Inviscid Limits, Uniqueness, and Anomalous Dissipation in Hydrodynamics

流体动力学中的无粘极限、唯一性和反常耗散

• 批准号: 2108573
• 负责人: Hakima Bessaih
• 金额: $ 24.9万
• 依托单位: University of Wyoming
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2021-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2108573&HistoricalAwards=false
• 关键词: Inviscid; Limits; Uniqueness; Anomalous; Dissipation

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.

湍流是流体的复杂，不规则和混乱的运动，是一种无处不在的基本机制，用于转移跨时空尺度的动量和能量。尽管描述流体动力学的确定性方程是完善的，并且其数值解决方案已经揭示了流体的湍流行为，但它们的分析具有挑战性，并且基本结果仍然难以捉摸。概率工具的使用提供了互补的方法，这些方法自然地包含了由于外力而导致的流体状态和随机扰动的不确定性。该项目将在概率，随机分析和动态系统方面利用最新工具，以研究与物理，工程和大气 - 海洋动力学相关的基本流体动力学模型及其某些变体。将重点是研究随机强迫对这些系统的影响，进一步了解跨量表的能量转移，并使用概率框架获得基本的存在和独特性结果，而这些概率框架在确定性方法中仍然难以捉摸。该项目还将为大学生和研究生提供参与研究的机会。该研究计划研究了源自湍流理论的挑战，涉及随机分析，动力学系统和部分微分方程。目的是了解Euler和Navier-Stokes方程，其随机版本以及湍流的现象学定律之间的联系。随机方法的优点是平均能量注入，耗散和通量之间平衡定律的主要简单性。由于这些随机模型的丰富结构，可以证明，在缺乏确定性的对应物的同时，可以证明一些关于三维粘性和无粘性流体流的近似值的结果。噪声对相关动力学系统的平滑效果将被使用。将解决对Inviscid限制问题的研究，以更好地了解三维情况的直接能量级联和二维情况的逆向能量级联。对于某些具有各向异性粘度的地球物理模型的一定限制也将被研究。理论问题（例如不变措施的存在，独特性和奇迹性）将通过数值模拟进行补充。该项目由数学科学司应用数学计划和启发竞争性研究的既定计划共同资助（EPSCOR）。该奖项反映了NSF的法定任务，并被认为值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来进行评估。