Geodesic
Hyperbolicity of two by two systems with two independent variables
Journées équations aux dérivées partielles (1998), article no. 10, 12 p.

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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.

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Nishitani, Tatsuo. Hyperbolicity of two by two systems with two independent variables. Journées équations aux dérivées partielles (1998), article  no. 10, 12 p. http://geodesic.mathdoc.fr/item/JEDP_1998____A10_0/

[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. | Zbl

[2] V. Ya. Ivrii, Linear Hyperbolic Equations, In Partial Differential Equations IV, Yu. V. Egorov, M.A. Shubin (eds.), Springer-Verlag 1993.

[3] P.D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 1957 627-646. | Zbl | MR

[4] W. Matsumoto, On the conditions for the hyperbolicity of systems with double characteristic roots I, J. Math. Kyoto Univ., 21 1981 47-84. | Zbl | MR

[5] W. Matsumoto, On the conditions for the hyperbolicity of systems with double characteristic roots II, J. Math. Kyoto Univ., 21 1981 251-271. | Zbl | MR

[6] S. Mizohata, Some remarks on the Cauchy problem, J. Math. Kyoto Univ., 1 1961 109-127. | Zbl | MR

[7] T. Nishitani, The Cauchy problem for weakly hyperbolic equations of second order, Comm. P.D.E., 5 1980 1273-1296. | Zbl | MR

[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. | Zbl | MR

[9] P.D'Ancona AND S. Spagnolo, On pseudosymmetric hyperbolic systems, preprint 1997. | Zbl | MR

[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. | Zbl | MR