Collaborative Proposal: The role of convection on dynamic stability of 3D incompressible Navier-Stokes equations.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本项目旨在研究对流对三维不可压缩欧拉方程和纳维-斯托克斯方程动力稳定性的影响。主要目的是证明对流和不可压缩性在研究不可压缩Euler和Navier-Stokes方程的动态稳定性中起着至关重要的作用。本课题的另一个目的是证明三维欧拉方程的全局正则性与三维纳维-斯托克斯方程的全局正则性之间存在着密切的联系。最后，利用拉格朗日方法对三维欧拉方程进行了新的正则性分析，以同时控制涡丝局部曲率和最大涡量的动态增长。本课题所建立的局部非线性稳定性分析方法，可以潜在地应用于研究其他学科中出现的一大类非线性动力问题。对流的动态稳定性和作用的理解对许多科学应用具有重要影响，这些应用可能从根本上影响人们的生活质量。这些应用包括天气预报、环境或全球气候变化、流体动力学应用、湍流建模和高性能计算。长期以来，许多专家认为对流是不稳定的。该项目揭示了对流实际上具有令人惊讶的稳定作用，它对三维不可压缩流动的大时间行为具有重要影响。这个项目的另一个影响是研究生和博士后研究人员的参与。这个项目在数学分析、物理建模和数值模拟方面提供了坚实的训练。他们在这个项目中接受的跨学科训练对他们未来在数学和科学领域的职业生涯非常重要。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project is to investigate the role of convection on dynamic stability of the three-dimensional incompressible Euler and Navier-Stokes equations. The main objective is to show that convection together with incompressibility plays an essential role in studying the dynamic stability of the incompressible Euler and Navier-Stokes equations. Another objective of this project is to show that there is a close connection between the global regularity of the three-dimensional Euler equations and that of the three-dimensional Navier-Stokes equations. Finally, a new regularity analysis using a Lagrangian approach for the three-dimensional Euler equations is developed to control the dynamic growth of the local curvature of vortex filaments and the maximum vorticity simultaneously. The local nonlinear stability analysis developed in this project can be potentially applied to study a large class of nonlinear dynamic problems arising from other disciplines.The understanding of the dynamic stability and the role of convection has a significant impact on many scientific applications which could affect the quality of people's life in a fundamental way. These applications include weather forecasting, environmental or global climate change, fluid dynamic applications, turbulence modeling and high performance computing. For a long time, many experts considered convection as destabilizing. This project reveals that convection actually has a surprising stabilizing effect which could affect the large time behavior of the three-dimensional incompressible flows in an essential way. An additional impact of this project is the involvement of graduate students and postdoctoral researchers. This project provides a solid training in mathematical analysis, physical modeling, and numerical simulation. The interdisciplinary training they receive in this project is very important for their future careers in mathematics and science.