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2001 Fiscal Year Final Research Report Summary

Theory of hyperloobic systems

Research Project

Project/Area Number 11440046
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

NISHITANI Tatsuo  Osaka University, Grad. Sch. of Sci., Professor, 大学院・理学研究科, 教授 (80127117)

Co-Investigator(Kenkyū-buntansha) KAJITANI Kunihiko  Tsukuba University, Fac. of Math., Professor, 数学系, 教授 (00026262)
OKAJI Takashi  Kyoto University, Grad. Sch, of Sci., Associate Professor, 大学院・理学研究科, 助教授 (20160426)
MATSUMURA Akitaka  Osaka University, Grad. Sch. of Sci., Professor, 大学院・理学研究科, 教授 (60115938)
SHIBATA Yoshihiro  Waseda University, Fac. of Sci., Professor, 理工学部, 教授 (50114088)
ICHINOSE Wataru  Shinshu University, Fac. of Sci., Professor, 理学部, 教授 (80144690)
Project Period (FY) 1999 – 2001
Keywordshyperbolicity / strong hyperbolicity / pseudosymmetric / noncommutative determinant / symmetrizability / reduced dimension / Cauchy problem / well posedness
Research Abstract

We have obtained a lot of results. We refer here some of the main results.
1. For 2 x 2 systems with two independent variables, we obtained a necessary and sufficient condition in order that the Cauchy problem is well posed. The condition is expressed using the Newton polyhedron. In this study we found a peculiar example which is strictly hyperbolic apart from the initial plane for that the Cauchy problem is not well posed for any lower order term.
2. We introduced a new notion "pseudo-symmetric hyperbolic systems" which extends the symmetrizable hyperbolic systems. We proved that the Cauchy problem for pseudo-symmetric hyperbolic systems with one space variable is well posed. The question is still open for pseudo-symmetric systems with several space variables.
3. We succeeded in obtaining a necessary condition on lower order terms for the Cauchy problem is well posed for general, hyperbolic systems using the determinant on a non commutative field where the localization lives : the. leading part of the non commutative determinant of the localization of the total symbol coincides with the principal part of the classical determinant of the principal symbol.
4. The symmetrizability of the frozen system at every space point implies the symmetrizability of the original systm if the reduced dimension is enough high. In particular if the every frozen system is stringly hyperbolic then the original system is also strongly hyperbolic if the reduced dimension is high.

  • Research Products

    (19 results)

All Other

All Publications (19 results)

  • [Publications] Nishitani, Tatsuo: "On pseudo symmetric systems with one space variable"Ann. Scu. Norm. Sup. Pisa. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Nishitani, Tatsuo: "Necessary conditions for local solvability for a class of differential systems"Comm. P.D.E.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Nishitani, Tatsuo: "Necessary conditions for hyperbolic systems"Bull. Sci. Math.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Nishitani, Tatsuo: "Hyperbolicity for systems"Proceeding 3rd ISSAC International Corgress.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Nishitani, Tatsuo: "Smoothly Symmetyizaled systems and the reduced dimeusion"Tsukaba J. Math.. 25. 165-177 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Nishitani, Tatsuo: "Regularity of solutions to non uniformly characteristic boundary value problems for symmetvic systems"Comm. P.D.E.. 25. 987-1018 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mishitani, Tatsuo: "Hyperbolic Equations with Double Characteristics"Istituti Editovialie Poligrafici Interraziorali, Pisa, Roma. 88 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Nishitani, tatsuo: "On pseudo symmetric systems with one space variable"Ann. Scu. Norm. Sup. Pisa. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Nishitani, Tatsuo: "Necessary conditions for local solvability for a class of differeutial systems"Comm. P. D. E.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Nishitani, Tatsuo: "Necessary conditions for hyperbolic systems"Bull. Sci. Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Nishitani, Tatsuo: "Hyperbolicity for systems"Proceedings 3rd ISSAC International Congress. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Nishitani, Tatsuo: "Smoothly symmetrizable systems and the reduced dimension"Tsukuba J. Math.. 25. 165-177 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Nishitani, Tatsuo: "On second order weakly hyperbolic equations and gevrey classes"Rend. Inst. Mat. Univ. Trieste. 31. 31-50 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Nishitani, Tatsuo: "Regularity of solutions to non uniformly characteristic boundary value problems for symmetric systems"Comm. P. D. E.. 25. 987-1018 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Matsumura, Akitaka: "Convergence to travelling fronts of solutions of the p-systems with viscosity"Arch. Rational. Mech. Anal.. 146. 1-22 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kajitani, kunihiko: "propagation of analyticity of solutions to the Cauchy problem"Ann. Scu. Novm. Sup. Pisa. 27. 1-17 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Okaji, Takashi: "Strong unique continuation property for the Divac equation"Publ. R. I. M. S. Kyoto Univ.. 35. 825-846 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shibata, Yoshihiro: "An exterior initial boundary value problem for Navier-Stokes equation"Qurt. Appl. Math.. 57. 117-155 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Doi, Shin-ichi: "Smoothing effects for Schrodinger evolution equation and global behavior of geodesic flow"Math. Ann.. 318. 355-389 (2000)

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

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Published: 2003-09-17  

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