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1 Introduction

The mathematical theory of hyperbolic systems of conservation laws were started by Eberhardt Hopf in 1950, followed in a series of studies by Olga Oleinik, Peter D. Lax and James Glimm [9] . The class of conservation laws is a very important class of partial differential equations because as their name indicates, they include those equations that model conservation laws of physics (mass, momentum, energy, etc). As important examples of hyperbolic systems of balance laws arising in continuum physics we have: Euler’s equations for compressible gas flow, the one dimensional shallow water equations [6], Maxwell’s equations in nonlinear dielectrics, Lundquist’s equations of magnetohydrodynamics and Boltzmann equation in thermodynamics [3] and equations of elasticity [11]. One of the main motivations of the theory of hyperbolic systems is that they describe for the most part real physical problems, because they are consistent with the fact that the physical signals have a finite propagation speed [11]. Such systems even with smooth initial conditions may fail to have a solution for all time, in such cases we have to extend the concept of classical solutions to the concept of a weak solution or generalized solution [6]. In the case of hyperbolic systems, the notion of weak solution based on distributions does not guarantee uniqueness, and it is necessary to devise ad- missibility criteria that will hopefully single out a unique weak solution. Sev- eral such criteria have indeed been proposed, motivated by physical and/or mathematical considerations. It seems that a consensus has been reached on this issue for such solutions, they are called entropy conditions [4]. Neverthe- less, to the question about existence and uniqueness of generalized solutions subject to the entropy conditions, the answer is, in general, open. For the scalar conservation law, the questions existence and uniqueness are basically settled [6]. For genuinely nonlinear systems, existence (but not uniqueness) is known for init