# American Institute of Mathematical Sciences

December  2017, 37(12): 6257-6289. doi: 10.3934/dcds.2017271

## Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions

 1 Department of Mathematics and Statistics, California State University, Long Beach, Long Beach, CA 90840, USA 2 Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA

* Corresponding author: Eun Heui Kim.

Received  January 2017 Revised  June 2017 Published  August 2017

Fund Project: The work of Kim was supported by the National Science Foundation under the Grants DMS- 1109202, 1615266. The work of Tsikkou was supported by the National Science Foundation under the Grant DMS-1400168.

We analyze rarefaction wave interactions of self-similar transonic irrotational flow in gas dynamics for two dimensional Riemann problems. We establish the existence result of the supersonic solution to the prototype nonlinear wave system for the sectorial Riemann data, and study the formation of the sonic boundary and the transonic shock. The transition from the sonic boundary to the shock boundary inherits at least two types of degeneracies (1) the system is sonic, and in addition (2) the angular derivative of the solution becomes zero where the sonic and shock boundaries meet.

Citation: Eun Heui Kim, Charis Tsikkou. Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6257-6289. doi: 10.3934/dcds.2017271
##### References:
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Kim, Free boundary problems for nonlinear wave systems: Mach stems for interacting shocks, SIAM J. Math. Anal., 37 (2006), 1947-1977. doi: 10.1137/S003614100342989X. Google Scholar [6] G.-Q. Chen, X. Deng and W. Xiang, Shock diffraction by convex cornered wedges for the nonlinear wave system, Arch. Ration. Mech. Anal., 211 (2014), 61-112. doi: 10.1007/s00205-013-0681-1. Google Scholar [7] S. Chen and B. Fang, Stability of transonic shocks in supersonic flow past a wedge, J. Differential Equations, 233 (2007), 105-135. doi: 10.1016/j.jde.2006.09.020. Google Scholar [8] S. Chen, Mixed type equations in gas dynamics, Quart. Appl. Math., 68 (2010), 487-511. doi: 10.1090/S0033-569X-2010-01164-9. Google Scholar [9] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer Verlag, New York, 1948. Google Scholar [10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, $4^{th}$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325. Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6. Google Scholar [11] 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 [12] V. Elling and T.-P. Liu, Supersonic flow onto a solid wedge, Comm. Pure Appl. Math., 61 (2008), 1347-1448. doi: 10.1002/cpa.20231. Google Scholar [13] 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 2D compressible Euler equations, SIAM J. Appl. Math., 69 (2008), 720-742. doi: 10.1137/07070632X. Google Scholar [14] K. Jegdic, B. L. Keyfitz and S. Čanić, Transonic regular reflection for the nonlinear wave system, Journal of Hyperbolic Differential Equations, 3 (2006), 443-474. doi: 10.1142/S0219891606000859. 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, 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 [17] E. H. Kim and C.-M. Lee, Transonic shock reflection problems for the self-similar two-dimensional nonlinear wave system, Nonlinear Anal., 79 (2013), 85-102. doi: 10.1016/j.na.2012.11.002. Google Scholar [18] A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Methods Partial Differential Equations, 18 (2002), 584-608. doi: 10.1002/num.10025. Google Scholar [19] P. D. Lax and X.-D. Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), 319-340. doi: 10.1137/S1064827595291819. Google Scholar [20] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253. Google Scholar [21] R. J. LeVeque et al., CLAWPACK 4. 3, http://www.amath.washington.edu/claw/.Google Scholar [22] C. S. Morawetz, Potential theory for regular and Mach reflection of a shock at a wedge, Comm. Pure Appl. Math., 47 (1994), 593-624. doi: 10.1002/cpa.3160470502. Google Scholar [23] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43 (1981), 357-372. doi: 10.1016/0021-9991(81)90128-5. Google Scholar [24] C. W. Schulz-Rinne, J. P. Collins and H. M. Glaz, Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. Sci. Comput., 14 (1993), 1394-1414. doi: 10.1137/0914082. Google Scholar [25] D. Serre, Shock reflection in gas dynamics, Handbook of Mathematical Fluid Dynamics, vol. 4. Eds: S. Friedlander, D. Serre. Elsevier, North-Holland, (2007), 39–122.Google Scholar [26] D. Serre and H. Freistühler, The hyperbolic/elliptic transition in the multi-dimensional Riemann problem, Indiana Univ. Math. J., 62 (2013), 465-485. doi: 10.1512/iumj.2013.62.4918. Google Scholar [27] M. Sever, Admissibility of self-similar weak solutions of systems of conservation laws in two space variables and time, J. Hyperbolic Differ. Equ., 6 (2009), 433-481. doi: 10.1142/S0219891609001897. Google Scholar [28] W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137 (1999), ⅷ+77 pp. doi: 10.1090/memo/0654. Google Scholar [29] K. Song, The pressure-gradient system on non-smooth domains, Comm. Partial Differential Equations, 28 (2003), 199-221. doi: 10.1081/PDE-120019379. Google Scholar [30] K. Song and Y. Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system, Discrete Contin. Dyn. Syst., 24 (2009), 1365-1380. doi: 10.3934/dcds.2009.24.1365. Google Scholar [31] A. M. Tesdall, R. Sanders and B. L. Keyfitz, The triple point paradox for the nonlinear wave system, SIAM J. Appl. Math., 67 (2006/07), 321-336. doi: 10.1137/060660758. Google Scholar [32] A. M. Tesdall, R. Sanders and B. L. Keyfitz, Self-similar solutions for the triple point paradox in gasdynamics, SIAM J. Appl. Math., 68 (2008), 1360-1377. doi: 10.1137/070698567. Google Scholar [33] 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 [34] T. Zhang and Y. Zheng, Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Math. Anal., 21 (1990), 593-630. doi: 10.1137/0521032. Google Scholar [35] Y. Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 177-210. doi: 10.1007/s10255-006-0296-5. Google Scholar [36] Y. Zheng, Systems of Conservation Laws. Two-dimensional Riemann Problems, Progress in Nonlinear Differential Equations and their Applications, 38. Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0141-0. Google Scholar

