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Integrated study of curves of maximal slope for the energy functionals on a functional space

Research Project

Project/Area Number 18K03381
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionNihon University

Principal Investigator

YAMAURA Yoshihiko  日本大学, 文理学部, 教授 (90255597)

Co-Investigator(Kenkyū-buntansha) 三村 与士文  日本大学, 文理学部, 助教 (30646427)
SVADLENKA KAREL  京都大学, 理学研究科, 准教授 (60572188)
Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2021: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywordsエネルギー勾配流 / 離散モース流法 / 時間発展 p-ラプラシアン方程式 / 準線形熱型方程式 / Gehring 理論 / 高可積分性 / 離散モース流 / 双曲型偏微分方程式 / 高可積分性解析 / 弱解正則性解析 / p-ラプラシアン方程式 / Curves of maximal slope / p-Laplacian / Fractional p-Laplacian / Moser iteration / Legendre-Hadamard 条件 / p-Laplacian system / 双曲型方程式の離散弱解に対する高可積分性 / Transformation trick / 準線型熱方程式の Weissler 指数 / 弱解の高可積分性 / Discrete Morse flow / Variational method / Keller-Segel 系 / 自由境界問題
Outline of Final Research Achievements

We aim at the study of time evolutional partial differential equations by invoking the two discrete approximation methods based on the variational analysis. One is the method of local minimizing method originated by Marino, Saccon and Tosques. The other is the Discrete Morse Flow method originated by DeGiorgi and N.Kikuchi, independently. We can point out that the analysis of the regularity of weak solutions is realized by the thory of elliptic, not parabolic, PDE. For the former method, we investigate the semilinear heat equations. In particular, we obtain that the Weissler's exponent which is well-known in this theory is significant in the framework of variational analysis. For the latter method, we construct a weak solution to a p-Laplacian parabolic equations by an approximation method. The characteristic feature of this result is to attain the uniform regularity estimation to the approximate solutions independently on the approximation parameter.

Academic Significance and Societal Importance of the Research Achievements

【学術的意義】通常、放物型偏微分方程式の弱解の正則性解析は、時間微分を含めた解析になる。本研究では、2つの方法を基にして研究を進めたが、そのいずれの方法でも正則性は楕円型偏微分方程式の正則性理論を適用することになる。さらに、変分解析に基づく弱解の構成を実現することにより、数値解析への応用の可能性を模索することが可能になる。時間発展問題への変分解析からのアプローチは、偏微分方程式の研究に新しい視点を与えるものと期待される。
【社会的意義】 自然現象、社会現象を数値解析的に視覚化することはコンピュータの発達により実現可能になってきている。本研究はその基礎理論構築と位置づけられる。

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (24 results)

All 2023 2022 2021 2020 2019 2018

All Journal Article (9 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 9 results,  Open Access: 1 results) Presentation (15 results) (of which Int'l Joint Research: 7 results,  Invited: 7 results)

  • [Journal Article] Variational solution construction methods for dissipative and conservative systems2022

    • Author(s)
      Yoshifumi Mimura
    • Journal Title

      日本大学文理学部自然科学研究所紀要

      Volume: 57

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem2022

    • Author(s)
      Yoshiho Akagawa, Elliott Ginder, Syota Koide, Seiro Omata, Karel Svadlenka
    • Journal Title

      Discrete and Continuous Dynamical Systems Series B

      Volume: 27(5) Pages: 2661-2681

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A numerical algorithm for modeling cellular rearrangements in tissue morphogenesis2022

    • Author(s)
      Rhudaina Z. Mohammad, Hideki Murakawa, Karel Svadlenka, Hideru Togashi
    • Journal Title

      Communications Biology 5

      Volume: 239 Issue: 1 Pages: 239-239

    • DOI

      10.1038/s42003-022-03174-6

    • Related Report
      2022 Annual Research Report 2021 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Elastoplastic deformations of layered structures2022

    • Author(s)
      D. Drozdenko, M. Knapek, M. Kruzik, K. Mathis, K. Svadlenka, J. Valdman
    • Journal Title

      Milan Journal of Mathematics

      Volume: 90 Pages: 691-706

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Variational solution construction methods for dissipative and conservative systems2022

    • Author(s)
      Yoshifumi MIMURA
    • Journal Title

      日本大学文理学部自然科学研究所紀要

      Volume: 57

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Dewetting dynamics of anisotropic particles: A level set numerical approach2021

    • Author(s)
      Gavhale Siddharth、Svadlenka Karel
    • Journal Title

      Applications of Mathematics

      Volume: 1 Issue: 5 Pages: 1-29

    • DOI

      10.21136/am.2021.0040-21

    • Related Report
      2022 Annual Research Report 2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem2021

