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

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Yunnan University, Kunming, 650091, China

    HanChun Yang & JinJing Liu

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  1. HanChun Yang
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  2. JinJing Liu
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Correspondence to HanChun Yang.

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Yang, H., Liu, J. Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation. Sci. China Math. 58, 2329–2346 (2015). https://doi.org/10.1007/s11425-015-5034-0

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  • DOI: https://doi.org/10.1007/s11425-015-5034-0

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