2003 Fiscal Year Final Research Report Summary
Study on non-liner partial differential equations by means of besov tpe norms
| Project/Area Number |
14540186
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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 |
Basic analysis
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| Research Institution | CHUO UNIVERSITY |
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Principal Investigator |
MURAMATSU Toshinobu Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (60027365)
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| Co-Investigator(Kenkyū-buntansha) |
MITSUMATSU Yoshihiko Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70190725)
MATSUYAMA Yosshio Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (70112753)
OHHARU Shinnosuke Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (40063721)
MOCHIZUKI Kiyoshi Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (80026773)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Besov type norm / Besov type norm / Fourier restiction norm / semilinear Schrodunger equation / Besov type norm / trilinear estimates / KdV equtatiion / initial value problem / well-posednessT |
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| Research Abstract |
We defined the Besov type norms which are generalizations of the Fourier restriction norm due to Bourgain, appilled them to the initial value problem of nonlimear partial differential equations, and obtained the following results :
1. The intial value problem for the semilinear Schrodinger equation.
(1)Quadratic nonlimearity case.
For the case whetre the space dimension is 1 or 2 we proved that the intial value problem in the Sobolev space of the critical order is well-posed.
We also obtained the results for the case where the space dimension is greater than 2. The key method is bilinear estimates by meas of Besov type norms.
(2)Cubic nonlinearity case
We proved that the initial value problem is well-posed in the Besov space of critical order when the space is 1, and this result is better than that obtained by the Fourier restriction norm. The key method is trilinear estimates by means of Besov type norms.
(3) We find that the initial value problem is well-posed in the space of square integrable functions when the nonliniarity is the derivative of the squrer of the complex conjugate of the unknown function.
2. The initial value problem for KdV equation.
We proves that the initial value problem is well-posed in the Sobolev space which is very closed to that of order -3/4 (the critical order).
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