Mathematical Research of Nonlinear Phenomena with Phase Transitions

相变非线性现象的数学研究

• 批准号: 09640154
• 负责人: KENMOCHI Nobuyuki
• 金额: $ 1.92万
• 依托单位: Chiba University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640154/
• 关键词: phase transition; subdifferential; evolution equation; Stefan Problem; free boundary; convex function; obstacle problem; variational inequality; 非線形発展方程式; 自由境界問題

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.

这个项目的主要目标是从跨学科的角度处理具有相变的各种非线性现象。我们的研究是利用非线性泛函分析中的工具，例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-固相变模型能够在希尔伯特空间中由次微分产生的非线性发展方程的微扰理论中处理。此外，我们提出了一类足够大的动力过程，包括我们考虑的典型的非线性现象，并发展了它的稳定性理论；事实上，我们成功地构造了全局吸引子。在我们的模型中，最重要的特征之一是动力过程的区域依赖于时间。此外，作为我们在该项目中所获得的结果的联合产物，我们解决了一个关于非线性椭圆算子的重要问题，这个问题已经悬而未决了30年。这在这个领域也是一个很大的贡献。

项目成果

N.Kenmochi, M.Kubo: "Weak solutions of nonlinear systems for nonisothermal phase transitions" Adv. Math.Sci. Appl.9. 499-521 (1999)

N.Kenmochi、M.Kubo：“非等温相变非线性系统的弱解”Adv。

蔵野正美(共著): "A fuzzy relational equation in dynamic fuzzy systems" Fuzzy Sets and Systems. 101. 439-443 (1999)

Masami Kurano（合著者）：“动态模糊系统中的模糊关系方程”Fuzzy Sets and Systems 101. 439-443 (1999)。

N.Kenmochi, T.Koyama, G.Meyer: "Parabolic PDEs with hysteresis and quasivariational inequalities" Nonlinear Anal.34. 665-686 (1998)

N.Kenmochi、T.Koyama、G.Meyer：“具有滞后和拟变分不等式的抛物线偏微分方程”非线性分析.34。

A.Damlamian(共著): "Evolution equations generated by subdifferentials in the dual space of H^2 (Ω)" Discrete and Continuous Dynamical Systems. 5. 269-278 (1999)

A.Damlamian（合著者）：“H^2 (Ω) 对偶空间中的次微分生成的演化方程”离散和连续动力系统。5. 269-278 (1999)

伊藤昭夫: "Asymptotic stability of Allen-Calin model for nolinear Laplecian with constraints" Advances in Mathematical Sciences and Applications. 9. 137-161 (1999)

Akio Ito：“带约束的非线性拉普勒斯 Allen-Calin 模型的渐近稳定性”数学科学与应用进展 9. 137-161 (1999)。