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動く薄膜領域上のナヴィエ・ストークス方程式の解析

Research Project

Project/Area Number 16J02664
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section国内
Research Field Mathematical analysis
Research InstitutionThe University of Tokyo

Principal Investigator

三浦 達彦  東京大学, 数理科学研究科, 特別研究員(PD)

Project Period (FY) 2016-04-22 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2018: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2017: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2016: ¥700,000 (Direct Cost: ¥700,000)
Keywordsナヴィエ・ストークス方程式 / 薄膜領域 / 特異極限 / 表面流 / 滑り境界条件 / 強解 / 動く曲面
Outline of Annual Research Achievements

今年度は静止した2次元閉曲面に退化する3次元薄膜領域上のナヴィエ・ストークス(NS)方程式について、膜の厚さゼロ極限に関する特異極限問題の研究を行い閉曲面上の極限方程式を導出した。また本研究の極限方程式と先行研究で導出された極限方程式の比較を行った。
1.特異極限問題の研究:3次元空間内に与えられた2次元閉曲面(極限曲面)に対し、極限曲面上の十分小さな関数のグラフとして表される2つの閉曲面の間の領域として定義される薄膜領域上で「流体が領域の内外に出入りせず、境界上で摩擦力を受けながら滑る」という(ナヴィエの)滑り境界条件を課したNS方程式を考える。ただし境界条件としては摩擦力のない完全滑り境界条件も含む。このとき、薄膜方程式の解に対して膜の厚さ方向への積分平均を考え、膜の厚さゼロの極限で解の積分平均が極限曲面上の適切な関数空間内で収束し、極限関数が極限方程式の唯一の解であることを示すことによって極限曲面上の極限方程式を数学的に厳密に導出した。さらに、薄膜領域の膜の厚さが一定で薄膜NS方程式に摩擦力のない完全滑り境界条件を課す場合、本研究で導出した極限方程式は曲面上のNS方程式となることを示した。曲面上のNS方程式を薄膜極限により数学的に厳密に導出したのは本研究が最初の結果である。
2.先行研究で導出された極限方程式との比較:Temam-Ziane (1997)は2次元単位球面に退化する薄い球殻上で「流体が領域の内外に出入りせず、速度場の回転が境界上で法線方向に平行である」というホッジ境界条件の下でNS方程式を考え単位球面上の極限方程式を導出した。本研究ではこの極限方程式と本研究で完全滑り境界条件の下で導出した極限方程式を比較し、その構造に大きな変化が現れることやその違いが薄膜NS方程式に課す境界条件の違いと薄膜領域の境界の曲率がゼロでないことに起因することを発見した。

Research Progress Status

平成30年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

平成30年度が最終年度であるため、記入しない。

Report

(3 results)
  • 2018 Annual Research Report
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • Research Products

    (28 results)

All 2019 2018 2017 2016

All Journal Article (5 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 5 results,  Open Access: 1 results,  Acknowledgement Compliant: 2 results) Presentation (23 results) (of which Int'l Joint Research: 12 results,  Invited: 13 results)

  • [Journal Article] Hamilton-Jacobi equations on an evolving surface2019

    • Author(s)
      K. Deckelnick, C. M. Elliott, T.-H. Miura, and V. Styles
    • Journal Title

      Mathematics of Computation

      Volume: 印刷中 Issue: 320 Pages: 2635-2664

    • DOI

      10.1090/mcom/3420

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] An energetic variational approach for nonlinear diffusion equations in moving thin domains2018

    • Author(s)
      T.-H. Miura, Y. Giga, and C. Liu
    • Journal Title

      Advances in Mathematical Sciences and Applications

      Volume: 27 Pages: 115-141

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On singular limit equations for incompressible fluids in moving thin domains2018

    • Author(s)
      T.-H. Miura
    • Journal Title

      Quarterly of Applied Mathematics

      Volume: 76 Issue: 2 Pages: 215-251

    • DOI

      10.1090/qam/1495

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On analyticity of the L^p-Stokes semigroup for some non-Helmholtz domains2017

    • Author(s)
      M. Bolkart, Y. Giga, T.-H. Miura, T. Suzuki and Y. Tsutsui
    • Journal Title

      Math. Nachr.

      Volume: 印刷中 Issue: 16 Pages: 2524-2546

    • DOI

      10.1002/mana.201600016

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Zero width limit of the heat equation on moving thin domains2017

    • Author(s)
      T.-H. Miura
    • Journal Title

      Interfaces Free Bound.

