Two dimensional relativistic Euler equations in a convex duct

In this paper, we investigate the motion of supersonic flow governed by 2-D steady isentropic irrotational relativistic Euler system entered into a convex duct. We will give the propagation and interactions of rarefaction waves between the upward and downward wall and obtain the global existence of piecewise continuous solutions. Furthermore, in some cases the vacuum state may appear and we also give a sufficient condition to the appearance of the vacuum in finite domain.