• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Vortex dynamics on surfaces exploring new fluid phenomena brought by geometry

Research Project

Project/Area Number 18H01136
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 InstitutionKyoto University

Principal Investigator

坂上 貴之  京都大学, 理学研究科, 教授 (10303603)

Co-Investigator(Kenkyū-buntansha) 横山 知郎  岐阜大学, 工学部, 准教授 (30613179)
米田 剛  東京大学, 大学院数理科学研究科, 准教授 (30619086)
Project Period (FY) 2018-04-01 – 2022-03-31
Project Status Granted (Fiscal Year 2021)
Budget Amount *help
¥16,640,000 (Direct Cost: ¥12,800,000、Indirect Cost: ¥3,840,000)
Fiscal Year 2021: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2020: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2019: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2018: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Keywords応用数学 / 流体力学 / 力学系 / 流体方程式 / 微分幾何学
Outline of Annual Research Achievements

研究実施計画に基づいてサブプロジェクト(P1)-(P3)について以下の研究を行った.(P1)では(P3)に関連して超流動流体の作る量子渦パターンを理解するためのトーラス上のLiouville方程式の解の構成をウィーン大学のV. Krishnamurthy博士と行った.現在投稿準備中である.次に昨年度得た有限長の長さを持つ線分渦層の一意性と安定性の解析をS. Llewellyn-Smith教授およびB.Protas教授と共同研究で行った.現在本成果は査読中である.(P2)では昨年度開発した反応拡散系のスポットダイナミクスを記述する方程式を使って定常スポット解の存在やその安定性に成果を得た.本成果はSIAM J. Appl. Dyn. Sysに掲載が決定している.C. Green博士と榊原博士との共同研究である多様体上のHarmonic measureの解析解と数値解の構成にも成功して,本成果は論文としてまとめている.横山は一般的な2次元ハミルトン流のトポロジカルな遷移距離の下限を調べる手法を構成した.これらの手法はデータ解析をする際に活用できるものであり,幅広い活用の可能性がある.米田は,体積保存する微分同相写像群の中心拡大から導かれるコリオリ力付き非圧縮オイラー流に対する断面曲率・Misiolek曲率を考え,コリオリ項の幾何学的構造が、帯状流(無限次元多様体上の測地線、コリオリ力付きEuler方程式の定常解)における断面曲率・Misiolek曲率を正値へ誘導する作用を有することを示した.論文として採録決定済みである.今年度はコロナウィルス感染症があったが,日本数学会や米国物理学会などでの研究成果の発表や国際研究集会でもオンライン発表した.

Current Status of Research Progress
Current Status of Research Progress

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

Reason

コロナ感染症の蔓延下で対面での研究打ち合わせが大きく制限される中でも昨年度末に繰り越した研究費を使ってオンライン研究環境を構築した.その結果,定期的に国内外の共同研究者との議論ができるようになったことが功を奏し,研究計画通りに研究が進捗し順調に成果が出ている.また,王氏をRAとして雇用することで数値計算を多数実施する体制の構築にも成功した.トーラス上のスポットダイナミクスの数学的理解が予想以上に進んだことは意義がある.推進した国際共同研究は,その成果が論文として多くまとまると同時に,論文投稿後も,これらの成果を踏まえた新しい問題について共同研究が継続している.横山は理論面で進展があり様々な拡張方向があることがわかってきた.実際,従来の2次元ハミルトン流の文字化を任意のコンパクト曲面上の文字化に拡張できることがわかってきた.米田は令和元年度では回転楕円面上の帯状流に対する断面曲率やMisiolek曲率の洞察を進め、令和二年度ではその発展版としての「自転を加えた場合」の洞察を進めた.これは当初に計画していた通りの進捗を意味するものである.

