# American Institute of Mathematical Sciences

January  2017, 16(1): 295-310. doi: 10.3934/cpaa.2017014

## The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term

 1 College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, China 2 Department of Mathematics, University of Iowa Iowa, City, IA 52242, USA

Received  May 2016 Revised  July 2016 Published  November 2016

Fund Project: This work is partially supported by National Natural Science Foundation of China (11401508,11461066), China Scholarship Council, the Scientific Research Program of the Higher Education Institution of XinJiang (XJEDU2014I001).

We study the vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. The phenomena of concentration and cavitation to Chaplygin gas equations with a friction term are identified and analyzed as the pressure vanishes. Due to the influence of source term, the Riemann solutions are no longer self-similar. When the pressure vanishes, the Riemann solutions to the inhomogeneous Chaplygin gas equations converge to the Riemann solutions to the pressureless gas dynamics model with a friction term.

Citation: Lihui Guo, Tong Li, Gan Yin. The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. Communications on Pure & Applied Analysis, 2017, 16 (1) : 295-310. doi: 10.3934/cpaa.2017014
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##### References:
Riemann solution in the phase plane
Riemann solution when $(u_+, \rho_+)\in \rm{Ⅳ}(u_-, \rho_-).$
Riemann solution when $(u_+, \rho_+)\in \rm{Ⅰ}(u_-, \rho_-).$
Riemann solution when $(u_+, \rho_+)\in \rm{Ⅲ}(u_-, \rho_-).$
Riemann solution when $(u_+, \rho_+)\in \rm{Ⅱ}(u_-, \rho_-).$
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