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Mathematical analysis for nonlinear partial differential equations with singular solutions

Research Project

Project/Area Number 15340041
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKanazawa University

Principal Investigator

OMATA Seiro  Kanazawa univ., Dept. of Natural Sciences, Prof., 自然科学研究科, 教授 (20214223)

Co-Investigator(Kenkyū-buntansha) MIYAKAWA Tetsuro  Kanazawa univ. Dept. of Natural Sciences, Prof., 自然科学研究科, 教授 (10033929)
JIMBO Shuichi  Hokkaido Univ. Dept. of Mathematics, Prof., 大学院・理学研究科, 教授 (80201565)
WEISS Georg  The Univ. of Tokyo. Dept. of Mathematical Science, Associate Prof., 大学院・数理科学研究科, 助教授 (30282817)
KIMURA Masato  Kyushu Univ. Dept. of Mathematics, Associate Prof., 大学院・数理学研究院, 助教授 (70263358)
KIKUCHI Koji  Shizuoka Univ. Dept. of Engneering, Professor, 工学部, 教授 (50195202)
田村 博  金沢大学, 自然科学研究科, 助教授 (80188440)
岩崎 宏  金沢大学, 自然科学研究科, 助教授 (30242514)
後藤 俊一  金沢大学, 理学部, 助教授 (30225651)
一瀬 孝  金沢大学, 理学部, 教授 (20024044)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥11,000,000 (Direct Cost: ¥11,000,000)
Fiscal Year 2005: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2004: ¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2003: ¥4,600,000 (Direct Cost: ¥4,600,000)
Keywordsvariational problem / nonlinear PDEs / singularity / numerical analysis / minimizing method / free boundary problem / 特異点数値解析
Research Abstract

The aim of this research was to solve nonlinear Partial Differential Equations whose solution is expected to have singularities depending on time. The candidates of singularities are defects in harmonic mapping, vortex in Ginzburg-Landau problem and free boundaries.
We have solved the following problems;
(1)On a Soap film vibration with free boundary, we have established the method to treat wave type free boundary problems
(2)We developed a numerical method via the discrete Mores flow for volume constraint conditions
(3)We constructed a weak solution to a hyperbolic equation with volume constraint
(4)We constructed a weak solution to a parabolic equation with volume constraint and showing Hoelder continuity of the
solution
Moreover we have developed solvers for parallel machine with minimizing algorithm via the discrete Morese flows. This works very well especially for volume constraint problems. This is also very nice for a weak connected parallel machines, because it uses direct method of variational principle.
Finally, we would like to express pur special thanks to all participants of this project.

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (28 results)

All 2006 2005 2004 2003 Other

All Journal Article (23 results) Publications (5 results)

  • [Journal Article] Construction of approximate solution to a hyperbolic free boundary problem with volume constraint and its numerical computation2006

    • Author(s)
      T.Yamazaki, S.Omata et.el.
    • Journal Title

      Adv. Math. Sci. Appl. 16

      Pages: 57-67

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Construction of approximate solution to a hyperbolic free boundary problem with volume constraint and its numerical computation2006

    • Author(s)
      T.Yamazaki, S.Omata, K.Svadlenka, K.Ohara
    • Journal Title

      Adv. Math. Sci. Appl. 16, 1

      Pages: 57-67

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Numericalsolution of film vibration with obstacle2006

    • Author(s)
      H.Yoshiuchi, S.Omata, K.Svadlenka, K.Ohara
    • Journal Title

      Adv. Math. Sci. Appl. 16, 1

      Pages: 33-43

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Construction of approximate solution to a hyperbolic free boundary problem with volume constraint and its numerical computation2006

    • Author(s)
      T.Yamazaki, S.Omata, K.Svadlenka, K.Ohara
    • Journal Title

      Adv.Math.Sci.Appl. 16, No.1(掲載決定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article]2006

    • Author(s)
      H.Yoshiuchi, S.Omata, K.Svadlenka, K.Ohara
    • Journal Title

      Adv.Math.Sci.Appl., 16, No.1(掲載決定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A wave equation in the curl-free space related to asmectics liquid crystal2005

    • Author(s)
      S.Omata
    • Journal Title

      Differential Integral Equations 18-1

      Pages: 61-70

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Bubble Motion on water surface2005

    • Author(s)
      T.Yamazaki, S.Omata et.el.
    • Journal Title

      Gakuto Intern. Ser. Math. Sci. Appl. 23

      Pages: 209-216

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A wave equation in the curl-free space related to a smectics liquid crystal2005

