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

Navier-Stokes方程用于模拟洋流、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，通过理解纳维-斯托克斯方程的解，可以找到对微风和湍流的解释和预测。 虽然大多数物理学家和工程师认为，如果没有外力，Navier-Stokes方程的光滑解就不能分解，但目前还没有理论保证这确实是事实。 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光滑的流动开始，对应于Navier-Stokes方程无粘极限的欧拉方程可能会产生灾难性的行为。 这种情况就像一场完美风暴，所有可能出错的事情都真的出错了。 由Navier-Stokes方程描述的流体流动的潜在奇异行为可以否定准确预测流体系统的行为的能力。 研究人员研究的条件下，欧拉或Navier-Stokes方程可能会开发一个潜在的奇异行为。 该项目的最终目标是开发有效的分析和计算工具，提高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。 研究生和博士后学者都包括在该项目的工作。 他们接受的跨学科培训对他们未来的数学和科学事业非常重要。该项目旨在了解不可压缩的3D Euler和Navier-Stokes方程是否可以从具有有限能量的光滑初始条件发展出有限时间奇点。 该项目的一个主要方法是研究潜在的自相似奇异性的解决方案，这可以通过解决一个非线性特征值问题的空间轮廓。 该项目的一个值得注意的方面是高度分辨率的数值模拟和严格的数学分析相结合。 数值计算首先进行检测潜在的有限时间奇异性的情况下，并获得有关奇异性的形成的初步了解。 然后，空间轮廓的演变方程的解决方案进行了研究，使用动态重标度制定的解析和数值。 在这个项目中开发的理论框架引入了一个适当的概念，通过动态重标度制定的自相似配置文件的稳定性。 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异性的关键步骤。 该项目的另一个有趣的方面是，动态重标度公式提供了一个自然的框架，以调查是否有限时间奇点在欧拉方程可能导致一个潜在的有限时间奇点的Navier-Stokes方程的某些类型的奇点。 欧拉方程的自相似轮廓的稳定性再次发挥了至关重要的作用，在这项研究中。

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.