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Well-posedness for partial differential equations with time-dependent constraints

Research Project

Project/Area Number 26400138
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionShizuoka University (2015-2016)
Hiroshima University (2014)

Principal Investigator

MATSUMOTO Toshitaka  静岡大学, 理学部, 教授 (20229561)

Co-Investigator(Kenkyū-buntansha) 小林 良和  中央大学, 理工学部, 教授 (80092691)
渡邉 紘  大分大学, 工学部, 准教授 (30609912)
Co-Investigator(Renkei-kenkyūsha) TANAKA Naoki  静岡大学, 理学部, 教授 (00207119)
Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords準線形方程式 / 適切性 / リプシッツ発展作用素 / 変異方程式 / 弱連続作用素
Outline of Final Research Achievements

We consider Cauchy problems for abstract evolution equations in Banach spaces where nonlinear operators are weakly continuous in some sense, and provide a necessary and sufficient condition for existence of weakly continuously differentiable solutions. We also discuss abstract quasilinear problems where the domain of quasilinear operators are neither dense nor constant, and prove a unique existence of continuously differentiable solutions. Under a generalised dissipativity condition, time local well-posedness for Cauchy problems to mutational equations are established. An existence and uniqueness result for entropy solutions to 1-D strongly degenerate parabolic equations is given. Finally phase field models related to grain boundary motions are considered. Existence of weak solutions which reproduce the energy-dissipation is shown and asymptotic behaviours of weak solutions are investigated.

Report

(4 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (37 results)

All 2017 2016 2015 2014

All Journal Article (7 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 5 results,  Open Access: 6 results,  Acknowledgement Compliant: 3 results) Presentation (30 results) (of which Int'l Joint Research: 4 results,  Invited: 7 results)

  • [Journal Article] Solvability for the non-isothermal Kobayashi-Warren-Carter system2017

    • Author(s)
      K. Shirakawa and H. Watanabe
    • Journal Title

      AIMS Mathematics

      Volume: 2 Issue: 1 Pages: 161-194

    • DOI

      10.3934/math.2017.1.161

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Mathematical analysis for a Warren-Kobayashi-Lobkovsky-Carter type system2016

    • Author(s)
      K. Shirakawa, H. Watanabe and N. Yamazaki
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 1997 Pages: 64-85

    • Related Report
      2016 Annual Research Report
    • Open Access
  • [Journal Article] Abstract Cauchy problem for weakly continuous operators2016

    • Author(s)
      Toshitaka Matsumoto and Naoki Tanaka
    • Journal Title

      J. Math. Anal. Appl.

      Volume: 435 Issue: 1 Pages: 267-285

    • DOI

      10.1016/j.jmaa.2015.10.027

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with variable coefficients2016

    • Author(s)
      H. Watanabe
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 1984 Pages: 23-45

    • Related Report
      2015 Research-status Report
    • Open Access
  • [Journal Article] Large-time behavior for a PDE model of isothermal grain boundary motion with a constraint2015

    • Author(s)
      Ken Shirakawa, Hiroshi Watanabe
    • Journal Title

      Dynamical Systems and Differential Equations, AIMS Proceedings 2015, Proceedings of the 10th AIMS International Conference (Madrid, Spain)

      Volume: 1 Pages: 1009-1018

    • DOI

      10.3934/proc.2015.1009

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Nonautonomous differential equations and Lipschitz evolutionoperators in Banach spaces2015

    • Author(s)
      Y. Kobayashi, N. Tanaka and Y.Tomizawa
    • Journal Title

      Hiroshima Math. J.

      Volume: 45 Pages: 267-307

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system2014

    • Author(s)
      H. Watanabe and K. Shirakawa
    • Journal Title

      Mathematica Bohemica

      Volume: 139

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Presentation] 非斉次Dirichlet型境界条件を組み入れた結晶粒界運動のフェーズ・フィールドモデル2017

    • Author(s)
      白川健, 渡邉紘, S.Moll
    • Organizer
      日本数学会
    • Place of Presentation
      首都大学東京(東京都八王子市)
    • Year and Date
      2017-03-24
    • Related Report
      2016 Annual Research Report
  • [Presentation] 変数係数を持つ非局所的強退化放物型方程式系に対する時間大域解2017

