Existence, Uniqueness, and Regularity for Equations in Mathematical Fluid Mechanics

数学流体力学方程的存在性、唯一性和正则性

• 批准号: RGPIN-2019-05410
• 负责人: Yu, Xinwei
• 金额: $ 1.53万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712118
• 关键词: Existence; Uniqueness; Regularity; Equations; Mathematical

This proposal is an effort to address the three major obstacles, nonlinearity, nonlocality, and coupling, in the mathematical study of partial differential equations arising from fluid mechanics. These equations not only govern fluid flows, but also model many phenomena in science and technology, even everyday life. The methods and techniques developed in the proposed research will shed new light on the study of such equations and may lead to resolution of major open problems. The proposal involves three main parts. 1.The method of pressure moderation. Together with students and co-authors, I have proposed a novel approach to the study of fluid mechanical equations. In this project I will further develop this new method and apply it to major open problems in mathematical fluid mechanics, in particular the regularity of solutions to the 3D Navier-Stokes equations. Goals: - Prove existence theorems for pressure moderators with desired properties; - Prove new regularity criteria for fluid mechanical equations through the application of the method of pressure moderation. 2.Wild solutions for coupled systems. I will develop a new convex integration framework that is both convenient and powerful in the study of coupled systems. Goals: - Construct pathological solutions with optimal regularity for coupled systems such as the MHD and Boussinesq equations. - Develop a new convex integration method for the study of coupled systems. 3.Regularity criteria with weak time integrability. In the past two years, I have proposed a new method to prove regularity criteria with weak time integrability for fluid mechanical equations. In this project, I will further develop this new method into a versatile and powerful framework for the study of regularity problems in mathematical fluid mechanics. Goals: - Prove improvements and generalizations of the weakly nonlinear Gronwall inequality which is a key ingredient in our new method. - Prove new regularity criteria for fluid mechanical equations. - Synthesize our method with the classical epsilon regularity theory into a systematic method for the proof of local regularity criteria with weak time integrability. These three parts will provide thorough training for three PhD students. The students will gain expertise in partial differential equations and related fields, and will be well-prepared for future careers in academia. I expect many interesting problems to “spin-off” from the proposed research to serve as motivating first projects for undergraduate summer research, the best opportunity to improve equity, diversity, and inclusion in mathematics.

该提案旨在解决流体力学偏微分方程数学研究中的三大障碍：非线性、非局域性和耦合。这些方程不仅控制流体流动，还模拟科学技术甚至日常生活中的许多现象。拟议研究中开发的方法和技术将为此类方程的研究提供新的思路，并可能解决主要的开放问题。 该提案包括三个主要部分。 1、压力调节法。 我与学生和合著者一起提出了一种研究流体力学方程的新方法。在这个项目中，我将进一步开发这种新方法，并将其应用于数学流体力学中的主要开放问题，特别是 3D 纳维-斯托克斯方程解的规律性。 目标： - 证明具有所需特性的压力调节器的存在定理； - 通过应用压力调节方法证明流体力学方程的新规律性标准。 2.耦合系统的多种解决方案。 我将开发一个新的凸积分框架，它在耦合系统的研究中既方便又强大。 目标： - 为 MHD 和 Boussinesq 方程等耦合系统构建具有最佳规律性的病理解。 - 开发一种新的凸积分方法来研究耦合系统。 3.具有弱时间可积性的正则准则。 近两年，我提出了一种证明流体力学方程弱时间可积正则性准则的新方法。在这个项目中，我将进一步将这种新方法发展成为一个通用且强大的框架，用于研究数学流体力学中的规律性问题。 目标： - 证明弱非线性 Gronwall 不等式的改进和推广，这是我们新方法的关键要素。 - 证明流体力学方程的新规律性标准。 - 将我们的方法与经典的 epsilon 正则理论综合成一种系统方法，用于证明具有弱时间可积性的局部正则准则。 这三个部分将为三名博士生提供全面的训练。学生将获得偏微分方程和相关领域的专业知识，并为未来的学术生涯做好充分准备。我预计拟议的研究会“衍生”出许多有趣的问题，以激励本科生暑期研究的第一个项目，这是提高数学公平性、多样性和包容性的最佳机会。