Project/Area Number  05452009 
Research Category 
GrantinAid for Scientific Research (B).

Research Field 
解析学

Research Institution  Hokkaido University 
Principal Investigator 
GIGA Yoshikazu Hokkaido University, Graduate School of Sciences, Professor, 大学院理学研究科, 教授 (70144110)

CoInvestigator(Kenkyūbuntansha) 
OZAWA Tohru Hokkaido University, Graduate School of Sciences, Professor, 大学院・理学研究科, 教授 (70204196)

Project Fiscal Year 
1993 – 1995

Project Status 
Completed(Fiscal Year 1995)

Budget Amount *help 
¥6,300,000 (Direct Cost : ¥6,300,000)
Fiscal Year 1995 : ¥1,800,000 (Direct Cost : ¥1,800,000)
Fiscal Year 1994 : ¥1,800,000 (Direct Cost : ¥1,800,000)
Fiscal Year 1993 : ¥2,700,000 (Direct Cost : ¥2,700,000)

Keywords  Motion of phase boundaries / Crystalline energy / Selfsimilar solution / Anisotropy / Uniqueness / Dispersive phenomena / Nonlinear Schrodinger equation / Asymptotic behavior / 相境界の運動 / クリスタライン・エネルギー / 自己相似解 / 異方性 / 一意性 / 分散現象 / 非線型シュレディンガー方程式 / 漸近挙動 / 非線形シュレディンガー方程式 / 非線形放物型方程式 / 曲面の発展方程式 / 非線形散乱理論 / ソボレフの不等式 / ザハロフ方程式 / 半線形楕円型方程式 / アブリオリ評価 / 曲線の発展方程式 / 粘性解 / 波動方程式 / シュレディンガー方程式 / 散乱問題 / 漸近展開 
Research Abstract 
Motion of crystal surface in crystal growth is a typical example of phaseboundaries (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 FukuiGiga 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 selfsimilar 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 selfsimilar 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 selfsimilar 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.
