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非圧縮流体の力学的諸問題の数理的研究

Research Project

Project/Area Number 18H01137
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 12040:Applied mathematics and statistics-related
Research InstitutionGakushuin University

Principal Investigator

岡本 久  学習院大学, 理学部, 教授 (40143359)

Project Period (FY) 2018-04-01 – 2022-03-31
Project Status Granted (Fiscal Year 2021)
Budget Amount *help
¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000)
Fiscal Year 2021: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2020: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2019: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2018: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Keywords流体力学 / 非線形偏微分方程式 / 水面波 / Prandtl-Batchelor theory / 精度保証計算 / 水面波の解析 / ナヴィエ・ストークス方程式 / Navier-Stokes方程式 / 数値計算法 / 数理流体力学 / 数値解析学 / Navier-Stokes 方程式
Outline of Annual Research Achievements

ウィーン大学のヘルマン教授と共同で、Fornberg-Whitham方程式の数値解析を行った。同方程式は解の中に不連続性が現れることが予想されているが、その決定的とも言える数値例を計算できた。また、同方程式の弱解の存在はここで初めて証明できた。この結果はさらに複雑な偏微分方程式にも使えそうで、応用を考えているところである。
京都大学准教授の宮地智行氏と共同で、Proudman-Johnson 方程式の解が高レイノルズ数の場合に単峰となることを証明した。これは2次元ナヴィエ-ストークス方程式の解が高レイノルズ数の場合に大規模な渦が発生することの、ある意味で傍証とも解釈できる結果である。証明には多倍長計算に対するの精度保証計算法を用いて、shooting methodによる計算を使った。
韓国のチュンアン大学教授キム・ソンチョル氏と共同で、コルモゴロフ流の高レイノルズ数での解を精査し、解が円形渦を持つ理由を研究した。それをPrandtl-Batchelor理論を援用することによって、数値解を理論的に再現することができた。これは現在、一度目の査読を経て、論文の改訂版を提出し終わったところである。
東海林まゆみ教授とともに渦あり流れを数値計算した。2層の異なる渦の層の上を伝わる波の形を数値計算した。流れのパターンは多種多様であり、興味深い現象が多く見つかっている。論文は一度目の査読を経て、改訂版を提出したところである。

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

論文は投稿済みのものが3編ある。

Strategy for Future Research Activity

当初予定に変更はない。ただ、Covid-19のせいで共同研究はやりづらくなっている。

Report

(2 results)
  • 2019 Annual Research Report
  • 2018 Annual Research Report

Research Products

(18 results)

All 2019 2018 Other

All Int'l Joint Research (1 results) Journal Article (4 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 4 results,  Open Access: 1 results) Presentation (12 results) (of which Int'l Joint Research: 5 results,  Invited: 11 results) Remarks (1 results)

  • [Int'l Joint Research] Chung-Ang University(韓国)

    • Related Report
      2019 Annual Research Report
  • [Journal Article] Existence proof of unimodal solutions of the Proudman?Johnson equation via interval analysis2019

    • Author(s)
      Miyaji T, Okamoto H
    • Journal Title

      Japan J. Indust. Appl. Math.

      Volume: 36 Issue: 1 Pages: 287-298

    • DOI

      10.1007/s13160-018-00339-x

    • NAID

      210000173644

    • ISSN
      0916-7005, 1868-937X
    • Related Report
      2019 Annual Research Report 2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Weal periodic solutions and numerical case studies of the Fornberg-Whitham equation2019

    • Author(s)
      G. Hoermann & H. Okamoto
    • Journal Title

      Discrete & Continuous Dynamical Systems - A

      Volume: 39 Pages: 4455-4469

    • DOI

      10.3934/dcds.2019182

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Models and Special Solutions of the Navier-Stokes Equations2018

    • Author(s)
      Hisashi Okamoto
    • Journal Title

      Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

      Volume: 2 Pages: 729-780

    • DOI

      10.1007/978-3-319-13344-7_14

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Unimodal solutions of the generalized Constantin?Lax?Majda equation with viscosity2018

    • Author(s)
      Kim S-C, Miyaji T, Okamoto Hi
    • Journal Title

      Japan J. Indust. Appl. Math.

      Volume: 35 Issue: 3 Pages: 1065-1083

    • DOI

      10.1007/s13160-018-0320-1

    • NAID

      210000172939

    • ISSN
      0916-7005, 1868-937X
    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] An application of the Prantdl-Batchelor theory to the Kolmogorov flows2019

    • Author(s)
      Hisashi Okamoto
    • Organizer
      East Asia SIAM Conference
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] An application of the Prandtl-Batchelor theory to the Kolmogorov flows2019

    • Author(s)
      Hisashi Okamoto
    • Organizer
      Int. Conference on Modeling, Computations, Theoretical Analysis on Fluid Dynamics and Related Problems
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Numerical experiments for weak solutions of the Fornberg-Whitham equation2019

    • Author(s)
      Hisashi Okamoto
    • Organizer
      Kyoto univ.--Chung-Ang Univ. Conference on Nonlinear PDEs
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Numerical experiments for weak solutions of the Fornberg-Whitham equation2019

    • Author(s)
      Hisashi Okamoto
    • Organizer
      Modeling, applied analysis and computation of fluids, SUSTech, China
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Mathematical issues on high-Reynolds number flows2019

    • Author(s)
      Hisashi Okamoto
    • Organizer
      Int. Congress on Industrial and Applied Mathematics, Valencia, Spain
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Numerical experiments for weak solutions of the Fornberg-Whitham equation2019

    • Author(s)
      Hisashi Okamoto
    • Organizer
      American Mathematical Society Regional conference, University of Hawaii
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Existence proof of solutions of fluid mechanics problems via interval analysis2018

    • Author(s)
      Hisashi Okamoto
    • Organizer
      Conference
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Numerical computations of water waves with piecewise constant vorticity2018

    • Author(s)
      Hisashi Okamoto
    • Organizer
      Taiwan SIAM Annual Conference
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Two-dimensional Large Eddies : a Mathematical Challenge from Fluid Mechanics2018

    • Author(s)
      Hisashi Okamoto
    • Organizer
      IBS Workshop for Nonlinear PDE, Korea
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Numerical computation of water waves on two vortical layers2018

    • Author(s)
      Hisashi Okamoto
    • Organizer
      A3 Workshop on mathematic of Biology, Fluid Dunamics, and Material Sciences Gangneung, Korea
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Unimodal solutions of the equations of 2D incompressible fluid motion2018

    • Author(s)
      Hisashi Okamoto
    • Organizer
      CAMIS International Conference on Partial Differential Equations Theories, Numerics and Applications
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation2018

    • Author(s)
      Hisashi Okamoto
    • Organizer
      International Conference in Mathematics and Applications
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Remarks]

    • URL

      https://www-cc.gakushuin.ac.jp/~20170001/index.htm

    • Related Report
      2018 Annual Research Report

URL: 

Published: 2018-04-23   Modified: 2021-08-30  

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