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Singular limit problem for the Navier-Stokes equations in a curved thin domain2019

    • Author(s)
      三浦達彦
    • Organizer
      若手による流体力学の基礎方程式研究集会
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Singular limit problem for the Navier-Stokes equations in a curved thin domain2019

    • Author(s)
      T.-H. Miura
    • Organizer
      Surface, Bulk, and Geometric Partial Differential Equations: Interfacial, stochastic, non-local and discrete structures
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 曲がった薄膜領域上のナヴィエ・ストークス方程式に関する特異極限問題2019

    • Author(s)
      三浦達彦
    • Organizer
      第15回数学総合若手研究集会
    • Related Report
      2018 Annual Research Report
  • [Presentation] 曲面上の接ベクトル場に作用する種々のラプラス作用素の関係について2019

    • Author(s)
      三浦達彦
    • Organizer
      FMSP院生集中講義
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Global existence of a strong solution to the Navier-Stokes equations in a curved thin domain2018

    • Author(s)
      三浦達彦
    • Organizer
      PDE実解析研究会
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Singular limit problem for the Navier-Stokes equations in a curved thin domain2018

    • Author(s)
      T.-H. Miura
    • Organizer
      The 11th Mathematical Society of Japan Seasonal Institute "The Role of Metrics in the Theory of Partial Differential Equations"
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Singular limit problem for the Navier-Stokes equations in a curved thin domain2018

    • Author(s)
      T.-H. Miura
    • Organizer
      The 43rd Sapporo Symposium on Partial Differential Equations
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Singular limit problem for the Navier-Stokes equations in a curved thin domain2018

    • Author(s)
      T.-H. Miura
    • Organizer
      Mathematical Aspects of Surface and Interface Dynamics 16
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Singular limit problem for the Navier-Stokes equations in a curved thin domain2018

    • Author(s)
      T.-H. Miura
    • Organizer
      Theoretical Developments to Phenomenon Analyses based on Nonlinear Evolution Equations
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On singular limit equations for the Navier-Stokes equations in moving thin domains2018

    • Author(s)
      三浦達彦
    • Organizer
      日本数学会2018年度年会
    • Related Report
      2017 Annual Research Report
  • [Presentation] Zero width limit of the heat equation on moving thin domains2017

    • Author(s)
      三浦達彦
    • Organizer
      日本数学会2017年度年会
    • Place of Presentation
      首都大学東京(東京都八王子市)
    • Year and Date
      2017-03-26
    • Related Report
      2016 Annual Research Report
  • [Presentation] Singular limit problems for the heat equation and the Navier-Stokes equations in curved moving thin domains2017

    • Author(s)
      T.-H. Miura
    • Organizer
      Partial Differential Equations and their Applications Seminar
    • Place of Presentation
      Warwick Mathematics Institute, Coventry (UK)
    • Year and Date
      2017-02-15
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Singular limit problems for parabolic equations in curved moving thin domains2017

    • Author(s)
      T.-H. Miura
    • Organizer
      Emerging Developments in Interfaces and Free Boundaries
    • Place of Presentation
      Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach (Germany)
    • Year and Date
      2017-01-25
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research
  • [Presentation] On singular limits for the Navier-Stokes equations in curved moving thin domains2017

    • Author(s)
      T.-H. Miura
    • Organizer
      Geophysical Fluid Dynamics
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 動く曲面に退化する薄膜領域上の熱方程式に関する特異極限問題2017

    • Author(s)
      三浦達彦
    • Organizer
      広島数理解析セミナー
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] 動く曲面に退化する薄膜領域上の放物型方程式に関する特異極限問題2017

    • Author(s)
      三浦達彦
    • Organizer
      界面現象の数理・モデリング研究合宿2017
    • Related Report
      2017 Annual Research Report
  • [Presentation] Mathematical and numerical analysis of the Hamilton-Jacobi equation on an evolving surface2017

    • Author(s)
      T.-H. Miura
    • Organizer
      Mathematical Aspects of Surface and Interface Dynamics 14
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the Navier-Stokes equations with Navier boundary conditions in a curved thin domain2017

    • Author(s)
      T.-H. Miura
    • Organizer
      Princeton-Tokyo Fluid Mechanics Workshop
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the Navier-Stokes equations in a curved thin domain2017

    • Author(s)
      T.-H. Miura
    • Organizer
      Mathematical Analysis of Viscous Incompressible Fluid
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Finite volume scheme for the Hamilton-Jacobi equation on an evolving surface2017

    • Author(s)
      三浦達彦
    • Organizer
      数値解析セミナー
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] 動く曲面に退化する薄膜領域上の熱方程式に関する特異極限問題2016

    • Author(s)
      三浦達彦
    • Organizer
      第42回発展方程式研究会
    • Place of Presentation
      日本女子大学(東京都文京区)
    • Year and Date
      2016-12-27
    • Related Report
      2016 Annual Research Report
  • [Presentation] 動く薄膜領域上の熱方程式に関する特異極限問題2016

    • Author(s)
      三浦達彦
    • Organizer
      数学・数理科学専攻若手研究者のための異分野・異業種研究交流会2016
    • Place of Presentation
      明治大学(東京都中野区)
    • Year and Date
      2016-11-19
    • Related Report
      2016 Annual Research Report
  • [Presentation] Zero width limit of the heat equation on moving thin domains2016

    • Author(s)
      T.-H. Miura
    • Organizer
      The 10th Euro-Maghrebian workshop on Evolution Equations
    • Place of Presentation
      Heinrich Fabri Institut, Blaubeuren (Germany)
    • Year and Date
      2016-09-29
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research

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Published: 2016-05-17   Modified: 2024-03-26  

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