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

纳维-斯托克斯方程用于模拟洋流、天气模式以及飞机或船舶后面的湍流。 数学家和物理学家相信，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。 尽管大多数物理学家和工程师认为，如果没有外力，纳维-斯托克斯方程的光滑解就不会被破坏，但目前没有理论保证事实确实如此。 研究人员最近与合作者的研究表明，与纳维-斯托克斯方程的无粘极限相对应的欧拉方程，如果从高度对称但完全平滑的流动开始，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确实出错了。 纳维-斯托克斯方程描述的流体流动的潜在奇异行为可​​能会削弱准确预测流体系统行为的能力。 研究人员研究欧拉或纳维-斯托克斯方程可能出现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，以增强我们模拟和预测各种复杂流体流动的能力，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参与了该项目的工作。 他们接受的跨学科培训对于他们未来在数学和科学领域的职业生涯非常重要。该项目旨在了解不可压缩的 3D 欧拉和纳维-斯托克斯方程是否可以从具有有限能量的平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解的潜在自相似奇点的空间分布，这可以通过求解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高分辨率数值模拟和严格数学分析的结合。 首先进行数值计算以检测潜在的有限时间奇点场景并获得对奇点形成的初步了解。 然后使用动态缩放公式对解中空间剖面的演化方程进行分析和数值研究。 该项目开发的理论框架通过动态重新缩放公式引入了自相似轮廓的适当稳定性概念。 数值构造的自相似轮廓的稳定性是构造欧拉方程的有限时间奇点的关键步骤。 该项目的另一个有趣的方面是动态缩放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点是否可能导致纳维-斯托克斯方程对于某些类型的奇点的潜在有限时间奇点。 欧拉方程自相似曲线的稳定性在这项研究中再次发挥了至关重要的作用。

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.