Evolution equations with the coexistence of fractional derivatives and nonlinear structures
| Project/Area Number |
18K18715
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| Research Category |
Grant-in-Aid for Challenging Research (Exploratory)
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| Allocation Type | Multi-year Fund |
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| Review Section |
Medium-sized Section 12:Analysis, applied mathematics, and related fields
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| Research Institution | Tohoku University |
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Principal Investigator |
Akagi Goro 東北大学, 理学研究科, 教授 (60360202)
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| Project Period (FY) |
2018-06-29 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥6,240,000 (Direct Cost: ¥4,800,000、Indirect Cost: ¥1,440,000)
Fiscal Year 2020: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2019: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2018: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
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| Keywords | 非整数階微分 / 非線形問題 / 発展方程式 / 異常拡散 / 多重スケール構造 / 多重スケール / 非整数回微分 |
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| Outline of Final Research Achievements |
The main results of this research project consist of the following:
(1) We established a theory for fractional (in time) gradient flows in Hilbert spaces. This result extends the so-called Brezis-Komura theory, which is concerned with classical gradient flows and has been used to analyze various nonlinear partial differential equations, to fractional evolution equations and can provide a basic frame for studying time-fractional partial differential equations, whose well-posedness (e.g., existence and uniqueness of solutions) has been open for long time.
(2) We proved a gradient inequality for fractional Laplacians restricted on domains and non-analytic potentials and applied it to prove the convergence of solutions to equilibria for the fractional Cahn-Hilliard systems.
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| Academic Significance and Societal Importance of the Research Achievements |
拡散現象や相転移現象のような不可逆過程については、前世紀に物理学の古典論が確立し、その後、様々な現象の解析に適用されてきた。しかしその過程に於いて、古典論の予測から大きく逸脱する不可逆現象が多々発見されるようになり、古典論の見直しに対する強い動機をもたらした。ここで扱う非整数階微分や非線形拡散は古典論の修正に効果的であり、多くの注目を集めている。本研究課題の研究成果はそのような新しい構造を伴う数学的対象に対して、十分な汎用性と応用性を兼ね備えた理論を与え、この分野の研究に於いて１つの基盤となるものと考える。数学的基盤が整備されることで応用研究の土台が整い、広範な分野への波及効果が期待される。
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Report
(6 results)
Research Products
(49 results)
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[Presentation] Goro Akagi2021
Author(s)
Rates of convergence to non-degenerate asymptotic profiles for fast diffusion equations via an energy method
Organizer
BIRS-CMO Workshop ``New Trends in Nonlinear Diffusion: a Bridge between PDEs, Analysis and Geometry'', Banff International Research Station for Mathematical Innovation and Discovery (BIRS)
Related Report
Int'l Joint Research / Invited
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