Definition and weak stability of nonconservative products

Abstract

We propose a definition for the nonconservative product g(u)du/dx, where g : R(p) --> R(p) is a locally bounded Borel function and u :]a, b[ --> R(p) is a function of bounded variation. This definition generalizes the one previously given by Volpert [Vo] 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.