$L^2$ well-posed Cauchy problems and symmetrizability of first order systems

Métivier, Guy

Journal de l’École polytechnique - Mathématiques, Tome 1 (2014), pp. 39-70.

Le problème de Cauchy est bien posé dans $L^2$ pour les systèmes du premier ordre $L(t,x,{\partial }_t,{\partial }_x)$ qui admettent un symétriseur microlocal $S(t,x,\xi )$ $C^{\infty }$ en $\xi \ne 0$ et lipschitzien en $(t,x)$. Cet article contient trois principaux résultats. D’abord, il est montré qu’une régularité lipschitzienne globale en $(t,x,\xi )$ pour le symétriseur est suffisante. Ensuite, il est établi que l’existence de symétriseurs microlocaux est équivalente à l’existence de symétriseurs complets $\Sigma (t,x,\tau ,\xi )$ de même régularité, notion introduite dans [FL67]. Cette étape est le point clé dans la démonstration du troisième résultat qui affirme que l’existence de symétriseurs microlocaux est préservée par changement de variable de temps. Un corollaire en est l’unicité locale et la vitesse finie de propagation.

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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