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November  2016, 15(6): 2373-2400. doi: 10.3934/cpaa.2016041

Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas

1. 

College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China, China

Received  March 2016 Revised  July 2016 Published  September 2016

The Riemann solutions for the relativistic Euler equations for generalized Chaplygin gas are considered. It is rigorously proved that, as the pressure vanishes, they tend to the two kinds of Riemann solutions to the zero-pressure relativistic Euler equations, which include a delta shock formed by a weighted $\delta$-measure and a vacuum state.
Citation: 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
References:
[1]

M. C. Bento, O. Bertolami and A. A. Sen, Generalized Chaplygin gas, accelerated expansion and dark energy-matter unification,, \emph{Phys. Rev. D}, 66 (2002). doi: 10.1103/PhysRevD.66.043507. Google Scholar

[2]

N. Bilic, R. J. Lindebaum, G. B. Tupper and R. D. Viollier, Nonlinear evolution of dark matter and dark energy in the Chaplygin gas cosmology,, \emph{J. Cosmol. Astropart. Phys.}, 57 (2004), 1238. doi: 10.1088/1475-7516/2004/11/008. Google Scholar

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Y. Brenier, Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations,, \emph{J. Math. Fluid Mech.}, 7 (2005). doi: 10.1007/s00021-005-0162-x. Google Scholar

[5]

A. Bressan, Hyperbolic Systems of Conservation Laws,, Oxford University Press, (2000). doi: 0-19-850700-3 . Google Scholar

[6]

T. Chang and L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics,, Pitman Monographs and Surveys in Pure and Applied Mathematics, (1989). doi: 0-582-01378-X . Google Scholar

[7]

S. Chaplygin, On gas jets,, \emph{Sci. Mem. Moscow Univ. Math. Phys.}, 21 (1904), 1. doi: http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:33.0789.01. Google Scholar

[8]

G.-Q. Chen and Y. Li, Stability of Riemann solutions with large oscillation for the relativistic Euler equations,, \emph{J. Differential Equations}, 202 (2004), 332. doi: 10.1016/j.jde.2004.02.009. Google Scholar

[9]

G.-Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids,, \emph{SIAM J. Math. Anal.}, 34 (2003), 925. doi: 10.1137/S0036141001399350. Google Scholar

[10]

G.-Q. Chen and H. Liu, Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids,, \emph{Phys. D}, 189 (2004), 141. doi: 10.1016/j.physd.2003.09.039. Google Scholar

[11]

H. Cheng and H. Yang, Riemann problem for the relativistic Chaplygin Euler equations,, \emph{J. Math. Anal. Appl.}, 381 (2011), 17. doi: 10.1016/j.jmaa.2011.04.017. Google Scholar

[12]

Norman Cruz, Samuel Lepe and Francisco Pena, Dissipative generalized Chaplygin gas as phantom dark energy Physics,, \emph{Phys. Lett. B}, 646 (2007), 177. doi: 10.1016/j.physletb.2006.12.070. Google Scholar

[13]

V. Gorini, A. Kamenshchik, U. Moschella and V. Pasquier, The chaplygin gas as an model for dark energy,, arXiv:gr-qc/0403062., (). doi: arXiv:gr-qc/0403062. Google Scholar

[14]

L. Guo, W. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system,, \emph{Commun. Pure Appl. Anal.}, 9 (2010), 431. doi: 10.2307/2152750. Google Scholar

[15]

C. H. Hsu, S. S. Lin and T. Makino, On the relativistic Euler equation,, \emph{Methods Appl. Anal.}, 8 (2001), 159. doi: 10.4310/MAA.2001.v8.n1.a7. Google Scholar

[16]

M. Huang and Z. Shao, Riemann problem for the relativistic generalized Chaplygin Euler equations,, \emph{Commun. Pure Appl. Anal.}, 15 (2016), 127. Google Scholar

[17]

M. Huang and Z. Shao, Riemann problem with delta initial data for the relativistic Chaplygin Euler equations,, \emph{J. Appl. Anal. Comput.}, 6 (2016), 376. Google Scholar

[18]

T. von Karman, Compressibility effects in aerodynamics,, \emph{J. Aeronaut. Sci.}, 8 (1941), 337. doi: http://dx.doi.org/10.2514/2.7046. Google Scholar

[19]

Jiequan Li, Note on the compressible Euler equations with zero temperature,, \emph{Appl. Math. Lett.}, 14 (2001), 519. doi: 10.1016/S0893-9659(00)00187-7. Google Scholar

[20]

