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2006 Fiscal Year Final Research Report Summary

Study of singularities of partial differential equations in the complex domain

Research Project

Project/Area Number 16540169
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

TAHARA Hidetoshi  Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (60101028)

Co-Investigator(Kenkyū-buntansha) OUCHI Sunao  Sophia Univ., Fac. of Science and Technology, Professor, 理工学部, 教授 (00087082)
UCHIYAMA Koichi  Sophia Univ., Fac. of Science and Technology, Professor, 理工学部, 教授 (20053689)
YOSHINO Kunio  Sophia Univ., Fac. of Sci. and Tech., Associate Prof., 理工学部, 助教授 (60138378)
OKADA Yasunori  Chiba Univ., Fac. of Science, Associate Prof., 理学部, 助教授 (60224028)
YAMANE Hideshi  Kwansei Gauin Univ., Fac. of Science and Technology, Associate Prof., 理工学部, 助教授 (80286145)
Project Period (FY) 2004 – 2006
Keywordscomplex domain / partial differential equation / holomorphic solution / singularity / formal solution / equation of transformation
Research Abstract

1.In the theory of ODEs, the method of analytic transformation is often used to show the equivalence of two nonlinear ODEs. In this research, this method is extended to the framework of PDEs, and a equivalence of two nonlinear PDEs of Kowalewskian type is proved. The result is applied to the problem of analytic continuation of the solution.
2.It is proved that the formal solution of some class of nonlinear PDEs is multi-summable. The PDE in this class is regarded as the one obtained by the perturbation of ODEs. The result is applied to the normal form theory of vector fields.
3.A uniqueness result of the solution is proved for a class of nonlinear totally characteristic PDEs. The result is applied to showing the nonexistence of singularities of the solution.
4.Singularities of solutions of general nonlinear PDEs are studied and solutions with logarithmic singularities are constructed.
5.The asymptotic behaviour of solutions near the singularity is investigated and it is proved that in some classes of PDEs the asymptotic expansion is given concretely by using Mellin transformation. Gevrey type estimate of the remainder term is also obtained.
6.Singularities of sphere-symmetric solutions of p-elliptic equation are studied and the form of singularities is given explicitly by the analysis of Briot-Bouquet type PDEs.

  • Research Products

    (19 results)

All 2006 2005 2004

All Journal Article (19 results)

  • [Journal Article] 複素領域における非線型偏微分方程式の正則解の存在について2006

    • Author(s)
      田原秀敏
    • Journal Title

      数理解析研究所講究録 1509

      Pages: 51-61

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Formal solutions of some nonlinear partial differential equations and their mutisummability2006

    • Author(s)
      S.Ouchi
    • Journal Title

      Asymptotic Analysis 47

      Pages: 187-226

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Nonlinear Cauchy problems with small analytic data2006

    • Author(s)
      H.Yamane
    • Journal Title

      Proc. Amer. Math. Soc. 134・11

      Pages: 3353-3361

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The structure of positive definite Fourier ultra hyperfunctions2006

    • Author(s)
      K.Yoshino, M.Suwa
    • Journal Title

      Complex Var. Elliptic Equ. 51・5-6

      Pages: 611-624

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The Bochner-Schwartz theorem for Fourier ultra hyperfunctions2006

    • Author(s)
      K.Yoshino, M.Suwa
    • Journal Title

      Integral Transforms Spec. Funct. 17・2-3

      Pages: 109-114

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Compact quotients of large domains in a complex projective 3-space2006

    • Author(s)
      M.Kato
    • Journal Title

      Tokyo J. Math. 29

      Pages: 209-232

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Nonlinear Cauchy problem with small analytic data2006

    • Author(s)
      H.Yamane
    • Journal Title

      Proc.Amer.Math.Soc. 134-11

      Pages: 3353-3361

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The structure of positive definite Fourier ultra hyperfunctions2006

    • Author(s)
      K.Yoshino, M.Suwa
    • Journal Title

      Complex Variables and Elliptic Equations 51-5&6

      Pages: 611-624

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The Bochner-Schwartz theorem for Fourier ultra hyperfunctions2006

    • Author(s)
      K.Yoshino, M.Suwa
    • Journal Title

      Integral Transforms Spec.Funct. 17-2&3

      Pages: 109-114

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Compact quotients of large domain in a complex projective 3-space2006

    • Author(s)
      M.Kato
    • Journal Title

      Tokyo J.Math. 29

      Pages: 209-232

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Removable singularities of solutions of nonlinear totally characteristic type partial differential equations2005

    • Author(s)
      H.Tahara
    • Journal Title

      数理解析研究所講究録 1431

      Pages: 116-122

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Uniqueness of the solution of nonlinear totally characteristic partial differential equations2005

    • Author(s)
      H.Tahara
    • Journal Title

      J. Math. Soc. Japan 57・4

      Pages: 1045-1065

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Borel summability of formal solutions of some first order singular PDEs and normal forms of vector fields2005

    • Author(s)
      S.Ouchi
    • Journal Title

      J. Math. Soc. Japan 57・2

      Pages: 415-460

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Normal forms of some vector fields by transformations with Borel summable functions2005

    • Author(s)
      S.Ouchi
    • Journal Title

      数理解析研究所講究録 1431

      Pages: 81-90

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Uniqueness of the solution of nonlinear totally characteristic partial differential equations2005

    • Author(s)
      H.Tahara
    • Journal Title

      J.Math.Soc.Japan 57-4

      Pages: 1045-1065

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields2005

    • Author(s)
      S.Ouchi
    • Journal Title

      J.Math.Soc.Japan 57-2

      Pages: 415-460

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the singularities of solutions of nonlinear partial differential equations in the complex domain2004

    • Author(s)
      H.Tahara
    • Journal Title

      数理解析研究所講究録 1397

      Pages: 102-111

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On holomorphic maps into compact non-Ka'hler manifolds2004

    • Author(s)
      M.Kato, N.Okada
    • Journal Title

      Ann. Inst. Fourier (Grenoble) 54・6

      Pages: 1827-1854

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On holomorphic maps into compact non-Keahler manifolds2004

    • Author(s)
      M.Kato, N.Okada
    • Journal Title

      Ann.Inst.Fourier (Grenoble) 54-6

      Pages: 1827-1854

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2008-05-27  

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