Counterexamples to the well posedness of the Cauchy problem for hyperbolic systems

Colombini, Ferruccio; Métivier, Guy

doi:10.2140/apde.2015.8.499

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Counterexamples to the well posedness of the Cauchy problem for hyperbolic systems

Ferruccio Colombini and Guy Métivier

Vol. 8 (2015), No. 2, 499–511

DOI: 10.2140/apde.2015.8.499

Abstract

This paper is concerned with the well-posedness of the Cauchy problem for first order symmetric hyperbolic systems in the sense of Friedrichs. The classical theory says that if the coefficients of the system and if the coefficients of the symmetrizer are Lipschitz continuous, then the Cauchy problem is well posed in $L^{2}$ . When the symmetrizer is log-Lipschitz or when the coefficients are analytic or quasianalytic, the Cauchy problem is well posed in $C^{\infty}$ . We give counterexamples which show that these results are sharp. We discuss both the smoothness of the symmetrizer and of the coefficients.

Keywords

hyperbolic systems, the Cauchy problem, nonsmooth symmetrizers, ill-posedness

Mathematical Subject Classification 2010

Primary: 35L50

Milestones

Received: 20 September 2014

Accepted: 9 January 2015

Published: 10 May 2015

Authors

Ferruccio Colombini
Dipartimento di Matematica Università di Pisa Largo B. Pontecorvo 5 I-56127 Pisa Italy

Guy Métivier
Institut de Mathématiques de Bordeaux, CNRS, UMR 5251 Université de Bordeaux 351 Cours de la Libération F-33405 Talence France

Vol. 8, No. 2, 2015