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Construction of new phase field methods for dynamical problems in the calculus of variations

Research Project

Project/Area Number 20K14343
Research Category

Grant-in-Aid for Early-Career Scientists

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

Principal Investigator

Takasao Keisuke  京都大学, 理学研究科, 准教授 (50734472)

Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2022: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords平均曲率流方程式 / 幾何学的測度論 / フェイズフィールド法 / 変分問題 / 弱解 / バリフォールド / 平均曲率流 / 特異極限問題 / 障害物問題
Outline of Research at the Start

平均曲率流方程式等の曲面の運動方程式を解析する。平均曲率流方程式の弱解の構成方法として、フェイズフィールド法、即ちAllen-Cahn方程式の解の特異極限を用いて弱解を構成する方法がよく知られている。本研究では、二相流体の方程式、細胞膜の運動方程式、金属粒界の方程式等に関して、フェイズフィールド法を用いて弱解の時間大域存在及びその正則性について解明することを目的とする。

Outline of Final Research Achievements

We studied the surface evolution equations and obtained the following. We proved the global existence of the Brakke flow of the obstacle problem, when the boundaries of obstacles are smooth. We used the Allen-Cahn equation with forcing term for the proof. We showed the global existence of the weak solution to the volume preserving mean curvature flow for any dimensions. For the proof, we used a new phase field model for the volume preserving mean curvature flow. We proved the short time existence of the classical solution to the geometric evolution equation studied by Epshteyn-Liu-Mizuno.

Academic Significance and Societal Importance of the Research Achievements

平均曲率流方程式等の曲線、曲面の発展方程式の解の近似方法の1つとして知られるフェイズフィールド法は、体積保存条件や外力項を加えることが容易であり、数値実験の分野で良く用いられている。しかし、フェイズフィールドモデルの特異極限が元の発展方程式に収束するか?という問題については、平均曲率流方程式の障害物問題や、任意の空間次元における体積保存平均曲率流方程式については未解決であった。本研究ではこれらの収束に関して数学的な正当性を与えた。Epshteyn-Liu-Mizunoの方程式については、金属粒界でよく見られる、トリプルジャンクションの形状を持つ解の時間局所存在を示すことが出来た。

Report

(5 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (32 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (3 results) Journal Article (5 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 5 results,  Open Access: 2 results) Presentation (24 results) (of which Int'l Joint Research: 10 results,  Invited: 22 results)

  • [Int'l Joint Research] ウィーン大学(オーストリア)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] ウィーン大学(オーストリア)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Scuola Normale Superiore(イタリア)

    • Related Report
      2020 Research-status Report
  • [Journal Article] The Existence of a Weak Solution to Volume Preserving Mean Curvature Flow in Higher Dimensions2023

    • Author(s)
      Takasao Keisuke
    • Journal Title

      Archive for Rational Mechanics and Analysis

      Volume: 247 Issue: 3

    • DOI

      10.1007/s00205-023-01881-w

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A curve shortening equation with time-dependent mobility related to grain boundary motions2021

    • Author(s)
      Mizuno Masashi、Takasao Keisuke
    • Journal Title

      Interfaces and Free Boundaries

      Volume: 23 Issue: 2 Pages: 169-190

    • DOI

      10.4171/ifb/453

    • NAID

      120007193023

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] On Obstacle Problem for Brakke's Mean Curvature Flow2021

    • Author(s)
      Takasao Keisuke
    • Journal Title

      SIAM Journal on Mathematical Analysis

      Volume: 53 Issue: 6 Pages: 6355-6369

    • DOI

      10.1137/21m1400432

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions2021

    • Author(s)
      Yoshikazu Giga, Fumihiko Onoue, Keisuke Takasao
    • Journal Title

      Differential and Integral Equations

      Volume: 34 Pages: 21-126

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Existence of weak solution for mean curvature flow with transport term and forcing term2020

    • Author(s)
      Keisuke Takasao
    • Journal Title

      Communications on Pure and Applied Analysis

      Volume: 19 Issue: 5 Pages: 2655-2677

    • DOI

      10.3934/cpaa.2020116

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Presentation] Brakkeの平均曲率流の障害物問題2024

    • Author(s)
      高棹圭介
    • Organizer
      室蘭工業大学応用解析セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Phase field method for mean curvature flow and vanishing of discrepancy measure2023

