[1] Agarwal R K, Halt D W. A modified CUSP scheme in wave/particle split form for unstructured grid Euler
flow//Caughey D A, Hafez M M, Eds. Frontiers of Computational Fluid Dynamics. New York: Wiley,
1994: 155–163
[2] Bang Seunghoon. Interaction of three and four rarefaction waves of the pressuregradient system. Journal
of Differential Equations, in press, 2008
[3] Ben-Artzi M, Falcovitz J, Li J -Q. Wave interactions and numerical approximation for two-dimensional
scalar conservation laws. Computational Fluid Dynamics Journal, 2006, 14: 401–418
[4] Ben-Artzi M, Li J -Q, Warnecke G. A direct Eulerian GRP scheme for compressible fluid flows. J Comput
Phys, 2006, 218: 19–43
[5] Ben-dor G, Glass I I. Domains and boundaries of non-stationary oblique shock wave reflection (1). J Fluid
Dynamics, 1979, 92: 459–496; (2). J Fluid Dynamics, 1980, 96: 735–756
[6] Bleakney W, Taub A H. Interaction of shock waves. Review of Moder Physics, 1949, 21: 584–605
[7] Bianchini S, Bressan A. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann Math, 2005,
161(1): 223–342
[8] Bouchut F. On zero pressure gas dynamics//Perthame B, ed. Advances in Kinetic Theory and Computing,
Series on Advances in Mathematics for Applied Sciences, Vol 22. Singapore: World Scientific, 1994: 171–
190
[9] Brenier Y, Grenier E. Sticky particles and scalar conservation laws. SIAM J Numer Anal, 1998, 35: 2317–2328
[10] Canic S, Keyfitz B L, Kim Eun Heui. A free boundary problem for a quasi-linear degenerate elliptic equation: regular reflection of weak shocks. Comm Pure Appl Math, 2002, 55: 71–92
[11] Canic S, Keyfitz B. Quasi-one-dimensional Riemann problems and their role in self-similar two-dimensional
problems. Arch Rational Mech Anal, 1998, 144: 233–258
[12] Chang T, Chen G Q. Diffration of planar shock along a compressive corner. Acta Mathematia Scientia,
1986, 6: 241–257
[13] Chang T, Chen G -Q, Yang S -L. On the 2-D Riemann problem for the compressible Euler equations, I.
Interaction of shocks and rarefaction waves. Discrete continuous dynamical systems, 1995, 1: 555–584; II.
Interaction of contact discontinuities. Discrete Contin Dynam Systems, 2000, 6: 419–430
[14] Chang T, Hsiao L. The Riemann problem and interaction of waves in gas dynamics//Pitman Monographs
and Surveys in Pure and Applied Mathematics 41. Harlow: Longman Scientific and Technical, 1989
[15] Chen G -Q, Feldman M. Global solutions of shock reflection by large-angle wedges for potential flow. Ann
Math, to appear, 2007
[16] Chen G -Q, Li D, Tan D -C. Structure of Riemann solutions for 2-Dimensional scalar conservation law. J
Diff Equat, 1996, 127: 124–147
[17] Chen G -Q, Liu H -L. Formation of -shocks and vacuum states in the vanishing pressure limit of solutions
to the Euler equations for isentropic fluids. SIAM J Math Anal, 2003, 34: 925–93
[18] Chen G -Q, Liu H -L. Concentration and cavitation in the vanishing pressure limit of solutions to the
Euler equations for nonisentropic fluids. Phys D, 2004, 189: 141–165
[19] Chen S -X. Mach configuration in pseudo-stationary compressible flow. J Amer Math Soc, 2008, 21: 63–100
[20] Chen X, Zheng Y -X. The interaction of rarefaction waves of the two-dimensional Euler equations. Indiana
Univ Math J, (in press)
[21] Cheng S, Li J -Q, Zhang T. Explicit construction of measure solutions of the Cauchy problem for the
transportation equations. Science in China (Series A), 1997, 40: 1287–1299
[22] Colella P, Henderson L F. The von Neumann paradox for the diffration of weak shock waves. J Fluid
Mech, 1990, 213: 71–94
[23] Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. New York: Interscience, 1948
[24] Ding X X, Chen G -Q, Luo P -Z. Convergence of the Lax-Friedrichs scheme for the system of equations of
isentropic gas dynamics I. Acta Math Sci (Chinese), 1987, 7: 467–480; II. 1988, 8: 61–94
[25] Dafermos C. Hyperbolic Conservation Laws in Continuum Physics (Grundlehren der Mathematischen
Wissenschaften). Berlin, Heidelberg, New York: Springer, 2000
[26] Dai Z, Zhang T. Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics.
