THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE ONE-DIMENSIONAL CHAPLYGIN GAS EQUATIONS

doi:10.1016/S0252-9602(12)60064-2

Abstract

Abstract:

In this article, we study the Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations. Under the generalized Rankine-Hugoniot conditions and the entropy condition, we constructively obtain the global existence of generalized solutions 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.

Key words: Chaplygin gas equations, Riemann problem, delta-shocks, generalized Rankine-Hugoniot conditions, generalized solutions

CLC Number:

35L65

WANG Zhen, ZHANG Qing-Ling. THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE ONE-DIMENSIONAL CHAPLYGIN GAS EQUATIONS[J].Acta mathematica scientia,Series B, 2012, 32(3): 825-841.

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References

[1] Bilic N, Tupper G B, Viollier R. Dark matter, dark energy and the Chaplygin gas. arXiv: astro-ph/0207423

[2] Born M, Infeld L. Foundations of the new field theory. Proc Roy Soc London, 1934, 144A: 425–451

[3] Bouchut F. On zero-pressure gas dynamics//Advances in Kinetic Theory and Computing. Ser Adv Math Appl Sci 22. River Edge, NJ: World Scientific, 1994: 171–190

[4] Brenier Y. Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equa-tions. J Math Fluid Mech, 2005, 7: S326–S331

[5] Chaplygin S. On gas jets. Sci Mem Moscow Univ Math Phys, 1904, 21: 1–121

[6] Rykov E W, Yu G, Sinai Ya G. Generalized varinational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm Math Phys, 1996, 177: 349–380

[7] Gorini V, Kamenshchik A, Moschella U, Pasquier V. The Chaplygin gas as a model for dark energy. arXiv: gr-qc/0403062

[8] Guo L H, Sheng W C, Zhang T. The two-dimensional Riemann problelm for isentropic Chaplygin gas dynamic system. Comm Pure Appl Anal, 2010, 9(2): 431–458

[9] Huang F. Weak solution to pressureless type system. Comm in Partial Differential Equations, 2005, 30(3): 283–304

[10] Huang F, Wang Z. Well posedness for pressureless flow. Comm Math Phys, 2001, 222: 117–146

[11] Jackson J D. Classical Electrodynamics (Section 1.3). Second Editon. John Wiley and Sons, 1975

[12] Korchinski D J. Solutions of a Riemann problem for a system of conservation laws possessing no classical weak solution. Thsis: Adelphi University, 1977

[13] Li J, Zhang T, Yang S. The Two-dimensional Riemann Prolem in Gas Dynamics//Pitman Monogr Surv Pure Appl Math 98. Longman Scientific and Technical, 1998

[14] Serre D. Multidimensional shock interaction for a Chaplygin gas. Arch Rational Mech Anal, 2009, 191: 539–577

[15] Setare M R. Holographic Chaplygin gas model. Phys Lett B, 2007, 648: 329–332

[16] Sheng W, Zhang T. The Riemann problem for transportation equations in gas dynamics. Mem Amer Math Soc, 1999, 137(654)

[17] Smoller J. Shock Waves and Reaction-Diffusion Equation. New York: Springer-Verlag, 1994

[18] Tan D, Zhang T, Zheng Y. Delta-shock wave as limits of vanishing viscosity for hyperbolic system of conservation laws. J Differential Equations, 1994, 112: 1–32

[19] Tsien H S. Two dimensional subsonic flow of compressible fluids. J Aeron Sci, 1939, 6: 399–407

[20] Wang Z, Ding X. Uniqueness of generalized solution for the Cauchy problem of transportation equations. Acta Mathematica Scientia, 1997, 17(3): 341–352

[21] Wang Z, Huang F, Ding X. On the Cauchy problem of transportation equations. Acta Math Appl Sinica, 1997, 13(2): 113–122

[22] Wang Z, Zhang Q L. Spiral solution to the two-dimensional transport equations. Acta Mathematica Scientia, 2010, 30(6): 2110–2128

[23] Yang H. Generalized plane delta-shock waves for n-dimensional zero-pressure gas dynamics. J Math Anal Appl, 2001, 260: 18–35

[24] Yang H, Sun W. The Riemann problem with delta initial data for a class of coupled hyperbolic system of conservation laws. Nonlinear Analysis: TMA, 2007, 67: 3041–3049