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The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas
1. | Department of Mathematics, Yunnan University, Kunming 650091 |
2. | Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, South Korea |
References:
[1] |
M. Brio and J. K. Hunter, Mach reflection for the two-dimensional Burgers equation, Phys. D, 60 (1992), 194-207.
doi: 10.1016/0167-2789(92)90236-G. |
[2] |
S. Čanić and B. L. Keyfitz, An elliptic problem arising from the unsteady transonic small disturbance equation, J. Differential Equations, 125 (1996), 548-574.
doi: 10.1006/jdeq.1996.0040. |
[3] |
S. Čanić, B. L. Keyfitz and E. H. Kim, Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection, Methods Appl. Anal., 7 (2000), 313-335. |
[4] |
S. Čanić, B. L. Keyfitz and E. H. Kim, Free boundary problems for nonlinear wave equations: Mach stems for interacting shocks, SIAM J. Math. Anal., 37 (2006), 1947-1977.
doi: 10.1137/S003614100342989X. |
[5] |
G.-Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc., 16 (2003), 461-494.
doi: 10.1090/S0894-0347-03-00422-3. |
[6] |
G.-Q. Chen and M. Feldman, Steady transonic shock and free boundary problems in infinite cylinders for the Euler equations, Comm. Pure Appl. Math., 57 (2004), 310-356.
doi: 10.1002/cpa.3042. |
[7] |
S. Chen and A. Qu, Two-dimensional Riemann problems for Chaplygin gas, SIAM J. Math. Anal., 44 (2012), 2146-2178.
doi: 10.1137/110838091. |
[8] |
H. Cheng and H. Yang, Riemann problem for the relativistic Chaplygin Euler equations, J. Math. Anal. Appl., 381 (2011), 17-26.
doi: 10.1016/j.jmaa.2011.04.017. |
[9] |
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948. |
[10] |
Z. Dai and T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Ration. Mech. Anal., 155 (2000), 277-298.
doi: 10.1007/s002050000113. |
[11] |
V. Elling and T.-P. Liu, The ellipticity principle for steady and selfsimilar polytropic potential flow, J. Hyperbolic Differential Equations, 2 (2005), 909-917.
doi: 10.1142/S0219891605000646. |
[12] |
J. Glimm, X. Ji, J. Li, X. Li, P. Zhang, T. Zhang and Y. Zheng, Transonic shock formation in a rarefaction Riemann problem for the 2-D compressible Euler equations, SIAM J. Appl. Math., 69 (2008), 720-742.
doi: 10.1137/07070632X. |
[13] |
Y. Hu and G. Wang, Semi-hyperbolic patches of solutions to the two-dimensional nonlinear wave system for Chaplygin gases, J. Differential Equations, 257 (2014), 1567-1590.
doi: 10.1016/j.jde.2014.05.020. |
[14] |
E. H. Kim, An interaction of a rarefaction wave and a transonic shock for the self-similar two-dimensional nonlinear wave system, Comm. Partial Differential Equations, 37 (2012), 610-646.
doi: 10.1080/03605302.2011.653615. |
[15] |
E. H. Kim, A global subsonic solution to an interacting transonic shock for the self-similar nonlinear wave equation, J. Differential Equations, 248 (2010), 2906-2930.
doi: 10.1016/j.jde.2010.02.021. |
[16] |
E. H. Kim, Subsonic solutions for compressible transonic potential flows, J. Differential Equations, 233 (2007), 276-290.
doi: 10.1016/j.jde.2006.10.013. |
[17] |
E. H. Kim and K. Song, Classical solutions for the pressure-gradient equations in non-smooth and non-convex domains, J. Math. Anal. Appl., 293 (2004), 541-550.
doi: 10.1016/j.jmaa.2004.01.016. |
[18] |
G. Lai, W. C. Sheng and Y. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions, Discrete Contin. Dyn. Syst., 31 (2011), 489-523.
doi: 10.3934/dcds.2011.31.489. |
[19] |
Z. Lei and Y. Zheng, A complete global solution to the pressure gradient equation, J. Differential Equations, 236 (2007), 280-292.
doi: 10.1016/j.jde.2007.01.024. |
[20] |
M. Li and Y. Zheng, Semi-hyperbolic patches of solutions to the two-dimensional Euler equations, Arch. Ration. Mech. Anal., 201 (2011), 1069-1096.
