Hyperbolicity of two by two systems with two independent variables

Tatsuo Nishitani

Journées équations aux dérivées partielles (1998)

  • page 1-12
  • ISSN: 0752-0360

Abstract

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We study the simplest system of partial differential equations: that is, two equations of first order partial differential equation with two independent variables with real analytic coefficients. We describe a necessary and sufficient condition for the Cauchy problem to the system to be C infinity well posed. The condition will be expressed by inclusion relations of the Newton polygons of some scalar functions attached to the system. In particular, we can give a characterization of the strongly hyperbolic systems which includes a fortiori symmetrizable systems.

How to cite

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Nishitani, Tatsuo. "Hyperbolicity of two by two systems with two independent variables." Journées équations aux dérivées partielles (1998): 1-12. <http://eudml.org/doc/93352>.

@article{Nishitani1998,
abstract = {We study the simplest system of partial differential equations: that is, two equations of first order partial differential equation with two independent variables with real analytic coefficients. We describe a necessary and sufficient condition for the Cauchy problem to the system to be C infinity well posed. The condition will be expressed by inclusion relations of the Newton polygons of some scalar functions attached to the system. In particular, we can give a characterization of the strongly hyperbolic systems which includes a fortiori symmetrizable systems.},
author = {Nishitani, Tatsuo},
journal = {Journées équations aux dérivées partielles},
keywords = { well-posedness; real analytic coefficients; Newton polygons},
language = {eng},
pages = {1-12},
publisher = {Université de Nantes},
title = {Hyperbolicity of two by two systems with two independent variables},
url = {http://eudml.org/doc/93352},
year = {1998},
}

TY - JOUR
AU - Nishitani, Tatsuo
TI - Hyperbolicity of two by two systems with two independent variables
JO - Journées équations aux dérivées partielles
PY - 1998
PB - Université de Nantes
SP - 1
EP - 12
AB - We study the simplest system of partial differential equations: that is, two equations of first order partial differential equation with two independent variables with real analytic coefficients. We describe a necessary and sufficient condition for the Cauchy problem to the system to be C infinity well posed. The condition will be expressed by inclusion relations of the Newton polygons of some scalar functions attached to the system. In particular, we can give a characterization of the strongly hyperbolic systems which includes a fortiori symmetrizable systems.
LA - eng
KW - well-posedness; real analytic coefficients; Newton polygons
UR - http://eudml.org/doc/93352
ER -

References

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  1. [1] V. YA. IVRII AND V.M. PETKOV, Necessary conditions for the Cauchy problem for non strictly hyperbolic equations to be well posed, Russian Math. Surveys, 29 1974 1-70. Zbl0312.35049
  2. [2] V. YA. IVRII, Linear Hyperbolic Equations, In Partial Differential Equations IV, Yu. V. Egorov, M.A. Shubin (eds.), Springer-Verlag 1993. 
  3. [3] P.D. LAX, Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 1957 627-646. Zbl0083.31801MR20 #4096
  4. [4] W. MATSUMOTO, On the conditions for the hyperbolicity of systems with double characteristic roots I, J. Math. Kyoto Univ., 21 1981 47-84. Zbl0471.35052MR82m:35096a
  5. [5] W. MATSUMOTO, On the conditions for the hyperbolicity of systems with double characteristic roots II, J. Math. Kyoto Univ., 21 1981 251-271. Zbl0487.35057MR82m:35096b
  6. [6] S. MIZOHATA, Some remarks on the Cauchy problem, J. Math. Kyoto Univ., 1 1961 109-127. Zbl0104.31903MR30 #353
  7. [7] T. NISHITANI, The Cauchy problem for weakly hyperbolic equations of second order, Comm. P.D.E., 5 1980 1273-1296. Zbl0497.35053MR82i:35107
  8. [8] T. NISHITANI, A necessary and sufficient condition for the hyperbolicity of second order equations with two independent variables, J. Math. Kyoto Univ., 24 1984 91-104. Zbl0552.35049MR85e:35075
  9. [9] P.D'ANCONA AND S. SPAGNOLO, On pseudosymmetric hyperbolic systems, preprint 1997. Zbl1014.35055MR99k:35113
  10. [10] J. VAILLANT, Systèmes hyperboliques à multiplicité constante et dont le rang peut varier, In Recent developments in hyperbolic equations, pp. 340-366, L. Cattabriga, F. Colombini, M.K.V. Murthy, S. Spagnolo (eds.), Pitman Research Notes in Math. 183, Longman, 1988. Zbl0723.35043MR90e:35106

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