    • Author(s)
      高棹圭介
    • Organizer
      RIMS 共同研究(公開型)調和解析と非線形偏微分方程式
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Phase field model for volume preserving mean curvature flow2023

    • Author(s)
      高棹圭介
    • Organizer
      東工大数理解析セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Convergence of Allen-Cahn equation with non-local term to volume preserving mean curvature flow2023

    • Author(s)
      Keisuke Takasao
    • Organizer
      第48回偏微分方程式論札幌シンポジウム
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Phase field model for volume-preserving mean curvature flow2023

    • Author(s)
      Keisuke Takasao
    • Organizer
      ICIAM 2023 Tokyo
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Monotonicity formula of Allen-Cahn equations for surface evolution equations2023

    • Author(s)
      Keisuke Takasao
    • Organizer
      Analysis, Geometry and Stochastics on Metric Spaces
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Phase field method for volume preserving mean curvature flow2023

    • Author(s)
      高棹圭介
    • Organizer
      第145回HMMCセミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] 重み付き平均曲率流に対するフェイズフィールド法について2023

    • Author(s)
      高棹圭介
    • Organizer
      研究集会「広島微分方程式研究会」
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Existence of weak solution to volume preserving mean curvature flow2023

    • Author(s)
      高棹圭介
    • Organizer
      金沢解析セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] フェイズフィールドモデルを用いた体積保存平均曲率流の弱解の存在について2023

    • Author(s)
      高棹圭介
    • Organizer
      南大阪応用数学セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Existence of weak solution to volume preserving mean curvature flow2023

    • Author(s)
      Keisuke Takasao
    • Organizer
      NCTS-Kyoto Mathematics Symposium
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Existence of weak solution to volume preserving mean curvature flow in higher dimensions2023

    • Author(s)
      Keisuke Takasao
    • Organizer
      Geometry and Analysis Seminar -- Mini Workshop 2023
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On volume preserving mean curvature flow in higher dimensions2023

    • Author(s)
      Keisuke Takasao
    • Organizer
      Kyoto-CAU Joint Meeting on Nonlinear PDEs
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 体積保存平均曲率流の弱解の時間大域存在について2022

    • Author(s)
      高棹圭介
    • Organizer
      鳥取 PDE 研究集会 2022
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] フェイズフィールド法を用いた体積保存平均曲率流方程式の弱解の存在について2022

    • Author(s)
      高棹圭介
    • Organizer
      第13回「解析学とその周辺」@野田
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] 任意の空間次元における体積保存平均曲率流方程式の弱解の存在について2022

    • Author(s)
      高棹圭介
    • Organizer
      2022 年度日本数学会秋季総合分科会
    • Related Report
      2022 Research-status Report
  • [Presentation] ある非局所項付きAllen-Cahn方程式を用いた体積保存平均曲率流の存在について2021

    • Author(s)
      高棹圭介
    • Organizer
      九州関数方程式セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] On obstacle problem for Brakke's mean curvature flow2021

    • Author(s)
      高棹圭介
    • Organizer
      第10回室蘭非線形解析研究会, 室蘭工業大学
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] On obstacle problem for Brakke's mean curvature flow2021

    • Author(s)
      Keisuke Takasao
    • Organizer
      The 22nd Northeastern Symposium on Mathematical Analysis, Tohoku University
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Phase field model for mean curvature flow with transport term and forcing term2021

    • Author(s)
      高棹圭介
    • Organizer
      第4回反応拡散方程式と非線形分散型方程式の解の挙動
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] フェイズフィールド法による外力項付き平均曲率流方程式の弱解の存在について2020

    • Author(s)
      高棹圭介
    • Organizer
      日本数学会2020年度秋季総合分科会, 実函数論分科会特別講演
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] On obstacle problem for Brakke flow2020

    • Author(s)
      高棹圭介
    • Organizer
      北海道大学偏微分方程式セミナー, 北海道大学
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Existence of weak solution for mean curvature flow with forcing term2020

    • Author(s)
      Keisuke Takasao
    • Organizer
      Workshop in Geometric Measure Theory and Applications, East China Normal University, Shanghai
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Phase field method for volume preserving mean curvature flow2020

    • Author(s)
      Keisuke Takasao
    • Organizer
      Asia-Pacific Analysis and PDE Seminar
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2020-04-28   Modified: 2025-01-30  

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