Arch Ration Mech Anal, 2000, 155: 277–298
[27] Yu W E, Rykov G, Sinai Ya G. Generalized variational principles, global weak solutions and behavior with
random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm Phys
Math, 1996, 177: 349–380
[28] Elling V. Regular reflection in self-similar potential flow and the sonic criterion. Preprint, 2007
[29] Elling V, Liu T -P. Supersonic flow onto a solid wedge. Comm Pure Appl Math, 2008, 61: 1347–1448
[30] Huang F -M, Wang Z. Well posedness for pressureless flow. Comm Math Phys, 2001, 222: 117–146
[31] Gelfand I M. Some problems in the theory of quasilinear equations. Usp Math Nauk, 1959, 14: 87–158
Amer Math Soc Transl, 1963, 29: 295–381
[32] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math,
1965, 18: 697–715
[33] Glimm J, Ji Xiaomei, Li Jiequan, Li Xiaolin, Zhang Peng, Zhang Tong, Zheng Yuxi. Transonic shock
formation in a rarefaction Riemann problem for the 2-D compressible Euler equations preprint. SIAM J
Appl Math, 2008, 69: 720–742
[34] Godunov S K. A difference scheme for numerical solution of discontinuous solution of dydrodynamic
equations. Math Sbornik, 1959, 47: 271–306
[35] Guckenheimer J. Shocks and rarefactions in two space dimensions. Arch Rational Mech Anal, 1975, 59:
281–291
[36] Kim E H, Song K. Classical solutions for the pressure-gradient equations in non-smooth and non-convex
domains. J Math Anal Appl, 2004, 293(2): 541–550
[37] Kirchinski D J. Solutions of a Riemann problem for a 2 × 2 system of conservation laws possessing no
classical solutions
[D]. Adelphi University, 1997
[38] Kofman L, Pogosyan D, Shandarin S. Structure of the universe in the two-dimensional model of adhesion.
Mon Nat R Astr Soc, 1990, 242: 200–208
[39] Kurganov A, Tadmor E. Solution of two-dimensional Riemann problems for gas dynamics without Riemann
problem solvers. Numer Methods Partial Differential Equations, 2002, 18: 584–608
[40] Lax P D. Hyperbolic system of conservation laws II. Comm Pure Appl Math, 1957, 10: 537–566
[41] Lax P D, Liu X D. Solution of two–dimensional Riemann problem of gas dynamics by positive schemes.
