Regularity, Blow Up and Mixing in Fluids

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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.The research covers three topics. The first topic concerns the Euler equation for incompressible inviscid fluid. It is nonlinear and nonlocal, which makes analysis difficult. Many key questions about behavior of solutions to the Euler equation remain open despite significant research efforts. Recently, the PI jointly with Vladimir Sverak have constructed examples of solutions to the 2D Euler equation which exhibit extremely fast formation of small scales. This work has been stimulated by the new scenario of potential singularity formation in the 3D Euler equation, proposed by Tom Hou and Guo Luo. The project aims to gain further rigorous insight into the possible singularity formation in three dimensions by analyzing a series of model equations. The second topic concerns properties of solutions to the surface quasi-geostrophic (SQG) and modified SQG equations. These equations model evolution of temperature on the surface of Earth. Recently, the PI and collaborators have constructed examples of singularity formation in modified SQG patches in presence of boundary for a part of the possible parameter range. These examples are the first available in this class of equations. The project will involve further work on the modified SQG patch solutions, in the absence of boundary. The methods to be deployed in the first two directions of the project involve novel analytic estimates, comparison principles, asymptotic analysis, partial differential equations (PDE) estimates, and Fourier analysis techniques. The third topic concerns mixing in fluid flow. The goal is to improve understanding of flows that are most efficient in speeding up mixing. Quite often, there are constraints on some aspects of mixing flow, and it is important to understand how to produce most effective mixing under these constraints. The problems here are at the interface of applied partial differential equations, dynamical systems, probability theory and functional analysis.