Asymptotic behavior of solutions to the dissipative equation with nonlocal operator
| Project/Area Number |
15K17566
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Mathematical analysis
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| Research Institution | Niigata University |
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Principal Investigator |
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| Research Collaborator |
Achleitner Franz University of Vienna
Jungel Ansgar Vienna Institute of Technology
Kato Keiichi 東京理科大学
Sugiyama Yuusuke 滋賀県立大学
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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 |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 函数方程式 / 函数解析 / 実解析 / 応用数学 / 数理物理学 / 関数方程式 / 関数解析 |
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| Outline of Final Research Achievements |
In this study, the structure with space-time variable of solutions of the dissipative equation with nonlocal operator is shown. An important example of nonlocal operators is the fractional Laplacian which is derived from the jumping process. This study treated well-posedness and behavior of solutions of the fractional diffusion equations which contain the drift-diffusion equation and the quasi geostrophic equation. For those equations, the well-posedness of solutions in the function space of the scale invariant and the global existence of the solution were proved. Moreover, asymptotic behavior of the solutions
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| Academic Significance and Societal Importance of the Research Achievements |
空間非局所な作用素の一例として跳躍過程に従う粒子の拡散を与える分数冪ラプラシアンが挙げられる。跳躍過程に従う粒子とは、半導体内部の荷電粒子や、田畑を飛び回る害虫のように、離れた点の間を瞬時に移動する粒子のことで、その拡散はブラウン運動では記述されない。本研究では、分数冪拡散方程式の解の時空変数に依存した挙動を明らかにした。その結果、熱の拡散などブラウン運動に従う粒子と比べて、空間遠方に分布する粒子の時間変動に大きな相違があることが明らかとなった。さらに当該研究では、Navier-Stokes方程式など非線形外力に空間非局所な作用素を含む方程式に対しても、解の構造を明らかにする手法を開発した。
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Report
(5 results)
Research Products
(17 results)
