Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation

• Published in: Science China. Mathematics Vol. 58; no. 11; pp. 2329 - 2346
• Main Authors: Yang, HanChun, Liu, JinJing
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01.11.2015
• Subjects: Applications of Mathematics; Approximation; Constants; Flux; Gas dynamics; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Perturbation methods; Shock waves; numerical simulations; Riemann problem; 35L67; 35L65; 76E19; transport equations; vacuum; 35Q35; delta shock wave; 35B30; zero-pressure flow; flux approximation; Euler equations of isentropic gas dynamics
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-015-5034-0

In this paper, firstly, by solving the Riemann problem of the zero-pressure flow in gas dynamics with a flux approximation, we construct parameterized delta-shock and constant density solutions, then we show that, as the flux perturbation vanishes, they converge to the delta-shock and vacuum state solutions of the zero-pressure flow, respectively. Secondly, we solve the Riemann problem of the Euler equations of isentropic gas dynamics with a double parameter flux approximation including pressure. Furthermore, we rigorously prove that, as the two-parameter flux perturbation vanishes, any Riemann solution containing two shock waves tends to a delta-shock solution to the zero-pressure flow; any Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum intermediate state in between tends to a vacuum state. Finally, numerical results are given to present the formation processes of delta shock waves and vacuum states.