Summary: We propose a definition for the nonconservative product
is a locally bounded Borel function and
is a function of bounded variation. This definition generalizes the one previously given by Vol’pert [A. I. Vol’pert
, Math. USSR, Sb. 2, 225-267 (1967); translation from Mat. Sb., n. Ser. 73(115), 255-302 (1967; Zbl 0168.07402
)] and is based on a Lipschitz continuous completion of the graphs of functions of bounded variation. We study the stability of this product for the weak convergence. As an application, the nonlinear hyperbolic systems in nonconservative form are considered: we give a notion of weak solution for the Riemann problem, and extend Lax’s construction.