Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves

Abstract

We construct solution of Riemann problem for a system of four conservation lawsadmitting $\delta$, $\delta'$ and $\delta''$-waves, using vanishing viscositymethod. The system considered hereis an extension of a system studied in [9] and [12] and admits moresingular solutions. We extend the weak formulation of [12] to the present case.For the rarefaction case, the limit is not yet fully understood, the limit givenin [12] is not correct and it does not satisfy the inviscid system. In factwe show that the limit of the third component contains $\delta$ measure andthe fourth component contains the measure $\delta$ and its derivative, for a specialRiemann data. We also solve Riemann type initial-boundary value problem in the quarter plane.