Advertisement

Introduction

  • Tatsuo Nishitani
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2202)

Abstract

In this chapter, after quickly reviewing the background which motivates to prepare this monograph we state basic facts on pseudodifferential operators without proofs, except for a few results. We then recall basic results on the Cauchy problem for differential operators with double characteristics, including basic notion and results about double characteristics of hyperbolic polynomials and hyperbolic quadratic forms which will be used throughout the monograph.

References

  1. 13.
    J. Chazarain, Opérateurs hyperboliques à caractéristiques de multiplicité constant. Ann. Inst. Fourier (Grenoble) 24, 173–202 (1974)MathSciNetCrossRefMATHGoogle Scholar
  2. 15.
    F. Colombini, E. De Giorgi, S. Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Sc. Norm. Super. Pisa 4, 511–559 (1979)MATHGoogle Scholar
  3. 16.
    F. Colombini, E. Jannelli, S. Spagnolo, Well posedness in the Gevrey class of the Cauchy problem for a non strictly hyperbolic equation with coefficients depending on time. Ann. Sc. Norm. Super. Pisa 10, 291–312 (1983)MATHGoogle Scholar
  4. 17.
    F. Colombini, T. Nishitani, N. Orrù, Some well-posed Cauchy problem for second order hyperbolic equations with two independent variables. Osaka J. Math. 48, 647–673 (2011)MathSciNetMATHGoogle Scholar
  5. 18.
    P. D’Ancona, Well posedness in C for a weakly hyperbolic second order equation. Rend. Sem. Mat. Univ. Padova 91, 65–83 (1994)MathSciNetMATHGoogle Scholar
  6. 23.
    H. Flaschka, G. Strang, The correctness of the Cauchy problem. Adv. Math. 6, 347–379 (1971)MathSciNetCrossRefMATHGoogle Scholar
  7. 24.
    L. Gårding, Solution directe du problème de Cauchy pour les équations hyperboliques, in La théorie des équations aux dérivées partielles, Nancy, 1956. Coll. Int. CNRS, vol. 71 (CNRS, Nancy, 1956), pp. 71–90Google Scholar
  8. 25.
    L. Gårding, Cauchy’s Problem for Hyperbolic Equations (University of Chicago, Chicago, 1957)Google Scholar
  9. 26.
    L. Gårding, Hyperbolic differential operators, in Perspectives in Mathematics (Birkhäuser, Basel, 1984), pp. 215–247Google Scholar
  10. 27.
    L. Gårding, Hyperbolic equations in the twentieth century, in Matériaux pour l’histoire des Mathématiques au XXe, Papers from the Colloquium, Nice, 1996, Semin. Congr., vol. 3 (Soc. Math. France, Paris, 1998), pp. 37–68Google Scholar
  11. 30.
    J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Dover Publications, New York, 1953)MATHGoogle Scholar
  12. 31.
    Q. Han, Energy estimates for a class of degenerate hyperbolic equations. Math. Ann. 347, 339–364 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 32.
    L. Hörmander, The Cauchy problem for differential equations with double characteristics. J. Anal. Math. 32, 118–196 (1977)MathSciNetCrossRefMATHGoogle Scholar
  14. 33.
    L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Grundlehren Math. Wiss., vol. 274 (Springer, Berlin, 1983)Google Scholar
  15. 34.
    L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Grundlehren Math. Wiss., vol. 274 (Springer, Berlin, 1985)Google Scholar
  16. 35.
