Partial differential equations with the total mass conservation and related topics of abstract approach

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K05321
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Kyoto University of Education
• Principal Investigator: Fukao Takeshi 京都教育大学, 教育学部, 教授 (00390469)
• Co-Investigator(Kenkyū-buntansha): 愛木 豊彦 日本女子大学, 理学部, 教授 (90231745)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: 動的境界条件 / カーン・ヒリアード方程式 / 発展方程式 / 総質量保存則

Outline of Final Research Achievements

Results, well-posedness, asymptotic behavior, optimal control, and numerical analysis, for the Cahn-Hilliard system under the dynamic boundray condition were obtained applying the framework of the abstract evolution equation.
It becames clear that the abstract theory can be applied by suitable usage of the conservation of total mass, to set up the framework of function space and preparing an appropriate framework such as Poincare's inequality corresponding to each problem. On the other hand, new models with different dynamic boundary conditions have been proposed one after another during the research period. On this project the well-posedness of the problem which has the mass conservation separately, could be studied. The same kind of problems related to dynamic boundary conditions, including this research, will be attracting more.

Free Research Field

発展方程式

Academic Significance and Societal Importance of the Research Achievements

動的境界条件下において、適切性を論じることができたいくつかの問題においては、内部の方程式より境界上の方程式の方が設定できる自由度が高い。今後は内部の方程式をある程度整え、一方で境界の方程式をより複雑にし、従来表現できなかった複雑な現象を記述しつつ、適切性が裏付けられた動的境界条件下での偏微分方程式が研究対象になることが予想される。動的境界条件というあらたな設定によって、古くから研究されてきた研究課題が再度見直され、分野の再開拓という意味で本研究によって得られた結果が応用できる道筋が構築された。