Mathematical Sciences: Heat Flow of Harmonic Maps

这个项目的重点是定义在曲面或流形上的偏微分方程解的数学分析。这项工作是直观的几何，因为它试图确定流形之间的光滑映射何时可以变形为相同流形之间的调和映射。这个问题可以用抛物型偏微分方程组来表述，其中变形成为解在时间上的演化。关于这种性质的问题，特别是在欧几里得空间中，已经做了大量的工作。众所周知，解决方案不需要一直进化，这种现象通常被称为有限时间爆炸。在目前的上下文中，防止爆破的已知条件与目标流形的尺寸和截面曲率有关。最好的结果是已知的二维流形。在本工作中，我们将致力于建立高维边界调和映射热流的整体存在性，并理解奇点出现时的性质。第二个目标是了解高维调和映射热流的边界规律性。有证据表明，热方程的不断发展的解不可能首先在边界处遇到奇点。这可能很难在更高的维度上表现出来，但有理由相信，在二维和三维流形上进行完整的分析是可能的。偏微分方程式是物理科学中数学建模的基础。人们知道，涉及运动、材料和能量等连续变化的现象遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不是描述关系，而是从它们中提取定性和定量的意义。

The focus of this project is the mathematical analysis of solutions of partial differential equations defined on surfaces or manifolds. The work is intuitively geometric in that it seeks to determine when smooth mappings between manifolds can be deformed into harmonic maps between the same manifolds. The question can be rephrased in terms of parabolic partial differential equations where the deformation becomes the evolution of the solution in time. Considerable work has been done on problems of this nature, especially in Euclidean space. It is known that solutions need not evolve for all time, a phenomenon commonly referred to a finite-time blow up. In the present context, the known conditions which preclude blow up are related to the dimension and sectional curvatures of the target manifold. The best results are known for two dimensional manifolds. In the present work, efforts will be made to establish global existence of the heat flow of harmonic maps with boundary in higher dimensions and to understand the character of singularities when they do occur. A second objective is that of understanding the boundary regularity of the heat flow for harmonic maps in higher dimensions. Evidence suggests that the evolving solutions to the heat equation cannot first encounter singularities at the boundary. This may be difficult to show in higher dimensions, but there is reason to believe that a complete analysis is possible in two and three dimensional manifolds. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.