界面方程式の自由境界問題における特異性をもつ進行波面の研究

界面方程自由边界问题中奇点行波前的研究

• 批准号: 22KJ2849
• 负责人: 森 龍之介
• 金额: $ 2万
• 依托单位: Meiji University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2023
• 资助国家: 日本
• 起止时间: 2023-03-08 至 2025-03-31
• 项目状态: 未结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-22KJ2849/
• 关键词: 外力項付き曲線短縮流; 自由境界問題; 特異点

今年度は，外力項付き曲線短縮流の自由境界問題について，時間発展の途中で解曲線の端点以外で生じる特異性の解析を行った．(1)解曲線の端点以外で生じる特異点の性質：まず，既存の結果を用いて弱解の時間大域的な存在を示し，弱解が局所時間内では古典解になることを示した．次に，最初の特異点発生時刻（以下では特異時刻と呼ぶことにする）において，特異点の個数が有限個であることと，特異時刻を過ぎると瞬時に解の正則性が回復し古典解になることを示した．さらに，特異時刻の集合が集積点をもたないことを示した．(2)角のある領域における自由境界問題の解の構成：開き角が180度よりも大きい扇型領域において，領域の境界と直交する半直線を初期値とする外力項付き曲線短縮流の自由境界問題の解を構成した．構成方法は，まず元の領域を近似するなめらかな近似領域において自由境界問題の解を構成する．次に近似領域を元の領域に近づける極限をとることで，元の領域における解を捉える．この方法で構成された解曲線は，その端点が領域の角以外にあるときは古典的な自由境界問題の解として振舞い，端点が領域の角にあるときには領域の境界とのなす角度が一定の範囲にある間は固定端の解として振舞うことが分かった．(3)角のある障害物に対する解の振舞：開き角が270度よりも大きい扇型領域において，領域の境界と交わらない直線を初期値とする外力項付き曲線短縮流の自由境界問題の解を上述の(2)の方法で構成した．解曲線は有限時間で領域の角に接触するが，それ以降は(2)の解と同様の振舞をすることがわかった．

This year, the external force project is responsible for the problem of the free boundary of the curve and the short flow, and the analysis of the characteristics outside the end of the curve during the time exhibition. (1) to analyze the characteristics outside the end of the curve. (1) to analyze the characteristics outside the end of the curve. During the weak solution of the office, the classical solution is displayed. Second, when the first special point is born (the following special time is called in the following special time), the number of special points is limited, and there are only a limited number of customs. in special time, there are only a limited number of listeners, and there are only a limited number of listeners. In this special time, the collection of positive points will be demonstrated. (2) the solution to the free realm problem is as follows: the solution to the free realm problem is as follows: open the 180-degree corner, open the 180-degree corner, expand the fan-shaped domain boundary, directly cross the domain boundary, at the initial stage of the semi-direct load cycle, pay for the curve and short-flow free realm problem, and form the method. The solution of the problem of the realm of freedom in the domain of the second approximation is limited, and the solution of the problem of the state of freedom in the domain of the second approximation is limited, and the solution of the problem of freedom in the domain of the second approximation is limited, and the solution of the problem of freedom is reduced to the solution of the problem of freedom. The end point, the field, the realm, the realm, the angle, the fixed end, the fixed end, the fixed end. The solution to the problem of the free boundary of the above-mentioned method (2) is a solution to the problem of the free boundary of the initial stage of the direct line. The solution to the above (2) method is a solution to the problem. The solution is limited to a limited period of time in the field of angular contact, so as to reduce (2) solve the problem of vibration and dance in the same direction.