Delta shock wave for a $3 \times 3$ hyperbolic system of conservation laws

TL;DR

This paper investigates delta-shock wave solutions for a specific 3x3 hyperbolic conservation law system, establishing existence and uniqueness under certain conditions, which simplifies a more complex viscoelastic fluid model.

Contribution

It introduces a delta-shock solution framework for a Temple class hyperbolic system, extending the understanding of shock waves in simplified viscoelastic fluid models.

Findings

Existence of delta-shock solutions is proven.

Uniqueness of these solutions is established.

The system's characteristic fields are linearly degenerate.

Abstract

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