Generalized Solutions to Conservation Laws

Abstract

this paper we study generalized solutions in the sense of Colombeau to the Cauchy problem for hyperbolic conservation laws and their parabolic approximations. We obtain existence and uniqueness results both in the scaler case and for systems. The relation of the generalized solution to the classical solution, when the latter exists, is established. An application of our results is to the $p$-system of gas dynamics with artificial viscosity.