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To global analysis for solutions of nonlinear partial differential equations tems

Research Project

Project/Area Number 20K03699
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionKyoto University

Principal Investigator

Nishida Takaaki  京都大学, 情報学研究科, 名誉教授 (70026110)

Project Period (FY) 2020-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2022: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords非線形編微分方程式系 / 熱対流問題 / Rayleigh-Benard 熱対流 / route to chaos / 擬圧縮性近似法 / 粘性流体の自由表面問題 / 非線形偏微分方程式 / 大域的解析 / 流体運動方程式系 / 非線形振動 / 計算機援用解析 / 非線形偏微分方程式系
Outline of Research at the Start

力学系、双曲型保存則、流体運動方程式系などを主な対象として、線形近似や弱非線形理論などによる扱いを越えているために未解決である重要な問題を明確に定式化する。 その非線形性と物理的に自然なパラメーター(解の大きさあるいは無次元化による Reynolds 数、Rayleigh 数など)に着目して、解空間における平衡解から離れた定常解、周期解のみならず、一般の解の時間的遷移過程を調べるために解空間の大域的な特異性、構造とその変化を解明する。 非線形系の解をパラメーターの値に応じて構成的に求める方法として、解析的に求められる特別の場合を除いて、計算機援用解析の方法を併用する事が必須である。

Outline of Final Research Achievements

Analysis on global behaviors of solutions according to change of physical parameters in the heat convection problems. When the Rayleigh number (the difference of temperature between the top surface and the bottom ) increases far across the critical number in the Rayleigh-Benard heat convection, the present analysis has no theory for the solutions. We made numerical simulations of the roll-type solutions for big Rayleigh numbers and found a route to chaos along bigger Rayleigh numbers.
There is a Chorin's method which makes pseudo-compressible approximations for the solution of incompressible viscous fluids. We made a justification of the method for the stationary solutions, stationary bifurcations and Hopf-bifurcation of Navier-Stokes equations and Rayleigh-Benard equations.

Academic Significance and Societal Importance of the Research Achievements

熱対流問題で上下の温度差が大きくなるに従って、熱伝導解から定常対流解に分岐し、パターンを形成した後、周期解分岐を起こし、周期倍分岐、周期4倍分岐を経て徐々に chaos 解に遷移する道を見つけた。長期的な予想の困難さの例になる。
非圧縮性粘性流体の解析をその擬圧縮性近似の方法によって行える事の正当性を示した。
非圧縮性条件の困難を避ける一つの方法である。

Report

(4 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (11 results)

All 2022 2021 2020

All Journal Article (5 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 5 results,  Open Access: 2 results) Presentation (6 results) (of which Int'l Joint Research: 2 results,  Invited: 3 results)

  • [Journal Article] Decay of solutions to the initial boundary value problems of the Stokes system in the half space2022

    • Author(s)
      Takaaki Nishida and Yoshiaki Teramoto
    • Journal Title

      Annali dell 'Universita' di Ferrara

      Volume: on-line Issue: 1 Pages: 245-262

    • DOI

      10.1007/s11565-022-00421-w

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Route to Chaos in Rayleigh-Benard Heat Convection2022

    • Author(s)
      C.-H. Hsia and T. Nishida
    • Journal Title

      Journal Mathematical Fluid Mechanics

      Volume: 24:38 Pages: 1-18

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Singular Limit in Hopf Bifurcation for Doubly Diffusive Convection Equations I: Linearized Analysis at Criticality2021

    • Author(s)
      Chun-Hsiung Hsia, Yoshiyuki Kagei, Takaaki Nishida, Yuka Teramoto
    • Journal Title

      J. Math. Fluid Mech.

      Volume: - Issue: 3 Pages: 1-27

    • DOI

      10.1007/s00021-021-00582-2

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Singular Limit in Hopf Bifurcation for Doubly Diffusive Convection Equations II: Bifurcation and Stability2021

    • Author(s)
      Chun-Hsiung Hsia, Yoshiyuki Kagei, Takaaki Nishida, Yuka Teramoto
    • Journal Title

      J. Math. Fluid Mech.

      Volume: 23 Issue: 3 Pages: 1-35

    • DOI

      10.1007/s00021-021-00583-1

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Decay of solutions of the Stokes system arising in free surface flow2020

    • Author(s)
      T. Beale, T. Nishida and Y. Teramoto
    • Journal Title

      RIMS Kokyuuroku Bessatsu

      Volume: B82 Pages: 137-157

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] A route to chaos in Rayleigh-Benard heat convection2022

    • Author(s)
      Takaaki Nishida
    • Organizer
      Japan-France joint workshop on interacting particles systems and fluid mechnics
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A route to chaos in Rayleigh-Benard heat convection2022

    • Author(s)
      Takaaki Nishida
    • Organizer
      Workshop on bifurcations governed by partial differential equations
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] 熱供給が一様でない時の熱対流問題2022

    • Author(s)
      西田 孝明
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2022 Annual Research Report
  • [Presentation] An example of thermal convection in the horizontal layer with non-uniform heat supply2022

    • Author(s)
      Takaaki Nishida
    • Organizer
      Taiwan-Japan Workshop on Numerical Analysis and Inverse Problems
    • Related Report
      2021 Research-status Report
  • [Presentation] Routes to chaos in Rayleigh-Benard heat convection2021

    • Author(s)
      西田 孝明
    • Organizer
      日本数学会 2021年度年会
    • Related Report
      2020 Research-status Report
  • [Presentation] Routes to chaos in Rayleigh-Benard heat convection2021

    • Author(s)
      Nishida Takaaki
    • Organizer
      Japan-Taiwan joint workshop on Numerical Analysis and Inverse problems
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2020-04-28   Modified: 2024-01-30  

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