The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system^*

Previous Article
Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system
CPAA Home
This Issue
Next Article
Spherically symmetric Navier-Stokes equations with degenerate viscosity coefficients and vacuum

2010, 9(2): 431-458. doi: 10.3934/cpaa.2010.9.431

The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system$^*$

Lihui Guo ^1, , Wancheng Sheng ^2, and Tong Zhang ^3,

1.	Department of Mathematics, Shanghai University, Shanghai, 200444, College of Mathematics and System Sciences, Urumqi, 830000, Xinjiang, China
2.	Department of Mathematics, Shanghai University, Shanghai, 200444, China
3.	Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100190, China

Received April 2009 Revised August 2009 Published December 2009

Abstract
Related Papers

The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system consists of interactions of four planar elementary waves. Different from polytropic gas, all of them are contact discontinuities due to the system is full linear degenerate, i.e., the three eigenvalues of the system are linear degenerate. They include compressive one ($S^\pm$), rarefactive one ($R^\pm$) and slip lines ($J^\pm$). We still call $S^\pm$ as shock and $R^\pm$ as rarefaction wave. In this paper, we study the problem systematically. According to different combination of four elementary waves, we deliver a complete classification to the problem. It contains 14 cases in all. The Riemann solutions are self-similar, and the flow is transonic in self-similar plane $(x/t,y/t)$. The boundaries of the interaction domains are obtained. Solutions in supersonic domains are constructed in no $J$ cases. While in the rest cases, the structure of solutions are conjectured except for the case $2J^++2J^-$. Especially, delta waves and simple waves appear in some cases. The Dirichlet boundary value problems in subsonic domains or the boundary value problems for transonic flow are formed case by case. The domains are convex for two cases, and non-convex for the rest cases. The boundaries of the domains are composed of sonic curves and/or slip lines.

Keywords: 2-D Riemann problem, full linear degenerate, delta wave., Chaplygin gas.

Mathematics Subject Classification: 35L65, 35L67, 76L05, 76N1.

Citation: Lihui Guo, Wancheng Sheng, Tong Zhang. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system$^*$. Communications on Pure & Applied Analysis, 2010, 9 (2) : 431-458. doi: 10.3934/cpaa.2010.9.431

[1]	Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 555-584. doi: 10.3934/dcds.1995.1.555
[2]	Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 419-430. doi: 10.3934/dcds.2000.6.419
[3]	Qin Wang, Kyungwoo Song. The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1661-1675. doi: 10.3934/dcds.2016.36.1661
[4]	Jianjun Chen, Wancheng Sheng. The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2018, 17 (1) : 127-142. doi: 10.3934/cpaa.2018008
[5]	Geng Lai, Wancheng Sheng, Yuxi Zheng. Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 489-523. doi: 10.3934/dcds.2011.31.489
[6]	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
[7]	Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127
[8]	Roberto Triggiani. Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D, wall-normal boundary controller. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 279-314. doi: 10.3934/dcdsb.2007.8.279
[9]	Ju Ge, Wancheng Sheng. The two dimensional gas expansion problem of the Euler equations for the generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2733-2748. doi: 10.3934/cpaa.2014.13.2733
[10]	Huahui Li, Zhiqiang Shao. Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2373-2400. doi: 10.3934/cpaa.2016041
[11]	Bingbing Ding, Ingo Witt, Huicheng Yin. Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 719-744. doi: 10.3934/cpaa.2017035
[12]	Chuandong Li, Fali Ma, Tingwen Huang. 2-D analysis based iterative learning control for linear discrete-time systems with time delay. Journal of Industrial & Management Optimization, 2011, 7 (1) : 175-181. doi: 10.3934/jimo.2011.7.175
[13]	Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 941-966. doi: 10.3934/dcds.2018040
[14]	Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917
[15]	Tian Ma, Shouhong Wang. Global structure of 2-D incompressible flows. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 431-445. doi: 10.3934/dcds.2001.7.431
[16]	Jean-françois Coulombel, Paolo Secchi. Uniqueness of 2-D compressible vortex sheets. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1439-1450. doi: 10.3934/cpaa.2009.8.1439
[17]	Nusret Balci, Ciprian Foias, M. S Jolly, Ricardo Rosa. On universal relations in 2-D turbulence. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1327-1351. doi: 10.3934/dcds.2010.27.1327
[18]	Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299
[19]	Yuri N. Fedorov, Luis C. García-Naranjo, Joris Vankerschaver. The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4017-4040. doi: 10.3934/dcds.2013.33.4017
[20]	Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi. Stability of the travelling wave in a 2D weakly nonlinear Stefan problem. Kinetic & Related Models, 2009, 2 (1) : 109-134. doi: 10.3934/krm.2009.2.109

2016 Impact Factor: 0.801

American Institute of Mathematical Sciences