Small Scale and Singularity Formation in Fluids

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This award focuses on several directions in fluid mechanics. Fluids are all around us, and we can witness the complexity and subtleness of their properties in everyday life, in ubiquitous technology, and in dramatic phenomena such as tornados or hurricanes. There has been an enormous wealth of knowledge accumulated in the broad area of fluid mechanics, yet it is quite remarkable that many of the most fundamental and relevant in applications questions remain poorly understood. Of particular interest is the question whether solutions to equations describing fluid motion can spontaneously form singularities. Understanding singularities is important because they often correspond to dramatic, highly intense fluid motion, can indicate the range of applicability of the model, and are very difficult to resolve computationally. Moreover, there are indications that mechanisms responsible for singularity and small-scale formation are also involved in production of turbulence. The project aims to analyze singularity and small scale formation process for some of the key equations of fluid mechanics. Some equations with related structure inspired by problems originating in biology will be considered as well. The award will provide opportunities and research experiences for undergraduate and graduate students, and postdoctoral scholars.The award is concerned with analysis of several equations in fluid mechanics and related models motivated by biological phenomena. The main theme is the analysis of singularity and small scale formation in solutions. The surface quasi-geostrophic equation comes from atmospheric science, and models evolution of temperature near the Earth's surface. The incompressible porous media equation describes flow of fluid of variable density through porous media. One research direction will focus on better understanding of nonlinear mechanisms responsible for fast generation of small scales and potentially singularities in the solutions of these equations. Another direction aims to further analyze the Hou-Luo scenario for singularity formation in the three-dimensional Euler equation. The scenario was proposed several years ago based on extensive numerical simulations, and so far all simplified models and settings that have been analyzed rigorously suggest finite time blow up. Here the main goal will be to better understand the boundary layer where intense growth of vorticity is observed, and to design and explore models that are increasingly close to the actual equation. The research will also include analysis of equations modeling collective behavior and chemotaxis in biology. Here the focus is on analysis of regularity, long time dynamics, and boundary effects.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.