## Analysis & PDE

- Anal. PDE
- Volume 8, Number 2 (2015), 499-511.

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

Ferruccio Colombini and Guy Métivier

#### 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.

#### Article information

**Source**

Anal. PDE, Volume 8, Number 2 (2015), 499-511.

**Dates**

Received: 20 September 2014

Accepted: 9 January 2015

First available in Project Euclid: 16 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.apde/1510843076

**Digital Object Identifier**

doi:10.2140/apde.2015.8.499

**Mathematical Reviews number (MathSciNet)**

MR3345635

**Zentralblatt MATH identifier**

1316.35172

**Subjects**

Primary: 35L50: Initial-boundary value problems for first-order hyperbolic systems

**Keywords**

hyperbolic systems the Cauchy problem nonsmooth symmetrizers ill-posedness

#### Citation

Colombini, Ferruccio; Métivier, Guy. Counterexamples to the well posedness of the Cauchy problem for hyperbolic systems. Anal. PDE 8 (2015), no. 2, 499--511. doi:10.2140/apde.2015.8.499. https://projecteuclid.org/euclid.apde/1510843076