Investigating Potential Singularities in the Euler and Navier-Stokes Equations Using an Integrated Analytical and Computational Approach

纳维尔-斯托克斯方程被用来模拟洋流、天气模式和飞机或船只后面的湍流。数学家和物理学家认为，通过对纳维-斯托克斯方程的解的理解，可以找到对微风和湍流的解释和预测。尽管大多数物理学家和工程师认为，如果没有外力，Navier-Stokes方程的光滑解不可能分解，但目前还不能从理论上保证这是真的。这位研究人员最近与合作者的研究表明，如果从高度对称但完全平滑的流动开始，对应于Navier-Stokes方程的无粘性极限的欧拉方程可能会发展成灾难性的行为。这样的情景就像一场完美的风暴，所有可能出错的事情都会出错。由Navier-Stokes方程描述的流体流动的潜在奇异行为可能会否定准确预测流体系统行为的能力。研究者研究了欧拉方程或纳维-斯托克斯方程可能产生潜在奇异行为的条件。该项目的最终目标是开发有效的分析和计算工具，以增强我们对各种复杂流体流动进行建模和预测的能力，例如工程、海洋学和天气预报中出现的流体流动。该项目的工作包括研究生和博士后学者。他们所接受的跨学科训练对他们未来的数学和科学生涯非常重要。该项目试图了解不可压缩的3D Euler方程和Navier-Stokes方程能否从有限能量的光滑初始条件发展成有限时间的奇点。该项目的一个主要方法是研究解的潜在自相似奇点的空间轮廓，这可以通过求解一个非线性特征值问题来获得。该项目的一个值得注意的方面是结合了高分辨率的数值模拟和严格的数学分析。首先进行数值计算，以检测潜在的有限时间奇点情景，并对奇点的形成有初步的了解。在此基础上，利用动态重定标法对解中空间轮廓的演化方程进行了解析和数值研究。这个项目中开发的理论框架通过动态重新标度公式为自相似轮廓引入了一个适当的稳定性概念。数值构造的自相似轮廓的稳定性是构造欧拉方程有限时间奇性的关键步骤。该项目的另一个有趣的方面是，动态重新标度公式提供了一个自然的框架来研究欧拉方程中的有限时间奇性是否会导致某些类型的奇点的N-S方程的潜在的有限时间奇性。欧拉方程的自相似轮廓的稳定性在本研究中再次起着至关重要的作用。

Navier-Stokes equations are used to model ocean currents, weather patterns, and turbulent flows behind a plane or ship. Mathematicians and physicists believe that an explanation for and the prediction of both breezes and turbulence can be found through an understanding of solutions to Navier-Stokes equations. Though most physicists and engineers believe that the smooth solutions of the Navier-Stokes equations cannot break down without external forcing, currently there is no theoretical guarantee that this is indeed the case. The investigator's recent study with collaborators indicates that the Euler equations, which correspond to the inviscid limit of the Navier-Stokes equations, could develop a catastrophic behavior if one starts with a highly symmetric but perfectly smooth flow. Such a scenario is like a perfect storm in which all things that could potentially go wrong indeed go wrong. A potentially singular behavior of the fluid flows described by the Navier-Stokes equations could negate the ability to forecast the behavior of fluid systems accurately. The investigator studies conditions under which the Euler or Navier-Stokes equations may develop a potentially singular behavior. The ultimate goal of the project is to develop effective analytical and computational tools that enhance our ability to model and predict various complex fluid flows, such as those arising in engineering, oceanography, and weather forecasting. Graduate students and postdoctoral scholars are included in the work of the project. The interdisciplinary training they receive is important for their future careers in mathematics and science.The project seeks to understand whether the incompressible 3D Euler and Navier-Stokes equations could develop a finite-time singularity from a smooth initial condition with finite energy. A major approach of the project is to study the spatial profiles in potential self-similar singularities of the solutions, which can be obtained by solving a nonlinear eigenvalue problem. A notable aspect of the project is the combination of highly resolved numerical simulations and rigorous mathematical analysis. Numerical computations are first conducted to detect potential finite-time singularity scenarios and gain primary understanding about the singularity formation. Then the evolution equations of spatial profiles in the solutions are studied using a dynamic rescaling formulation both analytically and numerically. The theoretical framework developed in this project introduces an appropriate notion of stability for the self-similar profiles through the dynamic rescaling formulation. Stability of the numerically constructed self-similar profile is a crucial step in constructing a finite-time singularity of the Euler equations. Another interesting aspect of the project is that the dynamic rescaling formulation provides a natural framework to investigate whether the finite-time singularity in the Euler equations may lead to a potential finite-time singularity of the Navier-Stokes equations for certain type of singularities. The stability of self-similar profiles of the Euler equations again plays a crucial role in this study.