Analysis and construction of interface equation without self-intersections

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K17595
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Okayama University (2016-2018); Meiji University (2015)
• Principal Investigator: MONOBE HARUNORI 岡山大学, 異分野基礎科学研究所, 特任准教授 (20635809)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 界面方程式 / 自由境界問題 / 進行波解

Outline of Final Research Achievements

In this research, I tried to construct an interface equation without self-intersections, but we did not establish it. Thus we changed the our purpose and analyzed the traveling waves composed of Jordan curve,
which is a solution of curvature flow equation with driving force. As a result, we showed that there exists a unique traveling solution for the equation, which is unstable, and the shape of it is strictly convex. Moreover, we analyzed the an interface equation with exponential curvature and a free boundary problem related to population dynamics.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

本研究で取り扱った、外力を持つ曲率流方程式は、反応拡散方程式系の特異極限や細胞運動や油滴運動などの数理モデルと深く関係を持っている。このため、本研究で得たJordan曲線によって構成される進行波解の存在は、それらの現象の運動を解析する上で重要な役割を果たすと考えられる。