| Project/Area Number |
13640226
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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 | Kyoto Sangyo University |
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Principal Investigator |
TSUJI Mikio Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (40065876)
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| Project Period (FY) |
2001 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2001: ¥800,000 (Direct Cost: ¥800,000)
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| Keywords | nonlinear hyperbolic equations / conservation laws / classical solution / weak solution / singularity / shock wave |
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| Research Abstract |
We have studied the Cauchy problem for nonlinear hyperbolic equations, especially the global theory for it. The difficulties of this problem is the appearance of singularities in finite time. Our problem is how to extend the solutions beyond the singularities. We explain our method. First we lift the equations into a higher dimensional space, and rewrite them as Pfaffian problems. They have sometimes smooth solutions in the large. We call them as "geometric solutions". Next we project them to the base space, and construct a reasonable "weak solution". This is our program. Therefore we do not change the equations essentially. We have expected that various kinds of results would be unified by our method. Though our program has been true for single first order partial differential equations, it has not generally been correct for higher order partial differential equations and systems. But, as our approach is very natural, we have continued our considerations by the same method. In this process we have solved many examples in explicit form and compared our solutions with another results obtained until now. Then we have had some question on the mathematical formulation of fluid mechanics. Refer to our "Research report". On the other hand we have considered several unsolved problems for single first order equations, and recognized that our program is correct. For example, an existence domain of solution is covered by a family of characteristic curves. Then we have constructed an example such that the boundary of the domain is obtained as an envelope of the characteristic curves, and that we can not extend the solution beyond the boundary. We have also studied an example where the equation is not convex with respect to p=(δu/δx_1, δu/δx_n) and showed that our program is correct.
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