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

[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.

[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.

[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.

[5]

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

[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 .

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[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 .

[22]

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

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

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

[33]

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

[34]

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

[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.

[36]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[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 .

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[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 .

[22]

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

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

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

[33]

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

[34]

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

[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.

[36]

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

[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.

[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.

[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.

[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.

[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.

[1]

Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127

[2]

Lihui Guo, Tong Li, Gan Yin. The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. Communications on Pure & Applied Analysis, 2017, 16 (1) : 295-310. doi: 10.3934/cpaa.2017014

[3]

Ju Ge, Wancheng Sheng. The two dimensional gas expansion problem of the Euler equations for the generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2733-2748. doi: 10.3934/cpaa.2014.13.2733

[4]

Yachun Li, Xucai Ren. Non-relativistic global limits of the entropy solutions to the relativistic Euler equations with $\gamma$-law. Communications on Pure & Applied Analysis, 2006, 5 (4) : 963-979. doi: 10.3934/cpaa.2006.5.963

[5]

Xingwen Hao, Yachun Li, Zejun Wang. Non-relativistic global limits to the three dimensional relativistic euler equations with spherical symmetry. Communications on Pure & Applied Analysis, 2010, 9 (2) : 365-386. doi: 10.3934/cpaa.2010.9.365

[6]

La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981

[7]

Geng Lai, Wancheng Sheng, Yuxi Zheng. Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 489-523. doi: 10.3934/dcds.2011.31.489

[8]

Philippe G. LeFloch, Seiji Ukai. A symmetrization of the relativistic Euler equations with several spatial variables. Kinetic & Related Models, 2009, 2 (2) : 275-292. doi: 10.3934/krm.2009.2.275

[9]

Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549

[10]

Yachun Li, Qiufang Shi. Global existence of the entropy solutions to the isentropic relativistic Euler equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 763-778. doi: 10.3934/cpaa.2005.4.763

[11]

Juan Calvo. On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1341-1347. doi: 10.3934/cpaa.2013.12.1341

[12]

Peng Zhang, Jiequan Li, Tong Zhang. On two-dimensional Riemann problem for pressure-gradient equations of the Euler system. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 609-634. doi: 10.3934/dcds.1998.4.609

[13]

Mapundi K. Banda, Michael Herty, Axel Klar. Coupling conditions for gas networks governed by the isothermal Euler equations. Networks & Heterogeneous Media, 2006, 1 (2) : 295-314. doi: 10.3934/nhm.2006.1.295

[14]

Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743

[15]

Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu. Existence and stability of periodic solutions for relativistic singular equations. Communications on Pure & Applied Analysis, 2017, 16 (2) : 591-609. doi: 10.3934/cpaa.2017029

[16]

Yonggeun Cho, Tohru Ozawa. On radial solutions of semi-relativistic Hartree equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 71-82. doi: 10.3934/dcdss.2008.1.71

[17]

Gui-Qiang G. Chen, Hairong Yuan. Local uniqueness of steady spherical transonic shock-fronts for the three--dimensional full Euler equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2515-2542. doi: 10.3934/cpaa.2013.12.2515

[18]

Marko Nedeljkov, Sanja Ružičić. On the uniqueness of solution to generalized Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4439-4460. doi: 10.3934/dcds.2017190

[19]

Chengchun Hao. Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2885-2931. doi: 10.3934/dcdsb.2015.20.2885

[20]

Yachun Li, Shengguo Zhu. On regular solutions of the $3$D compressible isentropic Euler-Boltzmann equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3059-3086. doi: 10.3934/dcds.2015.35.3059

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]