| Project/Area Number |
18K13430
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | The University of Tokyo (2022)
Waseda University (2018-2021) |
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Principal Investigator |
YOKOYAMA Satoshi 東京大学, 大学院数理科学研究科, 特任研究員 (70643774)
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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 |
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 確率偏微分方程式 / 確率解析 / 確率微分方程式 / 確率論 / 平均曲率方程式 / 偏微分方程式 |
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| Outline of Final Research Achievements |
The main subject of the study was a mathematical model that represents the time evolution of an equation with a stochastic element, noise, on the hypersurface that separates two materials with different properties. Ideally, the noise should be space-time white noise, which is considered physically natural, but in this study, the problem was captured with spatially correlated colored noise and the time evolution of the hypersurface driven by the noise was discussed. Specifically, a quasilinear second-order stochastic partial differential equation with multiplicative noise is derived as the equation satisfied by the signed distance function from the hypersurface. Due to technical difficulties arising from the poor regularity of the noise, the existence and uniqueness of local solutions are obtained, although conditions are required under which the coefficients on the colored noise are moderately transformed according to the value of the signed distance function above.
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| Academic Significance and Societal Importance of the Research Achievements |
確率項が付加された双安定で均衡条件を満たす反応項を持つ反応拡散方程式で、低温状態にするような極限操作では反応項の影響が強くなり２相を分ける界面の運動が現れる。界面の形状の時間発展を議論する事は重要である。２次元の空間での体積保存型の確率アレンカーン方程式では適切な条件のもと極限操作によって界面の時間発展を表す具体的な方程式が導かれるという結果を得た。また、２次元の空間で適切な条件を課した色付きノイズの場合の界面の運動も議論できた。数学的に議論を進めるための技術的な仮定の元、できる限り自然と思われるノイズを導入し確率偏微分方程式としてモデル化し、界面の運動の結果を得たことは意義深い。
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