Nonlinear Analysis and Stochastic Differential Equations
| Project/Area Number |
15K13450
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Mathematical analysis
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| Research Institution | Meiji University |
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Principal Investigator |
Nawa Hayato 明治大学, 理工学部, 専任教授 (90218066)
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| Research Collaborator |
AKAHORI Takafumi 静岡大学
KIKUCHI Hiroaki 津田塾大学
IBRAHIM Slim University of Victoria
IKOMA Norihisa 慶應義塾大学
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 非線形シュレーディンガー方程式 / 確率過程 / 爆発解 / 重対数法則 / 基底波解 / 散乱理論 / 非圧縮性オイラー方程式 / 乱流 / オイラー方程式 / 非線形偏微分方程式 / 確率過程論 / 解の爆発 / 基底波解の存在 / シュレーディンガー方程式 / 確率微分方程式 / 走化性方程式 / 古典乱流 / 乱流の可視化 |
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| Outline of Final Research Achievements |
We find that the loglog-law for the blowup solutions of the pseudo-conformally invariant nonlinear Schroedinger equations can be understood as a manifestation of the LIL for the Brownian motions consisting the Ito type stochastic differential equations which characterize the Nelson diffusions for the blowup solutions. In this study, we assume that the blowup speed is faster than self-similar one and slower than pseudo-conformal one, and assume further that the singularity is “very large” and the shoulder remains appropriately for the estimate from above. Thus, we may say that the loglog-law is a kind of universal nature of the blowup solutions.
Furthermore, we are trying to construct a new mathematical model of turbulence based on dissipative weak solutions of the incompressible Euler equations. This insight is brought by our previous work on the Kolmogorov’s scale law and the Onsager conjecture.
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| Academic Significance and Societal Importance of the Research Achievements |
決定論と非決定論の間にある様々な現象の理解に向けて，決定論的な非線形偏微分方程式の解の解析にその背後になる確率過程を利用したり，解の族がなす集団的な性質を統計力学的な視点を持ち込んで解析する方法論の確立を模索してきた。一般論の建設には至らなかったが，非線形シュレーディンガー方程式の様々な解や類似の現象を示す非線形偏微分方程式の解の解析，さらには新しい乱流の数学モデルの構築に向けて，さらなる理解と発展への新しい一歩を踏む出すことができたと思う。
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Report
(5 results)
Research Products
(16 results)
