Topics in Regularity Theory of Partial Differential Equations

本项目着重于物理和工程中自然出现的某些偏微分方程的正则性理论的数学研究。线性和非线性偏微分方程研究的一个中心问题是在系数和域的各种正则性假设下解的光滑性。物理和工程中的许多方程和系统涉及不连续系数或具有尖角或尖角的域，甚至是分形结构。例如，纤维增强材料可能具有间距非常紧密的纤维内含物，甚至可能相互接触。在这种情况下，偏微分方程的许多经典理论不再适用。另一个重要的研究课题是流体动力学和气体动力学中的流体动力学湍流。对湍流的研究为科学计算、微分方程和统计学带来了新的思想，并被认为是流体力学中最令人着迷的问题之一。首席研究员将开展与上述主题密切相关的研究，并试图解决这些领域的一些开放问题。他将把注意力集中在几个项目上，这些项目可以分为三个主要的主题领域。首先，该项目将开发新的方法来研究与复合材料有关的某些椭圆方程（例如，弹性问题，电导率问题）。首席研究员对这些方程的解的精细规律性特别感兴趣，包括高阶规律性和更定量的导数估计。其次，该项目将探索具有不规则系数或核的局部（或非局部）非线性椭圆和抛物方程的非标准Sobolev估计，可能在不规则域。最后，本项目将研究某些确定性和随机流体方程（如磁流体动力学方程）的适定性和规律性问题。

This project focuses on mathematical research in the regularity theory of certain partial differential equations that arise naturally in physics and engineering. A central topic in the research of linear and nonlinear partial differential equations is the smoothness of solutions under various regularity assumptions on coefficients and domains. Many equations and systems from physics and engineering involve discontinuous coefficients or domains with sharp corners or cusps, or even fractal structures. For instance, fiber-reinforced materials may have fiber inclusions that are very closely spaced and may even touch. In such cases, much of the classical theory of partial differential equations is no longer applicable. Another research topic of major importance is hydrodynamic turbulence in fluid and gas dynamics. The study of turbulence leads to new ideas in scientific computing, differential equations, and statistics, and it has been characterized as one of the most fascinating problems in fluid mechanics.The principal investigator will carry out research closely related to the aforementioned topics and will attempt to address some of the open problems in these areas. He will focus his attention on several projects that can be gathered into three main topical areas. First, the project will develop new methods to study certain elliptic equations associated with composite materials (e.g., elasticity problems, conductivity problems). The principal investigator is particularly interested in the fine regularity of solutions to these equations, including higher-order regularity and more quantitative derivative estimates. Second, the project will explore nonstandard Sobolev estimates for local (or nonlocal) nonlinear elliptic and parabolic equations with irregular coefficients or kernels, in possibly irregular domains. Finally, the project will investigate well-posedness and regularity issues for certain deterministic and stochastic fluid equations such as the magnetohydrodynamics equations.