Analysis of Singularity Formation in Three-Dimensional Euler Equations and Search for Potential Singularities in Navier-Stokes Equations

三维欧拉方程奇异性形成分析及纳维-斯托克斯方程潜在奇异性搜索

• 批准号: 2205590
• 负责人: Thomas Hou
• 金额: $ 54.37万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2205590&HistoricalAwards=false
• 关键词: Analysis; Singularity; Formation; Three; Dimensional

Navier-Stokes equations have been around for more than 150 years. Physicists use them to model ocean currents, weather patterns and turbulent flows behind a commercial jet or ship. Most physicists and engineers believe that the smooth solutions of the Navier-Stokes equations will not break down without external forcing. On the other hand, a recent study by the PI indicates that the Navier-Stokes equations could develop a catastrophic behavior if one starts with a highly symmetric but perfectly smooth flow. Such scenario corresponds to a "perfect storm" in which all things that could potentially go wrong indeed go wrong. Potentially singular behavior of the Navier-Stokes equations could post tremendous damage to our environment, affect the safety of our planes and ships, and our ability to do accurate weather forecasting. The purpose of this project is to investigate under what conditions the Euler and the Navier-Stokes equations may develop singular behavior. The understanding of this question would enable us to avoid catastrophic behavior of the fluid flows in nature. The ultimate goal of this research is to develop effective analytical and computational tools that would enhance our ability to model and predict various complex phenomena in nature so that we can have more confidence in the safety of commercial jets and ships, and weather forecasting. Additional impact of this project will be the involvement of graduate students. The interdisciplinary training they receive in this project will be very important for their future careers in mathematics and science.Whether the 3D incompressible Euler equations develop finite time singularities from smooth initial data has been a longstanding open question. Built upon the results obtained from the prior NSF support, this project aims at providing a rigorous proof of the potential finite time singularity in the 3D Euler equations with smooth initial data and boundary. A major approach of the research is to prove the existence and stability of an approximate self-similar profile with a small residual error for the 3D axisymmetric Euler equations. Numerical computations will first be conducted to construct an approximate self-similar profile with a very small residual error. A crucial step in the analysis is to obtain linear stability for the approximate self-similar profile through a dynamic rescaling formulation. Linear stability will be established by obtaining sharp functional estimates with appropriately chosen singular weights and using space-time estimates with computer assistance. The new techniques and tools developed during this project are likely to have an impact in the neighboring areas of mathematics. Another proposed project is to look for potential singular solutions of the 3D Navier-Stokes equations using specially designed initial data with periodic boundary conditions. The approach relies on using the dynamic rescaling formulation to solve the axisymmetric Navier-Stokes equations and to avoid the potential numerical instability induced by the frequent changes of the adaptive meshes in recent computations. The successful execution of this research would provide valuable insight to the Clay Millennium Problem on the 3D Navier-Stokes equations and become an important step towards its ultimate resolution.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.

Navier-Stokes方程已经存在了150多年。物理学家使用它们来建模洋流，天气模式和商用喷气机或船后的动荡流。大多数物理学家和工程师认为，如果没有外部强迫，Navier-Stokes方程的平滑解决方案将不会崩溃。另一方面，PI的最新研究表明，如果一个高度对称但完全光滑的流动开始，Navier-Stokes方程可能会发展出灾难性的行为。这种情况对应于一场“完美的风暴”，其中所有可能出错的事情确实出错了。 Navier-Stokes方程的潜在奇异行为可​​能会对我们的环境造成巨大破坏，影响飞机和船只的安全，以及我们进行准确的天气预报的能力。该项目的目的是在欧拉和Navier-Stokes方程的哪些条件下进行研究。对这个问题的理解将使我们能够避免流体在自然界流动的灾难性行为。这项研究的最终目的是开发有效的分析和计算工具，以增强我们在自然界建模和预测各种复杂现象的能力，以便我们可以对商业喷气机和船只的安全以及天气预报具有更大的信心。该项目的其他影响将是研究生的参与。他们在该项目中接受的跨学科培训对于他们在数学和科学领域的未来职业非常重要。是否3D不可压缩的Euler方程从平滑的初始数据中发展出有限的时间奇异性，这是一个长期的开放问题。基于从先前的NSF支持获得的结果，该项目旨在提供严格的证明，证明具有平滑的初始数据和边界的3D Euler方程中潜在的有限时间奇点。这项研究的一种主要方法是证明近似自相似轮廓的存在和稳定性，对于3D轴对称Euler方程，残余误差很小。将首先进行数值计算以构建一个近似的自相似轮廓，并具有很小的残留误差。分析的关键步骤是通过动态重新缩放公式获得近似自相似曲线的线性稳定性。线性稳定性将通过获得适当选择的奇异权重的尖锐功能估计来确定，并在计算机协助下使用时空估计。该项目期间开发的新技术和工具可能会影响数学的相邻领域。另一个建议的项目是使用具有周期性边界条件的特殊设计的初始数据来寻找3D Navier-Stokes方程的潜在奇异解决方案。该方法依赖于使用动态重新恢复公式来求解轴对称的Navier-Stokes方程，并避免在最近计算中自适应网格的频繁变化引起的潜在数值不稳定性。这项研究的成功执行将为3D Navier-Stokes方程中的Clay Millennium问题提供宝贵的见解，并成为迈向其最终解决方案的重要一步。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准来通过评估来支持的。

项目成果

Potential Singularity of the 3D Euler Equations in the Interior Domain

• DOI: 10.1007/s10208-022-09585-5
• 发表时间: 2021-07
• 期刊: Foundations of Computational Mathematics
• 影响因子: 3
• 作者: T. Hou

Potential Singularity Formation of Incompressible Axisymmetric Euler Equations with Degenerate Viscosity Coefficients

具有简并粘度系数的不可压缩轴对称欧拉方程的势奇异性形成

• DOI: 10.1137/22m1470906
• 发表时间: 2023
• 期刊: Multiscale Modeling & Simulation
• 影响因子: 1.6
• 作者: Hou, Thomas Y.;Huang, De
• 通讯作者: Huang, De

Exponentially Convergent Multiscale Methods for 2D High Frequency Heterogeneous Helmholtz Equations

二维高频异质亥姆霍兹方程的指数收敛多尺度方法

• DOI: 10.1137/22m1507802
• 发表时间: 2023
• 期刊: Multiscale Modeling & Simulation
• 影响因子: 1.6
• 作者: Chen, Yifan;Hou, Thomas Y.;Wang, Yixuan
• 通讯作者: Wang, Yixuan

Exponentially Convergent Multiscale Finite Element Method

指数收敛多尺度有限元法

• DOI: 10.1007/s42967-023-00260-2
• 发表时间: 2023
• 期刊: Communications on Applied Mathematics and Computation
• 影响因子: 1.6
• 作者: Chen, Yifan;Hou, Thomas Y.;Wang, Yixuan
• 通讯作者: Wang, Yixuan

Asymptotically self-similar blowup of the Hou-Luo model for the 3D Euler equations

• DOI: 10.1007/s40818-022-00140-7
• 发表时间: 2021-06
• 期刊: Annals of PDE
• 影响因子: 2.8
• 作者: Jiajie Chen;T. Hou;De Huang