Research topics in partial differential equations

这个项目关注偏微分方程组的几个主题，包括单方程和方程组，并考虑涉及完全非线性椭圆型方程的问题，流体力学中出现的发展方程，以及完全非线性椭圆型和抛物型方程解的数值逼近的误差界。该项目的第一部分涉及一族完全非线性退化椭圆型方程的正则性。这些方程出现在最优质量传输问题和几何学中。第二部分是流体力学中几个非线性抛物方程模型的整体正则性、爆破现象、局部正则性和正则性准则的研究。这些模型包括著名的Navier-Stokes方程、准地转方程和其他一些相关模型。第三部分研究完全非线性和可能退化的椭圆型和抛物型Bellman方程的有限差分近似解的误差估计。贝尔曼方程出现在许多数学领域(例如，控制论、数学金融学、微分几何)。因此，寻求数值方法来逼近这类方程的解是一个自然的问题。该项目的第四部分集中于二阶线性椭圆型和抛物型系统的理论。标量方程和系统的一个关键区别是经典的最大值原理和Harnack不等式不再适用于系统。上一段中描述的项目不仅从数学角度来看很有趣。它们也具有核心重要性，并在物理、经济和金融等领域有重要应用。例如，一种应用发生在流体力学和湍流领域，人们想要估计汽车、飞机、轮船和潜艇等各种交通工具上的外部流动，以了解飓风的形成，或预测地球的大气循环。这些例子都与三维Navier-Stokes方程有关，但其理论还很不完善。另一个重要的应用是在金融领域，比方说，用数字计算股票投资组合的预期表现。例如，金融学中的布莱克-斯科尔斯期权定价模型是一个著名的股票市场数学模型，它与控制心脏流动的偏微分方程式密切相关。此外，这些问题中的许多自然地被建模为偏微分方程组的耦合系统，而不是单个方程。这项研究的结果将有助于改进在此类应用中使用的数学模型。

This project focuses on several topics in partial differential equations, both single equations and systems of equations, and considers problems involving fully nonlinear elliptic equations, evolutionary equations that arise in fluid mechanics, and error bounds for numerical approximations to solutions of fully nonlinear elliptic and parabolic equations. The first part of the project deals with the regularity of a family of fully nonlinear degenerate elliptic equations. These equations turn up in the optimal mass transport problem and in geometry. The second part is the investigation of global regularity, blow-up phenomena, partial regularity, and regularity criteria for several models of nonlinear parabolic equations in fluid mechanics. These include the well-known Navier-Stokes equations, the quasi-geostrophic equations, and some other related models. The third component is a study of error estimates for finite-difference approximations to solutions of fully nonlinear and possibly degenerate elliptic and parabolic Bellman equations. Bellman equations surface in many areas of mathematics (e.g., control theory, mathematical finance, differential geometry). It is thus a natural problem to seek numerical methods for approximating solutions to such equations. The fourth portion of the project concentrates on the theory of second-order linear elliptic and parabolic systems. A crucial difference between scalar equations and systems is that the classical maximum principle and the Harnack inequality are no longer applicable for systems.The projects described in the previous paragraph are interesting not only from a mathematical perspective. They are also of central importance and have significant applications in areas such as physics, economics, and finance. For instance, one application occurs in the area of fluid mechanics and turbulence, where one would like to estimate external flow over all kind of vehicles such as cars, airplanes, ships and submarines, to understand the formation of hurricanes, or to predict the earth's atmospheric circulation. These examples are all related to the three-dimensional Navier-Stokes equations, the theory of which is far from complete. Another important application is in finance, say, to calculate numerically the expected performance of a stock portfolio. For example, the Black-Scholes option pricing model in finance, which is a well-known mathematical model of the market for equity, is closely related to the partial differential equation that governs heart flow. Moreover, many of these problems are naturally modeled as coupled systems of partial differential systems rather than as single equations. The results obtained under this research will help to improve the mathematical models that are used in such applications.