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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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