Two-dimensional Riemann problems for zero-pressure gas dynamics with three constant states. (English) Zbl 1139.35073

Summary: The Riemann problems for two-dimensional zero-pressure gas dynamics are solved completely when the initial data take three constant states having discontinuities on \(x,y\)-positive and \(x\)-negative axes. With the help of characteristic analysis, by studying interactions among delta-shocks, vacuums and contact discontinuities, the Riemann solutions constructed exhibit nine different explicit configurations. The Mach-reflection-like configurations appear in some solutions.


35L67 Shocks and singularities for hyperbolic equations
35L65 Hyperbolic conservation laws
76N15 Gas dynamics (general theory)
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[1] Agarwal, R. K.; Halt, D. W., A modified CUSP scheme in wave/particle split form for unstructured grid Euler flows, (Caughey, D. A.; Hafes, M. M., Frontiers of Computational Fluid Dynamics (1994), John Wiley and Sons) · Zbl 0977.76054
[2] Bouchut, F., On zero-pressure gas dynamics, (Advances in Kinetic Theory and Computing. Advances in Kinetic Theory and Computing, Ser. Adv. Math. Appl. Sci., vol. 22 (1994), World Scientific: World Scientific River Edge, NJ), 171-190 · Zbl 0863.76068
[3] Brenier, Y.; Grenier, E., Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35, 2317-2328 (1998) · Zbl 0924.35080
[4] Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock Waves (1948), Interscience Publishers Inc.: Interscience Publishers Inc. New York · Zbl 0041.11302
[5] Chen, Guiqiang; Liu, Hailiang, Formation of \(δ\)-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, SIAM J. Math. Anal., 34, 4, 925-938 (2003) · Zbl 1038.35035
[6] Chen, G.; Liu, H., Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Phys. D, 189, 141-165 (2004) · Zbl 1098.76603
[7] Weinan E.; Rykov, Yu. G.; Sinai, Ya. G., Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in Ashesion particle dynamics, Comm. Math. Phys., 177, 349-380 (1996) · Zbl 0852.35097
[8] Guckenheimer, J., Shocks and rarefactions in two space dimensions, Arch. Ration. Mech. Anal., 59, 281-291 (1975) · Zbl 0329.35043
[9] Li, Y.; Cao, Y., Large partial difference method with second accuracy in gas dynamics, Sci. Sinica A, 28, 1024-1035 (1985) · Zbl 0632.76082
[10] Li, Jiequan; Yang, Shuli; Zhang, Tong, The Two-Dimensional Riemann Problem in Gas Dynamics (1998), Longman Scientic and Technical · Zbl 0935.76002
[11] Li, J.; Zhang, T., Generalized Rankine-Hugoniot conditions of weighted Dirac delta waves of transportation equation, (Nonlinear PDE and Related Areas (1998), World Scientific: World Scientific Singapore)
[12] Shandarin, S. F.; Zeldovich, Ya. B., The large-scale structure of the universe: Turbulence, intermittency, structure in a self-gravitating medium, Rev. Mod. Phys., 61, 185-220 (1989)
[13] Sheng, Wancheng, Two-dimensional Riemann problem for scalar conservation laws, J. Differential Equations, 183, 239-261 (2002) · Zbl 1024.35073
[14] Sheng, W.; Zhang, Tong, The Riemann problem for transportation equation in gas dynamics, Mem. Amer. Math. Soc., 137, 654 (1999) · Zbl 0913.35082
[15] Tan, Dechun; Zhang, T., Two dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws. (I). Four-J cases, J. Differential Equations, 111, 203-254 (1994) · Zbl 0803.35085
[16] Tan, D.; Zhang, T.; Zheng, Yuxi, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conversation laws, J. Differential Equations, 112, 1-32 (1994) · Zbl 0804.35077
[17] Yang, Hanchun, Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differential Equations, 159, 447-484 (1999) · Zbl 0948.35079
[18] Yang, H., Generalized plane delta-shock waves for \(n\)-dimensional zero-pressure gas dynamics, J. Math. Anal. Appl., 260, 18-35 (2001) · Zbl 0985.35044
[19] Li, J.; Yang, H., Delta-shock waves as limits of vanishing viscosity for multidimensional zero-pressure gas dynamics, Quart. Appl. Math., LIX, 2, 315-342 (2001) · Zbl 1019.76040
[20] Zhang, Peng; Zhang, T., Generalized characteristic analysis and Guckenheimer structure, J. Differential Equations, 152, 409-430 (1999) · Zbl 0924.35079
[21] Zheng, Y., Systems of Conservation Laws: Two-Dimensional Riemann Problems, Progr. Nonlinear Differential Equations Appl., vol. 38 (2001), Birkhäuser: Birkhäuser Boston · Zbl 0971.35002
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