A system of two conservation laws with flux conditions and small viscosity

Abstract

We construct explicit solutions of a system of two conservation laws with small viscosity in the quarter plane {(x, t) : x> 0, t > 0}, with initial conditions at t = 0 and flux conditions at x = 0. We derive a formula for the limit as viscosity goes to zero which generally belongs to the space of locally bounded Borel measures. This limit satisfies the inviscid equation, in the sense of LeFloch [An existence and uniqueness result for two non-strictly hyperbolic systems, Springer, 1990]. We also treat more general initial and boundary datas and obtain solution in the algebra of generalized functions of Colombeau [C. R. Acad. Sci. Paris Ser. I Math. 317: 851-855, 1993, C. R. Acad. Sci. Paris Ser. I Math. 319: 1179-1183, 1994].