2001 Fiscal Year Final Research Report Summary
Studies on singular perturbation problems in nonlinear mechanics
| Project/Area Number |
11304005
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
OKAMOTO Hisashi KYOTO UNIVERSITY, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (40143359)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUMURA Akitaka Math. Dept., Osaka Univ., Professor, 大学院・理学研究科, 教授 (60115938)
NISHIDA Takaaki Dept. of Math., Kyoto Univ., Professor, 大学院・理学研究科, 教授 (70026110)
OHKITANI Koji KYOTO UNIVERSITY, Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (70211787)
NAKAKI Tatsuyuki Graduate School of Sciences, Kyushu Univ., Associate Professor, 大学院・数理学研究院, 助教授 (50172284)
KAWASHIMA Shuichi Graduate School of Sciences, Kyushu Univ., Professor, 大学院・数理学研究院, 教授 (70144631)
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| Project Period (FY) |
1999 – 2001
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| Keywords | singularity / blow-up of solutions / interior layer / self-similar solution / bifurcation / dynamical system / periodic solution / Navier-Stokes equations! |
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| Research Abstract |
(1) New phenomena on the Navier-Stokes equations were found. Among others, solutions having interior layers and those solutions having k-10 spectra are remarkable. (2) Bifurcation phenomena in surface waves were clarified. In particular, an accurate numerical method was developed for singular solitary waves. (3) dynamical systems viewpoints on the shell model of turbulence proposed by Ohkitani and Yamada were enhanced. (4) applications to reaction-diffusion systems, (5) vortex formation in the 2-dimensional decaying turbulence by Y. Kimura. (6) asymptotic behavior of shock wave solutions was clarified by Kawashima and Matsumura.
Okamoto, with the aid by Kim Sunchul, analyzed the bifurcating solutions arising in the rhombic periodic flows. It was demonstrated, by an elaborate numerical computations, that some solutions have k-10 spectra as the Reynolds number tends to infinity. Okamoto and A. Craik considered a three-dimensional dynamical system arising in fluid mechanics. Two different
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solutions, one with 90-degree bending and one without bending, were found and the mechanism of them was theoretically explained.
Y. Kimura, with J. Herring, successfully explained theoretical background of vortex structures arising in rotating fluid. S. Kawashima proved the well-posedness of radiating gases.
T. Ikeda considered models for combustion synthesis. With numerical experiments he demonstrated that the solutions of the model can reproduce the results of the laboratory experiments.
H. Ikeda and H. Okamoto considered a special solution of the Navier-Stokes equations called Oseen flows. Some interior layers was rigorously proved. H. Ikeda also proved that a Hopf bifurcation occurs in the traveling wave solutions of a certain bi-stable system of reaction diffusion.
H. Fujita proved the existence of the solutions of the Navier-Stokes equations when they are subjected to a leak boundary condition. He also derived a new convergence rate of the domain-decomposition method.
M. Yamada and K. Ohkitani discovered, by a numerical experiments, a time-periodic solution, which simulate the turbulent motions of real flows. Less
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