    L. Hörmander, Quadratic hyperbolic operators, in Microlocal Analysis and Applications, ed. by L. Cattabriga, L. Rodino (Springer, Berlin, 1989), pp. 118–160Google Scholar
  17. 37.
    V.Ja. Ivrii, Sufficient conditions for regular and completely regular hyperbolicity. Tr. Mosk. Mat. Obs. 33, 3–65 (1975); English transl., Trans. Moscow Math. Soc. 33, 1–65 (1978)Google Scholar
  18. 42.
    V.Ja. Ivrii, Linear hyperbolic equations, in Partial Differential Equations IV, ed. by Yu.V. Egorov, M.A. Shubin. Encyclopaedia of Mathematical Sciences, vol. 33 (Springer, New York, 1988), pp. 149–235Google Scholar
  19. 43.
    V.Ja. Ivrii, V.M. Petkov, Necessary conditions for the correctness of the Cauchy problem for non-strictly hyperbolic equations. Uspehi Mat. Nauk 29, 3–70 (1974); English transl., Russ. Math. Surv. 29, 1–70 (1974)Google Scholar
  20. 44.
    N. Iwasaki, The Cauchy problem for effectively hyperbolic equations (a special case). J. Math. Kyoto Univ. 23, 503–562 (1983)MathSciNetCrossRefMATHGoogle Scholar
  21. 45.
    N. Iwasaki, The Cauchy problem for effectively hyperbolic equations (a standard type). Publ. Res. Inst. Math. Sci. 20, 551–592 (1984)MathSciNetCrossRefMATHGoogle Scholar
  22. 46.
    N. Iwasaki, The Cauchy problem for effectively hyperbolic equations. Sugaku Exposition 36, 227–238 (1984)MATHGoogle Scholar
  23. 47.
    N. Iwasaki, The Cauchy problem for effectively hyperbolic equations (general case). J. Math. Kyoto Univ. 25, 727–743 (1985)MathSciNetCrossRefMATHGoogle Scholar
  24. 49.
    N. Iwasaki, Bicharacteristic curves and well-posedness for hyperbolic equations with noninvolutive multiple characteristics. J. Math. Kyoto Univ. 34, 41–46 (1994)MathSciNetCrossRefMATHGoogle Scholar
  25. 50.
    K. Kajitani, T. Nishitani, The Hyperbolic Cauchy Problem. Lecture Notes in Mathematics, vol. 1505 (Springer, Berlin, 1991)Google Scholar
  26. 51.
    K. Kasahara, M. Yamaguti, Strongly hyperbolic systems of linear partial differential equations with constant coefficients. Mem. Coll. Sci. Univ. Kyoto, Ser. A. Math. 33, 1–23 (1960/1961)Google Scholar
  27. 53.
    A. Lax, On Cauchy’s problem for partial differential equations with multiple characteristics. Commun. Pure Appl. Math. 9, 135–169 (1956)MathSciNetCrossRefMATHGoogle Scholar
  28. 54.
    P.D. Lax, Asymptotic solutions of oscillatory initial value problem. Duke Math. J. 24, 627–646 (1957)MathSciNetCrossRefMATHGoogle Scholar
  29. 55.
    J. Leray, Hyperbolic Differential Equations (The Institute for Advanced Study, Princeton, NJ, 1953)Google Scholar
  30. 58.
    E.E. Levi, Carateristiche multiple e problema di Cauchy. Ann. Mat. Pura Appl. 16, 161–201 (1909)CrossRefMATHGoogle Scholar
  31. 60.
    A. Melin, Lower bounds for pseudo-differential operators. Ark. Mat. 9, 117–140 (1971)MathSciNetCrossRefMATHGoogle Scholar
  32. 61.
    R. Melrose, The Cauchy problem for effectively hyperbolic operators. Hokkaido Math. J. 12, 371–391 (1983)MathSciNetMATHGoogle Scholar
  33. 62.
    R. Melrose, The Cauchy problem and propagation of singularities, in Seminar on Nonlinear Partial Differential Equations, Papers from the Seminar, Berkeley, CA, 1983, ed. by S.S.Chern. Mathematical Sciences Research Institute Publications, vol. 2 (Springer, Berlin, 1984), pp. 185–201Google Scholar