    • Author(s)
      渡邉紘
    • Organizer
      日本数学会
    • Place of Presentation
      首都大学東京(東京都八王子市)
    • Year and Date
      2017-03-24
    • Related Report
      2016 Annual Research Report
  • [Presentation] 結晶粒界運動の非等方的フェーズ・フィールドモデルにおけるエネルギー消散性2016

    • Author(s)
      白川健, 渡邉紘, S.Moll
    • 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
      2016-12-25
    • Related Report
      2016 Annual Research Report
  • [Presentation] 非局所的放物型・双曲型保存則系に対する適切性2016

    • Author(s)
      渡邉紘
    • 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
      2016-12-25
    • Related Report
      2016 Annual Research Report
  • [Presentation] 変数係数を持つ非局所的強退化放物型方程式系2016

    • Author(s)
      渡邉紘
    • Organizer
      日本数学会
    • Place of Presentation
      関西大学(大阪府吹田市)
    • Year and Date
      2016-09-15
    • Related Report
      2016 Annual Research Report
  • [Presentation] Wellposedness for nonlocal parabolic-hyperbolic conservation laws2016

    • Author(s)
      渡邉紘
    • Organizer
      PDE Summer Seminar in Fukuoka
    • Place of Presentation
      福岡大学セミナーハウス(福岡県福岡市)
    • Year and Date
      2016-09-05
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Evolution problems in metric spaces and dissipativity conditions2016

    • Author(s)
      Y. Kobayashi and N. Tanaka
    • Organizer
      The fifth Asian conference on Nonlinear Analysis and Optimization
    • Place of Presentation
      Toki Messe (新潟県新潟市)
    • Year and Date
      2016-08-01
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Parabolic-hyperbolic conservation laws with variable and nonlocal type coefficients2016

    • Author(s)
      Hiroshi Watanabe
    • Organizer
      The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 78
    • Place of Presentation
      Orlando, Florida (アメリカ)
    • Year and Date
      2016-07-01
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research
  • [Presentation] サイズ構造モデルへの抽象準線形理論的アプローチ2016

    • Author(s)
      松本敏隆
    • Organizer
      第200回 広島数理解析セミナー
    • Place of Presentation
      広島大学理学部(広島県東広島市)
    • Year and Date
      2016-05-27
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] 異方性を加味した結晶粒界運動のフェーズ・フィールドモデルに対するエネルギー消散性2016

    • Author(s)
      白川 健, 渡邉 紘, S. Moll
    • Organizer
      日本数学会
    • Place of Presentation
      筑波大学(茨城県つくば市)
    • Year and Date
      2016-03-16
    • Related Report
      2015 Research-status Report
  • [Presentation] 変数係数を持つ強退化放物型方程式のBV-エントロピー解に対する連続的依存性2016

    • Author(s)
      渡邉 紘
    • Organizer
      日本数学会
    • Place of Presentation
      筑波大学(茨城県つくば市)
    • Year and Date
      2016-03-16
    • Related Report
      2015 Research-status Report
  • [Presentation] サイズ構造モデルへの準線形理論的接近法2016

    • Author(s)
      松本敏隆,田中直樹
    • Organizer
      日本数学会
    • Place of Presentation
      筑波大学(茨城県つくば市)
    • Year and Date
      2016-03-16
    • Related Report
      2015 Research-status Report
  • [Presentation] Mutational equation に対する適切性定理について2015

    • Author(s)
      小林良和, 田中直樹
    • Organizer
      第41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
      2015-12-25
    • Related Report
      2015 Research-status Report
  • [Presentation] 結晶粒界の非等温系モデルのエネルギー消散性と漸近挙動2015

    • Author(s)
      渡邉 紘, 白川 健
    • Organizer
      第41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
      2015-12-25
    • Related Report
      2015 Research-status Report
  • [Presentation] 弱連続作用素に対する抽象的Cauchy問題2015

    • Author(s)
      松本敏隆, 田中直樹
    • Organizer
      第13回浜松偏微分方程式研究集会
    • Place of Presentation
      静岡大学(静岡県浜松市)
    • Year and Date
      2015-12-22
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 弱連続作用素に対する抽象的Cauchy問題について2015

    • Author(s)
      松本敏隆, 田中直樹
    • Organizer
      ポテンシャル論セミナー
    • Place of Presentation
      名城大学(愛知県名古屋市)
    • Year and Date
      2015-12-18
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 変数係数を持つ放物型・双曲型単独保存則に対する適切性2015

