• Previous Article
    Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences
  • CPAA Home
  • This Issue
  • Next Article
    Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces
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]

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]

Commun. Pure Appl. Anal., 9 (2010), 431-458. doi: 10.2307/2152750.  Google Scholar

[15]

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

[16]

Commun. Pure Appl. Anal., 15 (2016), 127-138. Google Scholar

[17]

J. Appl. Anal. Comput., 6 (2016), 376-395. Google Scholar

[18]

J. Aeronaut. Sci., 8 (1941), 337-356. doi: http://dx.doi.org/10.2514/2.7046.  Google Scholar

[19]

Appl. Math. Lett., 14 (2001), 519-523. doi: 10.1016/S0893-9659(00)00187-7.  Google Scholar

[20]

Z. Angew. Math. Phys., 56 (2005), 239-253. doi: 10.1007/s00033-005-4118-2.  Google Scholar

[21]

J. Hyperbolic Differ. Equ., 4 (2007), 629-653. doi: 10.1142/S021989160700129X .  Google Scholar

[22]

Arch. Ration. Mech. Anal., 191 (2009), 539-577. Google Scholar

[23]

Phys. Lett. B, 648 (2007), 329-332. doi: doi:10.1016/j.physletb.2007.03.025.  Google Scholar

[24]

Phys. Lett. B, 654 (2007), 1-6. doi: doi:10.1016/j.physletb.2007.08.038.  Google Scholar

[25]

Z. Angew. Math. Phys., 67 (2016), 1-24. doi: 10.1007/s00033-016-0663-x.  Google Scholar

[26]

Appl. Math. Lett., 24 (2011), 1124-1129. doi: 10.1016/j.aml.2011.01.038.  Google Scholar

[27]

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

[28]

Nonlinear Anal. RWA, 22 (2015), 115-128. doi: doi:10.1016/j.nonrwa.2014.08.007.  Google Scholar

[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

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]

Commun. Pure Appl. Anal., 9 (2010), 431-458. doi: 10.2307/2152750.  Google Scholar

[15]

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

[16]

Commun. Pure Appl. Anal., 15 (2016), 127-138. Google Scholar

[17]

J. Appl. Anal. Comput., 6 (2016), 376-395. Google Scholar

[18]

J. Aeronaut. Sci., 8 (1941), 337-356. doi: http://dx.doi.org/10.2514/2.7046.  Google Scholar

[19]

Appl. Math. Lett., 14 (2001), 519-523. doi: 10.1016/S0893-9659(00)00187-7.  Google Scholar

[20]

Z. Angew. Math. Phys., 56 (2005), 239-253. doi: 10.1007/s00033-005-4118-2.  Google Scholar

[21]

J. Hyperbolic Differ. Equ., 4 (2007), 629-653. doi: 10.1142/S021989160700129X .  Google Scholar

[22]

Arch. Ration. Mech. Anal., 191 (2009), 539-577. Google Scholar

[23]

Phys. Lett. B, 648 (2007), 329-332. doi: doi:10.1016/j.physletb.2007.03.025.  Google Scholar

[24]

Phys. Lett. B, 654 (2007), 1-6. doi: doi:10.1016/j.physletb.2007.08.038.  Google Scholar

[25]

Z. Angew. Math. Phys., 67 (2016), 1-24. doi: 10.1007/s00033-016-0663-x.  Google Scholar

[26]

Appl. Math. Lett., 24 (2011), 1124-1129. doi: 10.1016/j.aml.2011.01.038.  Google Scholar

[27]

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

[28]

Nonlinear Anal. RWA, 22 (2015), 115-128. doi: doi:10.1016/j.nonrwa.2014.08.007.  Google Scholar

[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

[1]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675

[2]

Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021062

[3]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003

[4]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008

[5]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[6]

Andrey Kovtanyuk, Alexander Chebotarev, Nikolai Botkin, Varvara Turova, Irina Sidorenko, Renée Lampe. Modeling the pressure distribution in a spatially averaged cerebral capillary network. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021016

[7]

Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021069

[8]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409

[9]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392

[10]

Beixiang Fang, Qin Zhao. Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021066

[11]

Yangrong Li, Fengling Wang, Shuang Yang. Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3643-3665. doi: 10.3934/dcdsb.2020250

[12]

Yunjuan Jin, Aifang Qu, Hairong Yuan. Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021048

[13]

Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207

[14]

Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021

[15]

Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021100

[16]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002

[17]

Jinye Shen, Xian-Ming Gu. Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021086

[18]

Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258

[19]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[20]

Flank D. M. Bezerra, Rodiak N. Figueroa-López, Marcelo J. D. Nascimento. Fractional oscillon equations; solvability and connection with classical oscillon equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021067

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]