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Symmetric systems and strongly hyperbolic systems

Research Project

Project/Area Number 07454027
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

NISHITANI Tatsuo  Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80127117)

Co-Investigator(Kenkyū-buntansha) TAKAHASHI Satoshi  Osaka University, Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (70226835)
TAKEGOSHI Kensho  Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20188171)
SAKUMA Makoto  Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (30178602)
NAMBA Makoto  Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60004462)
USUI Sampei  Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90117002)
杉本 充  大阪大学, 理学部, 講師 (60196756)
榎 一郎  大阪大学, 理学部, 助教授 (20146806)
長瀬 道弘  大阪大学, 理学部, 教授 (70034733)
Project Period (FY) 1995 – 1996
Project Status Completed (Fiscal Year 1996)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1996: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 1995: ¥1,600,000 (Direct Cost: ¥1,600,000)
Keywordssymmetrizable system / strong hyperbolicity / involutive / non degenerate / hyperbolic perturbation / localization / 対称化 / 非退化特異点
Research Abstract

Our research project has been organized as follows :
(i) Clarify the structure of strongly hyperbolic systems which can not be symmetrizable.
(ii) Study the stability of symmetrizable systems under hyperbolic perturbations.
As for (i) we got the following results. Let L be a m*m system of patial differential operators of first order. Denoting by h the determinant of the principal symbol of L the general picture of our necessary condition for strong hyperbolicity of L could be stated as : if L is strongly hyperbolic then the Cauchy problem for h+k is correctly posed for every m-1-th minor k of L.Moreover if the reference characteristic z is involutive and the system is strongly hyperbolic then KerL (z) * ImL (z) = {0}. Thus the Taylor expansion of L along KerL starts with a linear term L_Z called the localization of L.Let z, w be characteristics of the original system and of the localization respectively. If (z, w) is involutive then KerL_z (w) * ImL_z (w) = {0}.
As for (ii) we formulated non degenerate characteristic for first order system. We say that z is non degenerate if KerL (z) * ImL (z) = {0}, the dimension of L_Z is maximal and L_Z (w) is diagonalizable for every w. Then the main result is that every hyperbolic system is symmetrizable near non degenerate characteristic. From this we can derive stability of non degenerate characteristics. Namely we can not remove non degenerate characteristics by hyperbolic perturbations.
We proceed this study and got the following result. Let L be a m*m sysmmetric first order hyperbolic system. Then if the dimension of L is greater than m (m+1) /2-m+2 then genericaly, every hyperbolic perturbation is trivial that is every hyperbolic system near L can be symmetrized.

Report

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

    (16 results)

All Other

All Publications (16 results)

  • [Publications] T. Nishitani: "Symmetrization of hyperbolic systems with non degenerate characteristic" J. Func. Analysis. 132・2. 251-272 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T. Nishitani: "On localization of a class of strongly hyperbolic systems" Osaka J. Math.32・1. 41-69 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T. Nishitani M. Takayama: "A characteristic initial bourdary value problem for a pmenetric positux pstem" Hokkaido Math. J.25・1. 167-182 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T. Nishitani: "Stubility of symmetric systems urder hyperbolic perturbations" Hokkaido Math. J.26・1. (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T. Nishitani, M. Takayama: "Regularity of solutions to characteristic boundary value problem" in Geometrical optics and related topics. (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Nishitani: "Symmetrization of hyperbolic systems with non degenerate characteristics" J.Func.Analysis. 132・2. 251-272 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Nishitani: "On localization of a class of strongly hyperbolic systems" Osaka J.Math.25・1. 41-69 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Nishitani, M.Takayama: "A characteristic initial boundary value problem for a symmetric positive system" Hokkaido Math.J.25・1. 167-182 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Nishitani: "Stability of symmetric systems under hyperbolic perturbations" Hokkaido Math.J.26・1. (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Nishitani, M.Takayama: "Regularity of solutions to characteristic boundary value problem" Geometrical optics and related topics. (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Nishitani: "Synmetrization of hyperbolic systems with non degenerate draraeterastics" J.Func.Analysis. 132・2. 251-272 (1995)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Nishitani: "On localization of a class of strongly hyperbolic systems" Osaka J.Math.32・1. 41-69 (1995)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Nishitani M.Takayama: "A characteristic initial boundary value problem for a synmetric positive systems" Hokkaido,Math.J.25・1. 167-182 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Nishitani: "Stability of synmetric systems under hyperbolic perturbations" Hokkaido,Math.J.26・1. (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Nishitani M.Takayama: "Regularity of solutions to characteristic boundary value problem" in Geometrical optics and related topics. (1977)

    • Related Report
      1996 Annual Research Report
  • [Publications] Tatsuo Nishitani: "On localizations of a class of strongly hyperbolic systems" Osaka Journal of Mathematics. 32. 41-69 (1955)

    • Related Report
      1995 Annual Research Report

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

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