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On the expansion of a wedge of van der Waals gas into a vacuum. II. (English) Zbl 1354.35078

Summary: We study the expansion of a wedge of rest gas into a vacuum. When the rest gas is a van der Waals gas, the gas away from the sharp corner of the wedge may expand into the vacuum as symmetrical planar rarefaction waves, planar fan-jump composite waves, or planar fan-jump-fan composite waves. So, in order to solve the expansion problem we need to study the interactions of these elementary waves. In a recent paper [the author, J. Differ. Equations 259, No. 3, 1181–1202 (2015; Zbl 1317.35203)], we obtained the existence of global in time classical solution to the interaction of the planar rarefaction waves. This paper studies the interaction of the planar fan-jump composite waves. To construct the flow in the interaction region of the fan-jump composite waves, we consider a discontinuous Goursat problem for the two-dimensional self-similar Euler system. The existence of global solution to the discontinuous Goursat problem is obtained constructively by using the characteristic method.

MSC:

35Q31 Euler equations
35L65 Hyperbolic conservation laws
35J70 Degenerate elliptic equations
35R35 Free boundary problems for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics

Citations:

Zbl 1317.35203
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References:

[1] Bang, S., Interaction of three and four rarefaction waves of the pressure-gradient system, J. Differential Equations, 246, 453-481 (2009) · Zbl 1161.35032
[2] Canic, S.; Keyfitz, B. L.; Kim, E. H., A free boundary problem for a quasilinear degenerate elliptic equation: regular reflection of weak shock, Comm. Pure Appl. Math., 55, 71-92 (2002) · Zbl 1124.35338
[3] Canic, S.; Keyfitz, B. L.; Kim, E. H., Free boundary problems for nonlinear wave systems: Mach stems for interacting shocks, SIAM J. Math. Anal., 37, 1947-1977 (2006) · Zbl 1107.35083
[4] Chen, G. Q.; Deng, X. M.; Xiang, W., Shock diffraction by convex cornered wedges for the nonlinear wave system, Arch. Ration. Mech. Anal., 211, 61-112 (2014) · Zbl 1286.35204
[5] Chen, G. Q.; Feldman, M., Global solutions of shock reflection by large-angle wedges for potential flow, Ann. of Math., 171, 1067-1182 (2010) · Zbl 1277.35252
[6] Chen, S. X.; Qu, A. F., Riemann boundary value problems and reflection of shock for the Chaplygin gas, Sci. China Math., 55, 671-685 (2012) · Zbl 1239.35094
[7] Chen, S. X.; Qu, A. F., Two dimensional Riemann problems for Chaplygin gas, SIAM J. Math. Anal., 44, 2146-2178 (2012) · Zbl 1257.35126
[8] Chen, X.; Zheng, Y. X., The interaction of rarefaction waves of the two-dimensional Euler equations, Indiana Univ. Math. J., 59, 231-256 (2010) · Zbl 1203.35167
[9] Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock Waves (1948), Interscience: Interscience New York · Zbl 0041.11302
[10] Dai, Z. H.; Zhang, T., Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Ration. Mech. Anal., 155, 277-298 (2000) · Zbl 1007.76072
[11] Elling, V., Regular reflection in self-similar potential flow and the sonic criterion, Commun. Math. Anal., 8, 22-69 (2010) · Zbl 1328.76038
[12] Elling, V., Non-existence of strong regular reflection in self-similar potential flow, J. Differential Equations, 252, 2085-2103 (2012) · Zbl 1375.76158
[13] Elling, V.; Liu, T. P., Supersonic flow onto a solid wedge, Comm. Pure Appl. Math., 61, 1347-1448 (2008) · Zbl 1143.76030
[14] Fossati, M.; Quartapelle, L., The Rimann problem for hyperbolic equations under a nonconvex flux with two inflection points
[15] Guo, L. H.; Sheng, W. C.; Zhang, T., The 2D Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal., 9, 431-458 (2010) · Zbl 1197.35164
[16] Hu, Y. B.; Li, J. Q.; Sheng, W. C., Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations, Z. Angew. Math. Phys., 63, 1021-1046 (2012) · Zbl 1259.35144
[17] Lai, G., On the expansion of a wedge of van der Waals gas into a vacuum, J. Differential Equations, 259, 1181-1202 (2015) · Zbl 1317.35203
[18] Lai, G., Interaction of jump-fan composite waves in a two-dimensional jet for van der Waals gas, J. Math. Phys., 56, Article 061504 pp. (2015) · Zbl 1338.76111
[19] Lai, G.; Sheng, W. C.; Zheng, Y. X., Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions, Discrete Contin. Dyn. Syst., 31, 489-523 (2011) · Zbl 1222.35121
