Mathematical Sciences: Heat Flow of Harmonic Maps

该项目的重点是数学分析， 曲面上偏微分方程的解 或歧管。 这件作品直观上是几何的， 以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。 的 这个问题可以用抛物线偏微分方程来重新表述。 微分方程，其中变形成为 解决方案在时间上的演变。 了大量工作 在这种性质的问题上，特别是在欧几里得空间。 众所周知，解决方案不需要一直发展， 这种现象通常被称为有限时间爆破。 在 在本文中，排除爆破的已知条件是 与目标的尺寸和截面曲率有关 歧管 最好的结果是已知的二维 流形 在目前的工作中，将努力 建立调和映射热流的整体存在性， 更高维度的边界，并了解 奇异性，当它们发生时。 第二个目标是 了解热流的边界规律， 更高维度的调和映射。 证据表明 热方程的演化解不可能首先遇到 边界上的奇点 这可能很难在 更高的维度，但有理由相信， 在二维和三维流形中分析是可能的。 偏微分方程是 物理科学中的数学建模。 的现象 包括连续变化，如运动、材料 和能量都服从某些普遍规律， 可以用相互作用和关系来表达 偏导数之间的关系 数学的关键作用不是 来陈述这些关系，而是为了提取定性的 和量化的意义。

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.