Project/Area Number  06640264 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Hokkaido Tokai University 
Principal Investigator 
CHEN YunGang Hokkaido Tokai University, Research Institute for Higher Education Programs, Associated Professor, 教育開発研究センター, 助教授 (50217262)

CoInvestigator(Kenkyūbuntansha) 
GIGA Yoshikazu Hokkaido University, Faculty of Science, Professor, 理学部, 教授 (70144110)

Project Fiscal Year 
1994 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥2,100,000 (Direct Cost : ¥2,100,000)
Fiscal Year 1996 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1995 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1994 : ¥700,000 (Direct Cost : ¥700,000)

Keywords  motion of surface / mean curvature / level surface method / continuous solution / singlular point / diffusion equation / extinction / existence / 曲面運動 / 平均曲率 / 等高面法 / 連続解 / 特異点 / 拡散方程式 / 解の消滅 / 解の存在 / 等高価法 / 界面現象 / 粘性解 / 消滅時刻 / 安定性 / 差分近似スキーム 
Research Abstract 
(1) A class of nonlinear partial differential equations concerning the interface (closed surface) moving problem are studied. In particular, the mean curvature flow problem is studied with a general consideration. The level surface approach with the viscosity solution theory are applied to the analysis for the nonlinear equations (generalized mean curvature flow equations) with singularities where the gradient of the solution vanishes. (2) The level surface method is valid for the moving surface model even when some singularity occurs, and the motion can be tracked through the time when singularity appears. This makes it possible to study the behavior near the singular points of a closed surface at the singular time, which denoting the cases when a surface extincts, or pinches, or divided into two or more parts. (3) The profile of the solution, before and after the time when a singularity occurs, was studied analytically and numerically. Some asymptotic property of the solution was found for a kind of related equations, nonlinear diffusion equations with very fast diffusion term. (4) Numerical methods, especially the finite difference method for the concerned nonlinear partial differential equations, are studied and a class of stable difference schemes are constructed.
