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。该项目旨在研究对流对三维不可压缩欧拉方程和Navier-Stokes方程动态稳定性的作用。主要目的是表明，对流与不可压缩性起着至关重要的作用，在研究不可压缩的欧拉和Navier-Stokes方程的动态稳定性。本项目的另一个目的是证明三维Euler方程的整体正则性与三维Navier-Stokes方程的整体正则性之间有着密切的联系。最后，一个新的规则性分析，使用拉格朗日方法的三维欧拉方程的发展，以控制动态增长的局部曲率的涡丝和最大涡量的同时。本项目所发展的局部非线性稳定性分析方法可以应用于研究其他学科产生的一大类非线性动力学问题，对动力学稳定性和对流作用的理解对许多科学应用产生重大影响，这些科学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效果，这可能会影响三维不可压缩流的大时间行为。该项目的另一个影响是研究生和博士后研究人员的参与。该项目提供了数学分析，物理建模和数值模拟方面的扎实培训。他们在这个项目中接受的跨学科培训对他们未来的数学和科学职业非常重要。

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.