Regularity, Stability, and Turbulence in Fluid Flows

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This project will advance the knowledge of physical models which are of fundamental importance in engineering, meteorology, or natural disaster, such as flooding and hurricanes. It will advance the development of powerful mathematical tools based on state of the art analysis, and the theory of linear and nonlinear partial differential equations. An important component of the project concerns the training and mentoring of a new generation of scientists and researchers. The research program provides ample opportunities for Ph.D. students and Postdocs to be trained in mathematical analysis and physical modeling applied to fluid mechanics. This project uses powerful mathematical tools to tackle several fundamental questions on the behavior of flows in fluid mechanic. A rigorous theory will be developed to study how incompressible flows become linearly unstable close to possible blow-up time, a typical signature of turbulence. The effect of boundaries on the dynamic of incompressible flows is of paramount importance. It will be thoroughly investigated on the 3D quasi-geostrophic equation, a model used in meteorology and oceanography.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.