Problems in Regularity Theory of Partial Differential Equations

This project focuses on understanding certain types of partial differential equations (PDE) commonly encountered in physics and engineering, such as those governing elasticity and conductivity. When we study how materials deform under stress or conduct electricity, we often use equations to describe these phenomena. However, some equations don't behave smoothly, especially when dealing with high contrast materials or complex shapes. These situations can lead to equations that are much harder to analyze, and traditional methods may not work. Another area of study is equations from fluid dynamics. Understanding these questions is crucial for practical applications like designing airplanes or predicting weather patterns, and it also inspires new ideas in mathematics and statistics. Finally, the Principal Investigator (PI) is interested in kinetic equations, which describe how particles move and interact in systems like nuclear fusion experiments. By studying these equations, scientists hope to improve our understanding of how plasmas behave in extreme conditions, such as inside a tokamak. The project provides research training opportunities for graduate students. As part of this project, the PI will carry out research closely related to the aforementioned topics and will attempt to address some of the open problems in these areas. The focus will be on several projects that can be gathered into three main topical areas. First, the project will develop new methods to study elliptic equations arising in composite materials (e.g., elasticity problems, conductivity problems). The PI is particularly interested in the blowup behaviors of solutions to PDE in domains with Lipschitz inclusions, equations involving the p-Laplacian, and the insulated problem for the Lamé system. Second, the project will explore the free boundary problem involving an incompressible fluid permeating a porous medium, often referred to as the one-phase Muskat problem. The focus will be on investigating the regularity of solutions to the two- and three-dimensional one-phase Muskat problem in the whole space, as well as on exploring the short-term and long-term smoothing effects of these solutions. Finally, the project will investigate boundary regularity of linear kinetic equations as well as the stability and global well-posedness of nonlinear kinetic equations, including the relativistic Vlasov-Maxwell-Landau system and the spatially inhomogeneous Boltzmann equations in general domains. This project is jointly funded by the Analysis Program in the Division of Mathematical Sciences (DMS) and the Established Program to Stimulate Competitive Research (EPSCoR). This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.