Delta shocks and vacuum states for the isentropic magnetogasdynamics equations for Chaplygin gas as pressure and magnetic field vanish

TL;DR

This paper analyzes the Riemann problem for isentropic Chaplygin gas magnetogasdynamics equations, showing how solutions evolve into delta shocks or vacuum states as pressure and magnetic field vanish.

Contribution

It provides an analytical solution to the Riemann problem and rigorously proves the formation of delta shocks and vacuum states in the vanishing pressure and magnetic field limit.

Findings

Delta shocks form as pressure and magnetic field vanish.

Vacuum states emerge between contact discontinuities in the limit.

Solutions converge to transport equations with specific singularities.

Abstract

This paper is concerned with the Riemann problem for the isentropic Chaplygin gas magnetogasdynamics equations and the formation of delta shocks and vacuum states as pressure and magnetic field vanish. Firstly, the Riemann problem of the isentropic magnetogasdynamics equations for Chaplygin gas is solved analytically. Secondly, it is rigorously proved that, as both the pressure and the magnetic field vanish, the Riemann solution containing two shock waves tends to a delta shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted $\delta$-measure which forms the delta shock; while the Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the transport equations, the termediate state between the two contact discontinuities is a vacuum state.