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

纳维尔-斯托克斯方程已经存在了150多年。物理学家使用它们来模拟洋流、天气模式和商业喷气式飞机或轮船背后的湍流。大多数物理学家和工程师认为，在没有外力的情况下，Navier-Stokes方程的光滑解不会破裂。另一方面，PI最近的一项研究表明，如果一个人从高度对称但完全平滑的流动开始，纳维尔-斯托克斯方程可能会产生灾难性的行为。这种情景相当于一场“完美风暴”，所有可能出错的事情都会真的出错。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，影响我们飞机和船只的安全，以及我们进行准确天气预报的能力。这个项目的目的是调查在什么条件下欧拉方程和纳维-斯托克斯方程可以发展出奇异行为。对这个问题的理解将使我们能够避免自然界中流体流动的灾难性行为。这项研究的最终目标是开发有效的分析和计算工具，增强我们模拟和预测自然界各种复杂现象的能力，从而使我们对商用喷气式飞机和船只的安全以及天气预报更有信心。这个项目的额外影响将是研究生的参与。他们在这个项目中所接受的跨学科训练对他们未来的数学和科学生涯将是非常重要的。三维不可压缩欧拉方程是否能从平滑的初始数据发展成有限时间奇点一直是一个悬而未决的问题。基于先前NSF支持的结果，该项目旨在提供具有光滑初始数据和边界的3D Euler方程潜在的有限时间奇性的严格证明。研究的一个主要方法是证明三维轴对称欧拉方程具有小残差的近似自相似轮廓的存在性和稳定性。首先将进行数值计算，以构造具有非常小的残差的近似自相似轮廓。分析中的一个关键步骤是通过动态重标度公式获得近似自相似轮廓的线性稳定性。线性稳定性将通过获得具有适当选择的奇异权重的精确函数估计以及在计算机辅助下使用时空估计来建立。在这个项目中开发的新技术和工具可能会对邻近的数学领域产生影响。另一个被提议的项目是使用特别设计的带有周期边界条件的初始数据来寻找三维Navier-Stokes方程的潜在奇性解。该方法依赖于使用动态重尺度格式来解轴对称的Navier-Stokes方程，避免了最近计算中频繁变化的自适应网格所带来的潜在的数值不稳定性。这项研究的成功实施将为3D Navier-Stokes方程上的Clay Millennium问题提供有价值的见解，并成为最终解决该问题的重要一步。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.