Mathematical Analysis of Materials Interfacial Motions by Surface Diffusion

研究人员专注于表面扩散的数学研究，这是一个四级几何演化。这一运动规律是材料科学中研究相界运动的一个非常有用的模型，相界运动在热槽、晶界迁移、烧结等许多情况下都会出现。尽管该模型是由穆林在20世纪50年代提出的，但由于缺乏足够的数学工具，对该模型的分析和理解仍然很不完整。许多问题，如解的存在性、定性行为、奇异性的形成等，仍然是悬而未决的。该项目从变分方法分析表面扩散，该方法已被证明对许多相似的运动规律具有相当的通用性。要研究的问题包括解的近似和构造，晶面运动的分析，以及对表现出自相似结构的解的理解。这一成果对数学和材料科学界都有好处。在材料科学现象的建模和分析中使用数学变得越来越重要，特别是在当前纳米技术不断发展的情况下。实际材料的性质往往与不均匀和缺陷的存在有关，如相界和三重结。成功地理解这些结构，特别是它们的动态响应行为，需要复杂的数学工具。这个项目旨在引入新的数学技术来研究表面扩散现象，这些现象已经被提出并在实验上观察了很长时间，尽管许多问题仍然没有得到解决。该项目的科学价值包括对材料及其在各种外部环境下的反应有更好的理解。该项目还为将数学和工程科学结合在一起的跨学科研究和教育活动提供了一个极好的机会，研究生参与了这项工作。

Yip0707926 The investigator concentrates on the mathematicalinvestigation of surface diffusion, which is a fourth ordergeometric evolution. This motion law is a very useful model inmaterials science for phase boundary motions, which appear inthermal grooving, grain boundary migration, sintering, and manyother instances. Even though the model was proposed by Mullinsin the 1950s, analysis and understanding of the model is stillfar from complete, due to the lack of sufficient mathematicaltools. Many issues such as existence of solutions, theirqualitative behaviors, and singularity formations are still wideopen. The project analyzes surface diffusion from a variationalapproach, which has proved to be quite versatile for many similarmotion laws. Questions to be investigated include approximationand construction of solutions, the analysis of crystalline facetmotions, and the understanding of solutions exhibitingself-similarity structures. The outcomes can benefit both themathematical and materials science communities. The use of mathematics in the modeling and analysis ofmaterials science phenomena has become more and more important,in particular with the current continued outgrowth ofnanotechnology. The properties of real materials very often arelinked to the presence of inhomogeneity and defects such as phaseboundaries and triple junctions. Successful understanding ofthese structures, especially their dynamical response behavior,requires intricate mathematical tools. This project aims tointroduce new mathematical techniques to study phenomena ofsurface diffusion that have been proposed and experimentallyobserved for a long time, though many questions remain unsettled. The scientific merit of the project includes a betterunderstanding of the materials and their response under variousexternal environments. The project also provides an excellentopportunity for interdisciplinary research and educationalactivities that can tie together mathematical and engineeringsciences, and graduate students are involved in the work.