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Necessary Conditions for Strong Hyperbolicity

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Hyperbolic Systems with Analytic Coefficients

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2097))

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Abstract

In this chapter we study the Cauchy problem for a first order differential operator defined near the origin of \({\mathbb{R}}^{n+1}\). We give necessary conditions on L for the Cauchy problem for \(P = L + B\) to be C ∞ well posed for any lower order term B, that is necessary conditions for L to be strongly hyperbolic. Denoting by h and M = (m ij ) the determinant and the cofactor matrix of L(x, ξ) respectively, this necessary condition for strong hyperbolicity is roughly stated that if L is strongly hyperbolic then the Cauchy problem for scalar operators h + m ij is C ∞ well posed for all m ij . In particular, from this condition we see that if L is strongly hyperbolic then at a multiple characteristic point (x, ξ) the maximal size of Jordan blocks in the Jordan canonical form of L(x, ξ) is at most two, which corresponds to a well known Ivrii–Petkov necessary condition for scalar strongly hyperbolic operators. We also see that if the multiple characteristic point (x, ξ) is involutive then L(x, ξ) is diagonalizable for L to be strongly hyperbolic which recovers the necessary condition when L is a system with characteristics of constant multiplicity.

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Authors and Affiliations

  1. Department of Mathematics Graduate School of Science, Osaka University, Toyonaka, Osaka, Japan

    Tatsuo Nishitani

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  1. Tatsuo Nishitani
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Nishitani, T. (2014). Necessary Conditions for Strong Hyperbolicity. In: Hyperbolic Systems with Analytic Coefficients. Lecture Notes in Mathematics, vol 2097. Springer, Cham. https://doi.org/10.1007/978-3-319-02273-4_2

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