    • Author(s)
      S.Omata
    • Journal Title

      Differential Integral Equations 18 No.1

      Pages: 61-70

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Bubble Motion on water surface2005

    • Author(s)
      T.Yamazaki, S.Omata, H.Yoshiuchi, K.Ohara
    • Journal Title

      Gakuto Intern. Ser. Math. Sci. Appl. 23

      Pages: 209-216

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] 速い移流を持つStrum-Liouville問題の第1固有値の精密な漸近挙動2005

    • Author(s)
      木村正人, 永田真一
    • Journal Title

      日本応用数理学会論文誌 15, No.3

      Pages: 209-220

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem.2005

    • Author(s)
      John Andersson, Georg S.Weiss
    • Journal Title

      J.Differential Equations, 電子版1016

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A wave equation in the curl-free space related to a smectics liquid crystal2005

    • Author(s)
      S.Omata
    • Journal Title

      Differential Integral Equations 18, No.1

      Pages: 61-70

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A numerical treatment of thin film motion with free boundary2004

    • Author(s)
      S.Omata
    • Journal Title

      Adv. Math. Sci. Appl. 14, 1,

      Pages: 129-137

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Numerical computations for interfaces motion governed by triple potential well problem2004

    • Author(s)
      H.Iwasaki, S.Omata et.el
    • Journal Title

      Adv. Math. Sci. Appl. 14, 2

      Pages: 457-464

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A numerical treatment of thin film motion with free boundary2004

    • Author(s)
      S.Omata
    • Journal Title

      Adv. Math. Sci. Appl. 14, 1

      Pages: 129-137

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A numerical treatment of thin film motion with free boundary2004

    • Author(s)
      S.Omata
    • Journal Title

      Adv.Math.Sci.Appl. 14, No.1

      Pages: 129-137

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Numerical computations for interfaces motion governed by triple potential well problem2004

    • Author(s)
      H.Iwasaki, K.Ohara, S.Omata T.Zhou
    • Journal Title

      Adv.Math.Sci.Appl. 14, No.2

      Pages: 457-464

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Singular perturbation of domains and semilinear elliptic equations III2004

    • Author(s)
      S.Jimbo
    • Journal Title

      Hokkaido Math.J. 33

      Pages: 11-45

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A Parabolic Free Boundary Value Problem with Double Pinning2004

    • Author(s)
      G.S.Weiss
    • Journal Title

      Nonlinear Analysis 57

      Pages: 153-172

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Boundary Monotonicity Formulae and Applications to Free Boundary Problems, I.The Elliptic Case2004

    • Author(s)
      G.S.Weiss
    • Journal Title

      Electronic Journal of Differential Equations 44

      Pages: 1-12

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A numerical computation to the American option pricing via the discrete Morse flow2003

    • Author(s)
      S.Omata et.el.
    • Journal Title

      Theoretical and Applied Mechanics 52

      Pages: 261-266

    • NAID

      130004463535

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A numerical computation to the American option pricing via the discrete Morse flow2003

    • Author(s)
      S.Omata.H.Iwasaki, K.Nakane, X.Xiong, M.Sakuma
    • Journal Title

      Theoretical and Applied Mechanics 52

      Pages: 261-266

    • NAID

      130004463535

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Construction of solutions to heat-type problems with volume constraint via the discrete Morse flow

    • Author(s)
      K.Svadlenka, S.Omata
    • Journal Title

      to appear in Funkcialaj Ekvacio

    • NAID

      130000140688

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Publications] S.Jimbo, S.Kosugi: "Approximation of eigenvalues of elliptic operators with discontinuous coefficients"Comm. Partial Differential Equations. 28. 1303-1323 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Jimbo, J.Zhai: "Instability in a geometric parabolic equation on convex domain"J.Differential equations. 188. 447-460 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Omata, H.Iwasaki, K.Nakane, X.Xiong, M.Sakuma: "A.numerical computation to the American option pricing via the discrete Morse flow"Theoretical and Applied Mechanics. 52. 261-266 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Jimbo: "Singular perturbation of domains and semilinear elliptic equations III"Hokkaido Math.J. 33. 11-45 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Omata: "A numerical treatment of thin film motion with free boundary"Adv.Math.Sci.Appl., 14,. 14(to appear). (2004)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2021-04-07  

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