Free Boundary Problems and Interfacial Dynamics

界面现象在自然界中很常见。只要有一种物质可以存在于至少两个不同的相中，并且存在某种机制使这些相在空间上分离，它们就会发生。对大多数物质来说，至少有三相（例如，水蒸气、液体和水的冰）。分隔不同相的界面可能随着时间的推移而演变。典型的两相转变是液体的凝固过程。经典的多相转变是由不同晶粒的原子排列在晶界处不匹配的多晶中晶粒的演化引起的。固体多晶体由液体生长到固体多晶体的过程包括液体和固体之间的两相转变和晶粒之间的多相转变。迄今为止，已经积累了大量关于两相转变的知识。然而，对于多相转变，严格的数学研究还远远没有完成。本项目的目的是对多相转变进行系统的研究。宏观的自由边界模型是基于可观测量的，需要处理由自由边界的拓扑变化引起的奇点；微观的连续体模型是在分子水平上推导出来的，并且（总是）适定的。该项目的一个目标是在自由边界模型变得不确定时，从连续统模型中获得关键信息。另一个是建立新的微观连续体模型，用于多晶体中固体/液体和不同晶粒之间的相变。这里使用的主要工具将是偏微分方程理论、奇异摄动、渐近展开、复分析、分岔、中心流形、全局分析、动力系统和几何测量。首先需要计算机辅助来模拟，然后验证，并进一步扩展理论结论。该项目包括通过参与研究对本科生和研究生进行培训。

Interfacial phenomena are common in nature. They occur whenever there is a substance that can exist in at least two different phases and there is some mechanism that generates a spatial separation of these phases. For most substances there are at least three phases (vapor, liquid, and ice for water, for example). The interfaces that separate different phases may evolve with time. A typical two-phase transition is the solidification process of a liquid. A classical multi-phase transition arises from evolutions of grains in a polycrystal where alignments of atoms in different grains do not match at grain boundaries. The growth of a solid polycrystal from a liquid involves a two-phase transition between liquid and solid and a multi-phase transition among grains. To date, a significant amount of knowledge has been accumulated on two-phase transitions. For multi-phase transitions, however, rigorous mathematical studies are far from complete. The objective of this project is to provide a systematic study of multi-phase transitions.There are macroscopic free boundary models, which are based on observable quantities and are needed to deal with singularities due to topological changes of free boundaries, and microscopic continuum models, derived at the molecular level and (always) well-posed. One goal of the project is to derive critical information from continuum models for free boundary models when the latter become indeterminate. Another is to develop new microscopic continuum models for phase transitions among phases such as solid/liquid and different grains in a polycrystal. The main tools used here will be theories of partial differential equations, singular perturbations, asymptotic expansions, complex analysis, bifurcation, center manifolds, global analysis, dynamical systems, and geometric measure. Computer assistance will be needed to first simulate, then validate, and further extend theoretical conclusions. The project includes training of undergraduate and graduate students through their participation in research.