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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] M. R. Setare, Interacting holographic generalized Chaplygin gas model,, \emph{Phys. Lett.}, B 654 (2007), 1.   Google Scholar [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.  Google Scholar [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.  Google Scholar [23] J. Smoller and B. Temple, Global solutions of the relativistic Euler equations,, \emph{Comm. Math. Phys}, 156 (1993), 67.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] M. R. Setare, Interacting holographic generalized Chaplygin gas model,, \emph{Phys. Lett.}, B 654 (2007), 1.   Google Scholar [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.  Google Scholar [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.  Google Scholar [23] J. Smoller and B. Temple, Global solutions of the relativistic Euler equations,, \emph{Comm. Math. Phys}, 156 (1993), 67.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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