# Preprint 2015-007

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

## Zhiqiang Shao

**Abstract:**
This paper is concerned with the Riemann problem
for the isentropic Chaplygin gas magnetogas-dynamics 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 δ-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 intermediate state between the two contact discontinuities
is a vacuum state.