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January  2016, 15(1): 127-138. doi: 10.3934/cpaa.2016.15.127

Riemann problem for the relativistic generalized Chaplygin Euler equations

1. 

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

2. 

Department of Mathematics, Fuzhou University, Fuzhou 350002, China

Received  April 2015 Revised  September 2015 Published  December 2015

The Riemann problem for the relativistic generalized Chaplygin Euler equations is considered. Its two characteristic fields are genuinely nonlinear, but the nonclassical solutions appear. The formation of mechanism for $\delta-$shock is analyzed, that is the one-shock curve and the two-shock curve do not intersect each other in the phase plane. The Riemann solutions are constructed, and the generalized Rankine-Hugoniot conditions and the $\delta-$entropy condition are clarified. Moreover, under the generalized Rankine-Hugoniot conditions and entropy condition, we constructively obtain $\delta-$shock waves.
Citation: 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
References:
[1]

Y. Brenier, Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations,, \emph{J. Math. Fluid Mech}, 7 (2005), 326. doi: 10.1007/s00021-005-0162-x.

[2]

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: 10.1007/978-1-4612-0873-0.

[3]

Guiqiang Chen and P. G. LeFloch, Existence theory for the isentropic Euler equations,, \emph{J Arch. Rational Mech. Anal}, 166 (2003), 81. doi: 10.1007/s00205-002-0229-2.

[4]

Guiqiang Chen and Yachun 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.

[5]

Guiqiang Chen and Hailiang 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.

[6]

Hongjun Cheng and Hanchun 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.

[7]

Norman Cruz, Samuel Lepe and Francisco Pena, Dissipative generalized Chaplygin gas as phantom dark energy Physics,, \emph{Phys. Lett}, B 646 (2007), 177. doi: 10.2307/2152750.

[8]

V. G. Danilov and V. M. Shelkovich, Dynamics of progation and interaction of $\delta-$shock waves in conservation law systems,, \emph{J. Differential Equations}, 221 (2005), 333. doi: 10.1016/j.jde.2004.12.011.

[9]

Yin Gan and Sheng Wancheng, 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.

[10]

V. Gorini, A. Kamenshchik and Moschella, The chaplygin gas as an model for dark energy,, \emph{gr-qc}, 23 (2004). doi: 10.2307/2152750.

[11]

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

[12]

B. T. Hayes and P. G. Lefloch, Measure solutions to a strictly hyperbolic system of conservation laws,, \emph{Nonlinearity}, 9 (1996), 1547. doi: 10.1088/0951-7715/9/6/009.

[13]

B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions,, \emph{J. Differential Equations}, 118 (1995), 420. doi: 10.1006/jdeq.1995.1080.

[14]

Y. C. Li, D. M. Feng and Z. J. 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: 10.2307/2152750.

[15]

G. Dal Maso, P. G. Lefloch and F. Murat, Definition and weak stability of nonconservative products,, \emph{J. Math. Pures Appl.}, 74 (1995), 483. doi: 10.2307/2152750.

[16]

M. Nedeljkov, Delta and singular delta locus for one dimensional systems of conservation laws,, \emph{Math. Methods Appl. Sci}, 27 (2004), 931. doi: 10.1002/mma.480.

[17]

E. Yu. Panov and V. M. Shelkovich, Shock waves as a new type of solutions to system of conservation laws,, \emph{J. Differential Equations}, 228 (2006), 49. doi: 10.1016/j.jde.2006.04.004.

[18]

D. Serre, Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservation,, \emph{J. Differential Equations}, 68 (1987), 137. doi: 10.1016/0022-0396(87)90189-6.

[19]

D. Serre, Multidimensional shock interaction for a Chaplygin gas,, \emph{Arch. Ration. Mech. Anal.}, 191 (2009), 539. doi: 10.1007/s00205-008-0110-z.

[20]

M. R. Setare, Interacting holographic generalized Chaplygin gas model,, \emph{Phys. Lett.}, B 654 (2007), 1.

[21]

V. M. Shelkovich, The Riemann problem admitting $\delta-$, $\delta^{'}-$ shocks and vacuum states (the vanishing viscosity approach),, \emph{J. Differential Equations}, 231 (2006), 459. doi: 10.1016/j.jde.2006.08.003.

[22]

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.

[23]

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

[24]

D. Tan and T. Zhang, Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws (I) Four-J cases,, \emph{J. Differential Equations}, 111 (1994), 203. doi: 10.1006/jdeq.1994.1081.

