Hyperbolic systems of balance laws via vanishing viscosity

Abstract

Global weak solutions of a strictly hyperbolic system of balance laws in one-space dimension are constructed by the vanishing viscosity method of Bianchini and Bressan. For global existence, a suitable dissipativeness assumption has to be made on the production term g. Under this hypothesis, the viscous approximations u ɛ , that are globally defined solutions to u t ɛ + A ( u ɛ ) u x ɛ + g ( u ɛ ) = ɛ u xx ɛ , satisfy uniform BV bounds exponentially decaying in time. Furthermore, they are stable in L 1 with respect to the initial data. Finally, as ɛ → 0 , u ɛ converges in L loc 1 to the admissible weak solution u of the system of balance laws u t + ( f ( u ) ) x + g ( u ) = 0 when A = Df .