A numerical study for complex blow-up solutions of nonlinear evolution equations

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K03427
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Tokyo University of Science
• Principal Investigator: USHIJIMA TAKEO 東京理科大学, 理工学部数学科, 教授 (30339113)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 解の爆発 / リスケーリング・アルゴリズム / 数値解析 / 曲率流

Outline of Final Research Achievements

Solutions to nonlinear evolution equations do not always exist globally in time, and singularities can occur at finite times. Such a phenomenon is called blow-up. The rate at which the norm of a blow-up solution diverges to infinity is called the blow-up rate. In this research, the following were conducted with respect to the blow-up rate.
1. to propose a numerical estimation method of the blow-up rate using a rescaling algorithm, to improve the method, and to expand its range of applications; 2.Numerical experiments on partial differential equations describing the motion of a plane curve whose normal velocity is proportional to the power of its curvature (hereinafter referred to as "curvature flow"); 3. theoretical analysis of the traveling wave solution of the rescaled curvature flow; 4. evaluation of the blow-up rate of the curvature flow from above using the traveling wave solution of 3.

Free Research Field

応用解析

Academic Significance and Societal Importance of the Research Achievements

解の爆発は，非線形偏微分方程式論における代表的な研究課題の一つであり，爆発解の様相の解明は重要な意義がある．有限な量しか扱うことのできない数値計算によって，解の発散する様相を捉えようという研究は長い歴史もあり，また本質的に困難な問題である．本研究はこの分野に新たな有用な手法を与えることになった．また曲率流の爆発解は複雑な爆発レートを持つのだが，本研究ではその様相について新たな理論的知見を与えた．