative hypoelliptic regularity and Lyapunov exponent estimation in high dimensional systems; and (3) study quantitative and nonlinear aspects of Lagrangian chaos, that is, how the chaotic dynamics of particles in a fluid can be translated into nonlinear dynamics of the fluid itself.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.