Regularity, Blow UP, and Mixing in Fluids
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 weather phenomena. Although there is an enormous wealth of knowledge accumulated in the broad area of fluid mechanics, many of the most fundamental and important questions remain poorly understood. Of particular interest is the question whether solutions to equations describing fluid motion can spontaneously form singularities - meaning that some quantity becomes infinite. 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. More generally, one can ask a related and broader question of creation of small scales in fluids - coherent structures that vary sharply in space and time, and contribute to phenomena such as turbulence. The project aims to analyze singularity formation process for some key equations of fluid mechanics, and to better understand the mechanisms that generate small scales in fluid motion. Another direction of the project research focuses on mixing in fluid flow. Mixing in fluids plays an important role in a wide range of settings, from marine ecology to internal combustion engines. Here the goal is to find and study fluid flows that are especially efficient mixers, as well as to produce bounds on mixing efficiency given some natural constraints. Such bounds can serve as benchmarks in evaluation of mixing processes.