Strategy for Future Research Activity

最終年度に向けて,坂上は各プロジェクトの研究として平面の有限渦層解の構成,トーラスや二重周期平面上の量子渦の解析解の構成,トーラス上の反応拡散系スポットダイナミクスのさらなる研究などの個別研究を発展させる.それとともに,これらのサブプロジェクトの成果を活かしてサブプロジェクトをまたぐような研究成果の創出を目指す.特に量子渦パターン形成と多様体の構造の問題,多様体上での渦分布を考える上でのヒントとなる調和測度の構成や点渦統計を実現するギブス測度の構成などに挑戦する予定である.(P2)で開発した非線形シュレーディンガー方程式の数値コードはスポット解を得るまでには至っていないが,数値計算を継続していく.横山は上記で述べた,任意のコンパクト曲面上の2次元ハミルトン流の文字化を行う.さらに,非有界な場合に既存の理論を拡張していく.米田は体積保存する微分同相写像群の中心拡大における共役点の存在性に関する洞察を進める。コリオリ力(自転)に限らず、歳差運動に対する安定性・不安定性の洞察も進める予定である.本研究では複素解析に基づく流体運動の理論的・数値的に果たす役割が明確になってきたので,関連する研究者による研究集会なども計画して,本成果をまとめると同時にその先にある研究展開を俯瞰していく.

Report

(3 results)
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report

Research Products

(51 results)

All 2021 2020 2019 2018 Other

All Int'l Joint Research (13 results) Journal Article (9 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 9 results,  Open Access: 3 results) Presentation (29 results) (of which Int'l Joint Research: 10 results,  Invited: 9 results)

  • [Int'l Joint Research] McMaster University(カナダ)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] Moscow University(ロシア連邦)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] University of California, San Diego/University of Wichita(米国)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] University of Vienna(オーストリア)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] McMaster University(カナダ)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Imperial College London/The University of Bath/Reading University(英国)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] University of California, San Diego/University of Wichita(米国)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Mcquire University(オーストラリア)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] University of Vienna(オーストリア)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] McMaster University(カナダ)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Macquarie University(オーストラリア)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] The University of Michigan(米国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Technical Univ. Dresden(ドイツ)

    • Related Report
      2018 Annual Research Report
  • [Journal Article] Positivity for the curvature of the diffeomorphism group corresponding to the incompressible Euler equation with Coriolis force2021

    • Author(s)
      T. Tauchi and T. Yoneda
    • Journal Title

      Progress of Theoretical and Experimental Physics

      Volume: -

    • DOI

      10.1093/ptep/ptab043

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Rotating equilibria of vortex sheets2020

    • Author(s)
      Protas Bartosz、Sakajo Takashi
    • Journal Title

      Physica D: Nonlinear Phenomena

      Volume: 403 Pages: 132286-132286

    • DOI

      10.1016/j.physd.2019.132286

    • NAID

      120006878491

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Complete transition diagrams of generic Hamiltonian flows with a few heteroclinic orbits2020

    • Author(s)
      Yokoyama Tetsuo、Yokoyama Tomoo
    • Journal Title

      Discrete Mathematics, Algorithms and Applications

      Volume: 13 Pages: 2150023-2150023

    • DOI

      10.1142/s1793830921500233

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Vortex crystals on the surface of a torus2019

    • Author(s)
      Sakajo Takashi
    • Journal Title

      Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

      Volume: 377 Pages: 2158-2158

    • DOI

      10.1098/rsta.2018.0344

    • NAID

      120006734880

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 乱流の数理をめぐる流体方程式の特異解2019

    • Author(s)
      坂上 貴之
    • Journal Title

      数学

      Volume: 71

    • NAID

      130008029694

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Existence and non-existence of length averages for foliations2019

    • Author(s)
      Nakano Yushi、Yokoyama Tomoo
    • Journal Title

      Communications in Mathematical Physics

      Volume: 372 Pages: 367-383

    • DOI

      10.1007/s00220-019-03490-9

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus2019

    • Author(s)
      Sakajo Takashi
    • Journal Title

      Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

      Volume: 475 Pages: 1-16

    • DOI

      10.1098/rspa.2018.0666

    • NAID

      120006630979

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Harnessing the Kelvin?Helmholtz instability: feedback stabilization of an inviscid vortex sheet2018

