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

本课题的研究内容如下：(1)相变现象的建模及其数学理论；(2)相关最优控制问题的建立与数值模拟；(3)研究成果的学校回馈。特别是，在该项目的最后两年(2003-2004年)，我们处理了一类考虑损伤过程的不可逆相变现象的模型，并发展了它的数学理论。在我们的研究中，已经积累了15年的次微分理论运行得很好，我们取得了比预期更大的进展。关于(2)，提出了用一类滞后函数来描述控制参数的最优控制问题的一些公式，我们完成了问题的几乎理论部分，并进行了部分数值试验。有人指出，目前仍有一些问题需要改进。课题(1)和(2)发表了许多论文，在本研究项目期间，我们组织了两次国际会议。关于(3)，我们每年组织的“相变的数学分析及其相关的数学教育”会议上报告了许多实践。我们注意到日常生活中容易注意到的时间依赖现象，非常仔细地验证了它们作为中学数学或科学教材的增值教育方面。我们期望这项研究能够为学生提供新的教材，让他们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.