A Computer-Assisted Analysis Framework for Studying Finite Time Singularities of the 3D Euler Equations and Related Models

用于研究 3D 欧拉方程及相关模型的有限时间奇异性的计算机辅助分析框架

• 批准号: 1907977
• 负责人: Thomas Hou
• 金额: $ 56.63万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1907977&HistoricalAwards=false
• 关键词: Computer; Assisted; Analysis; Framework; Studying

The 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. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to Navier-Stokes equations. Although most physicists and engineers believe that the smooth solutions of the Navier-Stokes equations will not break down without external forcing, currently there is no theoretical guarantee that this is indeed the case. Our recent study indicates that the Euler equations, a special case of the Navier-Stokes equations with zero viscosity, could develop a catastrophic behavior if one starts with a highly symmetric but perfectly smooth flow. Such a 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. This award will investigate under what conditions the Euler or Navier-Stokes equations may develop singular behavior. 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 important for their future careers in mathematics and science.The award 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 unique aspect of the research is the integration of highly resolved numerical simulations and rigorous mathematical analysis. Our strategy is to reformulate the problem of proving finite time self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile using the dynamic rescaling equation. We first construct a highly accurate self-similar profile using a high order numerical method. We then use the energy method with carefully chosen singular weight functions and take into account cancellation among various nonlinear terms to extract the inviscid damping effect from the linearized operator around the approximate self-similar profile. Our stability result enables us to prove that the dynamic rescaling solution converges to the steady state self-similar solution exponentially fast in time. Moreover, by introducing a cut-off to the self-similar profile, we obtain a smooth initial condition with finite energy that develops a self-similar blowup in finite time. The novel approach of investigating finite-time singularity formation by studying the stability of spatial profiles in the potential singular solutions also forms the basis of a novel analytical framework for other nonlinear nonlocal systems of partial differential equations and has the potential to be applied to study a larger class of nonlinear dynamic problems.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.

Navier-Stokes方程已经存在了150多年。物理学家使用它们来建模洋流，天气模式和商用喷气机或船后的动荡流。数学家和物理学家认为，可以通过了解Navier-Stokes方程的解决方案来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都认为，如果没有外部强迫，Navier-Stokes方程的平滑解决方案将不会崩溃，但是当前没有理论上保证确实如此。我们最近的研究表明，如果粘度为零的Navier-Stokes方程的特殊情况，则如果一个人开始使用高度对称但完全平滑的流动，则可能会发展出灾难性的行为。这种情况对应于一场完美的风暴，在这种风暴中，所有可能出错的事情确实出错了。 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，影响飞机和船只的安全，以及我们进行准确的天气预报的能力。该奖项将在Euler或Navier-Stokes方程可能发展单一行为的情况下进行调查。这项研究的最终目的是开发有效的分析和计算工具，以增强我们在自然界建模和预测各种复杂现象的能力，以便我们可以对商业喷气机和船只的安全以及天气预报具有更大的信心。该项目的其他影响将是研究生的参与。他们在该项目中接受的跨学科培训对于他们在数学和科学领域的未来职业至关重要。该研究的一个独特方面是整合高度解决的数值模拟和严格的数学分析。 我们的策略是将有限时间自相似奇点的问题重新重新调整为使用动态恢复方程式建立近似自相似概况的非线性稳定性的问题。我们首先使用高阶数值方法构造了高度准确的自相似轮廓。然后，我们将能量方法与精心选择的奇异重量函数一起使用，并考虑到各种非线性项之间的取消，以从线性化操作员围绕近似自相似的轮廓中提取无粘性阻尼效应。我们的稳定性结果使我们能够证明动态恢复解决方案会及时快速地收敛到稳态自相似解决方案。 此外，通过引入自相似轮廓的截止，我们获得了有限能量的平滑初始条件，从而在有限的时间内会产生自相似的爆炸。通过研究潜在奇异解决方案中的空间特征的稳定性来研究有限时间奇异性形成的新型方法，这也构成了针对部分偏微分方程的其他非线性非局部非定位系统的新分析框架的基础更广泛的影响审查标准。

项目成果

On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations

• DOI: 10.1002/cpa.21991
• 发表时间: 2019-05
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: Jiajie Chen;T. Hou;De Huang

Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblems

• DOI: 10.4310/cms.2022.v20.n2.a4
• 发表时间: 2019-12
• 期刊: ArXiv
• 作者: Ziyun Zhang

On the Slightly Perturbed De Gregorio Model on $$S^1$$

• DOI: 10.1007/s00205-021-01685-w
• 发表时间: 2020-10
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Jiajie Chen

Multiscale Invertible Generative Networks for High-Dimensional Bayesian Inference

• 发表时间: 2021-05
• 作者: Shumao Zhang;Pengchuan Zhang;T. Hou

Multiscale Elliptic PDE Upscaling and Function Approximation via Subsampled Data

• DOI: 10.1137/20m1372214
• 发表时间: 2022-02
• 期刊: Multiscale Model. Simul.
• 作者: Yifan Chen;T. Hou