An existence and uniqueness result for two nonstrictly hyperbolic systems

We prove a result of existence and uniqueness of entropy weak solutions for two nonstrictly hyperbolic systems, both a nonconservative system of two equations $$ {\partial_t}u + {\partial_x}f(u) = 0,\,{\partial_t}w + a(u){\partial_x}w = 0 $$ , and a conservative system of two equations $$ {\partial_t}u + {\partial_x}f(u) = 0,\,{\partial_t}v + {\partial_x}(a(u)v) = 0 $$ , where f: R → R is a given strictly convex function and \( a = \frac{d}{{du}}f \). We use the Volpert’s product ([19], see also Dal Maso — Le Floch — Murat [1]) and find entropy weak solutions u and w which have bounded variation while the solutions v are Borel measures. The equations for w and v can be viewed as linear hyperbolic equations with discontinuous coefficients.