Regularity, Blow Up and Mixing in Fluids

流体中的规律性、膨胀和混合

• 批准号: 1712294
• 负责人: Alexander Kiselev
• 金额: $ 35万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2018-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712294&HistoricalAwards=false
• 关键词: Regularity; Blow; Up; Mixing; 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.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.

流体无处不在，我们可以在日常生活、无处不在的技术和戏剧性的天气现象中看到它们特性的复杂性和微妙性。尽管在流体力学的广阔领域中积累了大量的知识，但许多最基本和最重要的问题仍然知之甚少。特别令人感兴趣的是描述流体运动的方程的解是否能自发地形成奇点——这意味着某些量变得无限。理解奇点是很重要的，因为它们通常对应于戏剧性的、高度激烈的流体运动，可以指示模型的适用范围，并且很难通过计算来解决。更一般地说，人们可以提出一个相关的更广泛的问题，即在流体中创造小尺度——在空间和时间上急剧变化的连贯结构，并导致湍流等现象。本项目旨在分析流体力学中一些关键方程的奇点形成过程，更好地理解流体运动中产生小尺度的机理。项目研究的另一个方向是流体流动中的混合。流体混合在从海洋生态到内燃机的广泛环境中发挥着重要作用。这里的目标是寻找和研究特别有效的混合流体流动，以及在某些自然约束下产生混合效率的界限。这样的界限可以作为评价混合过程的基准。这项研究涵盖了三个主题。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，这给分析带来了困难。尽管进行了大量的研究，但关于欧拉方程解的行为的许多关键问题仍然是开放的。最近，PI与Vladimir Sverak共同构造了二维欧拉方程的解的例子，这些解表现出极快的小尺度形成。这项工作受到了Tom Hou和Guo Luo提出的三维欧拉方程中潜在奇点形成的新场景的启发。该项目旨在通过分析一系列模型方程，进一步深入了解三维空间可能形成的奇点。第二个主题是关于曲面准地转（SQG）和修正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和合作者在可能参数范围的一部分存在边界的情况下，在改进的SQG补丁中构造了奇点形成的例子。这些例子是这类方程中最早出现的。在没有边界的情况下，该项目将涉及改进的SQG补丁解决方案的进一步工作。在项目的前两个方向上部署的方法涉及新的分析估计、比较原理、渐近分析、偏微分方程（PDE）估计和傅立叶分析技术。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。这里的问题是在应用偏微分方程，动力系统，概率论和泛函分析的接口。

项目成果

Global Regularity for the Fractional Euler Alignment System

• DOI: 10.1007/s00205-017-1184-2
• 发表时间: 2018-04-01
• 期刊: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
• 影响因子: 2.5
• 作者: Do, Tam;Kiselev, Alexander;Tan, Changhui
• 通讯作者: Tan, Changhui

Finite time blow up in hyperbolic Boussinesq system

双曲 Boussinesq 系统中的有限时间爆炸

• 发表时间: 2018
• 期刊: Advances in mathematics
• 影响因子: 1.7
• 作者: Kiselev, Alexander;Tan, Changhui
• 通讯作者: Tan, Changhui