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

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。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方程的光滑解不会崩溃，但目前还没有理论保证这确实是事实。我们最近的研究表明，欧拉方程，一个特殊的情况下的Navier-Stokes方程与零粘度，可以开发一个灾难性的行为，如果一个开始与一个高度对称，但完美的光滑流。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会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。该奖项将研究在什么条件下欧拉或Navier-Stokes方程可能发展奇异行为。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职业生涯非常重要。该奖项旨在了解不可压缩的3D欧拉和Navier-Stokes方程是否可以从有限能量的光滑初始条件发展出有限时间奇点。研究的一个独特方面是高度分辨率的数值模拟和严格的数学分析的集成。 我们的策略是将证明有限时间自相似奇异性的问题转化为利用动态重标度方程建立近似自相似轮廓的非线性稳定性问题。我们首先使用高阶数值方法构造了一个高精度的自相似轮廓。然后，我们使用的能量方法与精心选择的奇异权函数，并考虑到各种非线性项之间的抵消提取的无粘阻尼效应的近似自相似轮廓周围的线性化运营商。我们的稳定性结果使我们能够证明，动态重标度的解决方案收敛到稳态自相似的解决方案在时间上指数快速。 此外，通过引入截断的自相似轮廓，我们得到了一个光滑的初始条件，具有有限的能量，在有限的时间内发展的自相似爆破。研究有限-通过研究潜在奇异解中空间剖面的稳定性来形成时间奇异性，也形成了一个新的分析框架的基础，用于其他非线性非局部偏微分方程系统，并有可能应用于研究更大类的非线性动力学问题。这个奖项反映了美国国家科学基金会的法定使命，并被认为是值得通过评估来支持的使用基金会的知识价值和更广泛的影响审查标准。

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.