Vanishing magnetic field limits of solutions to the non-isentropic Chaplygin gas magnetogasdynamics equations

Jingye Zhao; Zonghua Wei; Jiahui Liu; Yongqiang Fan; Jingye Zhao; Zonghua Wei; Jiahui Liu; Yongqiang Fan

doi:10.3934/math.2025077

AIMS Mathematics

2025, Volume 10, Issue 1: 1675-1703. doi: 10.3934/math.2025077

Previous Article Next Article

Research article

Vanishing magnetic field limits of solutions to the non-isentropic Chaplygin gas magnetogasdynamics equations

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China

Received: 14 October 2024 Revised: 16 January 2025 Accepted: 20 January 2025 Published: 23 January 2025
MSC : 35L03, 35L65, 35L67, 35Q31

This paper studied the Riemann problem for the non-isentropic Chaplygin gas magnetogasdynamics equations and investigated the general asymptotic behavior of its Riemann solutions. Due to the influence of the source term in the equations, the Riemann solutions for the non-isentropic Chaplygin gas magnetogasdynamics equations are no longer self-similar. We performed the analysis after eliminating the source term by using a velocity transformation. When the Riemann initial data of density and velocity satisfied the condition $ v_–\frac{1}{\rho_-}\geq v_++\frac{1}{\rho_+} $, as the reciprocal of magnetic flux density $ \mu $ tended to zero, the Riemann solutions of the non-isentropic Chaplygin gas magnetogasdynamics equations converged to the delta shock solutions of the non-isentropic Chaplygin Euler equations. Otherwise, the Riemann solutions converged to a contact discontinuity of the non-isentropic Chaplygin Euler equations.
- non-isentropic Chaplygin gas magnetogasdynamics equations,
- source term,
- Riemann problem,
- vanishing magnetic field limits,
- delta shock
Citation: Jingye Zhao, Zonghua Wei, Jiahui Liu, Yongqiang Fan. Vanishing magnetic field limits of solutions to the non-isentropic Chaplygin gas magnetogasdynamics equations[J]. AIMS Mathematics, 2025, 10(1): 1675-1703. doi: 10.3934/math.2025077

Related Papers:

Abstract

This paper studied the Riemann problem for the non-isentropic Chaplygin gas magnetogasdynamics equations and investigated the general asymptotic behavior of its Riemann solutions. Due to the influence of the source term in the equations, the Riemann solutions for the non-isentropic Chaplygin gas magnetogasdynamics equations are no longer self-similar. We performed the analysis after eliminating the source term by using a velocity transformation. When the Riemann initial data of density and velocity satisfied the condition $ v_–\frac{1}{\rho_-}\geq v_++\frac{1}{\rho_+} $, as the reciprocal of magnetic flux density $ \mu $ tended to zero, the Riemann solutions of the non-isentropic Chaplygin gas magnetogasdynamics equations converged to the delta shock solutions of the non-isentropic Chaplygin Euler equations. Otherwise, the Riemann solutions converged to a contact discontinuity of the non-isentropic Chaplygin Euler equations.

