Mathematical Analysis of Materials Interfacial Motions by Surface Diffusion

研究者集中于表面扩散的数学研究，这是一个四阶几何演化。在材料科学中，这一运动规律对于热开槽、晶界迁移、烧结和许多其他情况下的相界运动是一个非常有用的模型。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远未完成。许多问题，如解的存在性、解的定性行为、奇点的形成等，仍有待解决。该项目从变分方法分析表面扩散，这已被证明是相当通用的许多类似的运动定律。要研究的问题包括溶液的近似和构造，晶体表面运动的分析，以及对表现自相似结构的溶液的理解。研究结果对数学界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已经变得越来越重要，特别是随着纳米技术的不断发展。实际材料的性能通常与不均匀性和缺陷（如相界和三重结）的存在有关。成功地理解这些结构，特别是它们的动力响应行为，需要复杂的数学工具。该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和实验观察了很长时间，尽管许多问题仍未解决。该项目的科学价值包括更好地理解材料及其在各种外部环境下的反应。该项目还为跨学科研究和教育活动提供了一个极好的机会，可以将数学和工程科学联系在一起，研究生也可以参与其中。

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.