2002 Fiscal Year Final Research Report Summary
Geometric invariant, propagation of singularity and asymptotic behavior for nonlinear wave equations
| Project/Area Number |
12440033
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Tohoku University |
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Principal Investigator |
TSUTSUMI Yoshio Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10180027)
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| Co-Investigator(Kenkyū-buntansha) |
ARISAWA Mariko Graduate School of Information Science, Associate Professor, 大学院・情報科学研究科, 助教授 (50312632)
NAGASAWA Takeyuki Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (70202223)
KOZONO Hideo Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00195728)
MIZUMACHI Tetsu Faculty of Science, Yokohama City University, Associate Professor, 理学部, 助教授 (60315827)
OHTA Masahito Faculty of Science, Saitama University, Associated Professor, 理学部, 助教授 (00291394)
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| Project Period (FY) |
2000 – 2002
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| Keywords | null condition / Dirac-Proca equations / Maxwell-Higgs equations / global existence in time / well-posedness / modified KdV equation / initial value problem / weak solution |
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| Research Abstract |
It is well known that there are close relations between the regularity of solutions and the geometric invariant for nonlinear wave equations. Especially, the null condition introduced by Klainerman and Christodoulou often plays an important role in the case of relativistic nonlinear wave equations. From this point of view, in the acadimic years of 2000 and 2001, we studied the global existence of solutions for the Cauchy problem of the Dirac-Proca equations and the Maxwell-Higgs equations. We first showed that the Proca equation has a null condition structure, which led to the global existence of solution for small and smooth initial data. We next discovered that the Maxwell-Higgs equations generically have a null condition structure, which enabled us to show the global existence of solution for small and smooth initial data
In the academic year of 2002, we studied the well-posedness of the Cauchy problem for the modified KdV equation in a weak class. It is known that when s< 1/2, the solution map is not in C^2, while it is in C^∞ for s > 1/2. We studied what kind of structure for the modified KdV equation breaks down the well-posedness in a weak class
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