Spaces of Weighted Measures for Conservation Laws with Singular Shock Solutions

We study a model system of two strictly hyperbolic conservation laws which is genuinely nonlinear but for which the Riemann problem has no global solution. Singular solutions are defined by means of a generalized Rankine-Hugoniot relation and an overcompressive condition on the discontinuity. We show that approximate solutions which can be constructed by several standard methods converge in a weighted measure space and that the error in the approximation converges to zero. Viscous approximations satisfy approximate entropy inequalities which imply the overcompressive condition.