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

Lihui Guo; Tong Li; Gan Yin

doi:10.3934/cpaa.2017014

Article Contents

2017, Volume 16, Issue 1: 295-310. Doi: 10.3934/cpaa.2017014

This issue Previous Article Quasineutral limit for the quantum Navier-Stokes-Poisson equations Next Article Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary

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

Author Bio: E-mail address: tong-li@uiowa.edu; E-mail address: ganyinxj@gmail.com
Lihui Guo, E-mail address: lihguo@126.com
Lihui Guo, E-mail address: lihguo@126.com

Received: May 2016

Revised: July 2016

Published: January 2018

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).

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Abstract

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.

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Mathematics Subject Classification: Primary: 35L65, 35L67; Secondary: 76N15.

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Figure 3.1. Riemann solution in the phase plane

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Figure 4.1. Riemann solution when $(u_+, \rho_+)\in \rm{Ⅳ}(u_-, \rho_-).$

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Figure 4.2. Riemann solution when $(u_+, \rho_+)\in \rm{Ⅰ}(u_-, \rho_-).$

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Figure 4.3. Riemann solution when $(u_+, \rho_+)\in \rm{Ⅲ}(u_-, \rho_-).$

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Figure 4.4. Riemann solution when $(u_+, \rho_+)\in \rm{Ⅱ}(u_-, \rho_-).$

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