| Project/Area Number |
18H01136
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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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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
横山 知郎 岐阜大学, 工学部, 准教授 (30613179)
米田 剛 一橋大学, 大学院経済学研究科, 教授 (30619086)
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| Project Period (FY) |
2018-04-01 – 2022-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥16,640,000 (Direct Cost: ¥12,800,000、Indirect Cost: ¥3,840,000)
Fiscal Year 2021: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2020: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2019: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2018: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
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| Keywords | 応用数学 / 力学系 / 渦力学 / 流体力学 / 微分幾何学 / トポロジー / 流体方程式 |
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| Outline of Final Research Achievements |
We have conducted mathematical and numerical analysis of vortex dynamics on surfaces such as a sphere and a curved torus, investigating the influence of curvature and topology on vortex motion. The project consists of three sub-topics. (P1) Theory of vortex dynamics on surfaces, (P2) Development of numerical methods of flows on surfaces, and (P3) Applications to physical problems. The results are summarized as follows. (P1) Point vortex equilibria on a curved torus, the analytic formula of incompressible flows with Liouville-type background vorticity on a curved torus, finding stationary finite-length vortex sheets and the derivation of point vortex dynamics in a doubly periodic domain. (P2) The method of fundamental solution for calculating harmonic measures on a sphere, analysis of spot dynamics on a curved torus surface. (P3) point-vortex statistics as a turbulent model on surfaces, and construction of quantized point-vortex equilibria on surfaces as a model of superfluids.
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| Academic Significance and Societal Importance of the Research Achievements |
これまでほとんど知られていなかったトーラス面上の点渦力学に多くの成果を得ることができた．特にLiouville背景渦を持つ流れは超流動乱流のモデルとして，また数学的には二次元閉曲面上のLiouville方程式の解析解としても重要なものであり，学術的に意義があった．またトーラス面や二次元二重周期領域における点渦力学ではグリーン関数の具体的な解析表示を得ることができたことで，スポットダイナミクスなどの多くの問題への応用が実際に可能になった．こうした解析解の表示は今後の閉曲面上の流体方程式のみならず非線型偏微分方程式の数学解析や数値解析に学術的な貢献が期待できる．
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