RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE RELATIVISTIC CHAPLYGIN EULER EQUATIONS

Meixiang Huang; Zhiqiang Shao

doi:10.11948/2016029

2016 Volume 6 Issue 2

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Meixiang Huang, Zhiqiang Shao. RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE RELATIVISTIC CHAPLYGIN EULER EQUATIONS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 376-395. doi: 10.11948/2016029

Citation:

Meixiang Huang, Zhiqiang Shao. RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE RELATIVISTIC CHAPLYGIN EULER EQUATIONS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 376-395. doi: 10.11948/2016029

RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE RELATIVISTIC CHAPLYGIN EULER EQUATIONS

College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China

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Abstract

Abstract

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function for the relativistic Chaplygin Euler equations. Under the generalized Rankine-Hugoniot conditions and entropy condition, we constructively obtain the global existence of generalized solutions including delta shock waves that explicitly exhibit four kinds of different structures. Moreover, we obtain the stability of generalized solutions by making use of the perturbation of the initial data.
- Relativistic Euler equations /
- Riemann problem /
- delta shock wave /
- generalized-Rankine-Hugoniot conditions /
- entropy condition
MSC: 35L65;35L67

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References

[1]	Y. Brenier, Solutions with concentration to the Riemann problem for onedimensional Chaplygin gas equations, J. Math. Fluid Mech, 7(2005), 326-331. 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, 41, New York:Longman Scientific and Technical, 1989. Google Scholar
[3]	G. Chen and P. G. LeFloch, Existence theory for the isentropic Euler equations, Arch. Rational Mech. Anal, 166(2003), 81-98. Google Scholar
[4]	G. Chen and Y. Li, Stability of Riemann solutions with large oscillation for the relativistic Euler equations, J. Differential Equations, 202(2004), 332-353. Google Scholar
[5]	G. Chen and H. Liu, Formation of -shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal, 34(2003), 925-938. Google Scholar
[6]	H. Cheng and H. Yang, Riemann problem for the relativistic Chaplygin Euler equations, J. Math. Anal. Appl., 381(2011), 17-26. Google Scholar
[7]	Y. Gan and S. Wancheng, Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases, J. Math. Anal. Appl., 355(2009), 594-605. Google Scholar
[8]	L. Guo, W. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal., 9(2010), 431-458. Google Scholar
[9]	L. Guo, Y. Zhang and G. Yin, Interactions of delta shock wave for the Chaplygin gas equations with split delta functions, J. Math. Anal. Appl, 410(2014), 190-201. Google Scholar
[10]	H. C. Kranzer and B.L. Keyfitz, A strictly hyperbolic system of conservation laws admitting singular shocks, In:Nonlinear Evolution Equations that Change Type, in:IMA Vol. Math. Appl., Springer, New York, 27(1990), 107-125. Google Scholar
[11]	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, Z Angew Math Phys, 56(2005), 239-253. Google Scholar
[12]	J. Li, S. Yang and T. Zhang, The Two-Dimensional Riemann Problem in Gas Dynamics, Longman Scientific and Technical, 1998. Google Scholar
[13]	J. Li and Z. Shao, The Riemann problem with delta initial data for the zeropressure relativistic Euler Equations, Acta Mathematica Scientia, 34A(2014), 1083-1092. Google Scholar
[14]	E.Yu. Panov and V. M. Shelkovich, δ-Shock waves as a new type of solutions to systems of conservation laws, J. Differential Equations, 228(2006), 49-86. Google Scholar
[15]	D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Ration. Mech. Anal., 191(2009), 539-577. Google Scholar
[16]	V. M. Shelkovich, The Riemann problem admitting δ-shocks and vacuum states (the vanishing viscosity approach), J. Differential Equations, 231(2006), 459-500. Google Scholar
[17]	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, J. Differential Equations, 249(2010), 3024-3051. Google Scholar
[18]	W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas Dynamics, in:Mem. Amer. Math. Soc., AMS, Providence, 137, 1999. Google Scholar
[19]	J. Smoller and B. Temple, Global solutions of the relativistic Euler equations, Comm. Math. Phys, 156(1993), 67-99. Google Scholar
[20]	D. Tan, T. Zhang and Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations, 112(1994), 1-32. Google Scholar
[21]	Z. Wang and Q. Zhang, The Riemann problem with delta initial data for the one-dimensional chaplygin gas equations, Acta Mathematica Scientia, 32B(2012), 825-841. Google Scholar
[22]	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, J. Math. Anal. Appl, 413(2014), 800-820. Google Scholar
[23]	H. Yang and Y. Zhang, New developments of delta shock waves and its application in system of conservation laws, J. Differential Equations, 252(2012), 5951-5993. Google Scholar
[24]	H. Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differential Equations, 159(1999), 447-484. Google Scholar
[25]	H. Yang and W. Sun, The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws, Nonlinear Anal, 67(2007), 3041-3049. Google Scholar