The investigator concentrates on the mathematical investigation of surface diffusion, which is a fourth order geometric evolution. This motion law is a very useful model in materials science for phase boundary motions, which appear in thermal grooving, grain boundary migration, sintering, and many other instances. Even though the model was proposed by Mullins in the 1950s, analysis and understanding of the model is still far from complete, due to the lack of sufficient mathematical tools. Many issues such as existence of solutions, their qualitative behaviors, and singularity formations are still wide open. The project analyzes surface diffusion from a variational approach, which has proved to be quite versatile for many similar motion laws. Questions to be investigated include approximation and construction of solutions, the analysis of crystalline facet motions, and the understanding of solutions exhibiting self-similarity structures. The outcomes can benefit both the mathematical and materials science communities.
The use of mathematics in the modeling and analysis of materials science phenomena has become more and more important, in particular with the current continued outgrowth of nanotechnology. The properties of real materials very often are linked to the presence of inhomogeneity and defects such as phase boundaries and triple junctions. Successful understanding of these structures, especially their dynamical response behavior, requires intricate mathematical tools. This project aims to introduce new mathematical techniques to study phenomena of surface diffusion that have been proposed and experimentally observed for a long time, though many questions remain unsettled. The scientific merit of the project includes a better understanding of the materials and their response under various external environments. The project also provides an excellent opportunity for interdisciplinary research and educational activities that can tie together mathematical and engineering sciences, and graduate students are involved in the work.
