1999 Fiscal Year Final Research Report Summary
STUDY OF DIFFERENTIAL EQUATIONS BY RENORMALIZATION
| Project/Area Number |
10640221
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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 |
Global analysis
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| Research Institution | SETSUNAN UNIVERSITY |
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Principal Investigator |
IKEBE Teruo SETSUNAN UNIVERSITY, MATHEMATICS DEPARTMENT, PROFESSOR, 工学部, 教授 (00025280)
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| Co-Investigator(Kenkyū-buntansha) |
WATARAI Seizou Setsunan University, Mathematics Department, Associate Professor, 工学部, 助教授 (20131500)
NAKAWAKI Yuuji Setsunan University, Mathematics Department, Professor, 工学部, 教授 (60207959)
ITO Keiichi r. Setsunan University, Mathematics Department, Professor, 工学部, 教授 (50268489)
SHIMADA Shin-ichi Setsunan University, Mathematics Department, Associate Professor, 工学部, 助教授 (40196481)
TERAMOTO Yoshiaki Setsunan University, Mathematics Department, Associate Professor, 工学部, 助教授 (40237011)
NAKABAYASHI Kouzaburo Setsunan University, Mathematics Department, Associate Professor (30207857)
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| Project Period (FY) |
1998 – 1999
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| Keywords | classical spin model / block spin transformation / random walks / Navier - Stokes eqs. / turbulence / scattering theory / Aharonov - Bohm effect |
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| Research Abstract |
(1) In the joint study with Prof. Tamura in Kanazawa University, Ito obtained several new results on the problems of phase transition of low dimensional O(N) spin models. Though it has been conjectured that he phase transition never occurs in such models, no one proves this rigourously. They could get a better estimate for the inverse critical temperature, by use of several steps of renormalization.
(2) Teramoto studied the stability of stationary solution to the problem of The Navier-Stokes flow down an inclined plane and showed that under certain conditions there occurs Hopf bifurcation. To apply the bifurcation theory he must show that the linear operator arising in this problem generates an analytic semigroup. Ito and Teramoto are now preparing a new method to study the interactions between the roll waves obtained above and try to give new insights to the mechanism of transition to turbulence.
(3) It is conjectured that the difficulty of diversity of the "antiderivatives" in the study of Nakawaki has something to do with renormalization group.
(4) Shimada studied the mathematical scattering theory on the Aharonov - Bohm effect in the quantum mechanics. He obtained the engenfunctions of the operator associated with this problem and constructed the wave operator, the scattering matrix and the representation of scattering amplitude. This work gives the mathematical foundation on the study of the Aharonov - Bohm effect.
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