2006 Fiscal Year Final Research Report Summary
Studies on a new class of hyperbolic systems
| Project/Area Number |
15340044
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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 | Osaka University |
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Principal Investigator |
NISHITANI Tatsuo Osaka University, Graduate School of Science, Professor, 理学研究科, 教授 (80127117)
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| Co-Investigator(Kenkyū-buntansha) |
HAYASHI Nakao Osaka University, Graduate School of Science, Professor, 理学研究科, 教授 (30173016)
DOI Shinichi Osaka University, Graduate School of Science, Professor, 理学研究科, 教授 (00243006)
SUGIMOTO Mitsuru Osaka University, Graduate School of Science, Associate Professor, 理学研究科, 助教授 (60196756)
MATSUMURA Akitaka Osaka University, Infirmation Science and Technology, Professor, 情報科学研究科, 教授 (60115938)
OKAJI Takashi Kyoto University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (20160426)
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| Project Period (FY) |
2003 – 2006
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| Keywords | double characteristic / null bicharacteristic / Gevrey space / well posedness / hyperbolic operator / initial value problem / elementary decomposition / Hamilton map |
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| Research Abstract |
We have obtained a definitive result about the classification of hyperbolic double characteristics.
A hyperbolic double characteristic is called non effectively hyperbolic characteristic if the Hamilton map at the reference point admits only pure imaginary eigenvalues. A remaining fundamental question was whether the Cauchy problem around non effectively hyperbolic characteristic is C-infty well-posed?
We classify hyperbolic double characteristics whether the behavior of null bicharacteristics around the reference double characteristic is stable with respect to the doubly characteristic manifold, that is whether there exists a null bicharacteristic with a limit point in the doubly characteristic manifold. We have obtained the following results:
If the behavior of null bicharacteristics around the reference double characteristic then the principal symbol is elementary decomposable and the Cauchy problem is C-infty well-posed. On the other hand, if the behavior of null bicharacteristic is unstable then the principal symbol is not elementary decomposable and the Cauchy problem is not C-infty well-posed. We obtained more detailed results. In this unstable case the Cauchy problem is Gevrey 5 well-posed and this index 5 is optimal in the following sense; if there is a null bicharacteristic with a limit point in the doubly characteristic manifold then the Cauchy problem is not Gevrey s well-posed for any s>5.
Based on the above results, we obtained the following result : assume that the codimension of the doubly characteristic manifold is 3 and the all eigenvalues of the Hamilton map remain to be pure imaginary then the Cauchy problem is Gevrey 5 well-posed.
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