Interaction of Rarefaction Waves and Vacuum in a Convex Duct

The motion of the supersonic compressible flow governed by the Euler system in a two-dimensional convex duct is studied. The rarefaction waves in the compressible flow propagate and reflect on the walls of the convex duct, so that interaction occurs and a vacuum may appear. The existence of the global piecewise smooth solution to the steady Euler system in the interaction region is established. Meanwhile, the appearance of a vacuum is carefully considered. It is found that a vacuum is always adjacent to one of the walls and the appearance of a vacuum depends on the limit of the slope of the wall at the infinity.