    • Author(s)
      Protas Bartosz、Sakajo Takashi
    • Journal Title

      Journal of Fluid Mechanics

      Volume: 852 Pages: 146-177

    • DOI

      10.1017/jfm.2018.523

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Universality of the Anomalous Enstrophy Dissipation at the Collapse of Three Point Vortices on Euler--Poincar? Models2018

    • Author(s)
      Gotoda Takeshi、Sakajo Takashi
    • Journal Title

      SIAM Journal on Applied Mathematics

      Volume: 78 Pages: 2105-2128

    • DOI

      10.1137/17m1127855

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Presentation] On "quantized" vortex crystals on closed surfaces2021

    • Author(s)
      坂上貴之
    • Organizer
      量子渦と非線形波動(OCAMIワークショップ)
    • Related Report
      2020 Annual Research Report
  • [Presentation] ある渦層の相対定常解の族について2021

    • Author(s)
      坂上貴之
    • Organizer
      日本数学会年会2021
    • Related Report
      2020 Annual Research Report
  • [Presentation] Hamiltonian flows on punctured surfaces2021

    • Author(s)
      Tomoo Yokoyama
    • Organizer
      2020 年度冬の力学系研究集会
    • Related Report
      2020 Annual Research Report
  • [Presentation] Topological characterizations of 2D gradient flows and 2D Morse-Smale flows and their generic inter mediate flows2021

    • Author(s)
      Tomoo Yokoyama
    • Organizer
      Seminar cathedral on Monday, Moscow State University, Moscow, Russia
    • Related Report
      2020 Annual Research Report
  • [Presentation] Space-time topology behind formation of micro-macro magneto-vortical structure manifested by Nambu mechanics2020

    • Author(s)
      Takashi Sakajo
    • Organizer
      OCAMI, Osaka-City Univ. Osaka
    • Related Report
      2020 Annual Research Report
  • [Presentation] トーラス面上の定常渦2020

    • Author(s)
      坂上貴之
    • Organizer
      日本数学会2020年度秋季総合分科会
    • Related Report
      2020 Annual Research Report
  • [Presentation] Vortex Dynamics on Sphere - a survey2020

    • Author(s)
      Takashi Sakajo
    • Organizer
      Workshop on "Mathematical Aspects of Geophysical Flows, Erwin Schrodinger Insitute
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological characterization of gradient flows and Morse-Smale flows on compact surfaces and their transition graphs2020

    • Author(s)
      Tomoo Yokoyama
    • Organizer
      2019 年度 冬の力学系研究集会
    • Related Report
      2019 Annual Research Report
  • [Presentation] Mathematical models of cascading phenomena in turbulence2019

    • Author(s)
      Takashi Sakajo
    • Organizer
      Workshop on Scientific Computing Across Scales: Extreme Events and Criticality in Fluid Mechanics, The Fields Institute
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Linear feedback stabilization of point vortex equilibria near a Kasper Wing2019

    • Author(s)
      Takashi Sakajo
    • Organizer
      International Congress of Industrial and Applied Mathematics (ICIAM2019)
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Vortex dynamics on the surface of a torus2019

    • Author(s)
      Takashi Sakajo
    • Organizer
      Complex analysis in mathematical physics and applications, Isaac Newton Institute
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Vortex dynamics on the surface of a torus2019

    • Author(s)
      Takashi Sakajo
    • Organizer
      Analysis Seminar, the University of Reading
    • Related Report
      2019 Annual Research Report
  • [Presentation] 非粘性渦層におけるKelvin-Helmholtz不安定のフィードバック制御2019

