Elucidation of new dissipative structure and exploration of general stability analysis method for symmetric hyperbolic system

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K13818
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Tokyo University of Marine Science and Technology
• Principal Investigator: Mori Naofumi 東京海洋大学, 学術研究院, 准教授 (10803413)
• Project Period (FY): 2021-04-01 – 2024-03-31
• Keywords: 非線形偏微分方程式 / 対称双曲系 / 対称双曲・放物系 / 消散構造 / 安定性理論 / Timoshenko 方程式系 / 記憶型消散効果をもつ数理モデル

Outline of Final Research Achievements

The dissipative structures of partial differential equations appearing in gas dynamics, fluid mechanics, and elastodynamics are complex and diverse, and many of the proofs of stability of solutions are individual and technical in nature and lack applicability. Therefore, it is important to elucidate the natural mechanisms that give rise to the dissipative structures and to provide a proof in the general case. Through this research, the dissipative structure of mathematical models for complex fluids and a unified method for deriving decay properties and linear decay estimates for symmetric hyperbolic and symmetric hyperbolic-parabolic systems with strongly positive definite memory kernels were clarified, and the stability of solutions was demonstrated in a unified manner as well. The results contributed to the unified presentation of stability of solutions to general symmetric hyperbolic and symmetric hyperbolic-parabolic systems.

Free Research Field

非線形偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

粘性的 Timoshenko 方程式系や記憶型 Laminated beams など、 Timoshenko 方程式系に関連する新モデルの開発が現在も盛んに行われている。しかし、 その減衰評価については、従来型の消散構造に対応するように物理係数に制約条件を仮定し て導出されていることが多く、一般的な減衰特性の解明はほとんどされていなかった。また、指数的に減衰する記憶核よりも一般的な記憶項をもつ対称双曲系や対称双曲・放物系に関する統一的な研究成果は、本研究の中で初めて得られた。新型の消散構造を持つ偏微分方程式に関する本研究成果は、川島秀一氏らによる安定性理論の拡張にも貢献することが期待できる。