2006 Fiscal Year Final Research Report Summary
Structure of Solutions and Geometric Symmetry for Nonlinear Evolution Equations
| Project/Area Number |
15204008
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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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 | Kyoto University (2004-2006)
Tohoku University (2003) |
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Principal Investigator |
TSUTSUMI Yoshio Kyoto University, Dep.of Mathematics, Professor, 大学院理学研究科, 教授 (10180027)
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| Co-Investigator(Kenkyū-buntansha) |
KOKUBU Hiroshi Kyoto University, Dep.of Mathematics, Professo, 大学院理学研究科, 教授 (50202057)
NAKANISHI Kenji Kyoto University, Dep.of Mathematics, Associate Prof., 大学院理学研究科, 助教授 (40322200)
OHTA Masahito Saitama University, Dep.of Mathematics, Associate Prof., 理学部, 助教授 (00291394)
MIZUMACHI Tetsu Kyushu University, Dep.of Mathematics, Associate Prof., 大学院数理学研究院, 助教授 (60315827)
TAKAOKA Hideo Kobe University, Dep.of Mathematics, Associate Prof., 理学部, 助教授 (10322794)
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| Project Period (FY) |
2003 – 2006
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| Keywords | Well-posedness / Modified KdV / Nonlinear scattering theory / Long range interaction / Unconditional uniqueness / Nonlinear Schrodinger equations / Fourier restriction norm method / Besov spaces |
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| Research Abstract |
In what follows, the main research results of this project are described.
In the academic years of 2003-4, we studied the time local well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition. In 1993, Bourgain proved that this Cauchy problem is time locally well-posed in $H^s$, s ≧1/2. Furthermore, in 1996, Bourgain also proved that the solution map fails to belong to $C^3$ in $H^s$, s<1/2, though it is analytic in $H^s$, s≧ 1/2. Takaoka and Tsutsumi made a close investigation into the problem of what is the difference between the cases $H^s$, s ≧1/2 and $H^s$, s<1/2. It was showed that the Cauchy problem is still locally well-posed in $H^s$, s>1/3, but the solution map is not uniformly continuous because of the occurrence of nonlinear oscillation.
In the academic year of 2005, Tsutsumi studied the asymptotic behavior of solution for the quadratic nonlinear Schrodinger equation in two space dimensions with Akihiro Shimomura. In two space dimensio
… More
ns, the quadratic nonlinearity is a boader between the short renge case and the long range case and the quadratic nonlinearity has a special interest from a viewpoint of nonlinear scattering theory. It was showed that for a nonlinearity of squared modulous of the unkown, the solution does not approach a free solution, while it is already known that the rest of other quadratic nonlinearities belong to the shourt range interaction. casse.
In the academic year of 2006, Tsutsumi studied the unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger equation. When the solution is constructed, one usually impose a condition that the solution belongs to an auxiliary space associated with the Stricahrtz estimate. The unconditional uniqueness means that the solution is unique even though it is not in this kind of auxiliary space. It was proved that the unconditional uniqueness holds for the solution in the critical Sobolev space associated with the scaling invariance of nonlinear SchrOdinger equations. This is a substantial improvement over the results by Furioli and Terraneo. Less
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