SIAM Sci Comp, 1998, 19: 319–340
[42] Lei Z, Zheng Y -X. A complete global solution to the pressure gradient equation. J Differential Equations,
2007, 236: 280–292
[43] Levine L E. The expansion of a wedge of gas into a vacuum. Proc Camb Philol Soc, 1968, 64: 1151–1163
[44] Li J -Q. Note on the compressible Euler equations with zero-temperature. Lett Appl Math, 2001, 14:
519–523
[45] Li J -Q. On the two-dimensional gas expansion for compressible Euler equations. SIAM J Appl Math,
2001/02, 62: 831–852
[46] Li J -Q. Global solution of an initial-value problem for two-dimensional compressible Euler equations. J
Differential Equations, 2002, 179: 178–194
[47] Li J -Q, Li W. Riemann problem for the zero-pressure flow in gas dynamics. Progr Natur Sci (English
Ed), 2001, 11: 331–344
[48] Li J -Q, Warnecke G. Generalized characteristics and the uniqueness of entropy solutions to zero-pressure
gas dynamics. Adv Differential Equations, 2003, 8: 961–1004
[49] Li J -Q, Yang H C. Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure flow in
gas dynamics. Q Appl Math, 2001, 59: 315–342
[50] Li J -Q, Yang Z -C, Zheng Y -X. Characteristic decompositions and interactions of rarefaction waves of
2-D Euler equations. Preprint, 2008
[51] Li J -Q, Zhang T. Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation
equations. Advances in nonlinear partial differential equations and related areas (Beijing, 1997). River Edge, NJ: World Sci Publ, 1998: 219–232
[52] Li J -Q, Zhang T, Yang S -L. The two-dimensional Riemann problem in gas dynamics//Pitman Monographs
and Surveys in Pure and Applied Mathematics 98. Longman, 1998
[53] Li J -Q, Zhang T, Zheng Y -X. Simple waves and a characteristic decomposition of the two dimensional
compressible Euler equations. Comm Math Phys, 2006, 267: 1–12
[54] Li J -Q, Zheng Y -X. Interaction of rarefaction waves of the two-dimensional self-similar Euler equations.
Arch Rat Mech Anal, (to appear), 2008
[55] Li J -Q, Zheng Y -X. Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional
Euler equations. submitted for publication, 2008
[56] Li M -J, Zheng Y -X. Semi-hyperbolic patches of solutions of the two-dimensional Euler equations. preprint,
2008
[57] Li Y, Cao Y. Second order “large particle” difference method (in Chinese). Sciences in China (Series A),
1985, 8: 1024–1035
[58] Liu T -P, Zeng Y -N. Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems
of conservation laws. Mem Amer Math Soc, 1997, 125(599)
[59] Mach E, Wosyka J. Uber die Fortpflanzungsgeschwindigkeit von Explosionsschallwellen. Sitzungsber Akad
Wiss Wien (II Abth), 1875, 72: 44–52
[60] von Neumann J. Oblique Reflection of Shocks. U S Dept Comm Off of Tech Serv, Washigton, D C
PB-37079, 1943
[61] Oleinik O A. Uniqueness and a stability of the generalized solutions of the Cauchy problem for a quasilinear
equations. USP Mat Nauk; Amer Math Soc Transl, Ser 2, 1964, 3: 285–290
[62] Pogodin I A, Suchkov V A, Ianenko N N. On the traveling waves of gas dynamic equations. J Appl Math
Mech, 1958, 22: 256–267
[63] Riemann B. Uber die forpflanzung ebener luftwellen von endlicher schwingungsweite. Abhandl Koenig
Gesell Wiss, Goettingen, 1860, 8: 43
[64] Schulz-Rinne C W. Classification of the Riemann problem for Two-dimensional Riemann problem. SIAM
J Math Anal, 1993, 24: 76–88
[65] Schulz-Rinne CW, Collins J P, Glaz H M. Numerical solution of the Riemann problem for two-dimensional
gas dynamics. SIAM J Sci Compt, 1993, 4: 1394–1414
[66] Shandarin S F, Zeldovich Ya B. The large-scale structure of the universe: Turbulence, intermittency,
structures in a self-gravitating medium. Rev Mod Phys, 1989, 61: 185–220
[67] Sheng W -C. Two-dimensional Riemann problem for scalar conservation laws. J Differential Equations,
2002, 183: 239–261
[68] Sheng W -C, Zhang T. The Riemann problem for transportation equations in gas dynamics. Memoirs of
AMS, 1997, 137(654)
[69] Sheng W -C, Zhang T. A cartoon for the climbing ramp problem of a shock and von Neumann paradox.
