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

Navier-Stokes方程已经存在了150多年。物理学家用它们来模拟洋流，天气模式和商业喷气式飞机或船只后面的湍流。大多数物理学家和工程师认为，如果没有外力，Navier-Stokes方程的光滑解不会崩溃。另一方面，PI最近的一项研究表明，如果从高度对称但完全光滑的流动开始，Navier-Stokes方程可能会产生灾难性的行为。这种情况相当于一场“完美风暴”，所有可能出错的事情都真的出错了。Navier-Stokes方程的潜在奇异行为可能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。这个项目的目的是研究在什么条件下欧拉方程和Navier-Stokes方程可能发展奇异行为。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，提高我们建模和预测自然界各种复杂现象的能力，以便我们对商用飞机和船舶的安全以及天气预报更有信心。该项目的其他影响将是研究生的参与。他们在这个项目中接受的跨学科培训将对他们未来的数学和科学事业非常重要。三维不可压缩欧拉方程是否从光滑的初始数据发展出有限时间奇点一直是一个长期悬而未决的问题。该项目建立在先前NSF支持的结果的基础上，旨在提供具有光滑初始数据和边界的3D Euler方程中潜在有限时间奇异性的严格证明。研究的一个主要途径是证明三维轴对称欧拉方程的一个具有小残差的近似自相似轮廓的存在性和稳定性。数值计算将首先进行构造一个近似的自相似轮廓与一个非常小的残留误差。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。线性稳定性将建立通过获得尖锐的功能估计与适当选择的奇异权重和使用空间时间估计与计算机辅助。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。另一个建议的项目是寻找潜在的奇异解的三维Navier-Stokes方程使用专门设计的初始数据与周期性的边界条件。该方法依赖于使用动态重标度制定解决轴对称Navier-Stokes方程，以避免潜在的数值不稳定性所引起的自适应网格在最近的计算中的频繁变化。这项研究的成功执行将为三维Navier-Stokes方程的粘土千年问题提供有价值的见解，并成为其最终解决方案的重要一步。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。

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.