A class of systems of quasilinear conservation laws

Hyperbolic systems of conservation laws with a functional-calculus operator on the right-hand side are considered in the space of second-order symmetric matrices. The entropies of such systems are described. The concept of a generalized entropy solution (g.e.s.) of the corresponding Cauchy problem is introduced, the properties of g.e.s.'s are analyzed, and the lack of their uniqueness in the general case is demonstrated. Using a stronger version of the defining entropy condition, the class of strong g.e.s.'s is distinguished. The Cauchy problem under discussion is shown to be uniquely soluble in this class.