1998 Fiscal Year Final Research Report Summary
Mathematical Open Problems of the Navier-Stokes Equations
| Project/Area Number |
09304023
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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) |
NAKAKI Tatsuyuki Hiroshima University, Faculty of Education, Associate Professor, 教育学部, 助教授 (50172284)
SHOJI Mayumi Nihon University, College of Science and Technology, lecturer, 理工学部, 講師 (10216161)
IMAI Hitoshi The University of Tokushima, Faculty of Engineering, Professor, 工学部, 教授 (80203298)
IKEDA Hideo Toyama University, Faculty of Science, Associate Professor, 理学部, 助教授 (60115128)
OHKITANI Koji Kyoto University, Research Institute for Mathematical Sciences, Associate Profes, 数理解析研究所, 助教授 (70211787)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Fast summation method for the vortex method / Kolmogorov flow / the Burgers equation / point vortex / internal iayer / bifurcation of surface wave / thermal convection / singularity of solutions |
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| Research Abstract |
Progress is made in the study of the Navier-Stokes equation, the Burgers equations, and the reaction-diffusion equations. Okamoto and Shoji performed numerical experiments on the bifurcation of surface waves. New bifurcation diagrams are found and will be published in a form of textbook by World Scientific Inc. Okamoto and Sakajo compute numerically two-dimensional and three-dimensional vortex sheet motion. T.Ikeda and H.Ikeda consider a certain system of reaction-diffusion equations for three competing species. They clarify the structure of steady-states and traveling pulses. Their stability is also determined. K.Ohkitani and M.Yamada consider what is called the shell model of the turbulence. By numerical methods, they compute the Lyapunov numbers of the system and they study the scaling properties of the numbers. They derive an asymptotic formula as the viscosity tends to zero. T.Nakaki consider the motion of vortex patches as well as point vortices. He finds that the motion of the patches are quite similar to that of point vortices if the size of the patches are small enough and that the motion of vortex patches are substantially different if the sizes are large. Y.Kimura considers the motion of point vortices on two-dimensional hyperbolic surfaces. Its Hamiltonian formalism are derived and the algebraic properties of the invariants are studied. T.Nishida numerically computes the Boussinesq equations, which are the master equations for the thermal convection. In particular he obtains numerically the bifurcation from the trivial solution to the stationary convective flow. He applies the numerical verification technique and derives new criteria.
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