Analysis for partial differential equations systems in non-homogeneous regions.

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03572
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Japan Women's University
• Principal Investigator: AIKI Toyohiko 日本女子大学, 理学部, 教授 (90231745)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 弾性体 / 非線形偏微分方程式 / 非線形歪み / 非線形応力関数

Outline of Final Research Achievements

At the beginning in this research project, our aim was to analyze dynamics for bulk materials, for example, concrete buildings. However, we considered that porous elastic materials like sponges are more suitable for examples of non-homogenous materials, since the character can be changed at each point by shrinking and stretching motions. Moreover, there were few mathematical results dealing with large deformations, which appears in a mathematical model for elasticity of porous materials. Hence, we have aimed to analysis the model in which elastic materials is regarded as the closed curve in the plane. In this model we faced a mathematical difficulty caused from nonlinear strains. In this research we could overcome the difficulty by introducing a new stress function having a singularity and obtained completely new results on elastic equations, for instance we give a lower bound for the strain from below.

Free Research Field

数理解析学

Academic Significance and Societal Importance of the Research Achievements

弾性的な性質をもつ多孔性物質に対する浸透現象を記述する微分方程式モデルを提案することができた。本研究で示した数理モデルは極めて単純化した現象しか扱えないが，物質の力学的変化と水分量の変化を同時に考慮した初の数理モデルである。特に，スポンジのような変位が大きな値を取り得る物質の変化を，特異性が仮定された応力関数によって大域的な解の存在を示したことができた。これは，これまでの弾性体方程式では得られなかった結果であり，今後，この方程式を元にすることで弾性体の運動をより現実的なモデルで考慮記できるようになるものと考えている。