show all references

##### References:
 [1] S. Bang, Interaction of three and four rarefaction waves of the pressure-gradient system, J. Differential Equations, 246 (2009), 453-481. doi: 10.1016/j.jde.2008.10.001. Google Scholar [2] S. Čanić, B. L. Keyfitz and E. H. Kim, Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection, Methods and Applications of Analysis, 7 (2000), 313-335. doi: 10.4310/MAA.2000.v7.n2.a4. Google Scholar [3] S. Čanić, B. L. Keyfitz and E. H. Kim, A free boundary problem for a quasi-linear degenerate elliptic equation: Regular reflection of weak shocks, Communications on Pure and Applied Mathematics, 55 (2002), 71-92. doi: 10.1002/cpa.10013. Google Scholar [4] S. Čanić, B. L. Keyfitz and E. H. Kim, Mixed hyperbolic-elliptic systems in self-similar flows, Boletim da Sociedade Brasileira de Matemática, 32 (2001), 377-399. doi: 10.1007/BF01233673. Google Scholar [5] S. Čanić, B. L. Keyfitz and E. H. Kim, Free boundary problems for nonlinear wave systems: Mach stems for interacting shocks, SIAM J. Math. Anal., 37 (2006), 1947-1977. doi: 10.1137/S003614100342989X. Google Scholar [6] G.-Q. Chen, X. Deng and W. Xiang, Shock diffraction by convex cornered wedges for the nonlinear wave system, Arch. Ration. Mech. Anal., 211 (2014), 61-112. doi: 10.1007/s00205-013-0681-1. Google Scholar [7] S. Chen and B. Fang, Stability of transonic shocks in supersonic flow past a wedge, J. Differential Equations, 233 (2007), 105-135. doi: 10.1016/j.jde.2006.09.020. Google Scholar [8] S. Chen, Mixed type equations in gas dynamics, Quart. Appl. Math., 68 (2010), 487-511. doi: 10.1090/S0033-569X-2010-01164-9. Google Scholar [9] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer Verlag, New York, 1948. Google Scholar [10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, $4^{th}$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325. Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6. Google Scholar [11] 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 [12] V. Elling and T.-P. Liu, Supersonic flow onto a solid wedge, Comm. Pure Appl. Math., 61 (2008), 1347-1448. doi: 10.1002/cpa.20231. Google Scholar [13] 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 2D compressible Euler equations, SIAM J. Appl. Math., 69 (2008), 720-742. doi: 10.1137/07070632X. Google Scholar [14] K. Jegdic, B. L. Keyfitz and S. Čanić, Transonic regular reflection for the nonlinear wave system, Journal of Hyperbolic Differential Equations, 3 (2006), 443-474. doi: 10.1142/S0219891606000859. 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, 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 [17] E. H. Kim and C.-M. Lee, Transonic shock reflection problems for the self-similar two-dimensional nonlinear wave system, Nonlinear Anal., 79 (2013), 85-102. doi: 10.1016/j.na.2012.11.002. Google Scholar [18] A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Methods Partial Differential Equations, 18 (2002), 584-608. doi: 10.1002/num.10025. Google Scholar [19] P. D. Lax and X.-D. Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), 319-340. doi: 10.1137/S1064827595291819. Google Scholar [20] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253. Google Scholar [21] R. J. LeVeque et al., CLAWPACK 4. 