On a degenerate mixed-type boundary value problem for the two-dimensional self-similar Euler equations

This paper is concerned with the semi-hyperbolic structures originated from the study of the two-dimensional Riemann problem for the compressible Euler equations in gas dynamics. Given two piece of smooth curves in the self-similar plane such that one is a sonic curve and the other is a characteristic curve, we establish the existence of classical supersonic solutions in the angular region near the corner point. The main difficulty arises from the coupling of nonlinearity and degeneracy at the corner. With the help of the characteristic decomposition technique, the problem is solved by transforming the self-similar Euler equations into a new degenerate hyperbolic system with explicitly singularity-regularity structures. Based on the solution in the partial hodograph plane, we construct a smooth sonic-supersonic solution of the original de-generate mixed-type boundary value problem in the self-similar plane.