Y. Li, D. Feng and Z. Wang, Global entropy solutions to the relativistic Euler equations for a class of large initial data,, \emph{Z. Angew. Math. Phys.}, 56 (2005), 239. doi: 10.1007/s00033-005-4118-2. Google Scholar

[21]

D. Mitrovic and M. Nedeljkov, Delta-shock waves as a limit of shock waves,, \emph{J. Hyperbolic Differ. Equ.}, 4 (2007), 629. doi: 10.1142/S021989160700129X . Google Scholar

[22]

D. Serre, Multidimensional shock interaction for a Chaplygin gas,, \emph{Arch. Ration. Mech. Anal.}, 191 (2009), 539. Google Scholar

[23]

M.R. Setare, Holographic Chaplygin gas model,, \emph{Phys. Lett. B}, 648 (2007), 329. doi: doi:10.1016/j.physletb.2007.03.025. Google Scholar

[24]

M. R. Setare, Interacting holographic generalized Chaplygin gas model,, \emph{Phys. Lett. B}, 654 (2007), 1. doi: doi:10.1016/j.physletb.2007.08.038. Google Scholar

[25]

Z. Shao, Riemann problem with delta initial data for the isentropic relativistic Chaplygin Euler equations,, \emph{Z. Angew. Math. Phys.}, 67 (2016), 1. doi: 10.1007/s00033-016-0663-x. Google Scholar

[26]

C. Shen, The limits of Riemann solutions to the isentropic magnetogasdynamics,, \emph{Appl. Math. Lett.}, 24 (2011), 1124. doi: 10.1016/j.aml.2011.01.038. Google Scholar

[27]

C. Shen and M. Sun, Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model,, \emph{J. Differential Equations}, 249 (2010), 3024. doi: 10.1016/j.jde.2010.09.004. Google Scholar

[28]

W. Sheng, G. Wang and G. Yin, Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes,, \emph{Nonlinear Anal. RWA}, 22 (2015), 115. doi: doi:10.1016/j.nonrwa.2014.08.007. Google Scholar

[29]

W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics,, in \emph{Mem. Amer. Math. Soc.}, (1999). doi: 10.1090/memo/0654. Google Scholar

[30]

J. Smoller and B. Temple, Global solutions of the relativistic Euler equations,, \emph{Comm. Math. Phys}, 156 (1993), 67. doi: 10.1007/BF02096733. Google Scholar

[31]

A. H. Taub, Approximate solutions of the Einstein equations for isentropic motions of plane-symmetric distributions of perfect fluids,, \emph{Phys. Rev.}, 107 (1957), 884. doi: 10.1103/PhysRev.107.884. Google Scholar

[32]

K. Thompson, The special relativistic shock tube,, \emph{J. Fluid Mech.}, 171 (1986), 365. doi: 10.1017/S0022112086001489. Google Scholar

[33]

K. S. Thorne, Relativistic shocks: the Taub adiabatic,, \emph{Astrophys. J.}, 179 (1973), 897. doi: 10.1086/151927. Google Scholar

[34]

H. S. Tsien, Two dimensional subsonic flow of compressible fluids,, \emph{J. Aeronaut. Sci.}, 6 (1939), 399. doi: 10.2514/8.916. Google Scholar

[35]

G. Wang, The Riemann problem for one dimensional generalized Chaplygin gas dynamics,, \emph{J. Math. Anal. Appl.}, 403 (2013), 434. doi: 10.1016/j.jmaa.2013.02.026. Google Scholar

[36]

S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,, Wiley, (1972). doi:  978-0-471-92567-5. Google Scholar

[37]

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,, \emph{J. Math. Anal. Appl.}, 413 (2014), 800. doi: 10.1016/j.jmaa.2013.12.025. Google Scholar

[38]

G. Yin and W. Sheng, Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases,, \emph{J. Math. Anal. Appl.}, 355 (2009), 594. doi: 10.1016/j.jmaa.2009.01.075. Google Scholar

[39]

G. Yin and K. Song, Vanishing pressure limits of Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas,, \emph{J. Math. Anal. Appl.}, 411 (2014), 506. doi: 10.1016/j.jmaa.2013.09.050. Google Scholar

[40]

G. Yin and K. Song, Limits of Riemann solutions to the relativistic Euler systems for Chaplygin gas as pressure vanishes,, \emph{Abstr. Appl. Anal.}, 2013 (2013). doi: 10.1155/2013/296361. Google Scholar

[41]

X. Zhang, F.-Q. Wu and J. Zhang, New generalized Chaplygin gas as a scheme for unification of dark energy and dark matter,, \emph{J. Cosmol. Astropart. Phys.}, 2006 (2006), 731. doi:  10.1088/1475-7516/2006/01/003. Google Scholar

show all references

References:
[1]

