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The Scientific World Journal
Volume 2014 (2014), Article ID 287256, 11 pages
http://dx.doi.org/10.1155/2014/287256
Research Article

Limit of Riemann Solutions to the Nonsymmetric System of Keyfitz-Kranzer Type

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Received 18 January 2014; Accepted 2 March 2014; Published 3 April 2014

Academic Editors: D. Baleanu and H. Jafari

Copyright © 2014 Lihui Guo and Gan Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type with a scaled pressure is considered for both polytropic gas and generalized Chaplygin gas. In the former case, the delta shock wave can be obtained as the limit of shock wave and contact discontinuity when and the parameter tends to zero. The point is, the delta shock wave is not the one of transport equations, which is obviously different from cases of some other systems such as Euler equations or relativistic Euler equations. For the generalized Chaplygin gas, unlike the polytropic or isothermal gas, there exists a certain critical value depending only on the Riemann initial data, such that when drops to , the delta shock wave appears as , which is actually a delta solution of the same system in one critical case. Then as becomes smaller and goes to zero at last, the delta shock wave solution is the exact one of transport equations. Furthermore, the vacuum states and contact discontinuities can be obtained as the limit of Riemann solutions when and , respectively.