Development of Random Dynamical Systems Techniques in the Pursuit of a Predictive Theory of Turbulence and Related Questions

湍流是指在运动物体的尾迹中流体或气体的无序、看似随机的运动，例如船舶在水中运动所产生的尾迹。Andrei Kolmogorov在20世纪40年代出版的一系列著名的理论著作（统称为K41理论）中建立了一些控制湍流的基本定律。尽管这个理论很有影响力，但它并不完整：尽管对K41的一些预测非常准确，但有些预测并不准确，这指出了支撑K41的假设中的问题。这些缺陷（通常被称为“间歇性”）是物理学和工程界研究的一个活跃课题。此外，K41理论是基于对湍流行为的经验观察得出的。从控制流体如何随时间演化的基本方程（纳维-斯托克斯方程）直接证实K41的任何预测仍然是一个巨大的挑战。这个项目的长远目标是通过提供一个数学上严谨的理论来解决这些缺点，这个理论弥合了控制流体的基本方程和最准确的K41预测之间的差距。这将通过引进和改进随机动力系统理论的最新进展来实现。提供这样一座数学桥梁，也有可能揭示湍流理论中不太为人所知的方面，比如间歇性。PI将培养动力系统专业的研究生，为这一跨学科研究做准备。更准确地说，这项工作的目的是开发数学工具，其长期目标是直接从Navier-Stokes方程中获得以下两个经验观察到的湍流性质的数学证明：(a)相对于初始条件的敏感性（即，正李雅普诺夫指数）和(b)弱反常耗散（即，即使在流体实验的所有其他参数保持不变的情况下，将粘度取为0，流体中的能量仍以恒定的正速率耗散）。性质(a)和(b)是相互关联的，但并不完全相同：性质(a)与相空间中附近轨迹的分离有关，而(b)与所谓的能量级联有关，这是一种非线性的“传送带”式效应，它将流体中的动能从低傅立叶模式传递到高傅立叶模式，直到由粘度引起的耗散效应接管。然而，在其中一个方面的进展可能会导致对另一个的更好理解，例如，性质(b)可能与Navier-Stokes方程的高维混沌SRB吸引子的存在有关，这只有在性质(a)成立时才有可能。对于目标(a)，一个主要的数学挑战是，通常很难证明给定的动力系统是混沌的——即使对于二维玩具模型（如Chirikov-Taylor标准映射）也是如此，该模型以有序椭圆型和混沌行为的基本共存为特征。另一方面，PI和其他人的进步在验证受随机驱动的系统的混沌行为方面取得了相当大的进展，这通常是在与湍流相关的模型中假设的。本提案的工作将扩展和增强这些理论工具，最终目的是建立(a)现实流体模型，如GOY和SABRA壳模型，以及在极限雷诺数取为无穷大时的超粘性三维Navier-Stokes方程的性质。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。

Turbulence refers to the unordered, seemingly random motion of fluids or gases in the wake of a moving body, e.g., the wake produced by a ship moving through water. Some basic laws governing turbulence were established by Andrei Kolmogorov in his now-famous series of theoretical works published in the 1940s (collectively known as K41 theory). Although highly influential, this theory is not complete: even though some predictions of K41 are remarkably accurate, some are not, pointing to problems in the assumptions underpinning K41. These defects (often referred to as "intermittency") are an active subject of study in the physics and engineering communities. Moreover, K41 theory was derived based on empirical observations regarding how turbulent fluids behave. It remains a grand challenge to substantiate any of the K41 predictions directly from the fundamental equations (the Navier-Stokes equations) governing how fluids evolve in time. The far-reaching goal of this project is to address these shortcomings by providing a mathematically rigorous theory that bridges the gap between the fundamental equations governing fluids and the most accurate of the K41 predictions. This will be done by importing and improving on recent advances in random dynamical systems theory. Providing such a mathematical bridge could also potentially shed light on less well-understood aspects of turbulence theory, such as intermittency. The PI will train graduate students specializing in dynamical systems in preparation for this cross-disciplinary research. More precisely, the aim of this work is to develop mathematical tools with the long-term goal of obtaining a mathematical proof, directly from the Navier-Stokes equations, of the following two empirically observed properties of turbulent fluids: (a) sensitivity with respect to initial conditions (i.e., a positive Lyapunov exponent) and (b) weak anomalous dissipation (i.e., energy in the fluid is dissipated at a constant positive rate even as viscosity is taken to 0 with all other parameters of the fluid experiment held constant). Properties (a) and (b) are interrelated but not exactly the same: property (a) has to do with the separation of nearby trajectories in phase space, while (b) has to do with the so-called energy cascade, a nonlinear "conveyor-belt"-type effect which transmits kinetic energy in the fluid from low to high Fourier modes until dissipation effects due to viscosity take over. However, progress on one could lead to a better understanding of the other, e.g., property (b) is likely related to the presence of a high-dimensional chaotic SRB attractor for the Navier-Stokes equations, which is only possible if property (a) is true. Towards goal (a), a major mathematical challenge is that it is typically intractably hard to prove that a given dynamical system is chaotic— this is true even for two-dimensional toy models such as the Chirikov-Taylor standard map, which features fundamental coexistence of ordered elliptic-type and chaotic behavior. On the other hand, advances by the PI and others have made considerable progress towards verifying chaotic behavior for systems subjected to stochastic driving, as is typically assumed in models relevant to turbulence. The work in this proposal will extend and enhance these theoretical tools with the ultimate aim of establishing property (a) for realistic fluids models such as the GOY and SABRA shell models as well as for the hyperviscous 3D Navier-Stokes equations in the limit as Reynolds number is taken to infinity.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.