2021 Fiscal Year Final Research Report
On the structure analysis of measure value solutions and singular sets for non-linear drift diffusion systems
| Project/Area Number |
19K03561
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | Osaka University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2022-03-31
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| Keywords | Keller-Segel系 / 測度値解 / 時間大域解 / 初期著問題の適切性 / 特異性解析 |
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| Outline of Final Research Achievements |
We consider not only linear diffusion but also nonlinear diffusion Keller-Sgel systems, and prove that the metric solution is described by the sum of the finite sum of the δ-functions and the regular part at every time step. Furthermore, we succeeded in proving the following：
(1) Whether or not the solution is constructed beyond the explosion time. Whether there exists an appropriate solution space for this purpose. (2) Whether the trajectory of the explosion point (the center of the aggregation of δ-functions) is regular as a time function. (3) Whether the size of the cohesion of the explosion point is regular and monotonic as a function of time.
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| Free Research Field |
非線形偏微分方程式論
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| Academic Significance and Societal Importance of the Research Achievements |
本研究では，爆発点の集合及び凝集サイズの時間発展を解析することで，解の有する特異構造を詳らかしている．初期値のサイズに依存しないKeller-Segel系の解構造について，統一理論を構築したことで，特異性を有する方程式系の解析に新たな数学的枠組みを構築することが出来た．
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