Interactions of Rarefaction Waves and Rarefaction Shocks of the Two-Dimensional Compressible Euler Equations with General Equation of State

We consider two types of wave interactions arising from the expansion of a wedge of gas to vacuum: interaction of rarefaction waves and interaction of shock-rarefaction composite waves. In order to construct the solutions to these wave interactions, we consider a standard Goursat problem and a discontinuous Goursat problem for the two-dimensional self-similar Euler system. Global solutions to these Goursat problems are constructed by the method of characteristics. The main difficulty to construct the global solutions is that the type of the 2D self-similar Euler system is a priori unknown. By constructing invariant regions of the characteristic angles, we obtain the hyperbolicity of the system in the interaction regions. The solutions constructed in this paper can be used as building blocks of solutions to 2D Riemann problems of the Euler equations with general equations of state.