M. C. Bento, O. Bertolami and A. A. Sen, Generalized Chaplygin gas, accelerated expansion and dark energy-matter unification,, \emph{Phys. Rev. D}, 66 (2002). doi: 10.1103/PhysRevD.66.043507. Google Scholar

[2]

N. Bilic, R. J. Lindebaum, G. B. Tupper and R. D. Viollier, Nonlinear evolution of dark matter and dark energy in the Chaplygin gas cosmology,, \emph{J. Cosmol. Astropart. Phys.}, 57 (2004), 1238. doi: 10.1088/1475-7516/2004/11/008. Google Scholar

[3]

N. Bilic, G. B. Tupper and R. D. Viollier, Dark matter, dark energy and the Chaplygin gas},, arXiv:astro-ph/0207423., (). doi: arXiv:astro-ph/0207423. Google Scholar

[4]

Y. Brenier, Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations,, \emph{J. Math. Fluid Mech.}, 7 (2005). doi: 10.1007/s00021-005-0162-x. Google Scholar

[5]

A. Bressan, Hyperbolic Systems of Conservation Laws,, Oxford University Press, (2000). doi: 0-19-850700-3 . Google Scholar

[6]

T. Chang and L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics,, Pitman Monographs and Surveys in Pure and Applied Mathematics, (1989). doi: 0-582-01378-X . Google Scholar

[7]

S. Chaplygin, On gas jets,, \emph{Sci. Mem. Moscow Univ. Math. Phys.}, 21 (1904), 1. doi: http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:33.0789.01. Google Scholar

[8]

G.-Q. Chen and Y. Li, Stability of Riemann solutions with large oscillation for the relativistic Euler equations,, \emph{J. Differential Equations}, 202 (2004), 332. doi: 10.1016/j.jde.2004.02.009. Google Scholar

[9]

G.-Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids,, \emph{SIAM J. Math. Anal.}, 34 (2003), 925. doi: 10.1137/S0036141001399350. Google Scholar

[10]

G.-Q. Chen and H. Liu, Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids,, \emph{Phys. D}, 189 (2004), 141. doi: 10.1016/j.physd.2003.09.039. Google Scholar

[11]

H. Cheng and H. Yang, Riemann problem for the relativistic Chaplygin Euler equations,, \emph{J. Math. Anal. Appl.}, 381 (2011), 17. doi: 10.1016/j.jmaa.2011.04.017. Google Scholar

[12]

Norman Cruz, Samuel Lepe and Francisco Pena, Dissipative generalized Chaplygin gas as phantom dark energy Physics,, \emph{Phys. Lett. B}, 646 (2007), 177. doi: 10.1016/j.physletb.2006.12.070. Google Scholar

[13]

V. Gorini, A. Kamenshchik, U. Moschella and V. Pasquier, The chaplygin gas as an model for dark energy,, arXiv:gr-qc/0403062., (). doi: arXiv:gr-qc/0403062. Google Scholar

[14]

L. Guo, W. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system,, \emph{Commun. Pure Appl. Anal.}, 9 (2010), 431. doi: 10.2307/2152750. Google Scholar

[15]

C. H. Hsu, S. S. Lin and T. Makino, On the relativistic Euler equation,, \emph{Methods Appl. Anal.}, 8 (2001), 159. doi: 10.4310/MAA.2001.v8.n1.a7. Google Scholar

[16]

M. Huang and Z. Shao, Riemann problem for the relativistic generalized Chaplygin Euler equations,, \emph{Commun. Pure Appl. Anal.}, 15 (2016), 127. Google Scholar

[17]

M. Huang and Z. Shao, Riemann problem with delta initial data for the relativistic Chaplygin Euler equations,, \emph{J. Appl. Anal. Comput.}, 6 (2016), 376. Google Scholar

[18]

T. von Karman, Compressibility effects in aerodynamics,, \emph{J. Aeronaut. Sci.}, 8 (1941), 337. doi: http://dx.doi.org/10.2514/2.7046. Google Scholar

[19]

Jiequan Li, Note on the compressible Euler equations with zero temperature,, \emph{Appl. Math. Lett.}, 14 (2001), 519. doi: 10.1016/S0893-9659(00)00187-7. Google Scholar

[20]

Y. Li, D. Feng and Z. Wang, Global entropy solutions to the relativistic Euler equations for a class of large initial data,, \emph{Z. Angew. Math. Phys.}, 56 (2005), 239. doi: 10.1007/s00033-005-4118-2. Google Scholar