    • Author(s)
      渡邉 紘
    • Organizer
      第6回「拡散と移流の数理」
    • Place of Presentation
      愛媛大学(愛媛県松山市)
    • Year and Date
      2015-12-05
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Abstract Cauchy problem for weakly continuous operators2015

    • Author(s)
      松本敏隆, 田中直樹
    • Organizer
      第616回『応用解析』研究会
    • Place of Presentation
      早稲田大学(東京都新宿区)
    • Year and Date
      2015-12-05
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Lipschitz Evolution Operators2015

    • Author(s)
      Y. Kobayashi, N. Tanaka and Y. Tomizawa
    • Organizer
      VII Symposium on Nonlinear Analysis
    • Place of Presentation
      Nicolaus Copernicus University, Torun (ポーランド)
    • Year and Date
      2015-09-15
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] 異方性を加味した結晶粒界運動のKobayashi-Warren-Carter型数学モデル2015

    • Author(s)
      白川 健, 渡邉 紘, S. Moll
    • Organizer
      日本数学会
    • Place of Presentation
      京都産業大学(京都府京都市)
    • Year and Date
      2015-09-13
    • Related Report
      2015 Research-status Report
  • [Presentation] 結晶粒界現象を記述する非等温系数学モデルに対する解の漸近挙動2015

    • Author(s)
      渡邉 紘, 白川 健
    • Organizer
      日本数学会
    • Place of Presentation
      京都産業大学(京都府京都市)
    • Year and Date
      2015-09-13
    • Related Report
      2015 Research-status Report
  • [Presentation] Solvability of some degenerate parabolic equations with convective terms2015

    • Author(s)
      Hiroshi Watanabe
    • Organizer
      EquaDiff 2015
    • Place of Presentation
      Universite Claude Bernard Lyon1, Lyon (フランス)
    • Year and Date
      2015-07-06
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] 非等温条件下でのWarren-Kobayashi-Lobkovsky-Carter型の結晶粒界モデル2015

    • Author(s)
      渡邉 紘
    • Organizer
      表面・界面ダイナミクスの数理IX
    • Place of Presentation
      東京大学大学院数理科学研究科(東京都文京区)
    • Year and Date
      2015-04-23
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 変数係数を持つ強退化放物型方程式の可解性2015

    • Author(s)
      渡邉紘
    • Organizer
      日本数学会
    • Place of Presentation
      明治大学
    • Year and Date
      2015-03-21 – 2015-03-24
    • Related Report
      2014 Research-status Report
  • [Presentation] 非等温条件下における結晶粒界現象のフェーズフィールドモデル2015

    • Author(s)
      白川健、渡邉紘
    • Organizer
      日本数学会
    • Place of Presentation
      明治大学
    • Year and Date
      2015-03-21 – 2015-03-24
    • Related Report
      2014 Research-status Report
  • [Presentation] Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with variable coefficients2014

    • Author(s)
      渡邉紘
    • Organizer
      RIMS研究集会「抽象発展方程式理論から見た偏微分方程式に関する評価方法の再考」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2014-10-22 – 2014-10-24
    • Related Report
      2014 Research-status Report
  • [Presentation] 弱連続作用素に対する抽象的Cauchy問題について2014

    • Author(s)
      松本敏隆、田中直樹
    • Organizer
      日本数学会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-25 – 2014-09-28
    • Related Report
      2014 Research-status Report
  • [Presentation] Allen-Cahn型方程式と結晶粒界モデルによる連立系の時刻無限大における解挙動2014

    • Author(s)
      渡邉紘、白川健
    • Organizer
      日本数学会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-25 – 2014-09-28
    • Related Report
      2014 Research-status Report
  • [Presentation] 凝固現象と結晶粒界現象の等温系カップリングモデルにおけるエネルギー消散解2014

    • Author(s)
      白川健、渡邉紘
    • Organizer
      日本数学会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-25 – 2014-09-28
    • Related Report
      2014 Research-status Report
  • [Presentation] Strongly degenerate parabolic equations with diffusion coefficients depending on the spatial variable2014

    • Author(s)
      Hiroshi Watanabe
    • Organizer
      The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      The Universidad Autonoma de Madrid (Spain)
    • Year and Date
      2014-07-07 – 2014-07-11
    • Related Report
      2014 Research-status Report

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Published: 2014-04-04   Modified: 2018-03-22  

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