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

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 shocks 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 form the $$\delta $$ -shock; while the Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the transport equations if we do not consider the virtual velocity in the vacuum region, the intermediate state between the two contact discontinuities is a vacuum state. Moreover, we also give some numerical simulations to confirm the theoretical analysis.