[21]

D. Mitrovic and M. Nedeljkov, Delta-shock waves as a limit of shock waves,, \emph{J. Hyperbolic Differ. Equ.}, 4 (2007), 629. doi: 10.1142/S021989160700129X . Google Scholar

[22]

D. Serre, Multidimensional shock interaction for a Chaplygin gas,, \emph{Arch. Ration. Mech. Anal.}, 191 (2009), 539. Google Scholar

[23]

M.R. Setare, Holographic Chaplygin gas model,, \emph{Phys. Lett. B}, 648 (2007), 329. doi: doi:10.1016/j.physletb.2007.03.025. Google Scholar

[24]

M. R. Setare, Interacting holographic generalized Chaplygin gas model,, \emph{Phys. Lett. B}, 654 (2007), 1. doi: doi:10.1016/j.physletb.2007.08.038. Google Scholar

[25]

Z. Shao, Riemann problem with delta initial data for the isentropic relativistic Chaplygin Euler equations,, \emph{Z. Angew. Math. Phys.}, 67 (2016), 1. doi: 10.1007/s00033-016-0663-x. Google Scholar

[26]

C. Shen, The limits of Riemann solutions to the isentropic magnetogasdynamics,, \emph{Appl. Math. Lett.}, 24 (2011), 1124. doi: 10.1016/j.aml.2011.01.038. Google Scholar

[27]

C. Shen and M. Sun, Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model,, \emph{J. Differential Equations}, 249 (2010), 3024. doi: 10.1016/j.jde.2010.09.004. Google Scholar

[28]

W. Sheng, G. Wang and G. Yin, Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes,, \emph{Nonlinear Anal. RWA}, 22 (2015), 115. doi: doi:10.1016/j.nonrwa.2014.08.007. Google Scholar

[29]

W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics,, in \emph{Mem. Amer. Math. Soc.}, (1999). doi: 10.1090/memo/0654. Google Scholar

[30]

J. Smoller and B. Temple, Global solutions of the relativistic Euler equations,, \emph{Comm. Math. Phys}, 156 (1993), 67. doi: 10.1007/BF02096733. Google Scholar

[31]

A. H. Taub, Approximate solutions of the Einstein equations for isentropic motions of plane-symmetric distributions of perfect fluids,, \emph{Phys. Rev.}, 107 (1957), 884. doi: 10.1103/PhysRev.107.884. Google Scholar

[32]

K. Thompson, The special relativistic shock tube,, \emph{J. Fluid Mech.}, 171 (1986), 365. doi: 10.1017/S0022112086001489. Google Scholar

[33]

K. S. Thorne, Relativistic shocks: the Taub adiabatic,, \emph{Astrophys. J.}, 179 (1973), 897. doi: 10.1086/151927. Google Scholar

[34]

H. S. Tsien, Two dimensional subsonic flow of compressible fluids,, \emph{J. Aeronaut. Sci.}, 6 (1939), 399. doi: 10.2514/8.916. Google Scholar

[35]

G. Wang, The Riemann problem for one dimensional generalized Chaplygin gas dynamics,, \emph{J. Math. Anal. Appl.}, 403 (2013), 434. doi: 10.1016/j.jmaa.2013.02.026. Google Scholar

[36]

S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,, Wiley, (1972). doi:  978-0-471-92567-5. Google Scholar

[37]

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,, \emph{J. Math. Anal. Appl.}, 413 (2014), 800. doi: 10.1016/j.jmaa.2013.12.025. Google Scholar

[38]

G. Yin and W. Sheng, Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases,, \emph{J. Math. Anal. Appl.}, 355 (2009), 594. doi: 10.1016/j.jmaa.2009.01.075. Google Scholar

[39]

G. Yin and K. Song, Vanishing pressure limits of Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas,, \emph{J. Math. Anal. Appl.}, 411 (2014), 506. doi: 10.1016/j.jmaa.2013.09.050. Google Scholar

[40]

G. Yin and K. Song, Limits of Riemann solutions to the relativistic Euler systems for Chaplygin gas as pressure vanishes,, \emph{Abstr. Appl. Anal.}, 2013 (2013). doi: 10.1155/2013/296361. Google Scholar

[41]

X. Zhang, F.-Q. Wu and J. Zhang, New generalized Chaplygin gas as a scheme for unification of dark energy and dark matter,, \emph{J. Cosmol. Astropart. Phys.}, 2006 (2006), 731. doi:  10.1088/1475-7516/2006/01/003. Google Scholar

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