Delta shock wave and interactions in a simple model case

TL;DR

This paper investigates the interactions of delta shock waves with classical solutions in a simplified magneto-hydrodynamics model, revealing complex behaviors like delta contact discontinuities and the emergence of $L_{loc}^{1}$-functions.

Contribution

It introduces a solution framework for delta shock interactions in a $2 imes 2$ system using measure-valued solutions, extending the understanding of singular wave interactions.

Findings

Interaction can produce delta contact discontinuities.

Interactions may generate $L_{loc}^{1}$-functions.

The model extends classical solutions with measure-valued delta shocks.

Abstract

The notion of a delta shock wave and a singular shock wave was introduced and employed by different authors, and it was shown that a large class of Riemann problems can be solved globally with these additional building blocks. The aim of this paper is to study the interaction of one type of these new solutions, the delta shock waves, with the classical types of solutions. Our model problem is $2 \times 2$ system derived from a simplified model of magneto-hydrodynamics. The solution concept used in the present paper can be simply described as analogous to the standard one but with $L^{\infty}$-functions substituted by measures in one component of a solution. Here, the delta function is represented by so called two sided delta function. We shall mention only two specific details from the complete interaction result. In some situation the interaction result may contain non-overcompressible delta shock wave called delta contact discontinuity. During delta shock and rarefaction wave interaction it may happen that some $L_{loc}^{1}$-function appear.