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Guy Métivier
$L^2$ well-posed Cauchy problems and symmetrizability of first order systems
(Problèmes de Cauchy bien posés dans $L^2$ et symétrisabilité pour les systèmes du premier ordre)
Journal de l'École polytechnique — Mathématiques, 1 (2014), p. 39-70, doi: 10.5802/jep.3
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Class. Math.: 35L
Keywords: Systems of partial differential equations, Cauchy problem, hyperbolicity, strong hyperbolicity, symmetrizers, energy estimate, local uniqueness, finite speed of propagation

Résumé - Abstract

The Cauchy problem for first order system $L(t, x, \partial _t, \partial _x)$ is known to be well-posed in $L^2$ when it admits a microlocal symmetrizer $S(t,x, \xi )$ which is smooth in $\xi $ and Lipschitz continuous in $(t, x)$. This paper contains three main results. First we show that a Lipschitz smoothness globally in $(t,x, \xi )$ is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of full symmetrizers having the same smoothness. This notion was first introduced in [FL67]. This is the key point to prove the third result saying that the existence of microlocal symmetrizer is preserved if one changes the direction of time, implying local uniqueness and finite speed of propagation.

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