1999 Fiscal Year Final Research Report Summary
Relations between geometric invariant and singularity of solution in nonlinear evolution equations related to nonlinear waves
| Project/Area Number |
09640159
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | TOHOKU UNIVERSITY (1999)
The University of Tokyo (1997-1998) |
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Principal Investigator |
TSUTSUMI Yoshio Graduate School of Science, Tohoku University, Professor, 大学院・理学研究科, 教授 (10180027)
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| Co-Investigator(Kenkyū-buntansha) |
TAKEDA Masayoshi Graduate School of Science, Tohoku University, Professor, 大学院・理学研究科, 教授 (30179650)
KOZONO Hideo Graduate School of Science, Tohoku University, Professor, 大学院・理学研究科, 教授 (00195728)
SHIMAKURA Norio Graduate School of Science, Tohoku University, Professor, 大学院・理学研究科, 教授 (60025393)
MIZUMACHI Tetsu Faculty of Science, Yokohama City University, Associated Professor, 理学部, 助教授 (60315827)
NAGASAWA Takeyuki Graduate School of Science, Tohoku University, Associated Professor, 大学院・理学研究科, 助教授 (70202223)
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| Project Period (FY) |
1997 – 1999
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| Keywords | nonlinear wave equations / KdV equations / Cauchy problem / well-posedness / white noise / stochastic differential equations |
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| Research Abstract |
We had studied the following two subjects for the period of July, 1997-March, 2000.
We first studied the well-posedness of the Cauchy problem for the system of nonlinear wave equations with different propagation speeds. One of the most important problems in the field of partial differential equations is to look for the largest possible function space in which the wave equations with quadratic nonlinearity is well-posed. This problem is closely related to the Lorentz invariant for the wave equation. When we consider the system of nonlinear wave equations with different propagation speeds, the discrepancy of propagation speeds breaks the Lorentz symmetry. We classified the quadratic nonlinear terms from a point of view of the time local well-posedness.
Second, we studied the unique solvability of the Cauchy problem for the Korteweg-de Vries equation with stochastic forcing term. The stochastic forcing term is regarded as a nonsmooth perturbation from a mathematical point of view. Especially, in general, the inverse scattering method is inapplicable to the Korteweg-de Vries equation with forcing term. We investigated the time local existence of solution for a natural class of stochastic forces.
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