    • Author(s)
      Yoshiho Akagawa, Elliott Ginder, Syota Koide, Seiro Omata, Karel Svadlenka
    • Journal Title

      Discrete and Continuous Dynamical Systems Series B

      Volume: - Issue: 5 Pages: 2661-2661

    • DOI

      10.3934/dcdsb.2021153

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The discrete Morse flow method for parabolic p-Laplacian systems2021

    • Author(s)
      Nobuyuki Kato, Masashi Misawa, Yoshihiko Yamaura
    • Journal Title

      Annali di Matematica Pura ed Applicata

      Volume: 19 Pages: 1245-1275

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Another interpretation for the Weissler's critical exponent in the theory of semilinear parbolic PDE2020

    • Author(s)
      Yoshifumi Mimura, Yoshihiko Yamaura
    • Journal Title

      International Journal of Differential Equations and Applications

      Volume: 19 Pages: 109-132

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] 構造材料の弾塑性有限変形の変分解析2023

    • Author(s)
      Karel Svadlenka
    • Organizer
      日本数学会2023年会,中央大学,2023年3月15日-18日
    • Related Report
      2022 Annual Research Report
  • [Presentation] 構造材料の均質化と剛性定理2023

    • Author(s)
      Karel Svadlenka
    • Organizer
      北陸応用数理研究会2023,2023年3月23日-25日,石川県政記念しいのき迎賓館
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Variational approach to modeling of elastoplastic deformation of structured materials2022

    • Author(s)
      Karel Svadlenka
    • Organizer
      IMI Workshop II: Geometry and Algebra in Material Science III, September 10, 2022, Kyushu University, Japan
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 感覚器官上皮における細胞配列ダイナミックスの数理的解明2022

    • Author(s)
      Karel Svadlenka
    • Organizer
      第3回 京大―ハイデルベルク大―理研 ワークショップ 「医学と数理」,2022年9月30日~10月1日(オンライン)
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Understanding of kink-band formation by means of a rate-independent model obtained by homogenization of mille-feuille structure2022

    • Author(s)
      Karel Svadlenka
    • Organizer
      The 5th International Symposium on Long-Period Stacking/Order Structure and Mille-feuille Structure, December 12~14, 2022, Shinagawa Prince Hotel, Tokyo, Japan
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Solution of hyperbolic equations through energy preserving approximations2021

    • Author(s)
      Karel Svadlenka
    • Organizer
      The 46th Sapporo Symposium on Partial Differential Equations, Hokkaido University
    • Related Report
      2021 Research-status Report
  • [Presentation] A convergence of the approximated free boundary of regularized functionals via Gamma-convergence2019

    • Author(s)
      Yoshihiko Yamaura
    • Organizer
      The 2nd International Conference on Numerical Modelling in Engineering
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] On a convergence for the approximation of the gradient flow curve to the non-convex and singular functional2019

    • Author(s)
      Yoshihiko Yamaura
    • Organizer
      2019 International Conference on Machine Learning and Intelligent Systems (MLIS2019)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Life and Natural Science Fusion2019

    • Author(s)
      Svadlenka Karel
    • Organizer
      Mathematical modeling of cellular dynamics in morphogenesis, HeKKSaGOn Meeting Work Group 1, Heidelberg University, Germany
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Numerical analysis of mechanics of layered structures2019

    • Author(s)
      Svadlenka Karel
    • Organizer
      EU-Japan Workshop on Mille-feuille Structured Materials, Prague, Czech Republic
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Approximation of motion of weighted interfacial network and its application to cellular patterns formation2019

    • Author(s)
      Svadlenka Karel
    • Organizer
      Kyoto Winter School 2019 "Quantifying Dynamics of Life”, Kyoto, Japan
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Numerical approach to filament dynamics2018

    • Author(s)
      Svadlenka Karel
    • Organizer
      Czech‐Japanese Seminar in Applied Mathematics
    • Related Report
      2018 Research-status Report
  • [Presentation] On some extensions of thresholding schemes2018

    • Author(s)
      Svadlenka Karel
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2018 Research-status Report
  • [Presentation] Numerical approach to hyperbolic motions of co-dimension 22018

    • Author(s)
      Svadlenka Karel
    • Organizer
      The Seventh China-Japan-Korea Joint Conference on Numerical Mathematics
    • Related Report
      2018 Research-status Report
  • [Presentation] Numerical simulation of sensory tissue formation2018

    • Author(s)
      Svadlenka Karel
    • Organizer
      A3 Soft matter workshop, Peking, China
    • Related Report
      2018 Research-status Report

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Published: 2018-04-23   Modified: 2024-01-30  

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