Mathematical Analysis of Interfacial Dynamics

该项目的目标是提供多相转变的系统研究。有宏观的自由边界模型，它基于可观察的量，并且需要处理由于自由边界的拓扑变化而引起的奇点；还有微观的连续体模型，在分子水平上推导出来，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。二是建立新的微观连续体模型，用于研究多晶材料中固/液相和不同晶粒之间的相变。这里使用的主要工具是偏微分方程理论、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。有时，计算机辅助首先用于刺激，然后验证，并进一步扩展理论结论。研究者研究界面现象，这种现象在自然界中很常见，只要有一个连续体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，就会发生界面现象。分隔这些相的空间边界被称为界面或自由边界，由于相之间的某种强制转换而随时间演变。非典型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上不匹配的多晶晶粒的演变引起的。从液体中生长出固体多晶体，包括液体和固体之间的两相转变和晶粒之间的多相转变。这些现象激发了自由边界问题、抛物型椭圆偏微分方程组或常微分方程组以及动力系统的新的理论发展。从该项目中获得的见解增加了数学家和材料科学家之间的联系。这里开发的新模型为科学家们理解和设计新材料提供了工具。

ChenDMS-1008905 The objective of this project is to provide a systematicstudy of multi-phase transitions. There are macroscopic freeboundary models, which are based on observable quantities and areneeded to deal with singularities due to topological changes offree boundaries, and microscopic continuum models, derived at themolecular level and (always) well-posed. One goal of the projectis to derive critical information from continuum models for freeboundary models when the latter become indeterminate. Another isto develop new microscopic continuum models for phase transitionsamong phases such as solid/liquid and different grains in apolycrystalline material. The main tools used here are theoriesof partial differential equations, singular perturbations,asymptotic expansions, complex analysis, bifurcation, centermanifolds, global analysis, dynamical systems, and geometricmeasure. From time to time, computer assistance is used to firststimulate, then validate, and further extend theoreticalconclusions. The investigator studies interfacial phenomena, which arecommonplace in nature and occur whenever there is a continuumthat can exist in at least two different phases and there is somemechanism that generates a spatial separation of these phases. The spatial boundaries that separate these phases, referred to asinterfaces or free boundaries, evolve with time due to someenforcement of transitions of phases from one to another. Atypical two-phase transition is a solidification process of aliquid. A classical multi-phase transition arises fromevolutions of grains in a polycrystal where alignments of atomsin different grains do not match at grain boundaries. The growthof a solid polycrystal from a liquid involves a two-phasetransition between liquid and solid and a multi-phase transitionamong grains. These phenomena motivate new theoreticaldevelopment in free boundary problems, systems of parabolic orelliptic partial or ordinary differential equations, anddynamical systems. Insights gained from the project increaseconnections between mathematicians and materials scientists. Newmodels developed here provide scientists with tools forunderstanding and designing new materials.