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.

流体就在我们周围，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见证它们特性的复杂性和微妙性。尽管在流体力学的广泛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。特别令人感兴趣的问题是描述流体运动的方程的解是否可以自发地形成奇点——这意味着某个量变得无穷大。了解奇点很重要，因为它们通常对应于剧烈的、高度强烈的流体运动，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，即在流体中产生小尺度——在空间和时间上急剧变化的相干结构，并导致湍流等现象。该项目旨在分析流体力学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制。该项目研究的另一个方向集中于流体流动的混合。流体混合在从海洋生态到内燃机的各种环境中发挥着重要作用。这里的目标是寻找和研究特别高效的混合器的流体流动，以及在给定一些自然约束的情况下产生混合效率的界限。这样的界限可以作为混合过程评估的基准。该研究涵盖三个主题。第一个主题涉及不可压缩无粘流体的欧拉方程。它是非线性和非局部的，这使得分析变得困难。尽管进行了大量的研究工作，但关于欧拉方程解的行为的许多关键问题仍然悬而未决。最近，PI 与 Vladimir Sverak 联合构建了二维欧拉方程的解示例，该方程显示出小尺度的形成速度极快。这项工作受到 Tom Hou 和郭罗提出的 3D 欧拉方程中潜在奇点形成的新场景的推动。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。 第二个主题涉及地表准地转 (SQG) 方程和修正的 SQG 方程的解的性质。这些方程模拟了地球表面温度的演变。最近，PI 和合作者在可能的参数范围的一部分存在边界的情况下，在修改的 SQG 补丁中构建了奇点形成的示例。这些例子是此类方程中的第一个例子。在没有边界的情况下，该项目将涉及修改后的 SQG 补丁解决方案的进一步工作。该项目前两个方向将采用的方法涉及新颖的分析估计、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程的理解。通常，混合流的某些方面存在限制，了解如何在这些限制下产生最有效的混合非常重要。这里的问题是应用偏微分方程、动力系统、概率论和泛函分析的交叉点。

项目成果

Global regularity for the fractional Euler alignment system

分数欧拉对齐系统的全局正则性

• 发表时间: 2018-01
• 期刊: Archive for rational mechanics and analysis
• 影响因子: 2.5
• 作者: Do, Tam;Kiselev, Alexander;Ryzhik, Lenya;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: 2018-01
• 期刊: Advances in mathematics
• 影响因子: 1.7
• 作者: Kiselev, Alexander;Tan, Changhui
• 通讯作者: Tan, Changhui