  34. 63.
    S. Mizohata, Systèmes hyperboliques. J. Math. Soc. Jpn. 11, 205–233 (1959)CrossRefMATHGoogle Scholar
  35. 64.
    S. Mizohata, Note sur le traitement par les opérateurs d’intégrale singulière du problème de Cauchy. J. Math. Soc. Jpn. 11, 234–240 (1959)CrossRefMATHGoogle Scholar
  36. 65.
    S. Mizohata, Some remarks on the Cauchy problem. J. Math. Kyoto Univ. 1, 109–127 (1961)MathSciNetCrossRefMATHGoogle Scholar
  37. 66.
    S. Mizohata, The Theory of Partial Differential Equations (Cambridge University Press, Cambridge, 1973)MATHGoogle Scholar
  38. 67.
    S. Mizohata, Y. Ohya, Sur la condition de E.E.Levi concernant des équations hyperboliques. Publ. Res. Inst. Math. Sci. 4, 511–526 (1968/1969)Google Scholar
  39. 68.
    S. Mizohata, Y. Ohya, Sur la condition d’hyperbolicité pour les équations à caractéristiques multiples. II. Jpn. J. Math. 40, 63–104 (1971)CrossRefMATHGoogle Scholar
  40. 69.
    S. Mizohata, Y. Ohya, M. Ikawa, Comments on the development of hyperbolic analysis, in Hyperbolic Equations and Related Topics (Katata/Kyoto, 1984) (Academic, Boston, MA, 1986), pp. ix–xxxivGoogle Scholar
  41. 70.
    T. Nishitani, The Cauchy problem for weakly hyperbolic equations of second order. Commun. Partial Differ. Equ. 5, 1273–1296 (1980)MathSciNetCrossRefMATHGoogle Scholar
  42. 71.
    T. Nishitani, On the finite propagation speed of wave front sets for effectively hyperbolic operators. Sci. Rep. College Gen. Ed. Osaka Univ. 32(1), 1–7 (1983)MathSciNetMATHGoogle Scholar
  43. 73.
    T. Nishitani, A necessary and sufficient condition for the hyperbolicity of second order equations with two independent variables. J. Math. Kyoto Univ. 24, 91–104 (1984)MathSciNetCrossRefMATHGoogle Scholar
  44. 74.
    T. Nishitani, Local energy integrals for effectively hyperbolic operators. I, II. J. Math. Kyoto Univ. 24, 623–658, 659–666 (1984)Google Scholar
  45. 77.
    T. Nishitani, Note on a paper of N.Iwasaki: “Bicharacteristic curves and well-posedness for hyperbolic equations with noninvolutive multiple characteristics”. J. Math. Kyoto Univ. 38, 415–418 (1998)Google Scholar
  46. 79.
    T. Nishitani, Effectively hyperbolic Cauchy problem, in Phase Space Analysis of Partial Differential Equations, vol. II, ed. F. Colombini, L. Pernazza. Publ. Cent. Ric. Mat. Ennio Giorgi (Scuola Norm., Pisa, 2004), pp. 363–449Google Scholar
  47. 88.
    O.A. Oleinik, On the Cauchy problem for weakly hyperbolic equations. Commun. Pure Appl. Math. 23, 569–586 (1970)MathSciNetCrossRefGoogle Scholar
  48. 91.
    I.G. Petrovsky, Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen. Mat. Sb. N.S. 2(44), 815–870 (1937)Google Scholar
  49. 94.
    G. Strang, On strong hyperbolicity. J. Math. Kyoto Univ. 6, 397–417 (1967)MathSciNetCrossRefMATHGoogle Scholar
  50. 96.
    S. Wakabayashi, On the Cauchy problem for hyperbolic operators of second order whose coefficients depend on the time variable. J. Math. Soc. Jpn. 62, 95–133 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Tatsuo Nishitani
    • 1
  1. 1.Department of MathematicsOsaka UniversityToyonakaJapan

Personalised recommendations