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March  2016, 36(3): 1661-1675. doi: 10.3934/dcds.2016.36.1661

The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas

Qin Wang 1, and  Kyungwoo Song 2,

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

Received  November 2014 Revised  March 2015 Published  August 2015

We study the regularity of sonic curves from a two-dimensional Riemann problem for the nonlinear wave system of Chaplygin gas, which is an essential step for the global existence of solutions to the two-dimensional Riemann problems. As a result, we establish the global existence of uniformly smooth solutions in the semi-hyperbolic patches up to the sonic boundary, where the degeneracy of hyperbolicity occurs. Furthermore, we show the $C^1$-regularity of sonic curves.
Keywords: bootstrap method., 2-D Riemann problem, Nonlinear wave system, Chaplygin gas, regularity.
Mathematics Subject Classification: 35L65, 35J70, 35R3.
Citation: Qin Wang, Kyungwoo Song. The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1661-1675. doi: 10.3934/dcds.2016.36.1661
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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