Problems in Nonlinear Partial Differential Equations

从广义上讲，该项目解决了椭圆型和抛物型非线性偏微分方程组的正则性理论中的开放性问题，以及不可压缩欧拉方程理论中的若干问题。大多数椭圆型和抛物型问题都是在不可压缩的Navier-Stokes方程的背景下表述的，但也将研究各种模型方程，并且从模型方程中获得的见解预计是重要的。主要的重点是发展的方法和技术的问题是超出了摄动理论的范围。对于某些类型的方程（例如，Navier-Stokes方程），这些问题具有相当大的理论和实践意义。偏微分方程是许多自然和技术现象建模的基本工具。例如，描述流体流动的方程（这是本项目的一个重要部分）用于天气预报、气候研究和各种技术应用，例如设计最佳空气动力学形状。众所周知，流体动力学方程很难解，即使有最大的计算机的帮助。这些困难有些是内在的，但有些是由于我们对方程本身的数学理解不完整。对偏微分方程的理论理解的进步可以导致求解偏微分方程的实际方法的实质性改进。归根结底，理论研究的主要任务是找到一些简单而自然的参数来控制解的行为。一旦知道了这些参数的集合，设计计算解的实用方法就容易多了。原因：有了这些参数，研究人员在跟踪解决方案时就知道什么是重要的，什么是不重要的。然后，研究人员可以将计算能力集中在真正重要的方面。所提出的研究可以在这个方向上产生重大影响。

Broadly speaking, the project addresses open problems in regularity theory of elliptic and parabolic nonlinear systems of partial differential equations, as well as several problems from the theory of incompressible Euler equations. Most of the elliptic and parabolic problems are formulated in the context of the incompressible Navier-Stokes equations, but various model equations will also be studied and the insight obtained from the model equations is expected to be important. The main emphasis is on developing methods and techniques for problems that are out of the reach of perturbation theory. For certain classes of equations (such as, for example, the Navier-Stokes equation) such problems are of considerable theoretical and practical importance. Partial differential equations represent a basic tools for modeling many natural and technological phenomena. For instance, the equations describing fluid flow (on which an important part of this project will concentrate) are used in weather prediction, climate research, and various technological applications, such as the design of optimal aerodynamical shapes. The equations of fluid dynamics are notoriously difficult to solve, even with the help of the largest computers. Some of these difficulties are intrinsic, but some are due to the fact that our mathematical understanding of the equations themselves is incomplete. Advances in theoretical understanding of partial differential equations can lead to substantial improvement in practical methods for solving them. In the final analysis, the main task of the theoretical investigation is to find some simple and natural parameters that control the behavior of the solutions. Once a good set of such parameters is known, it becomes much easier to design practical methods for calculating the solutions. Reason: armed with such parameters a researcher knows what is important and what is not when one tracks a solution. The researcher can then focus computational power on the aspects that really matter. The proposed research can have a significant impact in this direction.