COMPARISON OF RIEMANN SOLUTIONS TO THE EULER EQUATIONS FOR MODIFIED AND GENERALIZED CHAPLYGIN GAS DYNAMICS

Li Wang; Xiaomei Liu; Kun-Peng Jin

doi:10.11948/20240534

2025 Volume 15 Issue 6

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(6): 3290-3316

Next Article Previous Article

Li Wang, Xiaomei Liu, Kun-Peng Jin. COMPARISON OF RIEMANN SOLUTIONS TO THE EULER EQUATIONS FOR MODIFIED AND GENERALIZED CHAPLYGIN GAS DYNAMICS[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3290-3316. doi: 10.11948/20240534

Citation:

Li Wang, Xiaomei Liu, Kun-Peng Jin. COMPARISON OF RIEMANN SOLUTIONS TO THE EULER EQUATIONS FOR MODIFIED AND GENERALIZED CHAPLYGIN GAS DYNAMICS[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3290-3316. doi: 10.11948/20240534

COMPARISON OF RIEMANN SOLUTIONS TO THE EULER EQUATIONS FOR MODIFIED AND GENERALIZED CHAPLYGIN GAS DYNAMICS

1.
Department of Arts and Sciences, Shanghai Dianji University, Shanghai, China
2.
School of Mathematics, Physics and Statistics, Shanghai Polytechnic University, Shanghai, China
3.
Center for Applied Mathematics of Guangxi, Nanning Normal University, Nanning 530100, China

Author Bio: Email: wangli@sdju.edu.cn(L. Wang); Email: kunpengjin@126.com(K.-P. Jin)
Corresponding author: Email: xmliu@sspu.edu.cn(X. Liu)
Fund Project: The authors were supported by the Natural Science Foundation of Guangxi Province (No. 2023GXNSFAA026172), the National Natural Science Foundation of China (12361049), and the Guangxi Science and Technology Base and Special Talents Program (Guike AD23023003)

Abstract

Abstract

In this article, utilizing the method of characteristic analysis, we construct the Riemann solutions of the Euler equations for the modified and the generalized Chaplygin gas, which contains rarefaction waves, shock waves, contact discontinuities and $ \delta- $ shock waves. For the $ \delta- $ shock waves, we propose generalized Rankine-Hugoniot relation and entropy conditon. Moreover, by studing the limiting behaviour, we find that the Riemann solutions of modified Chaplygin gas is the same as generalized Chaplygin gas including the $ \delta- $shock waves. Moreover, we give some numerical simulations to verify the theoretical analysis.
- Characteristic analysis /
- δ−shock waves /
- entropy conditon /
- generalized Rankine-Hugoniot conditions
MSC: 35L65, 35L67

FullText(HTML)

