2006 Fiscal Year Final Research Report Summary
study of time periodic solutions to the equations of gas dynamics
| Project/Area Number |
17540161
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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 | Yamaguchi University |
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Principal Investigator |
MAKINO Tetu Yamaguchi University, Graduate School of Science and Engineering, professor, 大学院理工学研究科, 教授 (00131376)
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| Co-Investigator(Kenkyū-buntansha) |
OKADA Mari Yamaguchi University, Graduate School of Science and Engineering, associate professor, 大学院理工学研究科, 助教授 (40201389)
MATSUNO Yoshimasa Yamaguchi University, Graduate School of Science and Engineering, professor, 大学院理工学研究科, 教授 (30190490)
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| Project Period (FY) |
2005 – 2006
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| Keywords | gas dynamics / quasilinear wave equations / periodic solutions / analysis / functional equations / fluid |
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| Research Abstract |
Originally this study started with the problem of nonlinear stability of the equilibria governed by the spherically symmetric Euler-Poisson equation of barotropic gas provided that the adiabatic exponent is greater than 4/3. The conjecture is that there exist time periodic solutions around these equilibria.
In order to clarify the essential points of the problem, we studied 1-dimensional movement of gas under a constant gravitatuional force, and proved that the linearized equation of the perturbations from the equilibria admits time periodic solutions which are described by the Besssel functions. Moreover we clarified a property of smooth periodic solutions, if exist, of the fully nonlinear equation. The existence of periodic solutions to the fully nonlinear equations is still open.
The problem is a free boundary problem at the interface with the vacuum, we have great difficulty in the theoretical consideration,. So, we studied the 1-dimensional movement of gas without exterior forces. The equilibria are constant density, and the eqwuation of the perturbation from the equilibria is a quasilinear wave equation, whose coeeeficients are regular at both boundaries. We proved there are no smooth periodic solutions for this fully nonlinear equation. Special cases about time periodic solutions to quasilinear wave equations are done by Greenberg, Rascal et al., but the tipe of equations arising from the gasdynamicas has no results. But there are possibility to show the existence of periodic solutions to these quasilinear gave equations applyinf the discussion by Rabinowitz, Brezis, Coron, Nirebnberg, craig, Wayne on the semilinear wave equations. We are now in the step of the study.
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