    • Author(s)
      坂上貴之
    • Organizer
      第65回理論応用力学講演会・第22回土木学会応用力学シンポジウム
    • Related Report
      2019 Annual Research Report
  • [Presentation] Vortex Dynamics on the surface of a torus2019

    • Author(s)
      Takashi Sakajo
    • Organizer
      数理研研究集会 ``流体と気体の数学解析''
    • Related Report
      2019 Annual Research Report
  • [Presentation] Zeroth-lawからみる瞬間的な渦伸長と或る定常流について2019

    • Author(s)
      米田剛
    • Organizer
      日本流体力学会
    • Related Report
      2019 Annual Research Report
  • [Presentation] 全測地的部分多様体による共役点の特徴づけ、及び非粘性流体への応用2019

    • Author(s)
      米田剛
    • Organizer
      日本流体力学会,
    • Related Report
      2019 Annual Research Report
  • [Presentation] Instantaneous vortex stretching and energy cascade on the incompressible 3D Euler equations2019

    • Author(s)
      T. Yoneda
    • Organizer
      National Chiao Tung University
    • Related Report
      2019 Annual Research Report
  • [Presentation] Recent topics on well-posedness and stability of incompressible fluid and related topics2019

    • Author(s)
      T. Yoneda
    • Organizer
      Summer Graduate School, Mathematical Sciences Research Institute (MSRI)
    • Related Report
      2019 Annual Research Report
  • [Presentation] 瞬間的な渦伸長を生成する3次元Euler流・それに関連するzeroth lawについて2019

    • Author(s)
      T. Yoneda
    • Organizer
      京都大学応用数学セミナー
    • Related Report
      2019 Annual Research Report
  • [Presentation] Characterization of finiteness and countability of non-wandering sets of surface flows2019

    • Author(s)
      Tomoo Yokoyama
    • Organizer
      力学系 -新たな理論と応用に向けて,数理解析研究所
    • Related Report
      2019 Annual Research Report
  • [Presentation] Vortex dynamics on the surface of a torus2019

    • Author(s)
      Takashi Sakajo
    • Organizer
      Applied Interdisciplinary Mathematics (AIM) Seminar
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] 非粘性渦層における Kelvin-Helmholtz 不安定のフィードバック制御2019

    • Author(s)
      坂上貴之
    • Organizer
      日本数学会年会2019
    • Related Report
      2018 Annual Research Report
  • [Presentation] Control of vortex dominated flows through vortex dynamics2018

    • Author(s)
      Takashi Sakajo
    • Organizer
      Taiwan-Japan Joint Workshop on Numerical Analysis and Scientific Computation
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Control of vortex dominated flows through vortex dynamics2018

    • Author(s)
      Takashi Sakajo
    • Organizer
      KIAS workshop, Mathematics of Fluid Motion II: Theory and Computation
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 複翼まわりの定常点渦の安定化を実現する線型フィードバック制御2018

    • Author(s)
      坂上貴之
    • Organizer
      ODE-JP 2018
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Instantaneous vortex stretching and energy cascade on the incompressible 3D Euler equations2018

    • Author(s)
      Tsuyoshi Yoneda
    • Organizer
      KIAS workshop, Mathematics of Fluid Motion II: Theory and Computation
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Instantaneous vortex-stretching and anomalous dissipation on the 3D Euler equations2018

    • Author(s)
      Tsuyoshi Yoneda
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Instantaneous vortex-stretching and anomalous dissipation on the 3D Euler equations2018

    • Author(s)
      Tsuyoshi Yoneda
    • Organizer
      2018 International Conference on Mathematical Fluid Dynamics School of Mathematics and Information Science
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Decompositions of surfaces flows and their applications2018

    • Author(s)
      Tomoo Yokoyama
    • Organizer
      International Conference “Integrable Systems and Nonlinear Dynamics”
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research

URL: 

Published: 2018-04-23   Modified: 2021-12-27  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi