Representation of Weak Limits and Definition of Nonconservative Products

The goal of this article is to show that the notion of generalized graphs is able to represent the limit points of the sequence {g(un) dun} in the weak-⋆ topology of measures when {un} is a sequence of continuous functions of uniformly bounded variation. The representation theorem induces a natural definition for the nonconservative product g(u) du in a BV context. Several existing definitions of nonconservative products are then compared, and the theory is applied to provide a notion of solutions and an existence theory to the Riemann problem for quasi-linear, strictly hyperbolic systems.