1995 Fiscal Year Final Research Report Summary
Analysis on Nonlinear Partial Differential Equations
| Project/Area Number |
05452009
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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 | Hokkaido University |
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Principal Investigator |
GIGA Yoshikazu Hokkaido University, Graduate School of Sciences, Professor, 大学院理学研究科, 教授 (70144110)
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| Co-Investigator(Kenkyū-buntansha) |
OZAWA Tohru Hokkaido University, Graduate School of Sciences, Professor, 大学院・理学研究科, 教授 (70204196)
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| Project Period (FY) |
1993 – 1995
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| Keywords | Motion of phase boundaries / Crystalline energy / Self-similar solution / Anisotropy / Uniqueness / Dispersive phenomena / Nonlinear Schrodinger equation / Asymptotic behavior |
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| Research Abstract |
Motion of crystal surface in crystal growth is a typical example of phase-boundaries (interface). Such a phenomena attracts interdeciplinary interest as nonequilibriun nonlinear phenomena. Interface controlled model is an important class of evolution equations of phase boundaries. This is the case when heat and mass diffusion is negligible so that the evolution is determined by geometry of surface. Phenomena that facets appear on interface arises, for example, in the growth of Helium crystal growth in low temperature. In this situation, the governing equation has a nonlocal term and it is difficult to describe. So far the evolution law is described by restricting a class of evolving interfaces. The head investigator gave a formulation to this problem which is comparible with partial differential equations. It is based on the theory of nonlinear semigroups and nowadays it is called Fukui-Giga formulation. By this formulation curve evolution by crystalline energy can be understood as a limit of evolution by smooth anisotropic energy.
In motion of interfacial energy having anisotropy, it is important whether or not there is a self-similar shrinking solution. If interfacial energy is isotropic and there is no external force, the equation becomes the famous curve shortening equation. It is known that the only self-similar solution is a circle. However, the proof is rather complicated. Head investigator gave an elementary proof. For motion by anisotropic curvature be proved the existence of self-similar solution in an elementary way. However, uniqueness is shown only for evolution law that does not depend the orientation of curves.
The above research is a study of important example of nonlinear parabolic equations.Investigator studied large time asymptotic behaviors of solutions of nonlinear Schrodinger equation describing dispersive phenomena and discovered a nonlinear effect that is not tractable as a linear phenomena.
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