1989 Fiscal Year Final Research Report Summary
Study of Asymptotic Analysis of Systems of Quasilinear Partial Differential Equations
| Project/Area Number |
62460005
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Kyushu University |
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Principal Investigator |
YOSHIKAWA Atsushi Kyushu Univ. Fac. of Eng. Professor, 工学部, 教授 (80001866)
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| Co-Investigator(Kenkyū-buntansha) |
TANIGUCHI Setsuo Kyushu Univ. Fac. of Eng. Assistant Prof., 工学部, 講師 (70155208)
WATANABE Hisao Kyushu Univ. Fac. of Eng. Professor, 工学部, 教授 (40037677)
KUNITA Hiroshi Kyushu Univ. Fac. of Eng. Professor, 工学部, 教授 (30022552)
NISHINO Toshio Kyushu Univ. Fac. of Eng. Professor, 工学部, 教授 (30025259)
KAWASHIMA Shuichi Kyushu Univ. Fac. of Eng. Associate Prof., 工学部, 助教授 (70144631)
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| Project Period (FY) |
1987 – 1989
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| Keywords | Nonlinear geometrical Optics / Asymptotic expansion of weak solution / Implicit function theorem in real interpolation spaces |
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| Research Abstract |
The main purpose of the present study is to clarify effect of concurrence between quasilinearity and hyperboldicity through description of behaviors of asymptotic solutions of systems of quasilinear hyperbolic partial differential equations.
In the first place, we developed formal asymptotic theory for weak solutions. We showed that, even for higher dimensional systems of conservation laws, the structure of 1-dimensional conservation law is observed when restricted any particular phase surface. On the other hand, we noticed possibility of extending the notion of simple waves which are basic in the 1-dimensional study.
In the second place, we started rigorous asymptotic analysis of strong solutions with small data. Formal solutions can be constructed according to the method similar to the linear case. However, the conventional estimates only assure the same order for the presumed remainder term and the formal expansion. To overcome this difficulty, we are planning to apply a modified hard implicit function theorem which is valid in the Sobolev scale. The full details are yet to be written, and for the moment, several related estimates required for handling nonlinearity in the Sobolev scale are compiled. It is believed that the above mentioned modified implicit function theorem makes sense in the frame of real interpolation spaces.
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