Sonic–supersonic solutions to a degenerate Cauchy–Goursat problem for 2D relativistic Euler equations

In this paper, we consider the degenerate Cauchy–Goursat problem for 2D steady isentropic relativistic Euler equations. Prescribing the sonic curve and a positive characteristic curve as boundaries, the existence and uniqueness of sonic–supersonic solution in an angular region are obtained. Employing the characteristic decomposition of angle variables, 2D relativistic Euler equations are transformed into the first-order hyperbolic equations. In the partial hodograph plane, introducing the change variables W¯=1W,Z¯=-1Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{W}=\frac{1}{W},\overline{Z}=\frac{-1}{Z}$$\end{document}, associated with the iterative method in Li, Hu (2019) yields a linear equations and the existence and uniqueness of the smooth sonic–supersonic solutions are established. Finally, we return the solution in the partial hodograph plane to that in the original physical variables.