Mathematical Research of Nonlinear Phenomena with Phase Transitions

本计画的主要目标是从跨学科的观点来处理各种非线性现象与相变。我们的研究主要是利用非线性泛函分析中的工具，如凸函数的次微分算子理论，对它们进行建模和理论分析，在本项目期间，我们发现许多固-液或固-固相变模型都可以用Hilbert空间中次微分产生的非线性发展方程的微扰理论来处理。此外，我们还提出了一类足够大的动力学过程，包括我们所考虑的非线性现象作为典型例子，并发展了它的稳定性理论，实际上，我们在构造全局吸引子方面取得了进展。在我们的方案中，最重要的特点之一是动力学过程的区域依赖于时间，并且作为我们在这个项目中所得到的结果的副产品，解决了一个关于非线性椭圆算子的30年来一直悬而未决的重要问题。这在这个领域也是一个很大的贡献。

The main objective of this project is to treat various nonlinear phenomena with phase transitions from interdisciplinary points of view. Our research is concerned with their modellings and theoretical analysis by using tools in nonlinear functional analysis for instance the subdifferential operaors theory of convex functions.During the term of this project we found that many solid-liquid or solid-solid phase transition models are able to be handled within the perturbation theory of nonlinear evolution equations generated by subdifferentials in Hilbert spaces. Moreover, we pro- posed a class big enough of dynamical processes, including nonlinear phenomena in our consideration as typical examples, and evolved the stability theory for it ; in fact, we suc- ceeded in the construction of global attractors. In our set-up, one of the most important characteristics is that the domain of dynamical process depends upon time.Also, as co-products of our results obtained in this project, an important question, which had been remained open for 30 years, about nonlinear elliptic operators was solved. This is a big contribution in this field, too.