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金属結晶成長の数理モデルに現れる螺旋状進行波解の漸近解析的研究

金属晶体生长数学模型中螺旋行波解的渐近解析研究

基本信息

批准号：
12740058
负责人：
中村健一
金额：
$ 1.54万
依托单位：
The University of Electro-Communications
依托单位国家：
日本
项目类别：
Grant-in-Aid for Encouragement of Young Scientists (A)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

昨年に引き続き,ファセットと呼ばれる平らな結晶表面のらせん転位による成長を記述する数理モデルである反応拡散方程式,およびその方程式に含まれる微小パラメータを0に近づけることで得られる界面方程式の解の定性的性質を,漸近解析的手法,数値シミュレーション,および無限次元力学系の一般論を用いて理論的解析を行った.前年度は結晶表面に1つのらせん転位が存在し,しかも領域が対称である場合を考察したが,今年度は結晶表面に複数の転位が存在する一般領域の場合を取り扱った.具体的には以下のような成果を得ることができた.1.科学研究費補助金で購入した数値計算用ワークステーションによる数値シミュレーションを行い,一般領域の場合には複数のらせん状の解が相互作用しつつ時間周期的に形および速度を変化させながら成長することを数値的に確認することができた.また,初期転位の大きさ(ステップの段差)に違いがある2つのらせん転位が存在する場合には,初期転位の大きいものが小さいものをあたかも取り込んでいくかのような振る舞いをみせることも確認された.これらの数値計算結果は,シミュレーション前の予想を覆すものであり,大きな成果であった.2.数値シミュレーションの結果をもとに,モデル方程式の理論的解析を行った.モデル方程式の持つ性質を抽出することで,数値的に観察された時間周期的に形や速度を変えながら成長する解の存在および(時間シフトを除いた)一意性・時間に関する単調性・リャプノフ安定性・漸近安定性に関する理論的結果を順序保存力学系の枠組みで得ることができた.この一般化により,異方性のある結晶成長のモデルである曲率流方程式のらせん解の存在および安定性にも適用が可能になった.

In the last year, the growth of the temperature field on the surface of the crystal has been reported, and the mathematical equation is described in detail. The equation contains the qualitative property of the solution of the interface equation, the analytical method, and the analytical method. The Department of finite element Mechanics in the Department of Mechanics in general uses the analysis of theory. In the previous year, there was a high temperature on the surface of the crystal, and the field was called a combination of survey data. This year, there is a general field of data on the surface of the crystal. The specific results are as follows. 1. In the calculation of the number of scientific research grants, the calculation of the number of scientific research grants is based on the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number of scientific research grants, the calculation of the number In the early stage, there is a big one in the first place, and there is a big one in the early stage, and there is a big one in the early stage. Please tell me the results of the calculation. I would like to review the results. 2. The results show that there are many problems in the analysis of the theory of equations. The equation is consistent with the extraction of the equation, and the mathematical equation is used to observe the shape speed of the cycle of the cycle, the speed of the cycle, the growth of the equation, the solution of the equation, the equation of stability, the stability of the equation, the order of the theory of the Department of Mechanics, the results of the theory of stability and stability. Because of the generalization of the equation, the equation of curvature flow is solved in terms of crystal growth and stability.