Global existence to a diagonal hyperbolic system for any BV initial data

Abstract

In this paper, we study the existence of solutions for a diagonal hyperbolic system, that is not necessarily strictly hyperbolic, in one space dimension, considering discontinuous BV initial data without any restrictions on the size of its norm. This system appears naturally in various physical domains, particularly in isentropic gas dynamics and dislocation dynamics in materials. In the case of strictly hyperbolic systems, an existence and uniqueness of a discontinuous solution result is available for BV initial data with small norm, whereas several existence and uniqueness results have been presented for non-decreasing continuous solutions. In the present paper, we show the global in time existence of discontinuous viscosity solutions to a diagonal hyperbolic system for every initial data of bounded total variation, without the assumption that the system is strictly hyperbolic. Up to our knowledge, this is the first global existence result of large discontinuous solutions to this system.