# American Institute of Mathematical Sciences

• Previous Article
Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media
• CPAA Home
• This Issue
• Next Article
Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents
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
 [1] 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 [2] 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 [3] Jianjun Chen, Wancheng Sheng. The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2018, 17 (1) : 127-142. doi: 10.3934/cpaa.2018008 [4] 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 [5] Lihui Guo, Wancheng Sheng, Tong Zhang. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system$^*$. Communications on Pure & Applied Analysis, 2010, 9 (2) : 431-458. doi: 10.3934/cpaa.2010.9.431 [6] 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 [7] Anupam Sen, T. Raja Sekhar. Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2641-2653. doi: 10.3934/cpaa.2020115 [8] W. G. Litvinov. Problem on stationary flow of electrorheological fluids at the generalized conditions of slip on the boundary. Communications on Pure & Applied Analysis, 2007, 6 (1) : 247-277. doi: 10.3934/cpaa.2007.6.247 [9] Dmitry V. Zenkov, Anthony M. Bloch. Dynamics of generalized Euler tops with constraints. Conference Publications, 2001, 2001 (Special) : 398-405. doi: 10.3934/proc.2001.2001.398 [10] 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 [11] Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 555-584. doi: 10.3934/dcds.1995.1.555 [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] Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 419-430. doi: 10.3934/dcds.2000.6.419 [14] 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 [15] Qin Wang, Kyungwoo Song. The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1661-1675. doi: 10.3934/dcds.2016.36.1661 [16] Tatsien Li, Bopeng Rao, Yimin Wei. Generalized exact boundary synchronization for a coupled system of wave equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2893-2905. doi: 10.3934/dcds.2014.34.2893 [17] Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759 [18] Jacek Banasiak, Aleksandra Puchalska. Generalized network transport and Euler-Hille formula. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1873-1893. doi: 10.3934/dcdsb.2018185 [19] 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 [20] 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

2019 Impact Factor: 1.105