[25]

Guodong 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.

[26]

H. Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws,, \emph{J. Differential Equations}, 159 (1999), 447. doi: 10.1006/jdeq.1999.3629.

[27]

W. E. Yu, G. Rykov and Ya. G. Sinai, Generalized variational principles,global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics,, \emph{Comm. Math. Phys.}, 177 (1996), 349. doi: 10.2307/2152750.

show all references

References:
[1]

Y. Brenier, Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations,, \emph{J. Math. Fluid Mech}, 7 (2005), 326. doi: 10.1007/s00021-005-0162-x.

[2]

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: 10.1007/978-1-4612-0873-0.

[3]

Guiqiang Chen and P. G. LeFloch, Existence theory for the isentropic Euler equations,, \emph{J Arch. Rational Mech. Anal}, 166 (2003), 81. doi: 10.1007/s00205-002-0229-2.

[4]

Guiqiang Chen and Yachun 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.

[5]

Guiqiang Chen and Hailiang 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.

[6]

Hongjun Cheng and Hanchun 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.

[7]

Norman Cruz, Samuel Lepe and Francisco Pena, Dissipative generalized Chaplygin gas as phantom dark energy Physics,, \emph{Phys. Lett}, B 646 (2007), 177. doi: 10.2307/2152750.

[8]

V. G. Danilov and V. M. Shelkovich, Dynamics of progation and interaction of $\delta-$shock waves in conservation law systems,, \emph{J. Differential Equations}, 221 (2005), 333. doi: 10.1016/j.jde.2004.12.011.

[9]

Yin Gan and Sheng Wancheng, 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.

[10]

V. Gorini, A. Kamenshchik and Moschella, The chaplygin gas as an model for dark energy,, \emph{gr-qc}, 23 (2004). doi: 10.2307/2152750.

[11]

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

[12]

B. T. Hayes and P. G. Lefloch, Measure solutions to a strictly hyperbolic system of conservation laws,, \emph{Nonlinearity}, 9 (1996), 1547. doi: 10.1088/0951-7715/9/6/009.

[13]

B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions,, \emph{J. Differential Equations}, 118 (1995), 420. doi: 10.1006/jdeq.1995.1080.

[14]

Y. C. Li, D. M. Feng and Z. J. 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: 10.2307/2152750.

[15]

G. Dal Maso, P. G. Lefloch and F. Murat, Definition and weak stability of nonconservative products,, \emph{J. Math. Pures Appl.}, 74 (1995), 483. doi: 10.2307/2152750.

[16]

M. Nedeljkov, Delta and singular delta locus for one dimensional systems of conservation laws,, \emph{Math. Methods Appl. Sci}, 27 (2004), 931. doi: 10.1002/mma.480.

[17]

E. Yu. Panov and V. M. Shelkovich, Shock waves as a new type of solutions to system of conservation laws,, \emph{J. Differential Equations}, 228 (2006), 49. doi: 10.1016/j.jde.2006.04.004.

[18]

D. Serre, Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservation,, \emph{J. Differential Equations}, 68 (1987), 137. doi: 10.1016/0022-0396(87)90189-6.

[19]

D. Serre, Multidimensional shock interaction for a Chaplygin gas,, \emph{Arch. Ration. Mech. Anal.}, 191 (2009), 539. doi: 10.1007/s00205-008-0110-z.

[20]

M. R. Setare, Interacting holographic generalized Chaplygin gas model,, \emph{Phys. Lett.}, B 654 (2007), 1.

[21]

V. M. Shelkovich, The Riemann problem admitting $\delta-$, $\delta^{'}-$ shocks and vacuum states (the vanishing viscosity approach),, \emph{J. Differential Equations}, 231 (2006), 459. doi: 10.1016/j.jde.2006.08.003.

[22]

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.

[23]

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

[24]

D. Tan and T. Zhang, Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws (I) Four-J cases,, \emph{J. Differential Equations}, 111 (1994), 203. doi: 10.1006/jdeq.1994.1081.

[25]

Guodong 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.

[26]

H. Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws,, \emph{J. Differential Equations}, 159 (1999), 447. doi: 10.1006/jdeq.1999.3629.

[27]

W. E. Yu, G. Rykov and Ya. G. Sinai, Generalized variational principles,global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics,, \emph{Comm. Math. Phys.}, 177 (1996), 349. doi: 10.2307/2152750.

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