Delta shock wave for a 3 × 3 hyperbolic system of conservation laws

Abstract

We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. The system is a simplification of a recently propose system of five conservations laws by Bouchut and Boyaval that models viscoelastic fluids. An important issue is that the considered 3 × 3 system is such that every characteristic field is linearly degenerate. We study the Riemann problem for this system and under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established.

On the delta shockwave interactions for the isentropic Chaplygin gas system consisting of three scalar equations

“…The generalized Riemann problem with delta initial data was also considered by Wang and Zhang [46] with the help of the generalized Rankine-Hugoniot conditions. Let us also see such as [11,14,17,21,25,27,28,33,34,37,39,42,47] for some related results. In contrast to the traditional isentropic Chaplygin gas dynamic system, the solution to the Riemann problem (1.3) and (1.4) is either the combination of three contact discontinuities when u − − 1 ρ − < u + + 1 ρ + or the delta shock wave solution when u − − 1 ρ − > u + + 1 ρ + , which can be obtained by using more complicated generalized Rankine-Hugoniot conditions.…”