doi: 10.1007/s00205-011-0410-6. |
[21] |
D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Ration. Mech. Anal., 191 (2009), 539-577.
doi: 10.1007/s00205-008-0110-z. |
[22] |
K. Song, Semi-hyperbolic patches arising from a transonic shock in simple waves interaction, J. Korean Math. Soc., 50 (2013), 945-957.
doi: 10.4134/JKMS.2013.50.5.945. |
[23] |
K. Song, Q. Wang and Y. Zheng, The regularity of semi-hyperbolic patches near sonic lines for the 2-D Euler system in gas dynamics, SIAM J. Math. Anal., 47 (2015), 2200-2219.
doi: 10.1137/140964382. |
[24] |
K. Song and Y. Zheng, Semi-hyperbolic patches of the pressure gradient system, Disc. Cont. Dyna. Syst., Series A, 24 (2009), 1365-1380.
doi: 10.3934/dcds.2009.24.1365. |
[25] |
A. M. Tesdall, R. Sanders and B. L. Keyfitz, The triple point paradox for the nonlinear wave system, SIAM J. Appl. Math., 67 (2006), 321-336.
doi: 10.1137/060660758. |
[26] |
G. Wang, B. Chen and Y. Hu, The two-dimensional Riemann problem for Chaplygin gas dynamics with three constant states, J. Math. Anal. Appl., 393 (2012), 544-562.
doi: 10.1016/j.jmaa.2012.03.017. |
[27] |
Q. Wang and Y. Zheng, The regularity of semi-hyperbolic patches at sonic lines for the pressure gradient equation in gas dynamics, Indiana Univ. Math. J., 63 (2014), 385-402.
doi: 10.1512/iumj.2014.63.5244. |
[28] |
T. Zhang and Y. Zheng, Sonic-supersonic solutions for the steady Euler equations, Indiana Univ. Math. J., 63 (2014), 1785-1817.
doi: 10.1512/iumj.2014.63.5434. |
[29] |
Y. Zheng, Existence of solutions to the transonic pressure gradient equations of the compressible Euler equations in elliptic regions, Comm. Partial Differential Equations, 22 (1997), 1849-1868.
doi: 10.1080/03605309708821323. |
show all references
References:
[1] |
M. Brio and J. K. Hunter, Mach reflection for the two-dimensional Burgers equation, Phys. D, 60 (1992), 194-207.
doi: 10.1016/0167-2789(92)90236-G. |
[2] |
S. Čanić and B. L. Keyfitz, An elliptic problem arising from the unsteady transonic small disturbance equation, J. Differential Equations, 125 (1996), 548-574.
doi: 10.1006/jdeq.1996.0040. |
[3] |
S. Čanić, B. L. Keyfitz and E. H. Kim, Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection, Methods Appl. Anal., 7 (2000), 313-335. |
[4] |
S. Čanić, B. L. Keyfitz and E. H. Kim, Free boundary problems for nonlinear wave equations: Mach stems for interacting shocks, SIAM J. Math. Anal., 37 (2006), 1947-1977.
doi: 10.1137/S003614100342989X. |
[5] |
G.-Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc., 16 (2003), 461-494.
doi: 10.1090/S0894-0347-03-00422-3. |
[6] |
G.-Q. Chen and M. Feldman, Steady transonic shock and free boundary problems in infinite cylinders for the Euler equations, Comm. Pure Appl. Math., 57 (2004), 310-356.
doi: 10.1002/cpa.3042. |
[7] |
S. Chen and A. Qu, Two-dimensional Riemann problems for Chaplygin gas, SIAM J. Math. Anal., 44 (2012), 2146-2178.
doi: 10.1137/110838091. |
[8] |
H. Cheng and H. Yang, Riemann problem for the relativistic Chaplygin Euler equations, J. Math. Anal. Appl., 381 (2011), 17-26.
doi: 10.1016/j.jmaa.2011.04.017. |
[9] |
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948. |
[10] |
Z. Dai and T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Ration. Mech. Anal., 155 (2000), 277-298.
doi: 10.1007/s002050000113. |
[11] |
V. Elling and T.-P. Liu, The ellipticity principle for steady and selfsimilar polytropic potential flow, J. Hyperbolic Differential Equations, 2 (2005), 909-917.