[20] Li, F. B.; Xiao, W., Interactionn of four rarefaction waves in the bi-symmetric class of the pressure gradient system, J. Differential Equations, 252, 3920-3952 (2012) · Zbl 1242.35168
[21] Li, J. Q., On the two-dimensional gas expansion for the compressible Euler equations, SIAM J. Appl. Math., 62, 831-852 (2002) · Zbl 1103.76056
[22] Li, J. Q.; Yang, Z. C.; Zheng, Y. X., Characteristic decompositions and interaction of rarefaction waves of 2-D Euler equations, J. Differential Equations, 250, 782-798 (2011) · Zbl 1209.35079
[23] Li, J. Q.; Zhang, T.; Yang, S., The Two-Dimensional Riemann Problem in Gas Dynamics, Pitman Monogr. Surv. Pure Appl. Math., vol. 98 (1998), Longman
[24] Li, J. Q.; Zhang, T.; Zheng, Y. X., Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations, Comm. Math. Phys., 267, 1-12 (2006) · Zbl 1113.76080
[25] Li, J. Q.; Zheng, Y. X., Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Ration. Mech. Anal., 193, 623-657 (2009) · Zbl 1170.76021
[26] Li, J. Q.; Zheng, Y. X., Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations, Comm. Math. Phys., 296, 303-321 (2010) · Zbl 1193.35140
[27] Li, T. T., Global Classical Solutions for Quasilinear Hyperbolic System (1994), John Wiley and Sons
[28] Li, T. T.; Yu, W. C., Boundary Value Problem for Quasilinear Hyperbolic Systems (1985), Duke University
[29] Li, T. T.; Yu, W. C., The centered waves for quasilinear hyperbolic systems, J. Fudan Univ. Nat. Sci., 25, 195-206 (1986), (in Chinese) · Zbl 0601.35069
[30] Li, T. T.; Yu, W. C., The centered wave problem for quasilinear hyperbolic systems, Chin. Ann. Math., 7A, 423-436 (1986), (in Chinese) · Zbl 0642.35056
[31] Liu, T. P., The Riemann problem for general systems of conservation laws, J. Differential Equations, 18, 218-234 (1975) · Zbl 0297.76057
[32] Menikoff, R.; Plohr, B. J., The Riemann problem for fluid flow of real materials, Rev. Modern Phys., 61, 75-130 (1989) · Zbl 1129.35439
[33] Müller, S.; Voß, A., A Riemann solver for Euler equations with non-convex equation of state, (Warnecke, G., Analysis and Numerics for Conservation Laws (2005), Springer-Verlag)
[34] Serre, D., Multi-dimensional shock interaction for a Chaplygin gas, Arch. Ration. Mech. Anal., 191, 539-577 (2008) · Zbl 1161.76025
[35] Serre, D., Shock reflection in gas dynamics, (Friedlander, S.; Serre, D., Handbook of Mathematical Fluid Dynamics, vol. IV (2007), Elsevier: Elsevier North-Holland), 39-122
[36] Sheng, W. C.; Zhang, T., The Riemann problem for transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137, 564 (1999)
[37] Smith, R. G., The Riemann problem in gasdynamics, Trans. Amer. Math. Soc., 249, 1-50 (1979)
[38] Smoller, J., Shock Waves and Reaction-Diffusion Equations (1999), Springer: Springer Berin-Heidelbery-New York
[39] Suchkow, V. A., Flow into a vacuum along an oblique wall, J. Appl. Math. Mech., 27, 1132-1134 (1963)
[40] Wendroff, B., The Riemann problem for materials with nonconvex equation of state, I: Isentropic flow, J. Math. Anal. Appl., 38, 454-466 (1972) · Zbl 0264.76054
[41] Wendroff, B., The Riemann problem for materials with nonconvex equation of state, II: General flow, J. Math. Anal. Appl., 38, 640-658 (1972) · Zbl 0287.76049
[42] Ying, L. A.; Wang, C. H., The discontinuous initial value problem of a reacting gas flow system, Trans. Amer. Math. Soc., 266, 361-387 (1981) · Zbl 0482.76071
[43] Zhang, T.; Zheng, Y., Conjecture on the structure of solution of the Riemann problem for 2D gas dynamics system, SIAM J. Math. Anal., 21, 593-630 (1990) · Zbl 0726.35081
[44] Zhao, W. X., The expansion of gas from a wedge with small angle into a vacuum, Commun. Pure Appl. Anal., 12, 2319-2330 (2013) · Zbl 1270.35310
[45] Zheng, Y. X., Systems of Conservation Laws: 2-D Riemann Problems (2001), Birkhäuser: Birkhäuser Boston, 38 PNLDE
[46] Zheng, Y. X., Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Acta Math. Appl. Sin. Engl. Ser., 22, 177-210 (2006) · Zbl 1106.35034
[47] Zhou, X. L., The centered wave solution with large amplitude for quasilinear hyperbolic systems, Northeast. Math. J., 3, 439-451 (1987), (in Chinese) · Zbl 0692.35061
[48] Zhou, X. L., The local solution with large amplitude of the centered wave problem for quasilinear hyperbolic systems, Chin. Ann. Math., 10A, 119-138 (1989), (in Chinese) · Zbl 0669.35066
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