Research on the structure and stability of the dynamical system describing phase transition phenomena

本课题主要研究内容为：(1)相变现象的数学理论建模；(2)相关最优控制问题的表述及数值模拟；(3)研究成果回馈学校教育。从动力系统理论的观点看材料的原子排序。特别是，在项目的最后两年（2003-2004），我们处理了一类不可逆相变现象的模型，考虑了损伤过程，并发展了它的数学理论。在我们这15年积累的次微分理论的研究中，我们取得了很大的进展，超出了我们的预期。关于(2)，提出了一些用一类滞后函数描述控制参数的最优控制问题的公式，我们几乎完成了问题的理论部分，并进行了部分数值试验。指出还有一些问题需要改进。课题(1)和课题(2)发表了多篇论文，并组织了两次国际会议。就(3)而言，在我们每年组织的“相变的数学分析及其相关数学教育”会议上，报告了许多实践。我们关注我们日常生活中容易注意到的时变现象，非常仔细地验证它们作为中学数学或科学教材的教育价值。我们期望这项研究能够提供新的教材，让学生感受到数学的有用性。

The subjects of this project are the following :(1)Modelings of phase transition phenomena with their mathematical theory(2)Formulations of the related optimal control problems and numerical simulations(3)Return of the research results to the school educationConcerning (1), on the basis of the fundamental laws of the thermo-mechanics, we investigated the structure (phase change, component separation, damage evolution, ordering of atoms) of materials from the view-point of the theory of dymanical systems. Especially, during the last two years (2003-2004) of the project, we treated a class of models for irreversible phase change phenomena, taking account of the process of damage, and evolved its mathematical theory. In our research the subdifferential theory, which have been accumulated for these 15 years, worked very well and we had a big progress more than we expected. Concerning (2), proposing some formulations of optimal control problems in which control parameters are described by a class of hysteresis functions, we finished almost the theoretical part of the problems and tried partially their numerical tests. It was pointed out that there are still some questions which should be improved. Many papers treating subjects (1)and (2)were published and we organized two international conferences during this research project.As far as (3)is concerned, many practices were reported in the meetings "Mathematical analysis of phase transitions and their related mathematics education" which we organized every year. We paid our attention to time-dependent phenomena noticed easily in our daily lives, verifying very carefully their value-added educational aspects as teaching materials of mathematics or sciences for the secondary school. We expect that this research enables to provide new teaching materials by which students can perceive usefulness of mathematics.