The limits of Riemann solutions for the isentropic Euler system with extended Chaplygin gas

References

• Chaplygin S . On gas jets. Sci Mem Moscow Univ Math Phys. 1904;21:1–121.
   Google Scholar
• Tsien HS . Two dimensional subsonic flow of compressible fluids. J Aeronaut Sci. 1939;6:399–407.
   Google Scholar
• Karman T . Compressibility effects in aerodynamics. J Aeronaut Sci. 1941;8:337–365.
   Google Scholar
• Bento MC , Bertolami O , Sen AA . Generalized Chaplygin gas, accelerated expansion, and dark-energy-matter unification. Phys Rev D. 2002;66(4):043507.
   Web of Science ®Google Scholar
• Bilic N , Tupper GB , Viollier RD . Unification of dark matter and dark energy: the inhomogeneous Chaplygin gas. Phys Lett B. 2012;535:17–21.
   Google Scholar
• Debnath U , Banerjee A , Chakraborty S . Role of modified Chaplygin gas in accelerated universe. Class Quant Gravity. 2004;21:5609–5618.
   Web of Science ®Google Scholar
• Pourhassan B , Kahya EO . Extended Chaplygin gas model. Res Phys. 2014;4:101–102.
   Google Scholar
• Heydarzade Y , Darabi F , Atazadeh K . Einstein static universe on the brane supported by extended Chaplygin gas. Astrophys Space Sci. 2016;361: Article No.250, 1–11.
   Web of Science ®Google Scholar
• Kahya EO , Khurshudyan M , Pourhassan B , et al. Higher order corrections of the extended Chaplygin gas cosmology with varying G and Λ. Eur Phys J C. 2015;75: Article No.43, 1–12.
   PubMed Web of Science ®Google Scholar
• Naji J . Extended Chaplygin gas equation of state with bulk and shear viscosities. Astrophys Space Sci. 2014;350:333–338.
   Web of Science ®Google Scholar
• Pourhassan B . Extended Chaplygin gas in Horava-Lifshitz gravity. Phys Dark Univ. 2016;13:132–138.
   Web of Science ®Google Scholar
• Paul BC , Thakur P , Saha A . Observational constraints on extended Chaplygin gas cosmologies. Pramana J Phys. 2017;89: Article No.29, 1–11.
   Web of Science ®Google Scholar
• Brenier Y . Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations. J Math Fluid Mech. 2005;7:S326–S331.
   Web of Science ®Google Scholar
• Kong D , Wei C . Formation and propagation of singlarities in one-dimensional Chaplygin gas. J Geom Phys. 2014;80:58–70.
   Web of Science ®Google Scholar
• Nedeljkov M . Higher order shadow waves and delta shock blow up in the Chaplygin gas. J Differ Equ. 2014;256:3859–3887.
   Web of Science ®Google Scholar
• Shen C . The Riemann problem for the Chaplygin gas equations with a source term. Z Angew Math Mech. 2016;96:681–695.
   Web of Science ®Google Scholar
• Chen GQ , Liu H . Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J Math Anal. 2003;34:925–938.
   Web of Science ®Google Scholar
• Li J . Note on the compressible Euler equations with zero temperature. Appl Math Lett. 2001;14:519–523.
   Web of Science ®Google Scholar
• Mitrovic D , Nedeljkov M . Delta-shock waves as a limit of shock waves. J Hyperb Differ Equ. 2007;4:629–653.
   Web of Science ®Google Scholar
• Shen C . The limits of Riemann solutions to the isentropic magnetogasdynamics. Appl Math Lett. 2011;24:1124–1129.
   Web of Science ®Google Scholar
• Shen C , Sheng W , Sun M . The asymptotic limits of solutions to the Riemann problem for the scaled Leroux system. Commun Pure Appl Anal. 2018;17:391–411.
   Web of Science ®Google Scholar
• Shen C , Sun M . Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model. J Differ Equ. 2010;249:3024–3051.
   Web of Science ®Google Scholar
• Shen C , Sun M , Wang Z . Limit relations for three simple hyperbolic systems of conservation laws. Math Methods Appl Sci. 2010;33:1317–1330.
   Web of Science ®Google Scholar
• Sun M . Stability of Riemann solutions for a non-strictly hyperbolic system with flux approximation. Elect J Differ Equ. 2016;2016(126):1–16.
   Google Scholar
• Guo L , Li T , Yin G . The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. Commun Pure Appl Anal. 2017;16:295–309.
   Web of Science ®Google Scholar
• Guo L , Li T , Yin G . The limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term. J Math Anal Appl. 2017;455:127–140.
   Web of Science ®Google Scholar
• Li H , Shao Z . 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;15:2373–2400.
   Web of Science ®Google Scholar
• Sheng W , Wang G , Yin G . Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes. Nonlinear Anal RWA. 2015;22:115–128.
   Google Scholar
• Yang H , Wang J . 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;413:800–820.
   Web of Science ®Google Scholar
• Yang H , Wang J . Concentration in vanishing pressure limit of solutions to the modified Chaplygin gas equations. J Math Phys. 2016;57:111504.
   Web of Science ®Google Scholar
• Chen J , Sheng W . The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas. Commun Pure Appl Anal. 2018;17:127–142.
   Web of Science ®Google Scholar
• Zhang Q . Concentration in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equatons. 2017, preprint, arXiv:1707.05153v1.
   Google Scholar
• Tong M , Shen C . The asymptotic limits of Riemann solutions for the isentropic extended Chaplygin gas dynamic system with the vanishing pressure, preprint.
   Google Scholar
• Smoller J . Shock waves and reaction--diffusion equations. New York (NY): Springer-Verlag; 1994.
   Google Scholar
• Sun M . Singular solutions to the Riemann problem for a macroscopic production model. Z Angew Math Mech. 2017;97:916–931.
   Web of Science ®Google Scholar
• Danilov VG , Shelkovich VM . Dynamics of propagation and interaction of δ-shock waves in conservation law systems. J Differ Equ. 2005;211:333–381.
   Web of Science ®Google Scholar
• Danilov VG , Shelkovich VM . Delta-shock waves type solution of hyperbolic systems of conservation laws. Q Appl Math. 2005;63:401–427.
   Web of Science ®Google Scholar
• Danilov VG , Mitrovic D . Delta shock wave formation in the case of triangular hyperbolic system of conservation laws. J Differ Equ. 2008;245:3704–3734.
   Web of Science ®Google Scholar
• Kalisch H , Mitrovic D . Singular solutions of a fully nonlinear 2×2 system of conservation laws. Proc Edinburgh Math Soc. 2012;55:711–729.
   Web of Science ®Google Scholar
• Nedeljkov M . Shadow waves: entropies and interactions for delta and singular shocks. Arch Rat Mech Anal. 2010;197:489–537.
   Web of Science ®Google Scholar