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

Research Project

Project/Area Number 21K13818
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionTokyo 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
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2023: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2022: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2021: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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 / 記憶型消散効果 / 強正定値記憶核 / 減衰評価 / 強正定値性 / 対称双曲型保存則系 / 減衰特性 / 漸近安定性
Outline of Research at the Start

気体力学、流体力学、弾性体力学等に現れる偏微分方程式がもつ消散構造は複雑・多様で、解の安定性に関する証明の多くが個別・技巧的で応用性に欠く。そのため、消散構造が生じる自然のメカニズムの解明と、一般の場合に統一的な証明を与えることが重要である。そこで、本研究では典型例より広い範囲で消散構造の特徴を明らかにするともに、解の安定性を示す個別・技巧的な方法を一般化し、安定性理論の拡張を行う。

本研究を通じて、偏微分方程式のもつ消散構造の特徴が具体的に明らかになり、解の安定性を一般的な対称双曲系や対称双曲・放物系の場合でも統一的に示すことが期待できる。

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.

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。

Report

(4 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • Research Products

    (7 results)

All 2024 2022 2021

All Journal Article (4 results) (of which Peer Reviewed: 4 results) Presentation (3 results) (of which Int'l Joint Research: 1 results,  Invited: 1 results)

  • [Journal Article] Decay property for symmetric hyperbolic system with memory-type relaxation2024

    • Author(s)
      Mori Naofumi、Okada Mari、Kawashima Shuichi
    • Journal Title

      Analysis and Applications

      Volume: 未定(Online ready) Issue: 04 Pages: 1-26

    • DOI

      10.1142/s0219530523500367

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Decay property for a novel partially dissipative viscoelastic beam system on the real line2022

    • Author(s)
      N. Mori, M. A. Jorge Silva
    • Journal Title

      Journal of Hyperbolic Differential Equations

      Volume: 19 Issue: 03 Pages: 391-406

    • DOI

      10.1142/s0219891622500114

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Thermodynamically consistent modeling for complex fluids and mathematical analysis2021

    • Author(s)
      Suzuki Yukihito、Ohnawa Masashi、Mori Naofumi、Kawashima Shuichi
    • Journal Title

      Mathematical Models and Methods in Applied Sciences

      Volume: 31 Issue: 10 Pages: 1919-1949

    • DOI

      10.1142/s0218202521500421

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Decay property for symmetric hyperbolic systems with memory-type diffusion2021

    • Author(s)
      Mari Okada, Naofumi Mori and Shuichi Kawashima
    • Journal Title

      Journal of Differential Equations

      Volume: 276 Pages: 287-317

    • DOI

      10.1016/j.jde.2020.12.021

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Presentation] Decay property for symmetric hyperbolic system with memory-type diffusion2022

    • Author(s)
      森 直文、岡田 真理、川島 秀一
    • Organizer
      2022 日本数学会 秋季総合分科会
    • Related Report
      2022 Research-status Report
  • [Presentation] Decay property for symmetric hyperbolic system with memory-type relaxation2022

    • Author(s)
      森 直文、岡田 真理、川島 秀一
    • Organizer
      2022 日本数学会 秋季総合分科会
    • Related Report
      2022 Research-status Report
  • [Presentation] Difference in decay properties for symmetric hyperbolic system with memory-type diffusion and relaxation2022

    • Author(s)
      N. Mori, M. Okada, S. Kawashima
    • Organizer
      The 8th Japan-China Workshop on Mathematical Topics from Fluid Mechanics
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2021-04-28   Modified: 2025-01-30  

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