References

[1]	P. Lax, Hyperbolic systems of conservation laws Ⅱ, Commun. Pur. Appl. Math., 10 (1957), 537–566. https://doi.org/10.1002/cpa.3160100406 doi: 10.1002/cpa.3160100406
[2]	Y. Brenier, Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations, J. Math. Fluid Mech., 7 (2005), S326–S331. https://doi.org/10.1007/s00021-005-0162-x doi: 10.1007/s00021-005-0162-x
[3]	G. Wang, The Riemann problem for one dimensional generalized Chaplygin gas dynamics, J. Math. Anal. Appl., 403 (2013), 434–450. https://doi.org/10.1016/j.jmaa.2013.02.026 doi: 10.1016/j.jmaa.2013.02.026
[4]	V. M. Shelkovich, B. Nilsson, O. S. Rozanova, Mass, momentum and energy conservation laws in zero-pressure gas dynamics and $\delta$-shocks: Ⅱ, Appl. Anal., 90 (2011), 831–842. https://doi.org/10.1080/00036811.2010.524156 doi: 10.1080/00036811.2010.524156
[5]	Y. Pang, Delta shock wave in the compressible Euler equations for a Chaplygin gas, J. Math. Anal. Appl., 448 (2017), 245–261. https://doi.org/10.1016/j.jmaa.2016.10.078 doi: 10.1016/j.jmaa.2016.10.078
[6]	Y. Song, L. Guo, Behavior of Riemann solutions of extended Chaplygin gas under the limiting condition, Acta Appl. Math., 174 (2021), 4. https://doi.org/10.1007/s10440-021-00422-5 doi: 10.1007/s10440-021-00422-5
[7]	Y. Song, L. Guo, General limiting behavior of Riemann solutions to the non-isentropic Euler equations for modified Chaplygin gas, J. Math. Phys., 61 (2020), 041506. https://doi.org/10.1063/1.5144326 doi: 10.1063/1.5144326
[8]	Y. Pang, M. Hu, The Riemann problem for the one-dimensional compressible flow of a van der Waals gas, Z. Angew. Math. Phys., 70 (2019), 142. https://doi.org/10.1007/s00033-019-1177-0 doi: 10.1007/s00033-019-1177-0
[9]	Y. Pang, J. Ge, H. Yang, M. Hu, The Riemann solutions to the compressible ideal fluid flow, Eur. J. Mech. B-Fluid., 82 (2020), 54–60. https://doi.org/10.1016/j.euromechflu.2020.03.002 doi: 10.1016/j.euromechflu.2020.03.002
[10]	Z. Shao, Vanishing magnetic field limits of solutions to the isentropic Chaplygin gas magnetogasdynamics equations, Bull. Malays. Math. Sci. Soc., 45 (2022), 3101–3129. https://doi.org/10.1007/s40840-022-01362-5 doi: 10.1007/s40840-022-01362-5
[11]	Z. Shao, Delta shocks and vacuum states for the isentropic magnetogasdynamics equations for Chaplygin gas as pressure and magnetic field vanish, Anal. Math. Phys., 12 (2022), 85. https://doi.org/10.1007/s13324-022-00692-8 doi: 10.1007/s13324-022-00692-8
[12]	C. Shen, The limits of Riemann solutions to the isentropic magnetogasdynamics, Appl. Math. Lett., 24 (2011), 1124–1129. https://doi.org/10.1016/j.aml.2011.01.038 doi: 10.1016/j.aml.2011.01.038
[13]	J. Chen, W. Sheng, The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas, Commun. Pur. Appl. Anal., 17 (2018), 127–142. https://doi.org/10.3934/cpaa.2018008 doi: 10.3934/cpaa.2018008
[14]	Y. Zhang, Y. Pang, J. Wang, Concentration and cavitation in the vanishing pressure limit of solutions to the generalized Chaplygin Euler equations of compressible fluid flow, Eur. J. Mech. B-Fluid., 78 (2019), 252–262. https://doi.org/10.1016/j.euromechflu.2019.103515 doi: 10.1016/j.euromechflu.2019.103515
[15]	Y. Zhang, Y. Pang, Concentration and cavitation in the vanishing pressure limit of solutions to a simplified isentropic relativistic Euler equations, J. Math. Fluid Mech., 23 (2021), 8. https://doi.org/10.1007/s00021-020-00526-2 doi: 10.1007/s00021-020-00526-2
[16]	W. Peng, T. Wang, On vanishing pressure limit of continuous solutions to the isentropic Euler equations, J. Hyperbol. Differ. Eq., 19 (2022), 311–336. https://doi.org/10.1142/S0219891622500084 doi: 10.1142/S0219891622500084
[17]	Z. Lei, Z. Shao, Concentration and cavitation in the vanishing pressure limit of solutions to the relativistic Euler equations with the logarithmic equation of state, J. Math. Phys., 64 (2023), 071507. https://doi.org/10.1063/5.0157277 doi: 10.1063/5.0157277
[18]	G. Faccanoni, A. Mangeney, Exact solutions for granular flows, Int. J. Numer. Anal. Met., 37 (2013), 1408–1433. https://doi.org/10.1002/nag.2124 doi: 10.1002/nag.2124
[19]	C. Shen, The limits of Riemann solutions to the isentropic magnetogasdynamics, Appl. Math. Lett., 24 (2011), 1124–1129. https://doi.org/10.1016/j.aml.2011.01.038 doi: 10.1016/j.aml.2011.01.038
[20]	H. Yang, 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., 413 (2014), 800–820. https://doi.org/10.1016/j.jmaa.2013.12.025 doi: 10.1016/j.jmaa.2013.12.025
[21]	L. Guo, T. Li, G. Yin, The limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term, J. Math. Anal. Appl., 455 (2017), 127–140. https://doi.org/10.1016/j.jmaa.2017.05.048 doi: 10.1016/j.jmaa.2017.05.048
[22]	Y. Pang, M. Hu, J. Wang, Riemann problem for a compressible perfect fluid with a constant external force for the Chaplygin gas, Bound. Value Probl., 2018 (2018), 89. https://doi.org/10.1186/s13661-018-1009-8 doi: 10.1186/s13661-018-1009-8
[23]	S. Li, J. Zhao, Delta-shock for the nonhomogeneous pressureless Euler system, Bull. Korean Math. Soc., 61 (2024), 699–715. https://doi.org/10.4134/BKMS.b230304 doi: 10.4134/BKMS.b230304
[24]	C. Shen, M. Sun, Exact Riemann solutions for the drift-flux equations of two-phase flow under gravity, J. Differ. Equations, 314 (2022), 1–55. https://doi.org/10.1016/j.jde.2022.01.009 doi: 10.1016/j.jde.2022.01.009

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)