Regularity, Blow Up and Mixing in Fluids

流体就在我们身边，我们可以在日常生活中，在无处不在的技术中，在戏剧性的天气现象中见证它们的复杂性和微妙性。虽然在流体力学的广泛领域积累了大量的知识，但许多最基本和最重要的问题仍然知之甚少。特别感兴趣的是描述流体运动的方程的解是否可以自发地形成奇点-这意味着一些量变得无限大。理解奇点是很重要的，因为它们往往对应于戏剧性的，高强度的流体运动，可以指示模型的适用范围，并且很难通过计算解决。更一般地说，人们可以提出一个相关的和更广泛的问题，即在流体中产生小尺度--在空间和时间上急剧变化的相干结构，并有助于产生湍流等现象。该项目旨在分析流体力学的一些关键方程的奇异性形成过程，并更好地理解流体运动中产生小尺度的机制。该项目的另一个研究方向是流体流动中的混合。从海洋生态学到内燃机，流体混合在各种环境中发挥着重要作用。在这里，我们的目标是找到和研究流体流动，特别是有效的混合器，以及产生边界的混合效率给定一些自然的约束。这些界可以作为混合过程的评价基准。第一个主题是关于不可压缩无粘流体的欧拉方程。它是非线性和非局部的，这使得分析困难。关于欧拉方程解的行为的许多关键问题仍然是开放的，尽管重大的研究工作。最近，PI与弗拉基米尔Sverak一起构建了二维欧拉方程的解的例子，这些解表现出非常快的小尺度形成。这一工作受到Tom Hou和Guo Luo提出的三维欧拉方程中的势奇点形成的新情况的启发。该项目旨在通过分析一系列模型方程，进一步严格了解三维空间中可能的奇点形成。 第二个主题是关于地表准地转（SQG）方程和修正的SQG方程解的性质。这些方程模拟了地球表面温度的演变。最近，PI和合作者已经构建了在修改后的SQG补丁中的奇点形成的例子，在可能的参数范围的一部分的边界的存在下。这些例子是这类方程中第一个可用的例子。该项目将涉及在没有边界的情况下对修改后的SQG补丁解决方案的进一步工作。该项目的前两个方向部署的方法涉及新的分析估计，比较原则，渐近分析，偏微分方程（PDE）估计，傅立叶分析技术。第三个主题涉及流体流动中的混合。目标是提高对加速混合最有效的流动的理解。通常，混合流的某些方面存在约束，了解如何在这些约束下产生最有效的混合是很重要的。这里的问题是在接口的应用偏微分方程，动力系统，概率论和功能分析。

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.