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On Gevrey property and asymptotic analysis for divergent solutions of analytic partial differential equations

Research Project

Project/Area Number 07454024
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionNagoya University

Principal Investigator

MIYAKE Masatake  Graduate School of Polymathematics, Nagoya University, Professor, 大学院・多元数理科学研究科, 教授 (70019496)

Co-Investigator(Kenkyū-buntansha) KOZONO Hideo  Graduate School of Polymathematics, Nagoya University, Associate Professor, 大学院・多元数理科学研究科, 助教授 (00195728)
OGAWA Takayoshi  Graduate School of Polymathematics, Nagoya University, Associate, 大学院・多元数理科学研究科, 助教授 (20224107)
NAKAMURA Shu  Graduate School of Polymathematics, Nagoya University, Professor, 大学院・多元数理科学研究科, 教授 (50183520)
OHSAWA Takeo  Graduate School of Polymathematics, Nagoya University, Professor, 大学院・多元数理科学研究科, 教授 (30115802)
AOMOTO Kazuhiko  Graduate School of Polymathematics, Nagoya University, Professor, 大学院・多元数理科学研究科, 教授 (00011495)
大和 一夫  名古屋大学, 大学院・多元数理科学研究科, 助教授 (30022677)
Project Period (FY) 1995 – 1996
Project Status Completed (Fiscal Year 1996)
Budget Amount *help
¥4,700,000 (Direct Cost: ¥4,700,000)
Fiscal Year 1996: ¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 1995: ¥2,400,000 (Direct Cost: ¥2,400,000)
Keywordssingular point / divegent solution / Borel summability / Gevrey index / Goursat problem / Toeplitz operator / index theorem / Freholmness / 形式的巾級数 / 偏微分作用素 / テプリッツ作用素 / 熱方程式 / 形式巾級数 / ジュブレイ度 / 不確定度 / 特異偏微分方程式 / 漸近解析 / ボテル総和
Research Abstract

The purpose of this project is to study Gevrey property and asymptotic analysis for divergent power series solutions of analytic partial differential equations.
The first result is that we established the Toeplitz operator method in analytic partial differential equations which enables us to give a precise notion of Fredholmness in the Goursat problem in various Gevrey spaces which has not been studied in explicite way in early studies. We proved further that an index formula for ordinary differnetial operator on Gevrey space is nothing but the geometrical index fornmula for a Toeplitz sumbol associated with the Gevrey filtration n the ring of ordinary differential operators.
The second result is that we gave a necessary and sufficient condition for the Borel summability for divergent power series solution of the Cauchy problem of the heat equation, and we proved that the Borel sum is just expressed by an integral with the heat kernel. The condition for the C data we obtained is the well known condition for the uniqueness of solutions of the Cauchy problem. In proving this we made clear that the problem of Borel summability in partial differential equations is not local property of solutions, whereas the notion of the Borel summability is only local one. We also made clear that this problem provides a new kind of problems in partial differential equations.

Report

(3 results)
  • 1996 Annual Research Report   Final Research Report Summary
  • 1995 Annual Research Report
  • Research Products

    (26 results)

All Other

All Publications (26 results)

  • [Publications] Miyake, M. and Yoshino, M.: "Toeplitz operators and an index theorem for differntial operators on Geusey spaces" Funkcialaj Ekvac.38. 329-342 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Miyake, M. and Yoshino, M.: "Fredholm property of partial differential operators of irregular singular type" Ark. Mat.33. 323-341 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Aomoto, K. and Kato, K.: "Gauss decomposition of connection matrices for symmetric A-type Jackson integrals" Selecta Math. New Series.1. 623-666 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Dledrick, K. and Ohsawa, T.: "An estimate for the Bergman distance on psendoconvex domains" Ann. of Math.141. 181-190 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Ohsawa, T.: "On the analytic structure of certain infinite dimensional Teichmuller spaces" Nagoya Math. J.141. 143-155 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Kozono, H. and Nakao, M.: "On strong Periodic solutions of the Navier-Stokes equations in unbounded domain" Tohoku Math. J.48. 33-50 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Miyake, M.and Yoshino, M.: "Toeplitz operators and an index theorem for differential operators Gevrey apaces" Funckcialay Ekvac.38. 329-342 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Miyake, M.and Yoshino, M.: "Fredholm property of partial differential operators of irregular singular type" Ark.Mat.33. 323-341 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Aomoto.K,and Kato, Y: "Gauss decomposition matrices for symmetric A-type Jackson integrals" Selecta Math.New Series.1. 623-666 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Diedrich, K.and Ohsawa, T.: "An estimate for Bergman distance on pseudoconvex domains" Ann.of Math. 141. 181-190 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Ohsawa, T.: "On the analytic structure of certain in finite dimensional Teichmiller spaces" Nagoya Math.J.141. 143-155 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Nakamura, S.: "Gaussian decay estimates for eigenfunctions of magnetic Schrodinger oprators" Comm.P.D.E.21. 993-1006 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Ogawa, T.and Suzuki, T.: "Non linear elliptic equations with the critical growth related with Trudinger inequality" Asymptotic Anal.12. 25-40 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Kozono, H.and Shor, H.: "Remark on uniqueness of weak solution of the Navier-Stokes equations in unbounded domains" Analysis. 16. 25-271 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Miyake,M.and Yoshino,M.: "Necessary and sufficient conditions for Fredholmness of irregular singufar partial differential operators" Procec dings of the sixth international colloguim on diferestial equations,plo〓div,Balagaria. 401-408 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Miyake,M.: "Toeplitz operators in the analytic Goursat problem" 数理解析研究所講究録. (to appear). (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] 大沢健夫: "L^2評価式の複素幾何への応用" 数学(岩波書店). 48・2. 142-161 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Nakamura,S.: "Gaussian decay estimates for eigenfunctions of magnetic Schrodinger operators" Comm.P.D.E.21. 993-1006 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Ogawa,T.and Suzuki,T.: "Nonlinear elliptic equations with the critical growth related to the Trudinger in equality" Asynptotic Anal.12. 25-40 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Kozono,H.and Sohr,H.: "Remark on uniqueness of weak solution of the Novier-Stokes equations in unbounded domains" Analysis. 16. 255-271 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] M,Miyake and M,Yoshino: "Toeplitz operators and an index theorem for differential operators on Gevrey spaces" Funckcialaj Ekvacioj. 38. 329-342 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] M,Miyake and M,Yoshino: "Fredholm property of partial differential operators of irregular singular type" Arkiv for Matematik. (印刷中).

    • Related Report
      1995 Annual Research Report
  • [Publications] A,Jensen and S,Nakamura: "L^P-mapping rroperties of functions of Schrodunger operators and their applications to scattereng theory" Journal of the Mathematical Society of Japan. 47. 253-273 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] K,Yamato: "A class of Riemannian manifold with integrable geodesic flows" Journal of the Mathematical Society of Japan. 47. 719-733 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] H.Kozono,Y.Maeda and H.Naito: "Global solution for the Yang-Mills gradient flow on 4-mauifold" Nagoya Mathematical Journal. 139. 93-128 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] T.Ogawa and T.Suzuki: "Microscopic asymptotics for solutons of some semilinear elliptic equations" Nagoya Mathematical Journal. 138. 33-50 (1995)

    • Related Report
      1995 Annual Research Report

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Published: 1995-04-01   Modified: 2016-04-21  

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