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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

Huahui Li 1, and  Zhiqiang Shao 1,

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.
Keywords: generalized Chaplygin gas, vanishing pressure limits., vacuum state, delta shock wave, zero-pressure relativistic Euler equations, Relativistic Euler equations.
Mathematics Subject Classification: 35L65, 35L6.
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
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show all references

References:
[1]

Phys. Rev. D, 66 (2002), 043507. doi: 10.1103/PhysRevD.66.043507.  Google Scholar

[2]

J. Cosmol. Astropart. Phys., 57 (2004), 1238-1243. 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]

J. Math. Fluid Mech., 7 (2005), S326-S331. doi: 10.1007/s00021-005-0162-x.  Google Scholar

[5]

Oxford University Press, Oxford, 2000. doi: 0-19-850700-3 .  Google Scholar

[6]

Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 41, New York: Longman Scientific and Technical, 1989. doi: 0-582-01378-X .  Google Scholar

[7]

Sci. Mem. Moscow Univ. Math. Phys., 21 (1904), 1-121. doi: http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:33.0789.01.  Google Scholar

[8]

J. Differential Equations, 202 (2004), 332-353. doi: 10.1016/j.jde.2004.02.009.  Google Scholar

[9]

SIAM J. Math. Anal., 34 (2003), 925-938. doi: 10.1137/S0036141001399350.  Google Scholar

[10]

Phys. D, 189 (2004), 141-165. doi: 10.1016/j.physd.2003.09.039.  Google Scholar

[11]

J. Math. Anal. Appl., 381 (2011), 17-26. doi: 10.1016/j.jmaa.2011.04.017.  Google Scholar

[12]

Phys. Lett. B, 646 (2007), 177-182. 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]

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[15]

Methods Appl. Anal., 8 (2001), 159-207. doi: 10.4310/MAA.2001.v8.n1.a7.  Google Scholar

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[19]

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[27]

J. Differential Equations, 249 (2010), 3024-3051. doi: 10.1016/j.jde.2010.09.004.  Google Scholar

[28]

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[29]

in Mem. Amer. Math. Soc., 137, AMS, Providence, 1999. doi: 10.1090/memo/0654.  Google Scholar

[30]

Comm. Math. Phys, 156 (1993), 67-99. doi: 10.1007/BF02096733.  Google Scholar

[31]

Phys. Rev., 107 (1957), 884-900. doi: 10.1103/PhysRev.107.884.  Google Scholar

[32]

J. Fluid Mech., 171 (1986), 365-375. doi: 10.1017/S0022112086001489.  Google Scholar

[33]

Astrophys. J., 179 (1973), 897-907. doi: 10.1086/151927.  Google Scholar

[34]

J. Aeronaut. Sci., 6 (1939), 399-407. doi: 10.2514/8.916.  Google Scholar

[35]

J. Math. Anal. Appl., 403 (2013), 434-450. doi: 10.1016/j.jmaa.2013.02.026.  Google Scholar

[36]

Wiley, New York, 1972. doi:  978-0-471-92567-5.  Google Scholar

[37]

J. Math. Anal. Appl., 413 (2014), 800-820. doi: 10.1016/j.jmaa.2013.12.025.  Google Scholar

[38]

J. Math. Anal. Appl., 355 (2009), 594-605. doi: 10.1016/j.jmaa.2009.01.075.  Google Scholar

[39]

J. Math. Anal. Appl., 411 (2014), 506-521. doi: 10.1016/j.jmaa.2013.09.050.  Google Scholar

[40]

Abstr. Appl. Anal., 2013 (2013), 296361. doi: 10.1155/2013/296361.  Google Scholar

[41]

J. Cosmol. Astropart. Phys., 2006 (2006), 731-750. doi:  10.1088/1475-7516/2006/01/003.  Google Scholar

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