Gradient-Like Flow

9703497陈这项拟议的工作是对某些“类梯度”流动的进一步研究。特别是研究了调和映象弱流的部分正则性、唯一性和奇性，以及具有Dirichlet边界条件和自由边界条件的曲面上H-系统弱流的部分正则性。由于这些流动的几何性质，这些流动满足非线性偏微分方程组，这很少用标准的偏微分方程法或直接变分方法来攻击。这项工作的目的是为这些问题找到新的结果，并为研究一类结构相似的几何演化问题发展新的方法。我们相信，这种非线性分析方法也可以应用于材料科学，如连续体铁磁体中自旋场的演化，以及其他领域。

9703497 Chen This proposed work is a further investigation of certain "gradient-like" flows. In particular, the focus is on the study of the partial regularity, the uniqueness and the behavior of the singularities for the weak flow of harmonic maps, and the partial regularity for the weak flow of H-system from a surface with a Dirichlet boundary condition or with a free boundary condition. Due to the geometric nature, these flows satisfy nonlinear partial differential equations, which rarely can be attacked by standard PDE methods or direct variational method. The aim of this proposed work is to find new results for these problems and to develop new methods for the study of a whole class of geometric evolution problems with a similar structure. The belief is that the nonlinear analysis method developed here can also be applied to material science, such as the evolution of spin fields in continuum ferromagnets, and to other areas.