Citation: |
[1] |
S. Bang, Interaction of three and four rarefaction waves of the pressure-gradient system, J. Differential Equations, 246 (2009), 453-481. |
[2] |
Y. Brenier, Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations, J. Math. Fluid Mech., 7 (2005), 326-331.doi: 10.1007/s00021-005-0162-x. |
[3] |
S. X. Chen and A. F. Qu, Interaction of rarefaction waves in jet stream, J. Differential Equations, 248 (2010), 2931-2954. |
[4] |
G. Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanish pressure limit of solutions to the Euler equations for isentropic fliuds, SIAM J. Math. Anal., 34 (2003), 925-938.doi: 10.1137/S0036141001399350. |
[5] |
R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Interscience, New York, 1948. |
[6] |
Z. Dai ang T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Rat. Mech. Anal., 155 (2000), 277-298.doi: 10.1007/s002050000113. |
[7] |
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, Springer-Verlag, 2003. |
[8] |
L. H. Guo, W. C. Sheng and T. Zhang, The 2D Riemann problem for isentropic Chaplygin gas dynamic system, Comm. Pure Appl. Anal., 9 (2010), 431-458.doi: 10.3934/cpaa.2010.9.431. |
[9] |
Y. B. Hu, J. Q. Li and W. C. Sheng, Interaction of rarefaction waves for 2D isothermal Euler equations, submitted for publication, 2011. |
[10] |
F. Huang and Z. Wang, Well posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146.doi: 10.1007/s002200100506. |
[11] |
F. John, "Partial Differential Equations," Springer-Verlag, 1982. |
[12] |
B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations, 118 (1995), 420-451. |
[13] |
D. J. Korchinski, "Solutions of a Riemann Problem for A 2 $\times$ 2 System of Conservation Laws Prosssing No Classical Solutions," thesis, Adelphi University, Garden City, NY, 1977. |
[14] |
N. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation, In "Nonlinear Functional Analysis and its Applications," Proc. Symp. Pure Math., 45 (1986), 81-90, Providence, Amer. Math. Soc.. |
[15] |
G. Lai and W. C. Sheng, Simple waves for 2D isentropic ir-rotational self-similar Euler system, Appl. Math. Mech, 31 (2010), 1-12. |
[16] |
P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566.doi: 10.1002/cpa.3160100406. |
[17] |
Z. Lei and Y. Zheng, A complete global solution to the pressure gradient equation, J. Differential Equations, 236 (2007), 280-292. |
[18] |
J. Li, On the 2D gas expansion for compressible Euler euqations, SIAM J. Appl. Math., 62 (2001), 831-852.doi: 10.1137/S0036139900361349. |
[19] |
J. Li, Global solution of an initial-value problem for 2D compressible Euler equations, J. Differential Equations, 179 (2002), 178-194. |
[20] |
J. Li and H. Yang, Delta-shocks as limits of vanishing viscosity fo multidimensional zero-pressure gas dynamics, Quart. Appl. Math., 59 (2001), 315-342. |
[21] |
J. Li, Zhicheng Yang and Y. Zheng, Characteristic decompositions and interaction for rarefaction waves of the 2D Euler equations, to appear in J. Diff. Euqs., 2011. |
[22] |
J. Li and T. Zhang, Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations, in "Nonlinear PDE and Related Areas" (G. Q. Chen et al. Eds.), pp. 219-232, World Scientific, Singapore, 1998. |
[23] |
J. Li, T. Zhang and Y. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations, Commu. Math. Phys, 267 (2006), 1-12.doi: 10.1007/s00220-006-0033-1. |
[24] |
J. Li and Y. Zheng, Interaction of rarefaction waves of the 2D self-similar Euler equations, Arch. Rat. Mech. Anal., 193 (2009), 623-657.doi: 10.1007/s00205-008-0140-6. |
[25] |
M. Li and Y. Zheng, Semi-hyperbolic patches of solutions of the 2D Euler equations, to appear in Arch. Rat. Mech. Anal., 2011. |
[26] |
T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems," John Wiley and Sons, 1994. |
[27] |
T. Li and W. Yu, "Boundary Value Problem for Quasilinear Hyperbolic Systems," Duke University, 1985. |
[28] |
D. Serre, Multi-dimensional shock interaction for a Chaplygin gas, Arch. Rat. Mech. Anal., 191 (2008), 539-577.doi: 10.1007/s00205-008-0110-z. |
[29] |
V. M. Shelkovich, $\delta$ and $\delta'$ wave types of singular solutions of systems of conservation laws and transport and concentration process, Russian Math. Surveys, 63 (2008), 473-546.doi: 10.1070/RM2008v063n03ABEH004534. |
[30] |
W. Sheng and T. Zhang, The Riemann problem for transportation equations in gas dynamics, Mem. Amer. Math. Soc. 137, 564 (1999). |
[31] |
K. Song and Y. Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system, Disc. Cont. Dyna. Syst., 24 (2009), 1365-1380.doi: 10.3934/dcds.2009.24.1365. |
[32] |
J. H. Spurk and N. Aksel, "Fluid Mechanics," Spring-Verlag Berlin Heidelberg, 2008. |
[33] |
D. Tan and 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. |
[34] |
T. Zhang and Y. Zheng, Conjecture on the structure of solution of the Riemann problem for 2D gas dynamics system, SIAM J. Math. Anal., 21 (1990), 593-630.doi: 10.1137/0521032. |
[35] |
Y. Zheng, "Systems of Conservation Laws: 2D Riemann Problems," 38 PNLDE, Birkhäuser, Boston, 2001. |
[36] |
Y. Zheng, Absorption of characteristics by sonic curves of the 2D Euler equations, Disc. Cont. Dyna. Syst., 23 (2009), 605-616.doi: 10.3934/dcds.2009.23.605. |