doi: 10.1142/S0219891605000646. |
[12] |
J. Glimm, X. Ji, J. Li, X. Li, P. Zhang, T. Zhang and Y. Zheng, Transonic shock formation in a rarefaction Riemann problem for the 2-D compressible Euler equations, SIAM J. Appl. Math., 69 (2008), 720-742.
doi: 10.1137/07070632X. |
[13] |
Y. Hu and G. Wang, Semi-hyperbolic patches of solutions to the two-dimensional nonlinear wave system for Chaplygin gases, J. Differential Equations, 257 (2014), 1567-1590.
doi: 10.1016/j.jde.2014.05.020. |
[14] |
E. H. Kim, An interaction of a rarefaction wave and a transonic shock for the self-similar two-dimensional nonlinear wave system, Comm. Partial Differential Equations, 37 (2012), 610-646.
doi: 10.1080/03605302.2011.653615. |
[15] |
E. H. Kim, A global subsonic solution to an interacting transonic shock for the self-similar nonlinear wave equation, J. Differential Equations, 248 (2010), 2906-2930.
doi: 10.1016/j.jde.2010.02.021. |
[16] |
E. H. Kim, Subsonic solutions for compressible transonic potential flows, J. Differential Equations, 233 (2007), 276-290.
doi: 10.1016/j.jde.2006.10.013. |
[17] |
E. H. Kim and K. Song, Classical solutions for the pressure-gradient equations in non-smooth and non-convex domains, J. Math. Anal. Appl., 293 (2004), 541-550.
doi: 10.1016/j.jmaa.2004.01.016. |
[18] |
G. Lai, W. C. Sheng and Y. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions, Discrete Contin. Dyn. Syst., 31 (2011), 489-523.
doi: 10.3934/dcds.2011.31.489. |
[19] |
Z. Lei and Y. Zheng, A complete global solution to the pressure gradient equation, J. Differential Equations, 236 (2007), 280-292.
doi: 10.1016/j.jde.2007.01.024. |
[20] |
M. Li and Y. Zheng, Semi-hyperbolic patches of solutions to the two-dimensional Euler equations, Arch. Ration. Mech. Anal., 201 (2011), 1069-1096.
doi: 10.1007/s00205-011-0410-6. |
[21] |
D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Ration. Mech. Anal., 191 (2009), 539-577.
doi: 10.1007/s00205-008-0110-z. |
[22] |
K. Song, Semi-hyperbolic patches arising from a transonic shock in simple waves interaction, J. Korean Math. Soc., 50 (2013), 945-957.
doi: 10.4134/JKMS.2013.50.5.945. |
[23] |
K. Song, Q. Wang and Y. Zheng, The regularity of semi-hyperbolic patches near sonic lines for the 2-D Euler system in gas dynamics, SIAM J. Math. Anal., 47 (2015), 2200-2219.
doi: 10.1137/140964382. |
[24] |
K. Song and Y. Zheng, Semi-hyperbolic patches of the pressure gradient system, Disc. Cont. Dyna. Syst., Series A, 24 (2009), 1365-1380.
doi: 10.3934/dcds.2009.24.1365. |
[25] |
A. M. Tesdall, R. Sanders and B. L. Keyfitz, The triple point paradox for the nonlinear wave system, SIAM J. Appl. Math., 67 (2006), 321-336.
doi: 10.1137/060660758. |
[26] |
G. Wang, B. Chen and Y. Hu, The two-dimensional Riemann problem for Chaplygin gas dynamics with three constant states, J. Math. Anal. Appl., 393 (2012), 544-562.
doi: 10.1016/j.jmaa.2012.03.017. |
[27] |
Q. Wang and Y. Zheng, The regularity of semi-hyperbolic patches at sonic lines for the pressure gradient equation in gas dynamics, Indiana Univ. Math. J., 63 (2014), 385-402.
doi: 10.1512/iumj.2014.63.5244. |
[28] |
T. Zhang and Y. Zheng, Sonic-supersonic solutions for the steady Euler equations, Indiana Univ. Math. J., 63 (2014), 1785-1817.
doi: 10.1512/iumj.2014.63.5434. |
[29] |
Y. Zheng, Existence of solutions to the transonic pressure gradient equations of the compressible Euler equations in elliptic regions, Comm. Partial Differential Equations, 22 (1997), 1849-1868.
doi: 10.1080/03605309708821323. |
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