Arch Ration Mech Anal, 2007, 184: 243–255
[70] Sheng W -C, Yin G. Transonic shock and supersonic shock in the regular reflection of a planar shock. Z
Angew Math Phys, DOI 10.1007/s00033-008-8003-4, 2008, (in press)
[71] Song K. The pressure-gradient system on non-smooth domains. Comm Partial Differential Equations,
2003, 28: 199–221
[72] Song K, Zheng Y -X. Semi-hyperbolic patches of solutions of the pressure gradient system. Disc Cont
Dyna Syst, 2009, (inpress)
[73] Suchkov V A. Flow into a vacuum along an oblique wall. J Appl Math Mech, 1963, 27: 1132–1134
[74] Tan D -C, Zhang T. Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation
laws. I. Four-J cases. J Differential Equations, 1994, 111(2): 203–254; II. Initial data involving some
rarefaction waves. J Differential Equations, 1994, 111: 255–282
[75] Tan D -C, Zhang T, Zheng Y -X. Delta-shock waves as limits of vanishing viscosity for hyperbolic systems
of conservation laws. J Differential Equations, 1994, 112: 1–32
[76] Wagner D H. The Riemann problem in two space dimensions for a single conservation law. SIAM J Math
Anal, 1983, 14: 534–559
[77] Wang Z, Huang F -M, Ding X -Q. On the Cauchy problem of transportation equations. Acta Math Appl Sinica (English Ser), 1997, 13: 113–122
[78] Xin Z -P, Yin H -C. Transonic shock in a nozzle. I. Two-dimensional case. Comm Pure Appl Math, 2005,
58: 999–1050
[79] Yang H -C, Zhang T. On two-dimensional gas expansion for pressure-gradient equations of Euler system.
J Math Anal Appl, 2004, 298: 523–537
[80] Zakharian A R, Brio, Hunter J K, Webb G M. The von Neumann paradox in weak shock reflection. J
Fluid Mech, 2000, 422: 193–205
[81] Zhang P, Li J -Q, Zhang T. On two-dimensional Riemann problem for pressure-gradient equations of Euler
system. Discrete and Continuous Dynamical Systems, 1998, 4: 609–634
[82] Zhang P, Zhang T. Generalized characteristic analysis and Guckenheimer structure. J Diff Equ, 1999, 52:
409–430
[83] Zhang T, Zheng Y -X. Two-dimensional Riemann problem for a single conservation law. Trans AMS,
1989, 312: 589–619
[84] Zhang T, Zheng Y -X. Conjecture on the structure of solution of the Riemann problem for two-dimensional
gas dynamics systems. SIAM J Math Anal, 1990, 21: 593–630
[85] Zhang T, Zheng Y -X. Exact spiral solutions of the two-dimensional Euler equations. Discrete and Con-
tinuous Dynamical Systems, 1997, 3: 117–133
[86] Zhang T, Zheng Y -X. Axisymmetric solutions of the Euler equations for polytropic gases. Arch Rat Mech
Anal, 1998, 142: 253–279
[87] Zheng Y -X. Existence of solutions to the transonic pressure-gradient equations of the compressible Euler
equations in elliptic regions. Comm P D Es, 1998, 22: 1849–1868
[88] Zheng Y -X. Systems of conservation laws. Two-dimensional Riemann problems. Birkhauser, in the series
of Progress in Nonlinear Differential Equations, 2001
[89] Zheng Y -X. A global solution to a two-dimensional Riemann problem involving shocks as free boundaries.
Acta Math Appl Sin, Engl Ser, 2003, 19: 559–572
[90] Zheng Y -X. Two-dimensional regular shock reflection for the pressure gradient system of conservation
laws. Acta Math Appl Sin, Engl Ser, 2006, 22: 177–210
[91] Zheng Y -X. The Compressible Euler System in Two Dimensions. Lecture Notes of 2007 Shanghai Math-
ematics Summer School. International Press and Higher Education Press, (in press)
[92] Zheng Y -X. Absorption of characteristics by sonic curves of the two-dimensional Euler equations. Discrete
and Continuous Dynamical Systems, 2009, 23(1/2): 605–616
[93] Zheng Y -X. Shock reflection for the Euler system//Asakura F (Chief), Aiso H, Kawashima S, Matsumura
A, Nishibata S, Nishihara K, ed. Hyperbolic Problems Theory, Numerics and Applications (Proceedings
of the Osaka meeting 2004) Vol II. Yokohama Publishers (Japan), 2006: 425–432
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