Mathematical Sciences: Some Mathematical Problems Associatedwith Phase Transitions

研究人员研究了纯物质或二元合金中的各种相变模型。基本模型是奇异摄动的偏微分方程组，其解是演化的连续场，表示有序参数、衡量凝固程度或二元合金中某一物种的局部浓度。这三个基本模型被称为卡恩-艾伦、卡恩-希利亚德和相场方程。当奇异参数趋于零时，取某些可分辨极限，得到了尖锐界面的运动方程，并对这些方程进行了研究。材料科学已经变得更加数学化，尽管它已经成为一项更关键的技术。这项工作有助于更好地了解材料的行为和新材料的开发。涉及相变的类似数学问题也出现在物理和工程的其他领域。

The investigator studies various models of phase transitions in either a pure substance or a binary alloy. The fundamental models are singularly perturbed partial differential equations whose solutions are evolving continuous fields that represent either an order parameter, measuring degree of solidification, or local concentration of one of the species in a binary alloy. The three basic models are known as the Cahn-Allen, the Cahn-Hilliard, and the Phase Field equations. Taking certain distinguished limits as the singular parameter approaches zero leads to equations for the motion of sharp interfaces, which are also studied. Materials science has grown more mathematical even as it has become a more critical technology. This work can contribute to a better understanding of behavior of materials and the development of new materials. Similar mathematical problems involving phase transitions arise in other areas of physics and engineering also.