Inviscid Limits, Uniqueness, and Anomalous Dissipation in Hydrodynamics

湍流是一种复杂的、不规则的、混沌的流体运动，是一种普遍存在的、跨越时空尺度的动量和能量转移的基本机制。尽管描述流体动力学的确定性方程已经建立，并且它们的数值解已经揭示了许多关于流体湍流行为的信息，但它们的分析仍然具有挑战性，基本结果仍然难以捉摸。概率工具的使用提供了补充方法，自然地将流体实际状态的不确定性和外力引起的随机扰动结合起来。该项目将利用概率、随机分析和动力系统方面最先进的工具，研究与物理、工程和大气-海洋动力学相关的基本流体动力学模型及其一些变体。重点将放在研究随机强迫对这些系统的影响，进一步了解跨尺度的能量传递，以及利用概率框架获得基本存在性和唯一性结果的可能性，这在确定性方法中仍然难以捉摸。该项目还将为本科生和研究生提供参与研究的机会。本研究计划研究源自湍流理论挑战的几个数学问题，涉及随机分析、动力系统和偏微分方程。目标是理解欧拉方程和纳维-斯托克斯方程之间的联系，它们的随机版本，以及湍流的现象学定律。随机方法的优点主要是能量注入、耗散和通量的平均速率之间的平衡律的简单性。由于这些随机模型的丰富结构，一些三维粘性和无粘性流体流动的近似可以证明唯一性和稳定性的结果，而缺乏确定性的对应结果。将使用噪声对相关动力系统的平滑效应。为了更好地理解三维情况下的正能量级联和二维情况下的逆能量级联，我们将研究无粘极限问题。对一些具有各向异性黏度的地球物理模型的非黏性极限也进行了研究。理论问题，如不变量测度的存在性、唯一性和遍历性，将通过数值模拟得到补充。该项目由数学科学部应用数学项目和促进竞争研究的既定项目（EPSCoR）共同资助。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, the complex, irregular and chaotic motion of fluids, is a ubiquitous and fundamental mechanism for the transfer of momentum and energy across spatio-temporal scales. Although the deterministic equations describing the dynamics of fluids are well-established and their numerical solution has revealed much about the turbulent behavior of fluids, their analysis is challenging, and basic results remain elusive. The use of probabilistic tools offer complementary approaches that naturally incorporate the uncertainties of the actual state of the fluid and random perturbations due to external forces. This project will leverage state-of-the-art tools in probability, stochastic analysis, and dynamical systems, to study the basic fluid dynamics models and some of its variants, which are relevant to physics, engineering, and atmosphere-ocean dynamics. Emphasis will be given to study the effect of random forcing on these systems, gain further understanding on the energy transfer across scales, and the possibility of obtaining basic existence and uniqueness results using the probabilistic framework, which remain elusive in the deterministic approach. The project will also provide opportunities for undergraduate and graduate students to participate in the research. This research program studies several mathematical problems stemming from the challenges of turbulence theory and involves stochastic analysis, dynamical systems, and partial differential equations. The goal is to understand the link between the Euler and Navier-Stokes equations, their stochastic versions, and the phenomenological laws of turbulence. The advantage of the stochastic approach is the major simplicity of balance laws between mean rates of energy injection, dissipation, and flux. Due to the rich structure of these stochastic models, some results on uniqueness and stability of some approximations of three-dimensional viscous and inviscid fluid flows could be proved while their deterministic counterpart is lacking. The smoothing effect of the noise on the associated dynamical system will be used. The study of the inviscid limit problem will be tackled for a better understanding of the direct energy cascade for the three-dimensional case and the inverse energy cascade for the two-dimensional case. Some inviscid limits for some geophysical models with anisotropic viscosity will be investigated as well. Theoretical issues, such as existence, uniqueness, and ergodicity of invariant measures, will be complemented by numerical simulations. This project is jointly funded by the Division of Mathematical Sciences Applied Mathematics program and the Established Program to Stimulate Competitive Research (EPSCoR).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.