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

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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. **