2014 Fiscal Year Final Research Report
Development of Analysis on Evolving Pattern for Complicated Phenomena
| Project/Area Number |
21224001
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| Research Category |
Grant-in-Aid for Scientific Research (S)
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| Allocation Type | Single-year Grants |
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| Research Field |
Global analysis
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| Research Institution | The University of Tokyo |
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Principal Investigator |
GIGA Yoshikazu 東京大学, 大学院数理科学研究科, 教授 (70144110)
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| Co-Investigator(Kenkyū-buntansha) |
YAMAMOTO Masahiro 東京大学, 大学院数理科学研究科, 教授 (50182647)
MATSUI Shin'ya 北海道情報大学, 情報メディア学部, 教授 (50219367)
FUNAKI Tadahisa 東京大学, 大学院数理科学研究科, 教授 (60112174)
ISHII Hitoshi 早稲田大学, 教育・総合科学学術院, 教授 (70102887)
JIMBO Shuichi 北海道大学, 大学院理学研究院数学部門, 教授 (80201565)
TONEGAWA Yoshihiro 北海道大学, 大学院理学研究院数学部門, 教授 (80296748)
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| Co-Investigator(Renkei-kenkyūsha) |
NISHIKAWA Takao 日本大学, 理工学部数学科, 准教授 (10386005)
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| Research Collaborator |
ABE Ken
ISHII Katsuyuki
UMEDA Noriaki
ETO Tokuhiro
OHTSUKA Takeshi
GIGA Mi-Ho
SEKI Yukihiro
HAMAMUKI Nao
POZAR Norbert
MIURA Hideyuki
MITAKE Hiroyoshi
YONEDA Tsuyoshi
LIU Qing
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| Project Period (FY) |
2009-04-01 – 2015-03-31
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| Keywords | 非線形現象 / 非線形偏微分方程式 / 粘性解 / 平均曲率流方程式 / 全変動流方程式 |
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| Outline of Final Research Achievements |
Various phenomena in natural sciences, for example, crystal growth phenomena and fluid motion, etc. are often modeled by nonlinear partial differential equations. We describe evolution of complex shapes and patterns observed there as mathematical phenomena and analyze them by developing the theory of viscosity solutions, variational analysis and real analysis, etc. In evolution of shapes and patterns even if the evolution law is simple and the initial shape is smooth, the solution often develops singularities by forming corners after some time. It is necessary to extend notions of a solution in a suitable way to interpret non-differentiable functions as a solution of differential equations for further analysis. In this project we introduce several new notions of solutions of diffusion equations describing for example crystal growth or fluid motion. We prove the existence of such solutions and analyze their behavior. Moreover, we study relation between discrete and continuous models.
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| Free Research Field |
数物系科学・数学・数学解析
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