Classification of the behavior of solutions to degenerate quasi-linear wave equations

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K17631
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: The University of Shiga Prefecture (2017-2018); Tokyo University of Science (2016)
• Principal Investigator: Sugiyama Yuusuke 滋賀県立大学, 工学部, 准教授 (30712161)
• Research Collaborator: Lu Yun-guang 杭州師範大学
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: 準線形波動方程式 / 方程式の退化 / 解の爆発 / 衝撃波 / 希薄波 / 双曲型保存則系 / elastic system / 弾性体

Outline of Final Research Achievements

We classify the behavior of solutions to elastic system which describes the density wave in elastic body. The behavior of this equation can be classified as the global existence, shock wave, the degeneracy of equation in finite time. The occurrence of the shock wave has been studied by many papers. However there are few works on the degeneracy. We give sufficient conditions for physically important and related equations including p-system and also obtain a threshold of integral of initial velocity separating the occurrence and nonoccurence of the degeneracy of the equation. Furthermore, in a joint work with Professor Yun-guang Lu in Hangzhou normal university, we construct a global week solution to degenerate quasi-linear wave equations.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

双曲型保存則系は、非常に単純な偏微分方程式であるが故にその研究の歴史は古く、それにもかかわらず空間１次元の場合でさえも、現在に至るまで多くの未解決問題を残している。また、気体、弾性体、行きかう車両の渋滞現象などさまざまな物の流れを記述することができ、数学的や物理的だけでなく工学的、社会的にも重要な研究対象である。この研究では、双曲型保存則系における新しい解の特異性である「方程式の退化」の解析を行った。退化が起こるための条件に波の速度の積分量という保存量が現れることを発見した。