2019 Fiscal Year Final Research Report
Evolution equations describing non-standard irreversible processes
| Project/Area Number |
16H03946
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Mathematical analysis
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| Research Institution | Tohoku University |
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Principal Investigator |
Akagi Goro 東北大学, 理学研究科, 教授 (60360202)
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| Co-Investigator(Kenkyū-buntansha) |
梶木屋 龍治 佐賀大学, 理工学部, 教授 (10183261)
木村 正人 金沢大学, 数物科学系, 教授 (70263358)
岡部 真也 東北大学, 理学研究科, 准教授 (70435973)
小池 茂昭 早稲田大学, 理工学術院, 教授 (90205295)
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | 函数方程式 / 非線形解析 / 不可逆過程 / 発展方程式 / 函数解析 / 偏微分方程式 / 変分法 / 無限次元力学系 |
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| Outline of Final Research Achievements |
Irreversible phenomena represented by diffusion are major factors of important phenomena closely related to our life such as unidirectionality of time, aging of lives and fracture. Classical theories for irreversible phenomena have already been established in the last century. However, many important irreversible phenomena beyond the scope of the classical theories have been observed, and therefore, studies of mathematical analysis have been developed in order to analyze and understand those new phenomena. In this research project, we have developed a new framework on Evolution Equations for covering new mathematical models which describe various non-standard irreversible phenomena. In particular, principal methods of analysis have been established for phase-field equations with strong irreversibility arising from fracture and damage models as well as nonlocal evolution equations involving fractional Laplacians, and moreover, systematic research has been done for those equations.
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| Free Research Field |
解析学
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| Academic Significance and Societal Importance of the Research Achievements |
発展方程式論は吉田耕作による半群理論の誕生以来，我が国が世界をリードしてきた研究分野の１つである．本研究課題はそのような発展方程式の研究を不可逆過程という我々の生活と密接につながる現象の観点から発展させるものである．このことは古典論の枠組みを逸脱する重要な不可逆現象に対する数理解析を推進するばかりでなく，発展方程式論の研究に於いても理論を発展させる道標を与える．また，ここで得られた研究成果は破壊現象の理解にかかせない強い不可逆性（一方向性）を数学的に扱うための基盤を確立する他，昨今，環境工学や生命科学をはじめ様々な分野で注目が集まっている異常拡散現象の記述に現れる非局所作用素の解析法を与える．
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