SEMI-HYPERBOLIC PATCHES ARISING FROM A TRANSONIC SHOCK IN SIMPLE WAVES INTERACTION Song, Kyungwoo;
In this paper we consider a Riemann problem, in particular, the case of the presence of the semi-hyperbolic patches arising from a transonic shock in simple waves interaction. Under this circumstance, we construct global solutions of the two-dimensional Riemann problem of the pressure gradient system. We approach the problem as a Goursat boundary value problem and a mixed initial-boundary value problem, where one of the boundaries is the transonic shock.
Semi-hyperbolic patches of solutions to the two-dimensional nonlinear wave system for Chaplygin gases, Journal of Differential Equations, 2014, 257, 5, 1567
The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas, Discrete and Continuous Dynamical Systems, 2015, 36, 3, 1661
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), no. 4, 277-298.
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), no. 3, 720-742.
J. Li, T. Zhang, and S. Yang, The Two-dimensional Riemann Problem in Gas Dynamics, Pitman monographs and surveys in pure and applied mathematics 98. Addison Wesley Longman limited, 1998.
J. Li, T. Zhang, and Y. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations, Comm. Math. Phys. 267 (2006), no. 1, 1-12.
K. Song, A pressure-gradient system on non-smooth domains, Comm. Partial Differential Equations 28 (2003), 199-221.
K. Song and Y. Zheng, Semi-hyperbolic patches of the pressure gradient system, Disc. Cont. Dyna. Syst. Series A 24 (2009), 1365-1380.
R.-H. Wang and Z.-Q. Wu, Existence and uniqueness of solutions for some mixed initial boundary value problems of quasilinear hyperbolic systems in two independent variables, Acta Sci. Natur. Jilin University 2 (1963), 277-298 (in Chinese).
Y. Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, Acta Math. Appl. Sin. Engl. Ser. 22 (2006), no. 2, 177-210.
Y. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems, 38 PNLDE, Birkhauser, Boston, 2001.
Y. Zheng, The compressible Euler system in two space dimensions. Nonlinear conserva- tion laws, fluid systems and related topics, 301-390, Ser. Contemp. Appl. Math. CAM, 13, World Sci. Publishing, Singapore, 2009.