References

[1]	M. C. Bento, O. Bertolami and A. A. Sen, Generalized Chaplygin gas model: Dark energy-dark matter unification and cmbrconstraints, Gen. Relativ. Gravit, 2003, 35, 2063–2069. doi: 10.1023/A:1026207312105 CrossRef Google Scholar
[2]	N. Bilic, G. B. Tupper and R. Viollier, Dark matter, dark energy and the Chaplygin gas, Dark Matter in Astro- and Particle Physics, 2002, 306–311. DOI: 10.1007/978-3-642-55739-2-30. CrossRef Google Scholar
[3]	Y. Brenier, Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations, J. Math. Fluid Mech., 2005, 7, 326–331. doi: 10.1007/s00021-005-0162-x CrossRef Google Scholar
[4]	S. Chaplygin, On gas jets, Sci. Mem. Mosc. Univ. Math. Phys., 1904, 21, 1–121. Google Scholar
[5]	R. K. Chaturvedi, L. P. Singh and D. Zeidan, Delta shock wave solution of the Riemann problem for the non-homogeneous modified Chaplygin gasdynamics, Journal of Dynamics and Differential Equations, 2022, 34(2), 1067–1084. doi: 10.1007/s10884-020-09914-8 CrossRef Google Scholar
[6]	H. J. Cheng and H. C. Yang, Riemann problem for the relativistic Chaplygin Euler equations, J. Math. Anal. Appl., 2011, 381, 17–26. doi: 10.1016/j.jmaa.2011.04.017 CrossRef Google Scholar
[7]	Q. Ding and L. Guo, The vanishing pressure limit of Riemann solutions to the non-isentropic Euler equations for generalized Chaplygin gas, Adv. Math. Phys., 2019, 5253717. Google Scholar
[8]	V. Gorini, A. Kamenshchik, U. Moschella and V. Pasquier, The Chaplygin gas as a model for dark energy, The Tenth Marcel Grossmann Meeting, 2006. DOI: 10.1142/9789812704030-0050. CrossRef Google Scholar
[9]	L. H. Guo, W. C. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal., 2010, 9(2), 431–458. doi: 10.3934/cpaa.2010.9.431 CrossRef Google Scholar
[10]	W. F. Jiang, T. Li, Z. Wang and S. T. Fang, The limiting behavior of the Riemann solutions of non-isentropic modified Chaplygin gas dynamics, J. Math. Phys., 2021, 62, 041501. doi: 10.1063/5.0033806 CrossRef Google Scholar
[11]	W. F. Jiang, T. Li, Z. Wang and T. T. Zhang, The Riemann solutions comparison for compressible Euler equations in different Chaplygin gas states, Applicable Analysis, 2022, 101(3), 4759–4773. Google Scholar
[12]	D. X. Kong and C. Wei, Formation and propagation of singularities in one-dimensional Chaplygin gas, J. Geom. Phys., 2014, 80, 58–70. doi: 10.1016/j.geomphys.2014.02.009 CrossRef Google Scholar
[13]	A. Kumar and R. Radha, Riemann problem for the Chaplygin gas equations for several classes of non-constant initial data, European Journal of Mechanics B-Fluids, 2022, 91, 121–127. doi: 10.1016/j.euromechflu.2021.09.013 CrossRef Google Scholar
[14]	Z. T. Lei and Z. Q. Shao, The limit behavior of Riemann solutions to the Euler equations of compressible fluid flow for the modified Chaplygin gas, Journal of Mathematical Physics, 2022, 63(7). DOI: 10.1063/5.0090251. CrossRef Google Scholar
[15]	H. H. Li and Z. Q. Shao, Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas, Commun. Pure Appl. Anal., 2016, 15(6), 2373–2400. doi: 10.3934/cpaa.2016041 CrossRef Google Scholar
[16]	Y. Pang, Delta shock wave with Dirac delta function in multiple components for the system of generalized Chaplygin gas dynamics, Boundary Value Probl., 2016(1). DOI: 10.1186/s13661-016-0712-6. CrossRef Google Scholar
[17]	Y. Pang, Delta shock wave in the compressible Euler equations for a Chaplygin gas, J. Math. Anal. Appl., 2017, 448, 245–261. doi: 10.1016/j.jmaa.2016.10.078 CrossRef Google Scholar
[18]	Priyanka and M. Zafar, Delta shocks and vacuum states in the Riemann solutions of Chaplygin Euler equations as pressure and magnetic field drop to zero, Journal of Mathematical Physics, 2022. DOI: 10.1063/5.0132580. CrossRef Google Scholar
[19]	A. F. Qu and L. Wang, Dependence of perturbation on the limit of Riemann solutions in non-isentropic Chaplygin gas dynamics, Chin. Ann. Math., Ser. A, 2018, 39(2), 219–228. Google Scholar
[20]	A. F. Qu and Z. Wang, Stability of the Riemann solutions for a Chaplygin gas, J. Math. Anal. Appl., 2014, 409(1), 347–361. doi: 10.1016/j.jmaa.2013.07.018 CrossRef Google Scholar
[21]	D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch Rational Mech Anal, 2009, 191, 539–577. doi: 10.1007/s00205-008-0110-z CrossRef Google Scholar
[22]	M. R. Setare, Holographic Chaplygin gas model, Phys Lett B, 2007, 648, 329–332. doi: 10.1016/j.physletb.2007.03.025 CrossRef Google Scholar
[23]	W. C. Sheng, G. J. Wang and G. Yin, Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes, Nonlinear Analysis: Real World Applications, 2015, 22, 115–128. doi: 10.1016/j.nonrwa.2014.08.007 CrossRef Google Scholar
[24]	M. Singh and R. Arora, Generalized Riemann problem for the one-dimensional Chaplygin gas with a friction term, International Journal of Non-Linear Mechanics, 2023, 148. DOI: 10.1016/j.ijnonlinmec.2022.104298. CrossRef Google Scholar
[25]	Y. Song and L. Guo, General limiting behavior of Riemann solutions to the non-isentropic Euler equations for modified Chaplygin gas, J. Math. Phys., 2020, 61(4), 041506. doi: 10.1063/1.5144326 CrossRef Google Scholar
[26]	M. Tong, C. Shen and X. Lin, The asymptotic limits of Riemann solutions for the isentropic extended Chaplygin gas dynamic system with the vanishing pressure, Boundary Value Problems, 2018(1). DOI: 10.1186/s13661-018-1064-1. CrossRef Google Scholar
[27]	H. S. Tsien, Two dimensional subsonic flow of compressible fluids, J Aeron Sci, 1939, 6, 399-–407. Google Scholar
[28]	G. D. Wang, The Riemann problem for one dimensional generalized Chaplygin gas dynamics, J. Math. Anal. Appl., 2013, 403(2), 434–450. Google Scholar
[29]	Z. Wang and Q. L. Zhang, The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations, Acta Math. Sci., 2012, 32(3), 825–841. Google Scholar
[30]	H. Yang and J. Wang, Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas, J. Math. Anal. Appl., 2014, 413(2), 800–820. Google Scholar
[31]	J. Y. Zhu, M. X. Huang and Z. Q. Shao, The limiting behavior of Riemann solutions to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force, Physics of Fluids, 2024, 36(2). DOI: 10.1063/5.0185216. CrossRef Google Scholar
[32]	L. Zhu and W. C. Sheng, The Riemann problem of adiabatic Chaplygin gas dynamic system, Commun. Appl. Math. Comput., 2010, 24 (1), 9–16. Google Scholar