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

这一建议是一个努力解决三个主要障碍，非线性，非局域性，和耦合，在数学研究的偏微分方程产生的流体力学。这些方程不仅控制着流体的流动，而且还模拟了科学技术甚至日常生活中的许多现象。所提出的研究方法和技术将为这类方程的研究提供新的思路，并可能导致重大开放性问题的解决。该提案包括三个主要部分。1.压力调节的方法。与学生和合著者一起，我提出了一种研究流体力学方程的新方法。在这个项目中，我将进一步发展这种新方法，并将其应用于数学流体力学中的主要开放问题，特别是三维Navier-Stokes方程解的规律性。目标：-证明具有期望性质的压力调节器的存在性定理；-通过压力调节方法的应用，证明了流体力学方程的新的规律性准则。2.耦合系统的野性解决方案。我将开发一个新的凸集成框架，它在耦合系统的研究中既方便又强大。目标：-为耦合系统如MHD和Boussinesq方程构建具有最优正则性的病理解。-为耦合系统的研究开发了一种新的凸积分方法。3.弱时间可积性的正则性准则。在过去的两年中，我提出了一种新的方法来证明具有弱时间可积性的流体力学方程的正则性准则。在这个项目中，我将进一步发展这种新方法，使其成为一个通用的、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组成部分。-证明流体力学方程的新规则准则。-将本方法与经典的epsilon正则性理论综合成一个证明弱时间可积局部正则性准则的系统方法。这三个部分将为三位博士生提供全面的培训。学生将获得偏微分方程及相关领域的专业知识，并为未来的学术生涯做好准备。我希望从提议的研究中“衍生”出许多有趣的问题，作为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。

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.