Concentration in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equations with Coulomb-like friction

TL;DR

This paper analyzes the formation of delta shock waves in the flux approximation limit of Riemann solutions to extended Chaplygin gas equations with Coulomb-like friction, revealing how solutions converge to delta-shocks or vacuum states.

Contribution

It extends the analysis of vanishing pressure limits and delta shock formation to nonhomogeneous extended Chaplygin gas equations with Coulomb-like friction, including the generalized Chaplygin case.

Findings

Two-shock solutions tend to delta-shocks as pressure vanishes.

Two-rarefaction solutions tend to contact discontinuities or vacuums.

Results generalize previous homogeneous case findings to nonhomogeneous equations.

Abstract

In this paper, two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock waves are analyzed and identified in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equations with Coulomb-like friction, whose special case can also be seen as the model of the magnetogasdynamics with Coulomb-like friction. Firstly, by introducing a transformation, the Riemann problem for the extended Chaplygin gas equations with Coulomb-like friction is solved completely. Secondly, we rigorously show that, as the pressure vanishes, any two-shock Riemann solution to the nonhomogeneous extended Chaplygin gas equations tends to a {\delta}-shock solution to the corresponding nonhomogeneous transportation equations, and the intermediate density between the two shocks tends to a weighted {\delta}-measure that forms the {\delta}-shock; any two-rarefaction-wave Riemann solution to the nonhomogeneous extended Chaplygin gas equations tends to a two-contact-discontinuity solution to the correspongding nonhomogeneous transportation equations, and the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. At last, we also show that, as the pressure approaches the generalized Chaplygin pressure, any two-shock Riemann solution to the nonhomogeneous extended Chaplygin gas equations tends to a delta-shock solution to the correspongding nonhomogeneous generalized Chaplygin gas equations. In a word, we have generalized all the results about the vanishing pressure limit now available for homogeneous equations to the nonhomogeneous case.