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

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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.