| Project/Area Number |
18K03427
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Tokyo University of Science |
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Principal Investigator |
USHIJIMA TAKEO 東京理科大学, 理工学部数学科, 教授 (30339113)
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 解の爆発 / リスケーリング・アルゴリズム / 数値解析 / 曲率流 / 爆発解 / 数値計算 / 非線形発展方程式 / 爆発レート / 非線形方程式 / 非線型微分方程式 / 非線形微分方程式 / 数値手法 |
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| Outline of Final Research Achievements |
Solutions to nonlinear evolution equations do not always exist globally in time, and singularities can occur at finite times. Such a phenomenon is called blow-up. The rate at which the norm of a blow-up solution diverges to infinity is called the blow-up rate. In this research, the following were conducted with respect to the blow-up rate.
1. to propose a numerical estimation method of the blow-up rate using a rescaling algorithm, to improve the method, and to expand its range of applications; 2.Numerical experiments on partial differential equations describing the motion of a plane curve whose normal velocity is proportional to the power of its curvature (hereinafter referred to as "curvature flow"); 3. theoretical analysis of the traveling wave solution of the rescaled curvature flow; 4. evaluation of the blow-up rate of the curvature flow from above using the traveling wave solution of 3.
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| Academic Significance and Societal Importance of the Research Achievements |
解の爆発は，非線形偏微分方程式論における代表的な研究課題の一つであり，爆発解の様相の解明は重要な意義がある．有限な量しか扱うことのできない数値計算によって，解の発散する様相を捉えようという研究は長い歴史もあり，また本質的に困難な問題である．本研究はこの分野に新たな有用な手法を与えることになった．また曲率流の爆発解は複雑な爆発レートを持つのだが，本研究ではその様相について新たな理論的知見を与えた．
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