Measure-Valued Solutions to a Non-Strictly Hyperbolic System with Delta-Type Riemann Initial Data

Abstract

This paper is concerned with the construction of global measure-valued solutions to the extended Riemann problem for a non-strictly hyperbolic system of two conservation laws with delta-type initial data. The wave interaction problems have been extensively studied for all kinds of situations by using the initial condition consisting of constant states in three pieces instead of delta-type initial data under the perturbation method. The measure-valued solutions of the extended Riemann problem are achieved constructively when the perturbed parameter tends to zero. During the process of constructing solutions, a new and interesting nonlinear phenomenon is discovered, in which the initial Dirac delta function travels along the trajectory of either delta shock wave or contact discontinuity (or delta contact discontinuity). Moreover, a delta shock wave is separated into a delta contact discontinuity and a shock wave during the process of delta shock wave penetrating a composite wave composed of a rarefaction wave and a contact discontinuity. In addition, we further consider the constructions of global measure-valued solutions when the initial condition contains Dirac delta functions at two different initial points.

The Perturbed Riemann Problem for a Macroscopic Production Model with Chaplygin Gas

“…In addition, it is well known that delta shock wave should be introduced in order to solve Riemann problems for some weak hyperbolic conservation systems and the theory of delta shock wave has been well established and developed in recent years, see [22][23][24][25][26][27][28] for examples. However, as far as we know, our constructed composite wave solution with the singularity of delta function cannot be found in any Riemann solution but only appears in the study [29][30][31] of wave interaction between a delta shock wave with a rarefaction wave for some hyperbolic systems of conservation laws.…”

Four-Quadrant Riemann Problem for a 2 × 2 System Involving Delta Shock

“…This equation can also be considered to be a simplified gas-dynamics-like model because it can be derived from 2-D isentropic Euler equations [25]. Because this is an isotropic case ( f = g in Equation (2)), they considered interactions in the η > ξ plane by applying the transformation and symmetry.…”