Two-Dimensional Steady Supersonic Relativistic Euler Flows Past Lipschitz Wedges

Abstract

This paper is a review of wedge problems for supersonic relativistic Euler flows. We are mainly concerned with two-dimensional compressible supersonic relativistic Euler flows past Lipschitz wedges, sharp corners, or bending wedges from mathematical point of view. When the vertex angle of the upstream flow is less than the critical angle, a shock wave is generated from the wedge vertex. If the vertex angle is larger than $\pi$ and the angle of the flow is larger than the critical value, then a rarefaction wave will appear. In this paper, we employ modified wave front tracking method to establish the structural stability of such wave patterns under some small perturbations of both the upcoming supersonic flow and the tangent slope of the boundary. It is an initial-boundary value problem for two-dimensional steady compressible relativistic Euler system. Moreover, we study global non-relativistic limits of entropy solutions for relativistic Euler flows as well as the asymptotic behavior of the solutions at the infinity. In particular, we demonstrate the basic properties of nonlinear waves for the two-dimensional steady supersonic relativistic Euler flows, especially, the geometric structures of shock polar and rarefaction wave.