On the $L^2$ well posedness of Hyperbolic Initial Boundary Value Problems

Métivier, Guy

Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 1809-1863.

On montre que le problème mixte est bien posé dans $L^2$, sous l’hypothèse nécessaire de Lopatinski uniforme, pour une classe de systèmes hyperboliques qui contient les systèmes de multiplicités constantes, mais significativement plus large. On montre en outre que la vitesse de propagation du problème aux limites est la même que la vitesse de propagation à l’intérieur du domaine. Au contraire, on montre sur un exemple que, même pour les systèmes symétriques au sens de Friedrichs mais à multiplicités variables, le problème mixte peut être mal posé sous la seule condition de Lopatinski uniforme.

In this paper we give a class of hyperbolic systems, which includes systems with constant multiplicity but significantly wider, for which the initial boundary value problem (IBVP) with source term and initial and boundary data in $L^2$, is well posed in $L^2$, provided that the necessary uniform Lopatinski condition is satisfied. Moreover, the speed of propagation is the speed of the interior problem. In the opposite direction, we show on an example that, even for symmetric systems in the sense of Friedrichs, with variable coefficients and variable multiplicities, the uniform Lopatinski condition is not sufficient to ensure the well posedness of the IBVP in Sobolev spaces.

Classification : 35L04
Keywords: Hyperbolic first order systems, Cauchy problems, boundary value problems, Lopatinski conditions, semi-group estimates
Mot clés : Systèmes hyperboliques du premier ordre, problèmes de Cauchy, problèmes aux limites, conditions de Lopatinski, estimations de semi-groupe

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