3, http://www.amath.washington.edu/claw/.Google Scholar [22] C. S. Morawetz, Potential theory for regular and Mach reflection of a shock at a wedge, Comm. Pure Appl. Math., 47 (1994), 593-624. doi: 10.1002/cpa.3160470502. Google Scholar [23] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43 (1981), 357-372. doi: 10.1016/0021-9991(81)90128-5. Google Scholar [24] C. W. Schulz-Rinne, J. P. Collins and H. M. Glaz, Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. Sci. Comput., 14 (1993), 1394-1414. doi: 10.1137/0914082. Google Scholar [25] D. Serre, Shock reflection in gas dynamics, Handbook of Mathematical Fluid Dynamics, vol. 4. Eds: S. Friedlander, D. Serre. Elsevier, North-Holland, (2007), 39–122.Google Scholar [26] D. Serre and H. Freistühler, The hyperbolic/elliptic transition in the multi-dimensional Riemann problem, Indiana Univ. Math. J., 62 (2013), 465-485. doi: 10.1512/iumj.2013.62.4918. Google Scholar [27] M. Sever, Admissibility of self-similar weak solutions of systems of conservation laws in two space variables and time, J. Hyperbolic Differ. Equ., 6 (2009), 433-481. doi: 10.1142/S0219891609001897. Google Scholar [28] W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137 (1999), ⅷ+77 pp. doi: 10.1090/memo/0654. Google Scholar [29] K. Song, The pressure-gradient system on non-smooth domains, Comm. Partial Differential Equations, 28 (2003), 199-221. doi: 10.1081/PDE-120019379. Google Scholar [30] K. Song and Y. Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system, Discrete Contin. Dyn. Syst., 24 (2009), 1365-1380. doi: 10.3934/dcds.2009.24.1365. Google Scholar [31] A. M. Tesdall, R. Sanders and B. L. Keyfitz, The triple point paradox for the nonlinear wave system, SIAM J. Appl. Math., 67 (2006/07), 321-336. doi: 10.1137/060660758. Google Scholar [32] A. M. Tesdall, R. Sanders and B. L. Keyfitz, Self-similar solutions for the triple point paradox in gasdynamics, SIAM J. Appl. Math., 68 (2008), 1360-1377. doi: 10.1137/070698567. Google Scholar [33] 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 [34] T. Zhang and Y. Zheng, Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Math. Anal., 21 (1990), 593-630. doi: 10.1137/0521032. Google Scholar [35] Y. Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 177-210. doi: 10.1007/s10255-006-0296-5. Google Scholar [36] Y. Zheng, Systems of Conservation Laws. Two-dimensional Riemann Problems, Progress in Nonlinear Differential Equations and their Applications, 38. Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0141-0. Google Scholar
Riemann data and configuration.
Configuration with details.
Regions and characteristics in the supersonic region.
$\mathcal{R}_1\setminus \mathcal{R}_1[\delta]$, the region $\mathcal{R}_1$ excluding the small neighborhoods of $\Xi_1$ and $\Xi_3.$
Schematics of constructing the solution near $\Xi_3.$
Envelope formation by the simple wave.
Density plots: the contour plot of $\rho$.
Density plots: Left figure is the cross section in the radial direction for a fixed angle $\theta$ ranging from $0$ to $\pi/2$ and incrementing by $10$ degrees. Right figure is the enlargement of the left figure near which the shock changes to sonic, which appears to be in-between the angle $50$ and $60$ degrees in this configuration.
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