Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:413

Article

Yang, Hanchun¹  Wang, Jinhuan¹ 

[1] Yunnan Univ, Dept Math, Kunming 650091, Peoples R China

关键词: Isentropic Euler equations;    Modified Chaplygin gas;    Delta shock wave;    Vacuum state;    Vanishing pressure limit;    Transport equations;    Pressureless fluids;   

DOI  :  10.1016/j.jmaa.2013.12.025

来源: Elsevier

【 摘 要 】

The phenomena of concentration and cavitation and the formation of delta-shocks and vacuum states in solutions to the isentropic Euler equations for a modified Chaplygin gas are analyzed as the double parameter pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for a modified Chaplygin gas is solved analytically. Secondly, it is rigorously shown that, as the pressure vanishes, any two-shock Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a delta-shock solution to the transport 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 isentropic Euler equations for a modified Chaplygin gas tends to a two-contact-discontinuity solution to the transport equations, the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, some numerical results exhibiting the formation of delta-shocks and vacuum states are presented as the pressure decreases. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free