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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758372
• 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.

该提议是为了解决由流体力学引起的部分微分方程的数学研究，以解决三个主要障碍，非线性，非局部性和耦合。这些方程式不仅控制流体流动，而且还模拟了科学和技术中的许多现象，甚至是日常生活。拟议的研究中开发的方法和技术将为这些方程式的研究提供新的启示，并可能导致解决主要的开放问题。 该提案涉及三个主要部分。 1.压力适度的方法。 与学生和合着者一起，我提出了一种新的方法来研究流体机械方程。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 Navier-Stokes方程的解决方案的规律性。 目标： - 证明具有所需属性的压力主持人的存在定理； - 通过应用压力适中的方法来证明流体机械方程的新规则标准。 2.耦合系统的婚姻解决方案。 我将开发一个新的凸集成框架，该框架在耦合系统的研究中既方便又有力。 目标： - 构建具有最佳规律性的病理解决方案，用于耦合系统，例如MHD和BousSinesQ方程。 - 开发一种用于研究耦合系统的新凸集成方法。 3.时间的可整合性较弱的登记标准。 在过去的两年中，我提出了一种新方法，以证明流体机械方程的时间整合性较弱的规律性标准。在这个项目中，我将进一步将这种新方法开发为一个多功能且有力的框架，以研究数学流体力学的规律性问题。 目标： - 证明弱非线性毛衣不平等的改进和概括，这是我们新方法中的关键要素。 - 证明流体机械方程式的新规则标准。 - 将我们的方法与经典的Epsilon规律性理论合成为一种系统的方法，以证明具有弱时间可集成性的局部规则性标准。 这三个部分将为三个博士生提供彻底的培训。学生将获得部分微分方程和相关领域的专业知识，并为学术界的未来职业做好准备。我希望从拟议的研究中“分拆”许多有趣的问题，可以激发夏季研究的第一个项目，这是改善股权，多样性和纳入数学的最佳机会。