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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

 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.
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 - A, 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.jmaa.2011.04.017. Google Scholar [9] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, Interscience, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/140964382. Google Scholar [24] K. Song and Y. Zheng, Semi-hyperbolic patches of the pressure gradient system,, Disc. Cont. Dyna. Syst., 24 (2009), 1365. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.jmaa.2011.04.017. Google Scholar [9] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, Interscience, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/140964382. Google Scholar [24] K. Song and Y. Zheng, Semi-hyperbolic patches of the pressure gradient system,, Disc. Cont. Dyna. Syst., 24 (2009), 1365. 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. 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. 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. 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. 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. doi: 10.1080/03605309708821323. Google Scholar
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