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

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 δ-shock solution to the correspongding nonhomogeneous transportation equations, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-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.

The adiabatic exponent limits of Riemann solutions for the extended macroscopic production model

“…First of all, the vanishing pressure method was extensively used such as in [8,11,12,22,26,30,33,41]. Subsequently, the approach of the so-called flux function limit was also adopted such as in [29,40,[44][45][46]. Recently, the approach of the adiabatic exponent limit has been well carried out for example in [15, 31, 34-36, 48, 49].…”

Stability of initial-boundary value problem for quasilinear viscoelastic equations

“…For more related information on the stability of problem (1.1) and some related equations or systems, we refer the reader to [1,3,4,5,6,7,8,9,10,11,18,19,20,21,22,23,24] and references therein.…”