Delta shock waves as flux-approximation limit of solutions to the modified Chaplygin gas equations. (English) Zbl 1441.76050
Summary: By introducing a triple parameter flux perturbation including pressure in the modified Chaplygin gas equations, we prove that, as the triple parameter flux perturbation vanishes, any two-shock Riemann solution and any two-rarefaction-wave Riemann solution to the perturbed modified Chaplygin gas equations tend to a delta-shock and a vacuum solution to the transport equations, respectively. Then we show that, as a double parameter flux perturbation vanishes, any two-shock Riemann solution under a certain condition to the perturbed modified Chaplygin gas equations tends to a delta shock wave solution to the generalized Chaplygin gas equations. In addition, we also show that, as a single parameter flux perturbation vanishes, any two-shock solution satisfying certain initial data and any parameterized delta shock wave solution to the perturbed generalized Chaplygin gas equations tend to a delta shock wave solution to the generalized Chaplygin gas equations. Finally, we exhibit some representative numerical simulations to confirm the theoretical analysis.
MSC:
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
Keywords:
modified Chaplygin gas; generalized Chaplygin gas; delta shock wave; vacuum state; constant density state; transport equationsReferences:
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