Vol. 8, No. 2, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Counterexamples to the well posedness of the Cauchy problem for hyperbolic systems

Ferruccio Colombini and Guy Métivier

Vol. 8 (2015), No. 2, 499–511
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 L2. When the symmetrizer is log-Lipschitz or when the coefficients are analytic or quasianalytic, the Cauchy problem is well posed in C. 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